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THEEUMAEUSGUIDETOEQUITYRELEASEVALUATIONRestatingtheCaseforaMarketConsistentApproach

SECONDEDITION

DeanBucknerandKevinDowd

15June2020

ExecutiveSummaryAcknowledgementsDisclosureStatementPARTONE:NNEGBASICSChapter1:IntroductionChapter2:OriginandHistoryChapter3:TheBasicsofNNEGandERMValuationChapter3Appendix:TheChoiceofOptionPricingModelPARTTWO:KEYINPUTSChapter4:Loan-to-ValueRatioChapter5:Risk-FreeRateChapter6:LoanRateChapter7:NetRentalYieldandDefermentRateChapter8:DilapidationChapter9:Volatility(I)Chapter9Appendix1:ProofofApproximation(9.12)Chapter9Appendix2:ProofofEquation(9.14)Chapter10:MortalityChapter11:Long-TermCareChapter12:DelayedPossession

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Chapter13:CreditSpreadsChapter14:DrawdownChapter15:PrepaymentChapter16:FeesandChargesPARTTHREE:APPLICATIONSChapter17:ScenarioAnalysisandStressTestingChapter18:ThePRA’sGoodPracticePrinciplesAppendixtoChapter18:Proofof(18.14)PARTFOUR:COMPETINGAPPROACHESTOEQUITYRELEASEVALUATIONChapter19:TheMarketConsistentApproachChapter20:The‘DiscountedProjection’ApproachChapter21:TheTunaruReportChapter22:Just’sNNEGValuationModelPARTFIVE:PROFESSIONALSTANDARDSANDRECOMMENDATIONSChapter23:TechnicalActuarialStandardsChapter24:AccountingStandardsPARTSIX:RECOMMENDATIONSChapter25:RecommendationsforGoodValuationPracticeChapter26:RecommendationsforGovernance

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ExecutiveSummary

• TheUKequityreleasesectorispermeatedbypoorvaluationpractice:asfarasweareaware,notasingleequityreleasefirmisvaluingitsNo-NegativeEquityGuarantees(NNEGs)inascientificallyvalidmanner.

• This NNEG under-valuation problem is on a large scale and impliescorrespondinglylargeover-valuationsofEquityReleaseMortgages(ERMs).

• TheDiscountedProjectionor ‘RealWorld’approachusedbytheequityreleaseindustryisinherentlyflawedandproducesvaluationsthatviolateboundsthatareknowntobeinviolable.

• TheonlyscientificallyvalidvaluationapproachistheMarketConsistentapproach,

which is also the only approach compatible with accounting principles andtechnicalactuarialstandards.

• Thismanual provides a guide that explains howNNEGs andERM should (and

shouldnot)bevalued.

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Acknowledgements

Wethankthemanypeoplewhohavekindlycontributedtothisreportthroughhelpfuldiscussions on the topics coveredhere or through comments on earlier versions.Wethank inparticular:DavidBlake, ChrisCundy, JonathanFord, CharlesGoodhart, TonyJeffery,HowardMustoe,GordonKerr,CavinO’Driscoll,AndrewSmith,JohnSkar,CraigTurnbull,MarioWuthrich,membersofthePRA’sCP7.19team,seminarparticipantsatthe London School of Economics and QueenMary University of London, and variouscorrespondentswho have offered comments in private. The usual caveat applies:wealoneareresponsibleforanyviewsexpressedhere,andforanyremainingerrors.DisclosureStatementThisreporthasnotbeenfundedbyanyresearchorganisationorcommercialbody,andnorhavesoughtanysuchfunding.Dowd is currently involved as a co-investigator in the Actuarial Research Council’s“Modelling,MeasurementandManagementofLongevityandMorbidityRisk”researchprogramme.Formoredetails,see:https://www.actuaries.org.uk/learn-and-develop/research-and-knowledge/actuarial-research-centre-arc/research-programmes/modelling-measurement-and-management-longevity-and-morbidity-risk/research-team

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ChapterOne:Introduction

“Idourgeyoualltokeepanopenmind…becausewe[actuaries]don'talways- sometimeswe've got a very narrowway of thinking, but I do think thatthere'smoretogo[onNo-NegativeEquityGuaranteevaluation],andthismaygoonforlongerthantheBrexitdiscussions.”

GinaCraske1

IntheEquityReleaseCouncil’sSpring2018MarketReport,itschairmanDavidBurrowesstruckareassuringtone:

Annuallendingactivitybyourmembershassurpassed£3billionforthefirsttime and customer numbers reached 67,000 in 2017. Property wealth isincreasingly recognised by people as a safe and sought-after source ofretirementfinance,withthemarketattractingtwiceasmanynewcustomersasitwasfiveyearsago.…Therangeofproductoptionsavailabletoequityreleasecustomershasgrown25% year-on-year, providing more choice to underpin a robust andcompetitivemarket.Lookingforward,weexpecttheneedfornewsourcesofincomeinretirementwill continue to grow asmany peoplewill be unable to rely on pressuredpensionpots.(EquityReleaseCouncil,2018,p.2)

Mr.BurrowesomittedhowevertomentionanissuethathadbeencausingsomeworriesinEquityReleaseMortgage(ERM)circlesforsometimenow.Theproblemisthatfirmsare under-valuing the No Negative Equity Guarantees (NNEGs) that are a standardfeatureofmostERMproducts.Thisunder-estimationseemstobeonalargescaletoo.These concerns received somepublicitywith thepublicationon7August last yearofreportsbyBBCbusinessjournalistHowardMustoe2andtheAdamSmithInstitute3ontheissue,andwiththeairingthateveningofaBBCRadio4programme,“TheEquityReleaseTrap.”4 Since then, there has been considerable public controversy over the NNEGvaluation issue, and we provide a commentary on our blog, The Eumaeus Project(eumaeus.org/).5

1StapleInneventtranscript28Feb2019.2H.Mustoe,withadditionalreportingbyM.Keyworth(2018)“HomeEquityReleaseMayCostPensionFirmsBillions.”BBCnewswebsite(7August).3K.Dowd(2018)AsleepattheWheel:ThePrudentialRegulationAuthorityandtheEquityReleaseSector.London:AdamSmithInstitute(7August)(a)4https://www.bbc.co.uk/programmes/b0bd8h785 Contributionstothesediscussionsincludetwoseminarswehavepresented(LSEinOctober2018andQMUL in June2019), several further reports byus (including the first edition of ourEumaeusGuide),reportsbyRaduTunaru(Feburary2019)andTonyJefferyandAndrewSmithinMarch2019,andanIFoAERMworking paper in February 2020. See:D. Buckner andK. Dowd (2018) “Equity Release: AnotherEquitableintheMaking,”JohnsHopkinsInstituteforAppliedEconomics,GlobalHealth,andtheStudyof

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Consider the following 2017 quotes from UK equity release firms discussing themethodologiestheyusetovaluetheirNo-NegativeEquityGuarantees(NNEGs).Asyoudoso,askyourselfwhattheyallhaveincommon:

“When calculating the value of theno-negative equityguarantee on thelifetimemortgages, certain economic assumptions are required withinthevariantof theBlack-Scholes formula. […] In theabsenceof a reliable long-termforwardcurveforUKresidentialpropertypriceinflation,the[firm]hasmadeanassumptionaboutfutureresidentialpropertypriceinflation.…Thisresultsinasinglerateoffuturehousepricegrowthof4.25%.”6

“[Thevalueof theNNEG] iscalculatedusingavariantof theBlackScholesoptionpricingmodel.Thekeyassumptionsusedtoderivethevalueoftheno-negative equity guarantee include current property price,property growthandpropertyvolatility.”

“Stochastic modelling is used to capture the expected cost of [the NNEG],whichwilldependon theexpected rateandvolatilityof futurehousepricegrowth…“Equityreleaseandsecuritisedmortgageloans…arevaluedusinganinternalmodel.Inputstothemodelincludeprimarilypropertygrowthrates,mortalityandmorbidityassumptions,….”

“Thefairvalueoftheguaranteeisdeterminedusingastochasticmodel.Thefair value of the loans is determined using assumptions for interest rates,futurehousepriceinflationanditsvolatility…”

The answer is that they are all using incorrect valuation approaches. They all usepropertygrowthassumptionsintheirNNEGmodels,butnocorrectoptionpricingmodelsincludepropertygrowthvariables.TheiruseofanirrelevantvariablethenindicatesthattheyarenotvaluingtheirNNEGsproperly.Totheircredit,thePRAhavebeenawareofthisproblemforsometime.Referringtotheresultsofanearliersurvey,CP48/16states(p.25):

ManyrespondentsmentionedaversionoftheBlack-Scholesformulaknownas‘Black76’,wheretheunderlyingpriceisthe‘forwardprice’oftheproperty.Thisversionusesthecurrentpriceofaforwardcontract.Somerespondentsappearedtoconflate thiswiththeforecastfuturepriceoftheproperty,but

BusinessEnterprise,StudiesinAppliedEconomicsNo.129,November2018;R.TunaruandE.Quaye(2019)“UK Equity Release Mortgages: a review of the No Negative Equity Guarantee,” Actuarial ResearchCentre/Institute and Faculty of Actuaries (19 February); T. Jeffery and A. D. Smith “Equity ReleaseMortgages:Irish&UKExperience,”presentationtotheSocietyofActuariesinIreland,28March2019;and“ADiscussionNoteontheEconomicValueofEquityReleaseMortgagesasPartofthePRA’sEffectiveValueTest,”InstituteandFacultyofActuariesEquityReleaseWorkingParty,17February2020.6SeeF.BlackandM.Scholes(1973)“ThePricingofOptionsandCorporateLiabilities,”JournalofPoliticalEconomy81:637–654.

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providedno justification forwhy house price inflationwas relevant to thecurrentpriceofaforwardcontract.(Ouritalics)

The key word is “conflate”. The reason why these correspondents provided nojustificationforusingprojectionsoffuturehousepriceinflationtovaluetheseguaranteesisbecausenosuchjustificationexists.Tospellitout:somefirmssaythattheyareusingassumptionsaboutfuturehousepricegrowth,butthePRAcorrectlysaysthatthisisobviouslywrong.Fromwhichitfollows(1)thatsomefirmsareusingamethodwhollyatoddswiththeoneendorsedbythePRAand(2)thatthePRAwouldnotbebotheringtostatethispointatall,particularlythroughaprotractedconsultationperiodifithadnotexperiencedsubstantialpushbackfromfirms.Wecantheninfer(3)thatfirmswithequityreleaseexposurehavebeenundervaluingtheirnonegativeequityguarantees.WecanmakethisinferencebecausethePRAwouldnotbepublishingon the subjector seeking industry consultation if they thought thatthese guarantees were correctly valued. Consequently, some firms are presumablyundervaluing them. Also (4) by a similar logic, if firms are dedicating substantialresourcestopushingback,theymustthinkthatthevaluationofguaranteesisamaterialissue.Infact,wearenotawareofasinglefirmthathasdemonstratedthatitisvaluingitsNNEGsusingadefensiblemethodology.Ourimpressionisthattheyareallgettingitwrong.EquityReleaseandtheGhostofEquitableLifeWehaveseenthismoviebefore.Acoupleofdecadesago,therewasascandalsurroundingEquitableLife.Theworld’soldestmutualinsurer,EquitableLifewasfoundedin1762.Inthemiddleofthe20thcentury,italsopioneeredGuaranteedAnnuityRate(GAR)optionsthat offered guaranteed fixed returns. At its peak in the 1990s, it had 1.5 millionpolicyholderswith fundsworth £26 billion undermanagement. However, it failed tovaluetheseoptionsproperly,andinsomecases,itdidn’tvaluethematall.Asaresult,itfailedtoprovideforthemproperly.Equitablecametogriefin2000whenitwasnolongerable to keep its promises. Therewas then a big outcry and the insurance regulatorysystemwasoverhauledtomakesurethatanEquitable-stylefiasconeverhappenedagain.Sotheproblemwasthatthecompanyhadbeenunder-valuingopaqueandapparentlyinnocuouslong-termguaranteesandtheundervaluationoftheseguaranteeseventuallybroughtitdown.Inbothcases,therewasatoxiccombinationofintellectualerrorandshort-termthinking.In the Equitable case, there was an underlying presumption that the guarantees inquestion,Equitable’sGARoptions,didn’treallymatterandthatanyproblemsthattheymightentailwerewellintothefutureanyway.Intheequityreleasecase,theintellectualerror involves a profoundmisunderstanding of option pricing theory by professionalactuaries,combinedwithamindsetonthepartofindustryleadersthatputsshort-termprofitabilityand‘competitiveness’aheadofnotionsoflong-termsustainability.WhenitcomestoNNEGvaluation,thismindsetprioritiseslowNNEGvaluationsoversoundNNEGvaluationsandtherestisobvious.

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The intellectual error centres around the underlying variable in the option pricingformula. ANNEG involvesaportfolioofputoptionsandwearedealingwithputsonforwardcontracts.Forexample,ifacustomertakesoutanERMattheageof70,thereisaNNEGletputforthepossibilitythattheERMloanmightendwhenthecustomeris71,anotherNNEGletputforthepossibilitythattheERMloanmightendwhenthecustomeris 72 and so forth. Each of these put options is issued now, but has a horizon (ordecrement)ofone,two,etc.yearsinthefuture.Thepricethatentersintoeachputoptionpricingequationistheforwardpriceoftheunderlying,andthedeliverableisahouse.Sofortheputoptionthatendsinfutureyear𝑡,theunderlyingistheforwardhousepriceforyear𝑡, thepriceagreednowforthehousetobedeliveredandpaidfor inyear𝑡.Thisapproachisbasedonstandardoption-pricingtheoryandisexemplifiedbyBlack(1976).7In actuarial circles, this typeof approach is often referred to as a ‘MarketConsistent’approachtoNNEGvaluation.TheproblemisthatanumberofpractisingactuariesintheUKequityreleasesectorhaveconvincedthemselvesthattheunderlyingpricethatisrelevantforputoptionpricingisnottheforwardhousepriceforyear𝑡butthefuturehousepriceorexpectedfuturepriceforyear𝑡.However,forwardandfuturepricesareverydifferentandtoconfusethetwoistocommitamajorerror.Thiserrorisabigdealbecauseinputtingtheexpectedfuturehousepriceintotheoption-pricingequationgivesverylowNNEGvaluations,whereasinputtingforwardhousepricesintoitgivesmuchlargerNNEGvaluations.Thissecond,incorrect,approach iscommonlyreferredto inactuarialcirclesas the ‘RealWorld’or‘DiscountedProjection’approach.AdifferencehoweverbetweentheEquitableLifeandequityreleasecasesisthatwhenEquitable started issuing GARs in the 1950s, the valuation of options was not well-understood.Theoptionpricingbreakthroughonlyoccurredin1973withthepublicationofthefamousarticlesbyBlack,ScholesandMerton(BlackandScholes,1973;Merton,1973),followedshortlyafterwardsbyBlack(1976).Boththeprinciplesandthenuancesofoptionvaluationhavebeenwellknownfordecadesandaretaughtinuniversitiesallovertheworld.Itiscurious,too,thattheUKactuarialprofessionalassociation,theInstituteandFacultyofActuaries(IFoA),hasyettospeakoutagainsttheseunsoundNNEGvaluationpractices.Onthecontrary,itisontherecordasitselfendorsinganumberofmisconceptionsaboutNNEGvaluation.Despite copious protestations to the contrary, theUK actuarial profession appears tohavelearnednothingfromthelessonsofEquitableLife,andsowelcometoEquitable2.Thosewhofailtolearnthelessonsofthepasthavetotaketheclassagain.

7SeeF.Black(1976)“ThePricingofCommodityContracts,”JournalofFinancialEconomics3:167-179.

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PurposeofthisReportThepurposeofthisreportistosetouttheissuesinvolvedinthevaluationofNNEGsandEquityReleaseMortgages(ERMs),i.e.,itprovidesahow-tohandbookforpractitionersworking onNNEG andERMvaluation issues. In doing so, it examines both valid (i.e.,MarketConsistent)andinvalid(i.e..,DiscountProjectionandTunaru)approaches,andexplainswhytheformerisvalidbutthelatterarenot.Weemphasisethatthisreportfocusesonlyonthevaluationissuesanddoesnotaddressbroader issues raised by NNEG under-valuation, such as the implications for publicpolicy.Weintendtoaddressthoseissuesinafollow-onreport.OrganisationofthisReportThisarticleisorganisedasfollows.Chapter2providesanintroductiontoequityrelease:itexplainstheproductsandthesignificanceoftheNNEG,andgivesanoverviewofrecentdevelopmentsinthesector.Chapter3explainsthebasicsofNNEGandERMvaluation.Thenext14chaptersexaminekeyinputsandotherfactorsrelevanttoNNEGandERMvaluation:theloan-to-valueratio(Chapter4),therisk-freerate(Chapter5),theloanrate(Chapter6), the theoryand calibrationof thenet rental yieldand thedeferment rate(Chapter 7), dilapidation (Chapter 8), volatility (Chapters 9), mortality (Chapter 10),long-termcare(Chapter11),delayedpossession(Chapter12),creditspreads(Chapter13),theimpactofdrawdownfacilities(Chapter14),prepayment(Chapter15)andfeesandcharges(Chapter16).Chapter17examinesERM-relatedscenarioanalysesandstresstestsandChapter18discussesthePRA’sERMGoodPracticePrinciples.Chapter19setsout theMarket-Consistent approach, Chapter 20 debunks the ‘Discounted Projection’approach,Chapter21examinestherecentTunarureportandChapter22discussesJustGroup’s NNEG valuationmodel. Chapters 23 and 24 examine technical actuarial andaccounting standards, andChapters25and26offer some recommendations forgoodvaluationpracticeanditsgovernance.

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ChapterTwo:IntroductiontoEquityReleaseTheHomeEconomicsofEquityReleaseAnERMatypeofloancollateralisedbyaproperty(‘house’),andthetypeofERMweareinterestedingoesasfollows.8Theloanistakenoutbyacustomerlateinlifewhoownsthepropertytheylivein.Thecustomerusestheloantosupplementtheirincome,helptheirchildrengetonthepropertyladderorwhatever.Unlikeanormalloan,thisloanhasnofixedenddateandinvolvesnoregularinterestpayments.Instead,theloanendswhenthecustomerexitsthehouse,eitherbydeathorbygoingintoanursinghome,andtheamountowedontheloanaccumulatesovertimeuntiltheloanisrepaid.9Atthetimeofexit,thelendertakespossessionofthepropertyandsellsittorepaytheloan.Ifthereareanyproceedsleftover,thesearereturnedtothecustomerortotheirestate.Theterm“equityrelease”ismisleading,however.AsJefferyandSmith(2019,pp.6,55)pointout:

It cannot be repeated too often that ERM is a misnomer. The equity in apropertyisnotreleaseditisborrowedagainst.Thevalueofthepropertytotheownerbecomesgeared.....Ashousepricesmoveupanddown(!),theloanremainsunchangedinvalue.

JefferyandSmithareright,buttheterm“equityrelease”issowidelyusedthatwearestuckwithit,andtheAmericanalternative(“reversemortgage”)hasissuesofitsown.So“equityrelease”itis.TherearetwomaintypesofERM.Thefirst,knownasaLifetimeMortgage(LTM),isastraightforwardmortgageloan,wherethelenderhandsovertheloanamountatthetimethecontractismade.ThesecondisadrawdownERM,inwhichthecontractprovidesforamaximumpossible loanamount,but theborrowerhasdiscretionoverhowmuchtodrawdownagainstthismaximumandwhentodoso,subjecttotheconstraintthatthetotal amount drawn down cannot exceed the stipulated maximum. Typically, theborrowerinadrawdownERMwouldmakeadrawdownwhenthecontractismade,andthenmakemoreinlateryears,asandwhenheorshefeelstheneedtodoso.Inthisreport,wefocusmainlyonthevaluationissuesrelatingtoLTMERMs.Thesearesimpler,butwediscussdrawdowninChapter15.Sufficetonotethatthevaluationissuesaremuchthesame,exceptfortheadditionalcomplicationsintroducedbyadrawdownfacilityandhowthatmightbeusedbytheborrower.

8 Wearenotconcernedherewithothertypesofequityreleaseproductsuchhomereversions,inwhichtheborrowersellsallorpartoftheirpropertyatlessthanitsmarketvalueinreturnforatax-freelumpsum,aregularincome,orboth,butstaysonintheirhomeasatenantwhopaysnorent.NorareweconcernedwithERMloansthatdonotincorporateNNEGs,sufficetonotethatmostincorporateNNEGsandthatallERMsissuedbymembersoftheEquityReleaseCouncildo. 9 Insomecases,theloancanalsoendbyearlyrepayment,butwedeferthatissuetoChapter16below.

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TheERMloanwillbetakenoutassomelowproportionofthepropertyvalue–40%istypicalfora70yearold,butLoantoValueratios(LTVs)tendtobelowerforloweragesandhigherforhigherones–andthelenderisprotectedagainstanyriskoflossforaslongastheloanvalueisbelowthevalueofthehouse.Theloanratewillbefixedattheinceptionoftheloan.Thevalueofthecollateral,thehouse,willvarywiththehouse’smarketprice.Typically,housepriceshave risen in recent years andwemight (ormightnot!) expect them tocontinue to rise, but we would not usually expect the house price to rise at a rateexceedingtheloanrate.Inanycase,housepricesareuncertainandsometimesfall,soexpectationsoffuturehousepricesareunlikelytobeexactlyrealised.AtypicalcaseisshowninFigure2.1:

Figure2.1:LoanEquityinaTypicalEquityReleaseMortgage

Wewouldcertainlyexpecttheloanamount(showninblue)toriseovertime,andwewouldusuallyexpectthehouseprice(inblack)torisetoo,butevenso,wewouldexpecttheloanamounttoriseatafasterrateandeventually,ifthecustomerliveslongenough,theblueloanamountlinewillcrossovertheblackhousepriceline.Thereaftertheloanamountwillexceedthevalueofthehouse,i.e.,theloanwillgointonegativeequity.Ifthecustomerexitsthehousebeforethepointofnegativeequity(whichis21yearsinFigure2.1),thenthelenderwouldberepaidinfull.Ifthecustomerexitsafterthatpoint,theloanwouldexpireinnegativeequity, i.e.,thevalue of the property collateral would not be enough to cover the accumulated loanamount.IntheabsenceofaNNEG,thelendercouldsuetheborrowerortheirestate,buttheremighthavefewassetsleft,especiallyiftheborrowerwasmovingintoaretirementhome and any remaining assets were being used to finance their care. Most ERMsincorporateaNNEG,however,andinsuchcasesthenegativeequitybecomesalossbornebythelender.

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AnotherwaytothinkabouttheERM-with-NNEGcontractisthatitgivesthelendertheminimumofthehouseprice(black)andloanamount(blue)lines.Thefactthatthelendergetstheminimumoftwovaluesindicatesthatthelenderisgrantingaputoptiontotheborrower.Thelender’spotentiallosswiththeNNEGinplaceisillustratedinFigure2.2,andlet’shenceforth assume for themoment that exit is due solely to death (althoughwewillrevisitthisassumptionlater):

Figure2.2:ERMLoanExpiresinNegativeEquity

Inthiscase,theborrowerdiesafter25yearsandthelendermakesthelossgiveninred,thedifferencebetweentheloanvalueandthehousepriceafter25years,relativetowhatthelenderwouldhavereceivedhadtherebeennoNNEGandthelenderbeenrepaidinfull.Naturally,thisloss(andwhetheranylossoccursatall)isuncertainbeforetheevent.Thetimingofdeathisuncertainandifthecustomerdiesearlythentherewouldbenolosstothelender.Butifthecustomerdieslaterthelendersuffersalossthatdependsamongother factors on the timing of death. Thus, the NNEG potentially exposes ERMs tolongevityrisk–theriskthatthecustomermightlivetoolong.ERMsarealsoexposedtohousepricerisk.ThisriskisillustratedinFigure2.3:

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Figure2.3:TheImpactofaFallinHousePricesonNegativeEquity

Thehousepricemightbeloweratthetimeofdeaththanthelenderexpectedittobe.Figure2.3showsacasewherethehousepricedeclinesinsteadofrising.Ifthecustomerdiesafter25years,thenitisclearfromacomparisonofFigures2.2and2.3thatthelenderwillsufferalargerlossduetothehousepricefall.ERMsarethussubjecttohousepriceaswellaslongevityrisk.Fromtheborrower’sperspective,takingoutanERMloanmightbeasuitablechoiceforanolder individualorcouplewhoareassetrichbutcashpoor,e.g., theymighthaveaneedforcashorwishforahigherstandardoflivinginretirement.OnecanalsoimagineadditionalcircumstancesinwhichanERMmightbesuitable,e.g.,becausetheirchildrenmaybeaffluentorbecausetheydon’twant to leavetheirchildrenany inheritance,orbecausetheymayhavenochildrenanddon’twanttoleavetheirhousetoacats’home.Forsuchpeople,aregularmortgagewouldnotnormallybepracticalbecausetheywouldnolongerbeworkingandthereforenothavetheincometorepaysuchamortgage.From the lender’s perspective, an ERM loan offers a high loan rate and is highlycollateralised,at least tostartwith. Itsmaindownside is the impactof theNNEG, thevaluationofwhichisthecorefocusofthisreport.10Tohelpformanintuition,considerthatanequityreleasemortgagecanbebrokendownintotwocomponents.Thefirstcomponentistheloanamountwhichrollsupatahighloanorroll-uprate,whichcurrentlyaveragesatabout5.25%fornewERMloans,butthereisconsiderabledisparityaroundthataverage.Nowthisloaniscollateralisedbythehouse,andtheloanatinceptionwillbesomefractionofthehousevalue.Forexample,fora 70 year old, the loan to (house) value ratio will be around 40%. The highcollateralisationoftheloanmeansthatwecanregardtheloanasclosetorisk-free:whentheborrowerexitsthehome,thelenderwilltakepossessionandusethehometogetitsloanrepaid.Thereisnodangerofdefault.Thesecondcomponent istheNNEG.Sothe

10 ItisoftenclaimedthatERMportfolios(orexposuretoERMfirms)aresuitableforpensionfundsbecausetheyarelong-termassetsthatarecorrelatedwithlongevityrisk.However,thosewhomakesuchclaimsoftenoverlooktheexposureofERMportfoliostohousingrisk.Wewillhavemoretosayonthisissueinalaterreport.

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valueoftheERMloantothelenderisthevalueoftheloan(whichwouldincludeexpectedprofitsontheloan,andwhichisnottobeconfusedwiththeloanamount)minusthevalueoftheNNEG.SupposenowthattheNNEGvalueisverylow.ThenitfollowsthatthevalueoftheERMtothelenderwillbeclosetothevalueoftheloancomponentoftheERM,becausetheNNEGvalueisverylowandsodoesn’treallymatter.TheERMisthenclosetobeingahighinterestloanthatiscertaintoberepaidinfull,whichisanattractivepropositiontothelender.Thus,ERMsarehighlyprofitabletothelenderiftheNNEGvalueisverylow.Atthe other extreme, if the NNEG value is very high, then an ERM loan might not beprofitable at all. A correct valuation of the NNEG is thus essential to determine theprofitabilityoftheloan.EquityReleaseisaRapidlyGrowingSectorThegrowthoftheequityreleasesectorisapparentinthefollowingFigure,whichplotstheamountslentsince2000:

Figure2.4:GrowthoftheEquityReleaseSector

Source: Equity Release Council: http://www.equityreleasecouncil.com/document-library/equity-release-market-report-autumn-2018/equity-release-market-report-autumn-2018.pdf

Thebluelineplotstheamountslentandtheredlineplotsthenumberofnewcustomers.Thelattershouldbetreatedwithsuspicionasitincludesreturningcustomers,i.e.,thosewithdrawdownfacilitywhosubsequentlydrewdown.Thinking fromamacroperspective, the lendingbroadly reflects changes inUKhouseprices,i.e.,the2008peakisclearlyvisiblewithsubsequentdeclineandthenrecovery.Thefinalnewlendingamountiswellabovethepreviouspeakin2008.Wecouldattributethepost-crisisgrowthtoanumberoffactorsincludingpossiblystrongermarketingandTreasury‘approval’oftheproduct,orperhapsmorenaturallytoincreaseddemandfor

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theproductasboomersreachretirementageandrealise that theycan’taffordanewwashingmachinebuthaveahouseworth£500k.Recentgrowthinthesectorhasbeenremarkable:

Since2016,theequityreleasemarkethasbeengrowingattheaveragerateof7.1 per cent each quarter. The amount of equity unlocked from people’shomesmore than doubled from £514m in the second quarter of 2016 to£1.08bninthefinalquarterof2018.11

The Equity Release Council’s latest (Autumn 2018) Market Report provides furtherdetails:

Atotalof38,912householdsaged55andoverusedequityreleaseproductsfrommembersofTheCounciltoaccesssomeoftheirpropertywealthduringH12018.Thisincluded21,490newplansagreedbycustomers,upby28%from16,805ayearearlier.A further 15,709 returning drawdown customersmadewithdrawals fromtheiragreedreservefundsbetweenJanuaryandJune,up25%yearonyear.…Thenumberofnewplansagreedin[2018]H1exceededtheentiresizeofthemarket in 2014 and represented an 81% increase since H1 2016. Activityamong new customers has increased during every half-year period in theinterveningtwoyears,asequityreleaseandhousingwealthhavetakenupamainstreampositionamongtheproductsandassetsthatformpartofmodernretirementplanning.Twothirdsofnewcustomers(65%)optedfordrawdownlifetimemortgagesinH1,while35%chose lumpsum lifetimemortgagesanda smallnumber(<1%)agreedhomereversionplans.(ERC2018,p.6,ouritalics)

TheECRreportoffersfurtherdetailaboutthegrowthofproductchoices:

Thegrowingbaseofequityreleasecustomersinrecentyearshasbeenmetwithagreaternumberofproductchoicesandflexibilities–helpingtomeethomeowners’increasinglycomplexneedsinlaterlife.AsofAugust2018,139productoptionswereavailabletoconsumers,morethandoublethenumber(58)seentwoyearsago.Therangeofproductoptionshasgrownbyover78%inthelastyearalonefrom78inAugust2017.Today’s equity release products also offer greater flexibilities thanks toongoingcompetitionandinnovationinthesector.Fourinfive(80%)productoptionsofferconsumersthechoicetomakead-hoc,penalty-freevoluntaryorpartialrepaymentsoftheirloan,upfrom68%ayearago.Therehasalsobeen

11L.Warwick-Ching“DramaticRiseinEquityReleasebyOver-55stoFundRetirement.”FinancialTimes4April2019.

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anincreaseinproductsofferingfixedearlyrepaymentcharges(ERCs),from49%inAugust2017to51%inAugust2018.Almost half of product options (45%) offer downsizing protection whichallows customers to downsize to a smaller property and repay the loan,withoutincurringanyERC.Inheritanceprotection,whichallowscustomerstoring-fenceasectionoftheirhousingwealthasaguaranteedminimumamounttopassontothenextgeneration,regardlessofthetotalinterestaccrued–isofferedby46%ofproducts.(p.7)

Andaboutcustomerprofiles,e.g.:

TheaverageageofnewcustomersduringH12018wastheclosestseentodate across the twomain categories of lifetimemortgages.At just over68yearsold,theaveragenewlumpsumcustomerwasbroadlyinlinewiththatseenoverthelastthreeyears.Theaveragedrawdowncustomerwasalmosttwoyearsolderandjustshortoftheir70thbirthday.(p.9)

Sothesectorisdoingnicely,thankyou,butthereisstilltheissueaboutNNEGvaluation.

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ChapterThree:TheBasicsofNNEGandERMValuation 1.IntroductionThischapterexplainsthebasicsofNNEGandERMvaluation.2.ExitProbabilitiesThe NNEG valuation model has two key ingredients: a set of expected house-exitprobabilitiesandaputoptionpricingmodel.Thehouseexitprobabilities(orexitprobabilitiesforshort)refertotheprobabilitiesthattheborrowerwillexitthehouse(andhenceterminatetheloan)overeachofthenext1,2,3,…etcyears.Forthetimebeing,letusassumeawaythepossibilityofearlyrepaymentof the loanandassumethat theborrower isasinglemalewho isexpectedtoexit thehouseinabox.Undertheseconditions,theexitprobabilityforyear𝑡isequalto(3.1)𝑒𝑥𝑖𝑡𝑝𝑟𝑜𝑏! = 𝑞! × 𝑆!where𝑞!isthemortalityrateforyear𝑡and𝑆!istheprobabilitythatanindividualalivenowwillsurvivetoyear𝑡.Notethat𝑆" = 1and𝑆! = (1 − 𝑞!#$)𝑆!#$forall𝑡 > 0. 12Theexitprobabilitiesforamalejustturned70areshowninFigure3.1:

12 The‘𝑞’terminologyforthemortalityrateisstandardandalittleunfortunateintheNNEGcontext,where𝑞issometimesusedtorefertothedefermentrate.Thereadershouldbearthisambiguityinmind,butthecontextshouldmakeitclearwhetheritisthemortalityrateorthedefermentratethatwearereferringto,anditismostlythelatter.

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Figure3.1ExitProbabilitiesforMalesCurrentlyAged70

Notes: Exit probabilities are based on CBD-M5 model (Cairns et alia, 2006, 2009) cohortmortality rateprojectionsusingmaleEngland&Walesdeaths ratedata estimatedover ages55:89andyears1971:2017.Source:Life&LongevityMarketsAssociation.13

Thelefthand(low𝑡)exitprobabilitiesareclosetothelow𝑡mortalityratesandreflecttheearlyhighsurvivalprobabilities(i.e.,thatpeopleaged70haveahighprobabilityoflivingatleastafewyears),andthelater(high𝑡)exitprobabilitiesprimarilyreflectthefactthattheprobabilitiesoflivingtoextremeoldagearelowandapproachzerointhelimit.143.ValuationIssuesandthePutPricingModelThepresentvalue𝐸𝑅𝑀oftheEquityReleaseMortgageloancanbeconsideredtobethepresentvalue𝐿ofarisk-freeloan,onewhichisguaranteedtoberepaidinfull,minusthepresentvalue𝑁𝑁𝐸𝐺oftheNNEGguarantee:(3.2)𝐸𝑅𝑀 = 𝐿 − 𝑁𝑁𝐸𝐺Theoriginalloanamountgrowsattheloanrate(sometimescalledtherolluprate)𝑙fromitscurrentamountuntilthetimewhentheloanends.Therefore𝐿isgivenby(3.3)𝐿 = ∑ [𝑒𝑥𝑖𝑡𝑝𝑟𝑜𝑏! × 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑙𝑜𝑎𝑛𝑎𝑚𝑜𝑢𝑛𝑡 × 𝑒(&#')!]!

13 SeeA. J. G. Cairns,D. Blake andK.Dowd (2006) “ATwo-FactorModel for StochasticMortalitywithParameterUncertainty,”JournalofRiskandInsurance,73(4):687-718;andA.J.G.Cairns,D.Blake,K.Dowd,G.D.Coughlan,D.Epstein,A.Ong,andI.Balevich(2009)“AQuantitativeComparisonofStochasticMortalityModelsUsingDatafromEngland&WalesandtheUnitedStates,”NorthAmericanActuarialJournalVolume13(1):1-35.14WehavemoretosayonthederivationoftheexitprobabilitiesinChapter10.

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where𝑒𝑥𝑖𝑡𝑝𝑟𝑜𝑏!istheprobabilityofexitingthehouseinperiod𝑡and𝑟istherisk-freeinterestrate.15Thevaluationof𝐿isstraightforward.𝑁𝑁𝐸𝐺isthesumoftheproductsoftheexitprobabilitiesforeachfuturetime𝑡andthepresentvalueoftheNNEGguaranteeforeachfuturetime𝑡:(3.4)𝑁𝑁𝐸𝐺 = ∑ [𝑒𝑥𝑖𝑡𝑝𝑟𝑜𝑏! × 𝑁𝑁𝐸𝐺!]! where𝑁𝑁𝐸𝐺!isthepresentvalueoftheNNEGguaranteeforperiod𝑡.Thequestionisthenhowtovalueeachoftheseindividual𝑁𝑁𝐸𝐺!termsandthencetheNNEGguarantee.RecallthattheNNEGgivesthecustomer(orthepersonactingforthecustomer)therighttorepaytheloanbypayingthelendertheminimumoftheloanvalueorthehousepriceatthetimeofdeath.Theright to repay theminimumof two futurevalues (oneofwhich, the futurehouseprice,isuncertain)atsomegivenfuturetimeimpliesaEuropeanputoptiongrantedbythelendertotheborrower.Sincethetimeofexerciseisuncertain,wecanthinkoftheNNEGasinvolvingaportfolioofsuchputoptions.Inthecaseofourputoptionstheunderlyingvariableisaresidentialproperty(‘house’)ormoreprecisely,aforwardcontractonahouse,andweshouldthinkofahouseasanassetthatbearsacontinuousyieldintheformofa(net)rentalyield.Thisnetrentalyieldreflectstheusebenefitoflivinginthehouseortherentalincomewemightgetbyrentingthehouseout.A natural option pricingmodel to use in these circumstances is the Black ’76model(Black,1976).Black’76isanappropriatepricingmodelwhentheunderlyingisaforwardcontractwithamaturitycoterminouswiththatoftheoptionitself.Thismodelisanear-relativeofthefamousBlack-Scholesmodel(BlackandScholes,1973;Merton,1973).16TheBlack ’76 formula for theprice𝑝! of aEuropeanputoptionwithmaturity 𝑡 onaforwardcontractonacommoditybearingacontinuousyieldqisgivenbytheformula:(3.5)𝑝! = 𝑒#'![𝐾!𝑁(−𝑑)) − 𝐹!𝑁(−𝑑$)]where𝑟istherisk-freerateofinterest,𝐾!isthestrikeorexercisepriceformaturity𝑡,𝐹!istheforwardhousepricefortime𝑡,thefunction𝑁(… )isthevalueofthecumulative

15 Notetheimplicitdistinctionherebetweentheloanamountorrolleduploanamount,ontheonehand,and𝐿,the(economic)valueoftheloan,ontheother.Theformeristheamountloanedplustheinterestaccumulated since the inception of the loan,whereas the latter is the value of the loan to the lender,includingtheexpectedprofitontheloan.AconcreteexampleofthedistinctionbetweenthetwoisgiveninTable3.1.Notetoothattheeconomicvalueoftheloanisnottobeconfusedwiththeaccountingbookvalueoftheloan,whichisanotherissueagain. 16ThechoiceofoptionmodelisdiscussedinmoredetailintheAppendixtothischapter.

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standardnormaldistributionatthevaluespecifiedinbrackets,and𝑑$and𝑑)aregivenby:(3.6)𝑑$ = [𝑙𝑛(𝐹!/𝐾!) + 𝜎)𝑡/2]/(𝜎√𝑡)(3.7)𝑑) = 𝑑$ − 𝜎√𝑇where𝜎isthevolatilityoftheforwardhouseprice.Thestrikeprice𝐾!isthentherolleduporaccumulatedloanamountbyperiod𝑡:(3.8)𝐾! = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑙𝑜𝑎𝑛𝑎𝑚𝑜𝑢𝑛𝑡 × 𝑒&!andtheforwardprice𝐹! ,thepriceagreednowtobepaidonpossessioninperiod𝑡,is:(3.9)𝐹! = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡ℎ𝑜𝑢𝑠𝑒𝑝𝑟𝑖𝑐𝑒 × 𝑒('#*)!where𝑞isthedefermentrate,namelythediscountrateappliedtothecurrenthousepricetogivethedefermentprice,thepricewewouldagreetopaytodaytotakepossessionofthehousein𝑡years’time.Thusthedefermenthouseprice𝑅!isgivenby:(3.10)𝑅! = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡ℎ𝑜𝑢𝑠𝑒𝑝𝑟𝑖𝑐𝑒 × 𝑒#*! .Thedifferencebetweentheforwardhousecontractandthedefermenthousecontractisthatwiththeforwardwesettlewhenwetakepossessionin𝑡years’time,butwiththedeferment contractwe settle today.17 Therefore, the deferment house price𝑅! is thepresentvalueoftheforwardprice,wherethepresentvalueisobtainedbydiscountingattherisk-freerate𝑟.Itisimportanttonotethatthedefermenthousepricewillbelessthanthecurrenthouseprice𝑆"becausethedefermentrate𝑞 > 0.Theforwardhouseprice𝐹!shouldnotbeconfusedwithfuturehousepricesorexpectedfuturehouseprices:

• Forwardpricesforfutureperiod𝑡areknown(orcanbeapproximated)nowandweneedtobeabletopriceoptionsusinginformationavailablenow.

• Optionscannotbepricedusingfuturehousepricesbecausefuturehousepricesarecurrentlyunknown.

• ExpectedfuturepricesdonotappearintheBlack‘76optionpricingformula.WeshouldalsokeepinmindthatalthoughtheoriginalBlack‘76articlediscussedoptionsonfutures,futurespricesarethepricesoffuturescontracts,aformofforwardcontract,notactualorexpectedfuturepricesofanysort.Amistaketobeparticularlyavoided–theonecommonamongUKERMactuaries–istoconfuseforwardandexpectedfutureprices.Thismistaketypicallymanifestsitselfinthe

17 SeePRASS3/17(p.12,note2).

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inputting of an assumed expected house price inflation rate into (3.9) instead of theforwardrate𝑟 − 𝑞.To repeat: it is not the future or expected future price of a contract for immediatepossessionthatweuseintheoptionpricingequation,butratherthecurrentpriceofacontractforfuturepossession.Theapproachsetouthereisanexampleofwhatisknowninactuarialcirclesasa‘marketconsistent’ approach, which gives ‘market consistent’ valuations. We would define amarketconsistentvaluationasa‘fairvalue’valuationbasedontheIFRSdefinitionofafairvalueprice,namely

Thepricethatwouldbereceivedtosellanassetorpaidtotransferaliabilityinanorderlytransactionbetweenmarketparticipantsatthemeasurementdate18

Analternative(andforourpurposespracticallyequivalent)definitionisthatprovidedbyTimGordon:hedefinesamarketconsistentvaluationasonewhichisconsistentwithmodernfinancetheoryasthetermisusedinExley,MehtaandSmith(1997).19Weshalltakethevalidityofthisapproachasgivenforthetimebeing,butwewillprovideajustificationforitinlaterchapters.4.AValuationExampleWe now provide some illustrative valuations based on plausible input parametercalibrationvalues.Thebaselineparameterinputsare:

• Currentageofcustomer=70,atypicalageforERMs.20• Loantovalueratio=40%.21• Risk-freerate𝑟=1%.• ERMloanrate𝑙 = 5.25%.• Defermentrate𝑞 = 4.2%.• Volatility𝜎 = 20%formalesaged70.

Allratesarein%p.a.

18 See,e.g.,https://www.iasplus.com/en/standards/ifrs/ifrs13. 19 SeeT.Gordon(1999)“ThePriceofActuarialValues.”PaperpresentedtotheStapleInnActuarialSociety(16February)andC.J.Exley,S.J.B.MehtaandA.D.Smith(1997)“TheFinancialTheoryofDefinedBenefitPensionSchemes,”BritishActuarialJournal3(4):835-966.20Implicitly,weareassumingasinglemalejustturned70.Inthecaseofasinglefemale,wewouldexpectdeath/exittooccursomewhatlater,whichwouldincreasethevalueoftheNNEG.Inthecaseofacouple,wewouldexpectevenlaterexit,whenthelongestsurvivingmemberofthecoupleexitsthehouse.21A40%LTVratiofora70-yearoldappearstobeapproximatelyinlinewithcurrentindustrypracticefornewERMloans.Wewillhavemoretosayonthissubjectinthenextchapter.

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Wewilldiscussthesecalibrationsinlaterchapters.Weassumeanillustrativehousepriceof£100which,combinedwiththeassumedloantovalueratioof40%,impliesaloanamount=£40.The exit probabilities are derived fromprojections of futuremortality rates obtainedusingtheM5versionoftheCairns-Blake-Dowdmortalitymodel(seeCairnsetalia,2006,2009)calibratedonEngland&Walesmalemortalitydatafortheperiod1971to2017and spanningages55 to89.Thedata are taken from theLife andLongevityMarketsAssociationdatabase(llma.org).TheM5-CBDmodelisparticularlysuitableforoldageprojectionsanditsgoodnessoffitandperformanceevaluationareassessedinCairnsetalia(2011)andDowdetalia(2010a,b).OurbaselineNNEGvaluationresultsareshowninTable3.1:

Table3.1:BaselineERM/NNEGValuationsCurrentHousePrice LoanAmount 𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

£100 £40 £81.72 £41.15 £40.63Notes:𝐿isthepresentvalueoftheloancomponentoftheEquityReleaseMortgage,𝑁𝑁𝐸𝐺isthepresentvalueoftheNNEGguarantee,and𝐸𝑅𝑀isthepresentvalueoftheEquityReleaseMortgage.Basedonthebaselineassumptions:maleaged70,𝐿𝑇𝑉=40%,𝑟=1%,𝑙=5.25%,𝑞=4.2%and𝜎=20%.ExitprobabilitiesasperFigure3.1.Giventheageofthecustomer,theexpectedpresentvalue𝐿oftheperfectlycollateralisedloanis£81.72.𝑁𝑁𝐸𝐺isvaluedat£41.15andsothevalueoftheERM,𝐸𝑅𝑀,isequalto£81.72–£41.15=£40.63.Itissometimesconvenienttoreporttheseresultsintermsoftheratiosof𝐿,𝑁𝑁𝐸𝐺and𝐸𝑅𝑀totheloanamountasinTable3.2:

Table3.2:ValuationsDividedbyLoanAmount𝑳/Loanamount 𝑵𝑵𝑬𝑮/Loanamount 𝑬𝑹𝑴/Loanamount

204.4% 102.9% 101.6%Notes:AsperTable3.1.Weseethat𝐸𝑅𝑀,forexample,is106.6%oftheloanamount.5.SensitivitiesofValuationstoKeyInputParametersTable3showsthesensitivitiesof𝐿,𝑁𝑁𝐸𝐺and𝐸𝑅𝑀tochangesinkeyparameterinputs.Theseareexpressedinelasticityform,i.e.,wheretheelasticityoftherelevantoutputwithrespecttoachangeinaninputisthe%changeintheoutputdividedbythe%changeintheinput.

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Table3.3:SensitivitiesofValuationsinElasticityFormElasticitywrt 𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

𝑟 -0.18 -0.31 -0.05𝑙 0.96 1.66 0.24𝑞 0 0.40 -0.40𝜎 0 0.30 -0.30LTV 1 1.55 0.45

Notes:AsperTable3.1.TheseresultsindicatethatNNEGvaluationsarehighlysensitivetochangesinthe𝑙andLTVinputparametercalibrations.ItisalsointerestingtonotethattheERMvaluationsaremuchlessso,becauseoftheoffsettingimpactsontheloanvalueandNNEGvaluations.

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AppendixtoChapterThree:TheChoiceofOptionPricingModelInChapterThreewesuggestthatBlack’76isanappropriateputpricingmodelwhentheunderlying is a forward contractwithamaturity coterminouswith thatof theoptionitself.WhynotuseBlack-Scholes(BS)instead?AshallowbutcorrectresponseisthatBSisnotappropriatebecauseitisbasedontheassumptionthattheunderlyingdoesnotbearanyyield,whereasBlack’76isappropriatebecauseitallowsforsuchayield.Theyieldinthiscontextwouldbethenetrentalyield.However,thisdeficiencyofBSiseasilyfixed.Itiswell-knownthatBScanbeadaptedforanunderlyingthatpaysacontinuousdividendyield𝑞ifwereplace𝑆intheBSformulawith𝑆𝑒#*! .22If wemake this adaptation, then we can use BS because the underlying can also beinterpretedasa spotasset thatpaysacontinuousdividendyield. In thiscase,BSandBlack’76willbeequivalent.23Todemonstratethisequivalence,startwiththerelationshipoftheforwardtothespot:(3A.1)𝐹 = 𝑆𝑒('#*)!(3A.1)implies(3A.2)𝑆 = 𝐹𝑒(*#')!NowconsidertheBlack-Scholesputprice:(3A.3)𝑝 = 𝑒#'!𝐾 × 𝑁(−𝑑)) − 𝑒#*!𝑆 × 𝑁(−𝑑$)where(3A.4)𝑑$ = [ln(𝑆/𝐾) + (𝑟 − 𝑞 + 𝜎)/2)𝑡]/[𝜎√𝑡](3A.5)𝑑) = 𝑑$ − 𝜎√𝑡Substitute𝐹𝑒(*#')!for𝑆toobtain:(3A.6)𝑝 = 𝑒#'!𝐾 × 𝑁(−𝑑)) − 𝑒#*!𝐹 × 𝑒(*#')!𝑁(−𝑑$)

22See,e.g.,J.C.Hull(2003)Futures,OptionsandOtherDerivatives,fifthedition,p.268.23We could also price our puts by tweaking the Garman-Kohlhagen foreign currency option model(GarmanandKohlhagen,1983)orbyusinganappropriatespecialcaseoftheMargrabeoption,theoptiontoexchangeoneassetforanother(Margrabe,1978),buttheseapproachesarealsoequivalenttoBlack‘76.See M. B. Garman and S. W. Kohlhagen (1983) “Foreign Exchange Currency Options,” Journal ofInternational Money and Finance 2(3): 231-237; andW. Margrabe (1978) “The Value of an Option toExchangeOneAssetforAnother,”JournalofFinance33(1):177-186.

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= 𝑒#'𝑑$ = [ln(𝐹𝑒(*#')!/𝐾) + (𝑟 − 𝑞 + 𝜎)/2)𝑡]/[𝜎√𝑡]= [ln(𝐹/𝐾) + (𝜎)/2)𝑡]/[𝜎√𝑡](3A.8)𝑑) = 𝑑$ − 𝜎√𝑡whichistheBlack76model!Theargumenthasbeenputtousthatsincetheforwardismorevolatilethanthespot(onaccount of the volatility of r and q impacting the forward rate), weshould prefer Black Scholes, which is priced off the volatility of the spot.The Black Scholes price will then be cheaper, and the NNEG will be lessexpensive.This objection is incorrect. Let’s assume that by BS we are referring to BS with theadjustmentforacontinuousdividendyield,i.e.,BSwith𝑆𝑒#*!astheunderlyingratherthanBSwith𝑆astheunderlying.24Inthatcase,ourresponsewouldbethatthisobjectionmustbe incorrectbecausebothmodelsareequivalent.Consideralsothe fundamentalmechanicsofoptionpricing.Theoptionpricereflectsthepayoffatexpirylessthecostofhedging,andthecostofhedgingresultsfromtheconstantrebalancingdrivenbychangesindelta,buttheoptiondeltasarethesameanddrivenbythe𝑑$term:

(3A.9)BSdelta=B76delta=−𝑒#'!𝑁(−𝑑$)Forbothmodels,the𝑑$termisdrivenbothbychangesintheassetprice,andbychangesinrandq. In theBlackScholesmodel this impact isexplicit,whereas inBlack76thisimpactisimplicitinthedefinitionof𝐹.

24Asdiscussedalready,ifwearenotmakingthisassumption,thenwearedealingwithaversionofBSthatisnotappropriate.

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ChapterFour:Loan-to-ValueRatiosTheLoan-to-Valueratioistheratiooftheloanamount,ortheoriginalloanamountplusrolledupinterest,tothepriceofthepropertythathasbeenmortgaged.TheRelationshipofLTVtoAgeERMlendershavetodecidehowmuchtoloanagainstthemortgagedproperty,andakeyfactortobeconsideredistheageoftheborrower.Giventhatthe loanrate is likelytoexceed any future house price inflation (hpi) rate, then the longer the loan lasts, thegreaterthelikelihoodthattheloanwillgointonegativeequity.Thereforelenderswillofferlowerloanstoyoungerborrowers.Anexampleofarulegoverningthisage-dependencyisthe‘age–40’rulesetoutbyHostyetalia(2007,p.31)andusedintheirNNEGanalysis:“Maximuminitialloantovalueratio(MLTV)15%atage55increasingby1%foreachyearofageto50%atage90(youngerageforjointlifecases).”OneformstheimpressionfromHostyetaliathatthisrulewasinuseintheperiodbeforetheywrote.(Afterall,ifitwasnot,thenwhywouldtheyciteit?)Ifthisviewiscorrect,thenitwouldappearthatLTVs/ageratioshaverisensincethenandthat lendersnowroutinelyofferlargerLTVsthantheyusedto,tolendersofthesameage.Ourevidencecomesfromtwosources.ThefirstisfromarecentspeechbyDavidRule,thePRA’sExecutiveDirectorofInsuranceSupervision(Rule,2018).Mr.Rulemakesthefollowingstatement:

Chart5plotsloan-to-valueratiosagainstageforequityreleasemortgagessoldbylifeinsurersin2017.Thepinkswatheshowsinsurers’risklimits.Thegreatmajorityofmortgagesarewithinriskappetites,butsomeloansexceedthelimits.Thismayormaynotbeaproblem.Forexample,itcouldbeexplainedbymedicalunderwritingofthemortgagethatjustifiesahigherloan-to-valueratio for a younger customer if their life expectancy is impaired. (Ouremphasis)

TheChart5hereferstoisreproducedbelow:

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Hedoesn’tdefine“risklimits”butwemightinterprettheselimitsasthosesuggestedbyagoodpracticerulesi.e.,arecommendedltv/agerulewithsomediscretionaroundit.Thekeyphraseishereis“soldbylifeinsurersin2017,”sothesearenewlymintedERMsandnothistoricalonesThe relatively small number of points above the bounds seem to us to be thosewithimpairedlives,whowouldbeofferedbetterLTVtermsthanthoseinnormalhealth.Thelargenumberbelowwouldseemtoustoreflecttheimpactofun-drawdownbalancesindrawdownERMs.Thepinkriskboundssuggesttousthatfirmsareusingsomethingclosetoa‘modifiedage–30’rulethatgoesasfollows:

• Forages55to80:LTV=age/100–30.• Forages81to85:ΔLTV=Δage/100.• Forage86+:ΔLTV=0.

Wehavealsoseenexamplesfromcurrentindustrypracticethatareconsistentwiththis‘modifiedage–30’ruleandleadustobelievethatthisrule(orsomethingfairlyclosetoit)isnotabadapproximationtotheLTVsincurrentfreshmintedERMs.Thispointmade,therecanbeconsiderablevariationinLTVsacrossproducts(see,e.g.,LegalandGeneral(2019)orTunaru(2019,p.63,Table12)).25

25LegalandGeneral“FlexibleLifetimeMortgage”(2019):https://www.legalandgeneral.com/adviser/files/retirement/literature-and-forms/product-summary/Q0052431.pdf

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TheImpactofRollupandPastHousePriceGrowthonHistoricalLTVsOnemustalsoconsiderthatafteranERMloanismade,LTVswillevolveafterwardsinlinewiththeloanrateandsubsequentlyrealisedhousepriceinflation.If𝐿𝑇𝑉(0)istheinitialLTV,after1yeartheLTVwillbe(4.1)𝐿𝑇𝑉 = 𝐿𝑇𝑉(0) × 𝑒&#+,-($)where𝑙=(assumedconstant)loanrateandℎ𝑝𝑖(1)=housepriceinflationafterthefirstyear.After2years,theLTVwillbe(4.2)𝐿𝑇𝑉 = 𝐿𝑇𝑉(0) × 𝑒&#+,-($) × 𝑒&#+,-())andsoforth.WecanformsomeideaofhowtheLTVsfromloansmadeinthepastwouldhavemovedbyasimplehistoricalsimulation.Let’sassumethatERMloansweremadeinthepastineachoftheyearsfrom2000to2017,andlet’sassumethattheseweremadeonthebasisofa30%LTVandaloanrateof7%,notingthatthesecalibrationswouldbereasonablefor this period. The next figure shows how the LTVs would have subsequently‘performed’.

Figure4.1:HistoricallySimulatedLTVs:2000to2017

Notes:Basedonloanrate=6%andLTV=40%whentheloanistakenout.

Tointerpretthefigure:anERMwitha30%LTVtakenoutin2000wouldnowhaveanLTVofabout39%iftheloanwerestillactivein2017;a30%-LTVloantakenoutshortlybefore the crisis would have an LTV of about 51% if the loan were still active; and(naturally!)a30%-LTVloantakenoutin2017wouldstillhaveanLTVof30%in2017.Weseethattheloansmostatrisk(inLTVterms)werethosetakenoutshortlybeforethecrisis.

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ThedrivingfactorbehindtheseLTVmovementswasthehpi:thestrongerthehpi,thelowerthegrowthoftheLTV.We should however bear in mind that past performance is no guarantee of futureperformance.FutureLTVsDependonFutureHousePriceGrowthForanygivenLTV,itsfuturepathwilldependonfuturehouseprices.RemembertoothatiftheLTVexceeds100%whenexitoccurs,thentheNNEGwillbetriggeredandthelenderwilllosethedifferencebetweentherolled-uploanamountandthehousepriceatthattime.

Figure4.2:FutureHousePriceGrowthandthePathofLTVs

Notes:Basedonloanrate=6%andLTV=40%whentheloanistakenout.

TherapidratesofgrowthofthetoptwoLTVsplots(i.e.,thoseforℎ𝑝𝑖 = 0%inblueandforℎ𝑝𝑖 = 2%inred)arestriking.Inthefirstcase,LTVhits100%injustover15yearsandthesecondcaseithits100%in23years.Inthethird(ℎ𝑝𝑖 = 4%,black)case,LTVhits100%in46years.Themessage is that lendersneedahigh futureℎ𝑝𝑖 rate tokeep theirpotentialNNEGlossesdown.Forexample, ifhousepricesremainflatandtheborroweris70whenhetakesouthis loan, then there is a good chance thathewill live the15ormoreyearsrequiredtotriggertheNNEG.Bycontrast,ifhousepricesgrowat4%p.a.,thenthesameborrower is vanishingly unlikely to live the 46 years needed to trigger the NNEG.Consequently,anERMloanishighlyexposedtohousepricerisk.TheexposureofERMstohousepriceriskisfurtherillustratedbyasimplestresstest.Supposethat thesameborrowertakesout thesameERMloan,andthenthenextdayhousepricesfall40%.TheLTVis40%onthedaytheloanistakenout,butthe40%housepricefallimpliesthattheLTVjumpsto(1/(1-0.4))*40%=66.7%thenextday!Ifhouse

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pricesthengrowatzeropercent,theLTVwillhit100%in8years;iftheygrowat2%theLTVwillhit100%in12yearsandiftheyriseat4%,theLTVwillhit100%in22years.Thechancesofsuchanoutcomearepresumablysmall,butthepointistoillustratetheriskexposure.AmorelikelyadverseoutcomewouldbeaJapantypescenarioinwhichhousepricesdonotfallsuddenly,butinsteadfallsecularlyoveralongtime.UnderthisJapanscenario,LTVwouldhit100%injust13years.

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ChapterFive:TheRisk-FreeRateWecanobtainestimatesoftherisk-freeratefromtheimpliedyieldsongilts,theso-calledyieldcurve.Figure5.1showsaplotoftheBankofEnglandyieldcurve:

Figure5.1:BankofEnglandSpotRates

Source:FinancialTimes,downloaded12March2020.If onewished to use a single ‘representative’ risk-free rate, then our earlier baselinechoiceof1%isnotunreasonable.However,inamoresophisticatedanalysisthereisnogoodreasonnottousethewholeyieldcurve,withspotratesformaturitiesgreaterthan30yearsset,e.g.,asequaltothe30-yearyield.

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ChapterSix:TheLoanRateTheloanrateorrolluprateistheratethatthelenderchargesontheERMloan.Thisratewillbe fixed for the lifeof the loan,but therateappliedtodrawdownERMloanswilltypicallybe floating, in thesense that therateapplied toadrawdowntypicallyvariesdependingonwhenthedrawdownsaremadeonaloan.InhisNNEGreport,Tunaru(2019,p.29)providesaniceplotoftheevolutionofERMratessince1999:

Figure6.1:EvolutionofERMrates

Notes:Tunaru(2019,Figure8).

Thisplotshowsthatloanrateswereabout7%in1999,thenroseshortlythereaftertojustover8%,beforetrendingdownwardstoabout5%in2018.IntheirAutumn2018marketreport,theEquityReleaseCouncilreport(p.7)thatasofJuly2018,theaverageloanrateacrossproductswas5.22%andthisvalueisconsistentwiththelatestvalueintheTunaruplot.Howmuchvariationthereisacrossproductsisdifficulttosay,butwehaveseencurrentratesthatvaryfrom4.15%AERto6.78%AER.ThelatestavailableloanratesquotedbytheERCandshownintheTunaruplotsuggestthatabaselineloanrateof5.25%mightbeappropriateasadefaultvalue.IfonewantedtoobtainavaluationforaspecificERMloanorclassof loans, thenonewouldwanttousetheloanratethatappliestothatspecificloanorloanclass,Indoingso,wewouldalsodistinguishbetweenthetwocasesofanewloanandanhistoric(i.e.,old)one.Intheformercase,wewoulduseanappropriatecurrentlyofferedloanrate.Inthesecond,wewouldusethehistoric loanrate, i.e., the loanratethatprevailedwhenthe

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contractwasfirstmade.Sinceloanrateshavebeentrendingdown,thesehistoricalrateswouldusuallybehigherthancurrentones.ThefirstcasewouldberelevantifoneweretryingtovaluetheNNEGorERMofaloancurrentlybeingconsidered.ThesecondcasewouldberelevantifoneweretryingtovaluetheNNEGorERMofanERMfirm’sloanbook.

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ChapterSeven:NetRentalYieldandDefermentRateNetRentalYieldvsDefermentRateLet’s start by some clarifying definitions. The net rental yield is the amount that thelandlord receives after deducting for void,management costs andmaintenance costs,dividedbythepropertyprice.Thedefermentrate,asdefinedbythePRA,is“thediscountratethatappliestothespotpriceofanassetresultinginthedefermentprice,wherethedefermentpriceisthepricethatwouldbeagreedandsettledtodaytotakeownershipoftheassetatsomepointinthefuture”.26Thedefermentpricereflectstheforegoneincomeoruseduringthedefermentperiod.Thesetworatesarefrequentlyruntogether,andasweshallshowshortly,theyareinfactmathematicallyidentical,buttheyaredefineddifferently.Inthischapterweshallshowhowtoderiveonefromtheother,basedonthedefinitionsgiven.ProofthattheDefermentRateEqualstheNetRentalYieldFromthedefinitionabove,thedefermentrateisthediscountratethatwhenappliedtothefreeholdpriceofvacantpossessionresultsinthepriceofdeferredpossession.Thedeferment rate itself is not directly market observable, but it can be estimated as afunctionofmarketvariables.Themethodproposedhereusesnetrentalyields,asfollows.Let𝑑bethecurrentnetnominalannualrental,thecurrenttimebeingthebeginningoftheyear.(Weuse‘𝑑’herebecausetheapproachweareusingderivesfromthedividenddiscountmodel,where‘𝑑’isusedtoreferto(nominal)dividends.)‘Net’meansthegrossor headline rental paid by tenants, less the costs incurred by the lessor such asmanagement,maintenanceand theexpectedcostsofvoidoremptyperiodswhile thepropertyisbeingre-let.Thenweshallshowthat

(7.1)𝑞 = 𝑑/𝑆where:𝑞thedefermentrate;𝑑isthenetrentalasabove,withthecurrenttimebeingthebeginningoftherentalyear;and𝑆istheestimated‘spotprice’,i.e.,thefreeholdvalueofvacantpossessionestimatedasthemarketvalueofanidenticalorsimilarpropertynotencumberedbyaleasehold.Assumethatthevalue𝑆ofaperpetualincomeproducingassetisthesumofthepresentvalues of its individual cashflows. (This assumption is the bedrock of practically allfinancial theory.27) Assume that the discount factor is given by 1/(1 + 𝑟 + 𝜋 + 𝜆),where𝑟 istheriskfreerate,𝜋theriskpremiumdemandedbyinvestorsfortakingon

26 PRA SS 3/17, July 2018. 27Itunderlies,forexample,inthediscountdividendmodel(e.g.,Gordon,1959)withpropertypricesandrentals taking the place of stock prices and dividends. See, e.g.,https://en.wikipedia.org/wiki/Dividend_discount_model.

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riskycashflows,and𝜆theilliquiditypremiumdemandedbyinvestorsforcashflowsthatcannoteasilybeexchangedatmarket.Assumealsothatcashflowsgrowataconstantrate𝑔,sothatfuturecashflowsare𝑑(1 + 𝑔),𝑑(1 + 𝑔)),etc.Define𝑦asfollows(7.2) 1/(1 + 𝑦) = (1 + 𝑔)/(1 + 𝑟 + 𝜋 + 𝜆).Thenfromthedefinitionsandassumptionsabove(7.3) 𝑆 = 𝑑[1/(1 + 𝑦) + 1/(1 + 𝑦)) + 1/(1 + 𝑦).… ]and from the definition of the deferment price𝑅/ as the present value of the futurecashflowsminus the present value of the first 𝑛 cashflows, i.e. minus the “foregoneincome”(7.4)𝑅/ = 𝑑[1/(1 + 𝑦)/0$ + 1/(1 + 𝑦)/0) + 1/(1 + 𝑦)/0.… ].Divideeverytermin(7.4)by(1 + 𝑦)/,expressintermsof𝑆,andnotethat𝑦meetsthedefinitionof𝑞asthediscountrateweapplytothe‘spot’price𝑆togivethedefermentprice𝑅/ (7.5)𝑅/ = 𝑑[1/(1 + 𝑦) + 1/(1 + 𝑦)) + 1/(1 + 𝑦).… ]/(1 + 𝑦)/= 𝑆/(1 + 𝑦)/= 𝑆/(1 + 𝑞)/.Define𝑋as𝑆/𝑑.Then,substituting𝑋into(7.3),andsubstituting𝑞for𝑦(7.6)𝑋 = 1/(1 + 𝑞) + 1/(1 + 𝑞)) + 1/(1 + 𝑞).+...Multiplybothsidesby(1 + 𝑞)(7.7)𝑋(1 + 𝑞) = 1 + 1/(1 + 𝑞) + 1/(1 + 𝑞))+...= 1 + 𝑋.Dividebothsidesby𝑋andsubtract1,substitute𝑆/𝑑for𝑋andrearrange:(7.8)𝑞 = 1/𝑋 = 𝑑/𝑆whichwastobeproved.Observethat(7.8)holdswhateverisontherighthandsideof(7.2),i.e.,itholdswhateverconstant growth rate we choose, and whatever the interest rate, risk premium orilliquiditypremiumthatmightberequiredbyinvestors.Whichisreallystrange,too.Tunaru(2019,p.50)says“Forrisk-managementcalculationpurposesthen,itisveryimportanttohaveanaccuratemeasurementof𝑞.Lackofdataavailability and long-term horizon makes this exercise extremely difficult, if notpractically impossible.” This claim seems plausible.Who could predict unobservablessuch as dividend growth, risk premia and so on? Yetwe can calculate𝑞 by a simpleformulausingvariablesthatareeitherdirectlyobservedoreasilycalibrated!

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Togiveanexample,SheffieldCityCouncilrecentlyreportedthattheaveragehousepricein the city is £163,288 and the average privatemonthly house rental is £600.28 Thecorrespondingaverageannualhouserentalis£600times12=£7,200.Wethenhavetomakeajudgmentabouttherelationshipofnetrentaltogrossandlet’ssupposethatnetis75%ofgross(butseealsothenextsubsectionformoreonthisbreakdownofgrosstonet).Theaveragenetrentalwouldthenbe75%times£7,200=£5,400,givingusa𝑞=5,400/163,288=3.3%.Onecanthenenvisageequityreleaselendersasimilarapproachtodrilldownfurthertoobtain𝑞calibrationsforneighbourhood,propertyclassandevenindividualproperties.CalibratingtheNetRentalYieldWenowprovideamoreprecisecalibrationofthenetrentalyield.Wefirstdecomposethenetrentalyieldasfollows:(7.9)𝑛𝑒𝑡𝑟𝑒𝑛𝑡𝑎𝑙𝑦𝑖𝑒𝑙𝑑 = 𝑦 − 𝑣 − 𝑐 −𝑚where𝑦 isthegrossrentalyieldortheyieldpaidbythetenant,𝑣 isthevoidrate,𝑐 ismanagementcostand𝑚isthemaintenancecost.Definethemaintenancecost𝑚astherateofexpenditure(asapercentageofgrossrental)requiredtokeepthepropertyinperfectcondition(i.e.suchastoachievethebestsalesprice forapropertyof thatsize in thesamearea),anddefinethetenantmaintenanceshare(𝑠)astheproportionof𝑚thatthetenantislikelytospendonmaintenance.𝑠willtypicallyvarybetween0and100%.Forashortrental,𝑠willbeclosetozero,andforalongletwewouldexpect𝑠tobecloseto100%intheearlyyearsoftenancy,fallingovertime.Inthefinalyearsitmightfalltozero,evenforastandardtenancy,giventhelackofincentive tokeep in fullorder for the landlord’sbenefit.29ForanERM, itwouldseemunlikely that the ‘tenant’ at endof life, perhaps in the situationwhere theNNEGhadbitten,wouldhaveanyincentivetokeeptheproperty ingoodcondition,sowewouldexpect𝑠tofalltowardszerointhatcasetoo.Wenowusethefollowingcalibrations:

• Void:weusethestandard‘1monthin12’ruleofthumb,i.e.,𝑣 = (1/12) × 𝑦.30

28 Sheffield Housing Market Bulletin, January-March 2019, Sheffield City Council. 29Infact,wecanalsoimagine𝑠 < 0.Soif𝑠 = 0reflectsnoactiveefforttokeepthepropertyincondition,𝑠 < 0reflectsadeterminedeffortbyoccupierstostriptheproperty(e.g.,oflightfittings,marblefireplaces,etc.)ortrashtheproperty!Ithappens.30Analternativeistouseempiricalvoiddata.Averagevoidperiodforlandlordsinprivaterentedsectorin the United Kingdom (UK) have varied from 2.4 weeks to 2.9 weeks. (Source:https://www.statista.com/statistics/421102/rental-properties-void-periods-in-the-uk/. Accessed 19March2019.)Ifwetakethemid-point,2.65weeks,thentheaveragevoidratebythismeasurewouldbe2.65/52=5.1%,ascomparedtothe‘ruleofthumb’voidrateof11/12=8.3%.

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• Managementcost:followingTunaru(2019,p.32),weassumemanagementcost𝑐 = 10% × 𝑦.

• Maintenancecost:againfollowingTunaru(2019,p.32),weassumemaintenancecosts𝑚 = 15% × 𝑦.

Thus,themaintenancecostbornebythelandlordandtobesubtractedfromthegrossrentalyieldis𝑚 = 15% × 50% × 𝑦 = 7.5% × 𝑦.Wethenhave(7.10)𝑛𝑒𝑡𝑟𝑒𝑛𝑡𝑎𝑙𝑦𝑖𝑒𝑙𝑑 = 𝑦 × (11/12 − 0.1 − 0.075) = 𝑦 × 0.7417.Thus,thenetis74.17%ofthegross.AgainfollowingTunaru(2019,p.31),wetake(7.11)𝑦 = 5.6%.Therefore(7.12)𝑞 = 𝑛𝑒𝑡𝑟𝑒𝑛𝑡𝑎𝑙𝑦𝑖𝑒𝑙𝑑 = 74.17% × 5.6% = 4.15% ≈ 4.2%.Soweuse𝑞 = 4.2%asour‘bestestimate’ofthedefermentrate.31TermStructureofGrowthInthederivationofequation(7.1)itwasassumedthatthereisnotermstructuretothecost of equity (𝑟 + 𝑃) or to growth (𝑔). Assuming a term structure would make adifferencetothedefermentrate.Decreasingthediscountrateforearlyperiods(saythefirst20years)orincreasingthegrowthrateforthesameperiodwouldhavetheeffectofincreasing𝑞,becauseitwouldmakethevalueoflostincomehigherasaproportionofthedefermentprice.However,apronouncedtermstructuretoeitherriskpremiumorgrowthseemsunlikely.A forward rental rate is the rateonewouldpay to lease thepropertywitha forwardstartingdateforacertainperiod.Butwhywouldthemarketimaginethattheforwardratebetweenyears39-40 issignificantlydifferent fromthatbetweenyears40-41, forexample?Itisdifficulttoseewhatinformationwouldjustifysuchajump,andtheremustbeapresumptionthatweshouldn’t introduceadditionalcomplicating factorswithoutgoodreason.Inshort,thereislittlepointassuminganygrowthtermstructure.

31Ifweusetheempiricalvoidrateof5.1%,thennetis77.4%%ofgrossandwewouldobtain𝑞 = 4.3%.

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TermStructureofDefermentRateObservationsofdefermentratesusingleaseholdpricesshowatermstructure(ofwhichmoreinthenextchapter),withthedefermentrateforshortleaseshigherthanforlongleases.This effect arisesbecause thevalueof a short-term leaseapproaches thatof ashort-termrental,andtheshort-termrentalreflectsthegross,ratherthanthenetrentalyield.As the leasehold term increases, itsvaluewill approach that impliedby thenetrentalyield,whichcanbeshownasfollows.Let𝑞"betheshortterm(i.e.,annual)rentalyieldgrossofannualcosts𝑐suchasvoidrate,maintenance,shareofmanagementetc.Assumethefollowing(crude)model:(7.13)𝑞" = (1 − 𝑐)𝐴 = 𝑞!(1 − 𝑐/𝑡)𝐴where𝐴 = (𝑦 + 𝑦) + 𝑦.…)and𝑞!isthegrosseffectiverentalyieldoveraletof𝑡years.Then(7.14)𝑞! = 𝑞"(1 − 𝑐)/(1 − 𝑐/𝑡)Thusthegrossrentalconvergestothenetrentalastheleaseholdtermincreases.Figure7.1givesanillustrativeplotofthegrossrentalvsleaseholdterm.

Figure7.1:GrossRentalvsLeaseholdTerm

Notes:basedonillustratevaluesofinitialgrossrental=5.5%and𝑐(costs)=25%.Forthechosencalibration(𝑞"=5.5%and𝑐=25%), thegrossrentalgoes frominitialvalue,5.5%relativelyquicklytowardsalong-runvalueof4.1%.

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TheDefermentRateandtheRealRisk-FreeRateInConsultationPaperCP7/19,thePRAproposed“totakeaccountofmovementsinrealrisk-freerateswhensettingthedefermentrate,”’ inordertopreventvariability intherealrisk-freeratecausingvariabilityintheforwardrate:

The PRAwould increase (reduce) the deferment rate if the review showsthere has been a material increase (reduction) in long-term real risk-freeinterestratessincethelastupdate.32

Inourderivationofequation(7.1)above,however,weshowedthatthedefermentrateisequaltothecurrentnetrentalyield,i.e.thenominalnetrentalpaymentdividedbythecurrentnominalhouseprice.Therealrisk-freeratedoesnotenterintoit!Sowhatisgoingonhere?WellitwouldappearthatthePRAproposalimplicitlydependsonsomeassumedrelationshiporequivalencebetweenthedefermentrateandtherealrateofinterest,butnofurtherdetailsaregiven.Diggingdeeper,itturnsoutthatthereisanequivalence,butonlyunderconditionsthatdonothold.Westartwiththedividenddiscountequation:(7.15)𝑞 + 𝑔 = 𝑟 + 𝜋where𝑞isthedefermentrate,𝑔thegrowthofnominalnetrental,𝑟theriskfreerateand𝜋theriskpremium.Nowassumefirstthatthepropertyinvestmentisriskfree,i.e.thatthereisnoriskpremium𝜋.Then(A1) 𝜋 = 0Substitutinginto(7.15)weobtain(7.16)𝑞 + 𝑔 = 𝑟Second,assumethat𝑔,theimputedgrowthinnetrental,isequaltothegeneralinflationrate𝑖:(A2) 𝑔 = 𝑖Hence(7.17)𝑞 + 𝑖 = 𝑟.Finally,assume(asstatedinCP7/19para2.5thatthenominalrate𝑟isthesumoftheexpectedgeneralinflationrateandtherealrate𝑟𝑟,arelationshipknownastheFisherEffect:

32CP7/19S2.4

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(A3) 𝑟 = 𝑖 + 𝑟𝑟andwhere(A4)𝐸[𝑖] = 𝑖i.e., we assume that expected and actual inflation are the same. Substituting andsubtractingbothsidesof(A3):(7.18) 𝑞 + 𝑖 = 𝑟𝑟 + 𝑖Hence(7.19) 𝑞 = 𝑟𝑟Given those fourassumptions (A1-A4), it follows that thedefermentrateand therealinterestrateareidentical.Wewouldquestionthoseassumptions,however.First,apropertyportfolioisclearlynotriskfree. Aninvestmentinahousingportfolioiscommensuratetoaninvestmentinarisky index-linkedbond,asopposed toan investment inan index-linkedgilt,which isvirtuallyriskfree.Thisconsiderationsuggests𝜋 > 0,sothattheriskydefermentratewillbehigherthantherealrisk-freerate.Moreover𝜋mayvarythroughtime.Second,whilerentalinflationandgeneralinflationarelikelytobecorrelated,theyarenotthesame.Nominalrentalswilltendtogouplinewithinflation,indeedrentalcostsarepartoftheUKConsumerPriceIndex,33butasFigure7.2belowshows,rentalinflationandCPIarefarfrom100%correlated.Therearelongperiods,suchas1991-2004,whenrentalinflationisconsistentlyhigherthanCPI.Figure7.2:UKConsumerPriceInflationversusUKRentalInflation,1970-2017

Source:OECD

33 HistoricaldatafromONS1988-2004canbefoundhere.

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Third,assumption(A3)abovedependsontheunobservablequantity,theexpectedfuturerateofinflation.Thisvariable,astheBankofEnglandmustknow,isdifficulttopredictwithanycertainty,andhenceisdifficulttomonitor.34Bycontrast,thenetrentalyield,whichweprovedabovetobemathematicallyidenticaltothedefermentrate,isrelativelysimpletoobserve.Inshort,ifthePRAwantstomonitorthedefermentrate–whichwethinkisareasonableidea–thenitshouldmonitordevelopmentsinthenetrentalyield.35Butmonitoringanirrelevantvariableliketherealrisk-freeratemakesaboutasmuchsenseasmonitoringthefrogpopulationtoseehowthellamasaregettingon.

34ItispossiblethatthePRAintendstomonitormarketexpectedrealinterestusingthereturnonindex-linkedgilts.However,returnsonindex-linkedgiltshavebeennegativeinthe2010s,whereaswewouldnotexpect thedeferment rate tobenegative.TheCPalsonotes (S2.4) that thedeferment ratewill alwaysremainpositive,inordertocomplywithPrincipleIIIofSS3/17,butgivenorationaleofwhythisshouldbeso.35Notethatthemathematicalequivalenceofthedefermentrateandnetrentalyieldalsodependsonthedividenddiscountmodel,butwithoutadditionalassumptionslike(A1)-(A4)above.

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ChapterEight:DilapidationTheERMonomicsofDilapidation(I):BorrowerMoralHazard

It is usually the case that the owner has a long-term interest in the upkeep of theirproperty,anditisthenreasonabletopresumethatheorshewouldmaketheinvestmentsinthepropertythatarenecessarytocounterthedilapidationthatwouldotherwiseoccur.

Nowconsiderequityrelease.Naturally,thelenderwouldpreferthattheborrowerlookafterthepropertyasmuchastheywouldiftheystillhadalong-termvestedinterestinit,buttheborrowernolongerhasanysuchinterest.Inanycase,theborrowerwouldbecashpoor(becausewhyelsewouldheorshehavetakenoutanERMloan in the firstplace?)andashe/sheages,itwouldbecomemoredifficulttomaintainthepropertyeveniftheborrowerwishedtodoso.Thereisaclassicmoralhazardproblem.Ofcourse,thelendercanimposeconditionsaboutmaintenance,buthaslimitedmeansofmonitoringmaintenanceorenforcingthem.Soanysuchcovenantswouldhavelimitedeffectivenessandthereisstillamoralhazardproblem.Thelikelihoodthenisthattheborrowermightmakesomeinvestmentsinthepropertyearlyon(e.g.,anewkitchenorpatio)butastheyage,anysuchinvestmentswillbebasedon short-term considerations or become increasingly cosmetic (e.g., a paint job) orabsolutelynecessarytobeabletocontinuelivinginthehouse(e.g.,payingtofixawaterleak). By the time the lender takes possession of the property, it will likely havedepreciatedinvalue,relativetothevalueonemustassumeitwouldotherwisehavehad,and certainly relative to the value itwould have hadhad it been ownedby someoneyounger,withthemeans,abilityandincentivetomaintainitproperly.Consequently,anERMedpropertywilllikelydepreciateinvalueoverthelifetimeoftheERM,relativetothevalueofanotherwisesimilarpropertythathadnotbeenERMed.Thisdilapidationeffectisreasonablywellknown,butthereisnoconsensusonhowtodealwithit.ToquotePRACP48/16:

Opinionontheappropriateadjustment[neededforit]wasdivided,suggestionsincludedadjustmentstopropertyvalue,propertyvolatility,HPI,andamarginfordilapidation.Somefeltthatsystematicunderperformanceriskduetoadverseselectionshouldbeallowedforinthevaluation.(pp.25-26)

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TheERMonomics ofDilapidation (II): StochasticDilapidation and LenderMoralHazardTherearealsootherfactorsinvolvedthatarenotsowellknown.Considerthefollowingfigure.36

Figure8.1:NNEGClaims:AvivaERF4Dataset

Source:Aviva:https://www.erfunding.co.uk/literature-library/erf4.html

ThechartshowscumulativeNNEGclaimsonERF4,asetofAvivaERMswhichstartsin2004.MostERMsstartwithaloantovalueoflowerthan50%,andpropertypriceshavegoneupinmostareasoftheUKsince2004,soittakesafewyearsforthecompoundloanamounttoreachcurrentpropertyvalueandreachNNEGterritory.SolowNNEGclaimsintheearlyyearsaretobeexpected.Butthereisasurpriseinstore.TheexerciseoftheNNEGwasinnocaseduetothecompoundloanvaluehittingthehousepriceindex.Avivahelpfullyprovide(i)and‘indexedhousevalue’atexit,togetherwith(ii)theoriginalvaluationand(iii)therealisedsalevalue.TheindexedvalueistheoriginalvalueprojectedbytheincreaseintheHalifaxindexsincethevaluation.Itturnsoutthatif all properties had followed the index,noNNEG would have been exercised,andallpropertieswouldhavebeensafelyoutofthemoneywhentheloanswererepaid.Instead, the exercisewas in all cases due to the underperformance, often a dramaticunderperformance,ofthepropertiesusedascollateral.Asanextraordinaryexample,considerthepropertythatcausedthelargeblipin2016.Itwasoriginallyvaluedat£1.2m,withanestimatedLTVof45%,i.e.,aloanvalueofabout£540,000.(Avivadonotprovideanexplicitloanrate,butweestimateabout7%basedon

36WethankT.PocockforpointingoutthisAvivaERMdataset: https://www.erfunding.co.uk/literature-library/erf4.html.

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redemptionsandloanamountsatexit.)Theloanvalueatexitwas£1.4m,butthesalepriceofthehousewasonly£625,172,leavingaNNEGlossof£763,225.Inotherwords,whiletheHalifax indexwentup70%,withthe indexedhousevaluebeingover£2m–easilyenoughtocovertheloanvalueatexitof£1.4m–thepropertynotonlyfailedtofollowtheindex,butactuallyfellinvalue(byabout50%).ThispropertywaspresumablyownedbytheAddamsfamilyandmaintainedbyUncleFester.Soitwasalsowith44%ofthe properties where the NNEG was exercised: nearly half the properties used ascollateralforequityreleasenotonlyfailedtomatchtheindex,butwereworthlessthanwhentheyfirstcollateralisedtheloan.37Whatishappeninghere?Well,weknowthataspeoplegetolder,theyarelessinclinedandlessabletomaintaintheirpropertytothestandardtheymaintainedearlier.Wealsoknowthattheyhavelessincentiveto,becausetheyhavemortgageditaway.Wediscussedtheseissuesalready.Evenso,themagnitudeoftheeffectissurprising.Itseemsthattheexerciseoftheputoptionappearstobeduetotheunderperformance,oftenadramaticunderperformance,ofthepropertiesusedascollateral.Evenmoresurprisingisthatfromtheperspectiveofthelender,theunderperformanceseemstohaveaconsiderablerandomelement,i.e.,wearedealingwithstochasticdilapidation!38ThenextFigureshowsscatterplotofthe ‘achievedrates’ofERMedproperties, i.e., theamounts that the lenderwasable to realiseafter theborrowerexited,expressedasapercentageoftheindexedvalue,basedontheSharedAppreciationMortgageSecurities(SAMS)originatedbyHBOSinthelate1990s:

37 One of our correspondentswho understands these datamuch better thanwe do suggests that thisexampleis“quiteclearlyafraudcase.”Hebacksupthisclaimwithapersuasiveanalysisandconcludesthattheonlynon-fraudexplanationhecanthinkofisifthesitehadbecomecontaminatedbyradioactivity.Wedefer to his judgment. Needless to say, firms should then either be modelling thepossibleimpactoffraudortheyshouldputsuitablemitigatingcontrolsinplace.38SeealsoCP7/19p.8:“Akeypointtodrawoutisthatlendersofequityreleasemortgagesareexposedtotheriskofindividualpropertyprices.Thisisbecauseinsurersprovideano-negative-equityguaranteetoeveryborrower.Modellingapproachesfocusedonhousepriceindicesdonotcapturealltherisks–aportfolioofoptionsisaverydifferentthingtoanoptiononadiversifiedindex.IndeedUKinsurershaveexperiencedanumberoftheseguaranteescrystallisinginrecentyearsdespitetherapidriseinUKhousepriceindicesoverthepastdecades.OnereasonisthatdifferentlocalitiesoftheUnitedKingdomhaveseenwidelyvaryinghousepriceinflation–anationalindexmaskstherangeofoutcomes….Anotheristhatsome properties may become dilapidated if elderly borrowers are unable to maintain the property.Willingness tomaintainmay be lowerwhere borrowers have limited or no equity remaining in theirproperties.Equityreleasecontractsgenerallyrequirepropertiestobemaintained.But,inpractice,lossesdooccurandcannotnecessarilyberecovered.”

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Figure8.2:Indexedvs.AchievedHousePrices

Source:SAMS.Theachievementratesarerealdata;theindexedvalueistheoriginalvalueprojectedbytheincreaseintheHalifaxindexsincethevaluation.

Thereddotsarethescattershotoftheindividualachievementratesinthesample.Thedarkerbluerandom-lookinglineisasimulatedhousepriceindex.Whatjumpsoutisthattheachievedvaluesareallovertheplacerelativetotheindex.Onaverage the achieved value is about 94%of the index but there is a huge dispersionaround the index. The volatility of the index does not capture the volatility of thedispersionaroundtheindex!An insightful explanation is provided by one of our EUMAEUS correspondents. Toparaphrase:

Theachievementratesarelikebetasinstocks.Everyhouseisindividual.Butthedispersionofresultsaroundtheaverageandthevolatilityofthatdispersionarebothenormous.Theimpactofolderpeopleallowingtheirpropertiestosubsideinlaterlifeisthefirstreactionofmostpeople,butisinfactonlyasmallpartofwhatisgoingon.Now practical experience suggests that the difference in price between thecheapesthouseonastreetcomparedtothemostexpensivehouseonastreet–assumingthey’reallthesamesizeanddesign–shouldbenomorethanabout20%. That’s from our practical experience of taking ownership of propertiesarisingfromsecuritisationsandmanagingthesaleprocessourselvesratherthanrelyingonbankservicers.That’swhatitcoststofixanewbathroomandkitchen,take control of the gardens and generally paint the place before sale. So themaximumspreadofaseriesofsprucehousesandneglectedhousesshouldn’tbemorethanabout20%,andthisspreadshouldputa floorunderthe impactofdilapidation.Thetroubleisthatitdoesn’t.

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Theproblemisthatnoneofthisprocessoccursinbankmanagedsales–allofwhichhappen“asis”oncethepropertyfallsintotheservicer’sownership.Noone inthechainofsalessetupbygeneralservicershasany incentivetotakecontrolofthevalueatsaleintheinterestsoftheowner.Everyonejustwantsthequickestpossiblesaleandtheremovalof“in-hand”propertiesfromthebalancesheet.Generalservicingispassiveandentirelyacceptingofanypriceaslongasthepropertyissold.Thenetoutcomeisthattheachievementratesinsecuritisationsthatneedactualhousesalesinthefuturetoachievetheircashflowsisgenerallydisappointing.Butnotbecauseofoldpeoplebecomingdisinterestedintheproperty.Themainreasonisthedislocationofactiveinterest[i.e.,incentivetosecureagoodprice]arisingfromsecuritisation.Theagents,lawyers,servicersandcashmanagersdonothavetheactiveinterest inmanagingtheactualhousesaleprocessthatanownerwouldhave.Thesecuritisationprocess itselfdetachesownership fromeffectivesalesmanagement.Sothelowachievementratesweseeinthedataaremostlyattributabletotheinvolvementoftheservicers,mostofwhomhavenoeconomicinterestinmaximisingsalesproceeds.39

Thereis,thus,asecondmoralhazardthatworkstotheadvantageofthelenderrelativetotheborrower.ERMlendersappeartoworkonthepresumptionthatERMloanswillusuallyexpireinpositiveequity,andthisassumptionisconsistentwiththeperformanceofmostERMloanstodate.Giventhispresumption,thenlendershavelittleincentivetomanage the properties they take possession of in order to find a good selling price.Instead, theirconcern issimplyto turnthepropertiesaroundtorecovertheamountsowed,i.e.,tomakeaquicksale,andanyproceedsleftoverarereturnedtotheborroweror their estate. If theborroweror their estate fetch apoorprice, then that is not thelender’s concern and the borrower or their estate has no control over the disposalprocess.Thissecondmoralhazardisanadditionalreasonforthe lowforERMedpropertiestofetchlowsalesprices.Wecanthinkofthissecondmoralhazardascreatinganeconomicdilapidationeffectinadditiontothelowsalespricescreatedbyphysicaldilapidation.

Thereisalsoathirdmoralhazard.Toquoteanotherofourcorrespondents:

theissuefrommyexperienceisthateverybodyhadanincentivetoovervaluethe property from the start, it’s not an issue of dilapidation over time …overvaluation isoneof thebiggestrisksand insurerswhohaveno lendingexperience are just clueless to this. I have seen the same issue in debtconsolidationmortgagesacrossEurope:ifthepropertyvaluedoesnotcomefrom an actual transaction, when you sell the property you always get asurprise!…In anormalpurchase the valueof theproperty is quite certainbecauseitisdeterminedbetweenawillingbuyerandawillingseller,butinERMthereisnosale,onlyanappraisal.Thelender,thebroker,theappraiser

39AsecondcorrespondentconfirmsthisanalysisandrefersustoaFitchUKRMBScriteriaaddendum(18-May-2018)thatstatesthataquick-saleadjustmentof17%ofpropertyvalueisappliedtohousesand25%toflats.Forilliquidproperties(i.e.inthetop5%and1%byprice)afurther10%or15%discountmaybeapplied.

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andtheborrowerallhaveanincentivetoclosethedeal.Why?Tomeettheirbusiness plan volume. In these circumstances, history teaches that thepropertyvaluewillalwaysbeoverestimated,butthiswillnotappearinanyExcelspreadsheetandthereforeanymodellingwillbebasedoffunreliableLTVassumptions.

Inshort,theimpactofeconomicdilapidationassuchwouldappeartobedrownedoutbymultiplemoralhazards,whosenet impactmight easilybe confusedwithdilapidationunlessonedigsdeeperintotheunderlyingeconomiccausesoflowachievementrates.

Thesemoralhazards, in turn,reflectunderlyingagencyproblemswherebythosewithfiduciarydutiesofcaretowardsshareholdersputtheirowninterestsfirst.

StatisticalAnalysisoftheSAMSAchievementRatesTable 8.1 shows the main statistical features of the achievement rates in the SAMSdataset:

Table8.1:MainStatisticalFeaturesoftheSAMSAchievementRatesMean 94.3%

Standarddeviation 19%Skewness 0.30Kurtosis 3.82

Samplesize 1420Range [31%170%]

5%lowerbound 64%5%upperbound 125%

Source:SAMS.Weseethattheachievementratesarehighlydispersed,somewhatpositivelyskewedandsomewhatheavytailed.NotethattheresultsinTable8.1reflectthewholedataset,andinterpretationoftheseresultsismadedifficultbythefactthattheseloanshavedifferentdurations.Tomakeinterpretationeasier,Table8.2showsthesameresultsforthoseloansthatstartin1997andendin2017.Theseloanshaveanapproximatedurationof21years.

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Table8.2:MainStatisticalFeaturesoftheSAMSAchievementRatesforLoansSpanning1997to2017

Mean 91.0%Standarddeviation 23.4%

Skewness 0.01Kurtosis 2.13

Samplesize 69Range [47%138%]

5%lowerbound 48%5%upperbound 129%

Source:SAMS.CalibratingtheDilapidationRateWecanthenusetheseresultstoestimatethedilapidationrate.

Forthe1420loansintheentireSAMSdataset,theaveragedurationis10.74yearsandtheaverageachievementrateis94.3%.Theaverageannualdilapidationrate𝑑isthen

(8.1)𝑑 = −(1/10.74) × ln(0.943) = 0.54%.Alternatively,forthe69loansintheSAMSdatasetthatstartin1997andendin2017,theaverage duration is about 21 years and the average achievement rate is 91.0%. Theaverageannualdilapidationrate𝑑isthen(8.2)𝑑 = −(1/21) × ln(0.91) = 0.45%.

So for ERMed, properties, there is an additional physical/economic/stochasticdilapidationeffectthatisnotpresentwhendealingwithtypicalnon-ERMedproperties,andtheassociatedaveragedilapidationrateisabout0.5%ayear.40

Finally,Table8.3showsthesemeandilapidationresultsalongwiththe5%lowerboundandthe5%upperboundforthedilapidationrateforthesecondsetofloans.

40Toquoteafurthercommentfromthepreviouscorrespondent.“Isthedifferenceinabsolute£explainablewiththecostsofrepairsandinefficienciesofanauctionsale?Well,youcansaythatthesaleisinefficient,orthatthepropertywasneverworththatmuch,especiallysincetheoriginalvaluewasnotanarm'slengthprice.Thereissomuchfraudintothosenumbers,IMHO.Thatisnotastochasticprocess,rathersystematicover-valuation.IfIwasamortgagebroker,Iwouldsaytoacustomerthatanovervaluationof20%+shouldgototallyunnoticed,andmaybewecanpushitto30-40%.Beyondthat,thelenderprobablyhasautomatedchecksinplace(Zoopla)thoughnotinthe90swhentheSAMSwereoriginated.Thatfitswithyour0.5%pa,justfromadifferentperspective.”

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Table8.3:MainStatisticalFeaturesoftheSAMSAchievementRatesforLoansSpanning1997to2017WholeSAMSdataset

Mean𝑑 0.54%Samplesize 1420SAMSloansspanning1997to2017Mean𝑑 0.45%

5%upperbound 3.5%5%lowerbound -1.21%Samplesize 69

Source:SAMS.Theseboundsreflectthe90%confidenceboundsforthedilapidationrate,andspan3.5%attheupperendto-1.21%atthelowerend.Thesmallsamplesizetellsusnottorelytooheavilyontheseresults,buttheyareindicative.41

Wearetemptedtoconcludethatthereisalotmoregoingonherethanmerephysicaldilapidation, although doubtless therewould be that too. The ‘other things going on’wouldappeartobetheeconomicconsequencesofthevariousmoralhazardswehavementioned. If these other factors dominate, as they appear to do, then the term‘dilapidation’isonlyasmallpartofthepictureandisitselflookingalittle,er,dilapidated.

41 AfulleranalysisoftheSAMSdatabasemight,e.g.,lookattheachievementratesanddurationsforeachloan, estimate 𝑑 for each, and thence we produce a sample of 𝑑 rates, from which we can obtain astatisticallystrongersenseofthem.

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ChapterNine:VolatilityThe standard approach to volatility estimation takes the volatility to be the standarddeviationofthereturntotheunderlying,wherethelatterisoftentakentobeahousepriceindex(HPI)andsomeadjustmentismadefortheimpactofautocorrelationintheHPI.AgoodexampleisCP13/18(p.9),whichstates:

2.16ThePRAestimatedavalue for thepropertyvolatilityparameter fromanalysis of residential property price index data. Nationwide, Halifax andOfficeforNationalStatisticsindexdatawereanalysedandseveraltimeseriesmodelswerefittedtothequarterlylog-returnsofdatasetsoveravarietyofhistoricaltimeperiods.ThePRAselectedaparsimoniousmodelthatfittedthedata well, and extracted from the model the unconditional volatility forvarious holding periods, allowing for autocorrelation. Further adjustmentsweremadetoallowforconcentrationriskandbasisriskbetweenthechangesinpricesofindividualpropertiesandtheindex.ThePRA’scentralestimateisofa13%volatilityassumptionfortypicalholdingperiodsforERMs,althoughuse of alternative data choices gives a range of 13%-16%, andmaking anallowanceforparameteruncertaintygivesarangeof11%-18%.Estimatesforpropertyvolatilityprovidedto thePRAby firmsaregenerally in therange10%-15%.(Ouremphasis)

Wewouldsuggestthatthisapproachisunsatisfactoryfortworeasons.First,itimplicitlysupposesthatthereisonesingle‘onesizefitsall’volatilityinputtoNNEGvaluationor,morespecifically,thateachoftheput‘nneglets’thatmakeuptheNNEGshouldhavethesamenumericalvolatilityinput.Wesuggestthatthisimplicitassumptionismistaken,andthateachnnegletshould inprinciplehaveadifferentvolatility input.Second, it isnotnecessary,42andnotadvisable,43toadjustforautocorrelationintheunderlying.Weneedtogobacktofirstprinciples.Weproposeanewapproachthatstartswiththevolatilityoftheforwardreturnderivedfromahousepriceindex(HPI)andthengraduallyaddsinfurtherriskfactors.

42 ThatitisnotnecessarytotakeaccountofautocorrelationintheunderlyinghasalreadybeenestablishedbyCornalbaetal.(2002).Thispaperprovidesafairlygeneralanalysisoftheimpactoftemporal(i.e.,auto-correlation)onoptionpricingandtheirconclusionsareclear.“IntheGaussiancase[theoneconsideredinBlack-Scholes],wefindthattheeffectof[auto-]correlationscanbecompensatedbyachangeinthehedgingstrategyandthereforeoptionsshouldbepricedusingthestandarduncorrelatedBlack-Scholesmodel(ouritalics).” (See See L. Cornalba, J.-P. Bouchaud andM. Potters (2002) “Option Pricing andHedgingwithTemporalCorrelations,”InternationalJournalofTheoreticalandAppliedFinance5(3):307-320.)ThisclaimisconfirmedbyourrecentworkonthederivationoftheBSMformula.Takeyourfavouritedefinitionofvolatility and assume that the underlying is not autocorrelated. Estimate the volatility. Now introduceautocorrelationintheunderlyingbutadjustthenoiseprocesssothatthevolatilityisremainsasitwas.Ourderivationimpliesthattheintroductionofautocorrelationinthiswaymakesnodifferencetothevalueoftheoption.SeeD.BucknerandK.Dowd‘BSwithfewerassumptions,’EumaeusWorkingPaper,February2020. 43 Supposeoneestimatesthevolatilitycorrectlyignoringautocorrelation.Onethendeterminesthattheunderlyingisautocorrelatedandadjuststhevolatilityto ‘takeaccount’oftheautocorrelation,Thenthenewestimatemustbewrong,becausethefirstonewascorrectregardlessofanyautocorrelation.

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VolEstimation:AFirstPassWebeginbydefiningthevolatility𝜎ofthereturnonaforwardcontractasthesquarerootofthevarianceofthatreturn:(9.1)𝜎) = $

/∑ (𝐹𝑅! − 𝐹𝑅jjjj))./$

where𝐹𝑅!istheforwardreturnovertheperiodtotime𝑡.Morespecifically,weseektoobtainthevarianceofthereturnonaforwardwithmaturity𝑇,𝐹!,2 say,basedonobservationsofaspotprice𝑆!andwewouldtypicallytake𝑆!tobesomeHPI.Wealreadyknow(see(3.9)above)that(9.2) 𝐹!,2 = 𝑆!𝑒('#*)2 wherethevariableshavetheirusualinterpretations.ThereturnonaforwardwithmaturityTis(9.3) 𝐹𝑅!,2 = 𝑙𝑛𝐹!,2 − 𝑙𝑛𝐹!#$,2 wherevariableshavetheirobviousinterpretations.Substituting(9.2)into(9.3)(9.4) 𝐹𝑅!,2 = ln[𝑆!𝑒('#*)2] − ln[𝑆!#$𝑒('#*)2] = ln 𝑆! + (𝑟 − 𝑞)𝑇 − ln 𝑆!#$ − (𝑟 − 𝑞)𝑇 = ln 𝑆! − ln 𝑆!#$whichisthereturnonthespothouseprice.Therefore the variance of 𝐹𝑅!,2 must be equal to the variance of the spot return,regardlessofthematurityoftheforward.Sowhatshouldwetakethevalueofthespotreturnvariancetobe?WewouldsuggesttogowiththePRA’scentralestimateof13%mentionedearlier.IfitisgoodenoughforthePRAthenitisgoodenoughforus.Theirmistakewastothinkthatoneshouldthenpluganysuchestimateintothennegletputequationsandthat’sit.VolatilityAroundtheIndexTheestimated13%volatilityonlyreferstothevolatilityoftheindex,butthereisalsothevolatilityaroundtheindex.Thisadditionalvolatilitywouldincludetheimpactofregional

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variation around the index, but there are further contributory factors as well. Theseinclude,e.g., the impactofchanges inconsumerrelativedemandfordifferenttypesofproperty,expansionsofnearbyroads,theimpactofnewhousingestates,yuppification,middleclass flight, theopeningorclosingofagoodnearbyschool, and thestochasticdilapidationeffectsdiscussedinthepreviouschapter.OnewillrecallFigure8.2fromthepreviouschapter:

Figure8.2:Indexedvs.AchievedHousePrices

Source:SAMS

Recallthatthedarkerbluerandom-lookinglineisahousepriceindexandthereddotsareascattershotoftheindividualachievementratesinthesample.We immediately see that the achieved values aremuchmore volatile than the index.Aboveall,whenseekingtocalibratethevolatility,weneedtokeepinmindthatitisthatdispersionthatmatters,notthevolatilityoftheindexitself.Wecangetabettersenseofthedispersionaroundtheindexfromthenextfigure:

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Figure9.1:Indexedvs.AchievedHousePrices(II)

Source:SAMS

Itwould thenbehoveus to revise our earlier index-based volatility estimates to takeaccount of this additional source of volatility. We could start with our earlier indexvolatility.Letus label thisvolatilityas𝜎34567 .Wenowobtain𝜎89 , thevolatilityof theachievementrate,asfollows:takearollingstandarddeviationoftheachievementrateanddividebyroottimetogetanannualisedvalue.Wefoundthattheannualisedvolatilityvaluesvaryfrom7%to10%.Let’sworkwiththemiddlevalueof8.5%.Wethenhavetoassume a plausible correlation between the index vol and the achievement rate vol.Assumingzerocorrelation,whichisnotunreasonable,weobtaintheresultsreportedinTable9.1:

Table9.1:IllustrativeVolatilityIncludingImpactofAchievementRateVolatility

𝝈𝑰𝑵𝑫𝑬𝑿 𝝈𝑨𝑹 𝝈𝑰𝑵𝑫𝑬𝑿𝒂𝒏𝒅𝑨𝑹13% 8.5% 15.5%

Note:ThetermintherightmostcellareobtainedbyPythagoras.InterestRateRiskasaFurtherContributortoVolatilityWe have hitherto assumed (as per Black-Scholes/Black ’76) that the interest rate isconstant,i.e.,thatthereisnointerestraterisk.Infact,interestraterisknotonlyexists,butarisesfromtwodifferentsources.ConsiderthefollowingcomponentsoftheBlackmodel,reproducedhereinslightlysimplifiedfromChapter3:(9.5) 𝑝! = 𝑒#'![𝐾2𝑁(−𝑑)) − 𝐹2𝑁(−𝑑$)] (9.6)𝑑$ = [𝑙𝑛(𝐹2/𝐾2) + 𝜎2)𝑇/2]/(𝜎!𝑇)(9.7)𝑑) = 𝑑$ − 𝜎2√𝑇(9.8) 𝐹2 = 𝑆𝑒('#*)2

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whereasusual:𝑝!isthevalueofthe𝑡decrementput,𝐾2 thestrikeprice,𝐹2 theforwardprice,𝜎2 theannualised inputvolatility,𝑇 thetimetomaturity inyears,𝑟 the interestrate,𝑆thepriceof‘spot’possessionoftheproperty,and𝑁(…)isthecumulativenormaldistributionfunction.Theinterestrateterm𝑟appearsfirst(see(9.5))asadiscounttermwrappedaroundthetermsrepresentingthefuturevalueoftheputoption,whichbringsthefuturevalue(i.e.,[𝐾2𝑁(−𝑑)) − 𝐹2𝑁(−𝑑$)]) back to present value. Here 𝑟 plays the role of an outerdiscountfactor.𝑟thenappearsagain(see(9.8))asaprojectiontermorinnerdiscountfactortakingusfromthespotprice𝑆totheforwardprice𝐹2 .Eachappearancegivesrisetointerestraterisk,butindifferentways.DiscountrateriskThefirstcanbecalleddiscountinterestraterisk.Thisriskcanbehedgedrelativelyeasily,thegistofitbeingtoswapfloatingintofixed.Amoredetailedexplanationgoesasfollows.Whenatradingdesksellsanoption,itplacesthepremiuminanaccountcalledthe‘hedgingaccount’.Thisaccountearnsinterestfromthefirm’scentralfundingdeskandtheinterestearnedwilltypicallybeclosetothefirm’soverallfundingrate.Tohedgetheriskarisingfromchangesinthisouterdiscountfactor,thedeskshouldmakeaninternalorexternalIRswapintoafixedratewithmaturityattheoptionexpirydate.Itcanthenbeshownthatthisswapguaranteesthat,withnootherchangetakingplaceinthemarket,thehedgeaccountwillearnthefixedrate𝑟intheouterdiscountfactor𝑒#'! .Thedemonstrationgoesasfollows.Let(9.9)𝑃 = 𝐹𝑉 × 𝑒#'2 where𝑃istheoptionpremiumpaid,𝑟hereisthelongtermrateearnedontheoptionaccount,and𝐹𝑉isthefuturevalueoftheoptiongivenbytheundiscountedBlackformula.Nowsupposethelong-terminterestrate𝑟changes,butthereisnochangeintheforwardprice𝐹.Suchacircumstancewouldoccurwherethespotrate𝑆changedbyanamountΔ𝑆insuchawaythat𝐹remainedconstantundertheformulaconnecting𝑆with𝐹,i.e.(9.10)Δ𝑆 = 𝑆(𝑒#E'2 − 1)whereΔ𝑟isthechangeindiscountrate.Inthissituation𝐹willremainthesameandhencethefuturevalue𝐹𝑉oftheputoptionwillalsoremainthesame.ThechangeΔ𝑃 inthevalueoftheoptionpremiumwillthenbeasimplediscountfunction:(9.11)𝑃 + Δ𝑃 = 𝐹𝑉[𝑒#('0E')2 − 𝑒#'2]

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Assumingtheamount𝑃iscurrentlyheldinthehedgingaccount,wecouldreplicatethechangeΔ𝑃if 𝑃 were invested in a long dated zero-coupon bond with maturity 𝑇. Inpracticethesameeffectcanbeachievedbyinvesting𝑃atthefirm’sshort-termfundingrate,butswappingtheshort-termfloatingpaymentsintoazero-couponswap. ProjectionrateriskInthepreviousexampleweassumedthattheforwardpriceremainsconstantwhilethespotpricechanges,i.e.,ariseinlongterminterestrateswillforcethespotpricelower,whileafallintheinterestrateforcesthespotpricehigher.Thiseffectmightbeexplainedby a market expectation of unchanged future nominal rental cashflows, whosediscounted present value would fall or rise according to the long-term interest rateoperatingasadiscountfactor.Theoppositecasecanalsooccur,i.e.,wecouldhaveasituationwherethespotremainssteady,buttheforwardchangesduetothewayinwhichtheinterestrateoperatesasaprojectionfactor.Using the standard formula for the forward house price (i.e., (9.8)), and assumingconstant𝑞,thereturnontheforwardiscalculatedasfollows.(9.12)forwardreturn≈D𝐻𝑃 + (D𝐼𝑅 − D𝑞) × 𝑇 whereD𝐻𝑃 = ln((𝑆 + Δ𝑆)/𝑆) and𝑇 is thematurityof the forwardatanypoint in thehistoricaltimeseriesforthegivencombinationofinterestrate(𝐼𝑅),defermentrate(𝑞)andhousepriceindex(𝐻𝑃𝐼).44Aproofof(9.12)isprovidedinAppendix1tothisChapter.Threekeypointsfollowfrom(9.12).FourriskfactorsThefirstisthatwenowhavefourriskfactors(Index,achievementrate,interestrateanddefermentrate)impactingtheforwardprice.CorrelationsThe second is thatwe need to consider their correlations.We are now interested inparticularly(a)thecorrelationbetweenthehousepriceindexandtheinterestrate,and(b)thecorrelationbetweenthehousepriceindexandthedefermentrate.Let’sconsidereachoftheseinturn.

44 Notethatwecannothedgeawayinterestanddefermentratevolatilityontheassumptionthattheindexpriceislessvolatilethantheforward.AswehavepointedoutintheAppendixtoChapter3,therelevantvolatilityfortheBlack-Scholesoptionisnotthatofthespotreturnalone,butratherthatofthe𝑑!termusedtodeterminetheoptiondelta.Thenumeratorinthe𝑑!termincludesbothinterestrateanddefermentrate–explicitlyinBlackScholes,implicitlyinBlack76.

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(a)CorrelationbetweenhousepriceindexandinterestrateWehavealreadyassumedthatthecorrelationbetweentheIndexandtheachievementrate is zero.The tablebelowshows thecorrelationbetween the10year interest rate(whichwetakeasabenchmarkforthewholetermstructure)andthehousingindex,for10representativecountries.

Table9.2:Correlationbetween10YInterestRateandIndexCountry 𝝆𝑰𝑹,𝑰𝑵𝑫𝑬𝑿AUS10Y 0.22CAN10Y -0.05GER10Y 0.11ESP10Y 0.06FRA10Y 0.16GB10Y 0.10IRL10Y -0.03SWE10Y 0.14US10Y 0.03JP10Y 0.27

Source:OECD(10Yinterestrate)andDallasFed(HousePriceindices)ThetakeawaypointsfromthistablearethatthecorrelationsbetweeninterestratesandhousepriceindicesaregenerallylowandthatareasonablecorrelationfortheUKwouldbezero.45(b)Correlationbetweenhousepriceindexand𝑞Equation(9.12)indicatesthatthedefermentrate𝑞isafurthersourceofvolatility.Takeequation(7.1),whichsaysthatthedefermentrateisequaltotherentalyielddividedbythehouseprice,thenreplacethehousepricebytheHPindexandaddatime𝑡subscript.Wethenobtain:(9.13) 𝑞! = 𝑑!/𝐻𝑃! where𝑑! is the aggregate nominal rental.We have no time series data on aggregatenominalrentals,butwecanestimatetheirchangeusingrentalandhousepriceindices.DatafromOECDsuggestthatthedefermentrate𝑞isnotconstant(theannualvolatilityof𝑞fortheUKisoftheorderof0.3%)andthatchangesin𝑞arenegativelycorrelatedwithchangesintheindex.TheseeffectsareshowninFigure9.2:

45ThelowcorrelationsreportedinTable9.2areabitofasurprise,consideringthatalowerinterestrateimmediatelytransformsintohigheraffordabilityandtherefore-intheabsenceofnewsupply–intohigherhouseprices.OnereasoncouldbethatTable9.2looksat10yrrateswhereasthekeydrivermightbeshort-termrates.Werethecorrelationsbetweeninterestratesandhousepriceshigher,theresulting‘combined’volatilities(ofwhichmorebelow)wouldbehigheraswell.

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Figure9.2:UKNominalHouseandRentalIndicesandImpliedDefermentRate

Sources:OECD

AsDeancommentsinablogpostinginearly2019:46

WhenIworkedatthePRAonthepaperthatbecameCP13/18,Ihadassumedthatthedefermentratestaysroughlyconstant.Therationaleisthatifrentalsare expected to increase, this would increase the market value ofproperties,allotherthingsbeingequal.Butallotherthingsaren’tequal:thereisstrongevidencethatnominalrentalstrackpriceinflation,andalsostrongevidence that interest ratesanticipate inflation. So an increase in expectednominalrentalsshouldcorrelatestronglywithanincreaseintheinterestrateusedtodiscountthesamerentals,andtherentalyield,hencethedefermentrate,shouldremainroughlyconstant.47Iassumedthis,andIimaginethePRAassumedthistoo.

Apossibleexplanation for thevolatile𝑞 rateand thenegative correlationwithhousepricesmightthengoasfollows.Nominalhouseprices,whichintheoryshouldreflectthenetpresentvalueofallfuturenominal(net)rentalcashflows,tendquicklytoanticipate–perhapstoover-anticipate–futureupwardordownwardchangesinrentals.Nominalrentalsarestickyhoweverandrespondslowly.48Thusthelargefallinhousepriceswhich

46“ItMoves.”TheEumaeusProject(14January2019):http://eumaeus.org/wordp/index.php/2019/01/14/it-moves/47AccordingtotheDividendDiscountModel[seechapter7],thedefermentrate𝑞equalsthenetrentalyield𝑑/𝑆(see(7.1)).Weshowbelow(see(18.14))that𝑞alsoequals𝑟 + 𝜋 + 𝜆 − 𝑔.UsingthewellknownFisherEquation,wecandecomposethenominalinterestrate𝑟intothesumof𝑟"#$% + 𝑖𝑛𝑓,where𝑟"#$% isthe‘real’rateofinterestand𝑖𝑛𝑓istheexpectedinflationrate.However,wecanapplyasimilarreasoningto𝑔 as well, giving us𝑔 = 𝑔"#$% + 𝑖𝑛𝑓, where𝑔"#$% is the growth rate of ‘real’ or inflation-protecteddividends.Wethenobtain𝑞 = 𝑟"#$% + 𝑖𝑛𝑓 + 𝜋 + 𝜆 − (𝑔"#$% + 𝑖𝑛𝑓) = 𝑟"#$% + 𝜋 + 𝜆 − 𝑔"#$% fromwhichweseethattheeffectsofexpectedinflationcancelout.48 This effect is well known in the literature, although there is no consensus on the explanation. Forexample,CampbellandHercowitz(2009)findthat“movementsinU.S.houseprice-rentratioscannotbefullyexplainedbymovementsinsubsequentrentgrowth”(J.R.CampbellandZ.Hercowitz(2009)“WelfareImplicationsof theTransition toHighHouseholdDebt.” Journal ofMonetaryEconomics 56,1-16.For areview of the literature, see P. Gelain and K. Lansing “House Prices, Expectations, and Time-VaryingFundamentals,”FederalReserveBankofSanFranciscoWorkingPaper2013-03.

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occurredinthehousingrecessionoftheearly1990swasnotreflectedinrentalprices,whichcontinuedtoriseslowly,andso𝑞roseinthatperiod.Conversely,thesignificantrise in house prices from the late 1990s to 2007was notably higher than the rise inrentals,so𝑞felloverthislaterperiod.Asanaside,thiscombinationofavolatile𝑞thatisnegativelycorrelatedwithhousepriceshasaninterestingpolicyimplication.Ifhousepricesgoup,theloan-to-valueofanexistingequityreleasemortgagewillfall,whichwilldecreasethecostoftheNNEG.Atthesametime, the graph above suggests the deferment ratewill also fall,whichwillmake theNNEG even cheaper, given that the deferment rate is themain driver of NNEG cost.Conversely,afallinhousepriceswillmaketheNNEGmoreexpensivebecauseofthefallitself,andwillthenmaketheNNEGevenmoreexpensivebecauseoftheimpliedriseinthe𝑞rate.Thecostoftheembeddedguaranteeisthusdoublygearedtothestateofthehousingmarket.Ouch!ThePRAwouldappeartobestillunawareofthisdoubleexposure,whichhasimplicationsforhowitshoulddesignitscapitalrequirementregimeforequityrelease.Butwedigress.NoristhisnegativecorrelationeffectuniquetotheUK.Table9.3showsevidenceforastrongandconsistentnegativecorrelationbetween thedeferment rateand thehousepriceindexofourtencountries:

Table9.3:Correlationbetween𝑞andIndexCountry 𝝆𝒒,𝑰𝑵𝑫𝑬𝑿AUS -0.90CAN -0.95GER -0.79ESP -0.88FRA -0.84GB -0.82IRL -0.43SWE -0.80US -0.86JP -0.92

Source:OECD(10Yinterestrate)andDallasFed(HousePriceindices)VolatilitytermstructureThethirdimportantcorollaryofequation(9.12)isthatitimpliesthatthereisavolatilitytermstructure.Forexample,otherthingsbeingequal,theeffectofa10bpchangeonacontractwith3monthstomaturitywillbe10x3/12=2.5bp,whichistrivial.Buttheimpactofthesamebpchangeona30yearforwardwillbe10x30=300bp,whichisawholelotmore.Thischangingsensitivitythroughoutthelifeofthecontractmeansthattheimpactonvolatilitycausedbychangesintheinterestrateisnotconstant,andthesame applied to changes in the deferment rate. Instead, mapped against time, thevolatilitystartshighwhenthematurityisfarfrommaturityandfallstowardszeroasthecontractapproachesexpiry.Mappedagainstmaturity,thevolatilitystartsfromzeroandincreasesasthematuritygetslarger.

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Wenowwishtodeterminetheaveragelifetimevolatilityofthecontract.If𝑋istheseriesofmaturitiesand𝑌istheseriesofforwardreturns,itcanthenbeshownthatthevolatilityoftheproduct𝜎(𝑋𝑌)isthefollowingsimplefunction:(9.14) 𝜎(𝑋𝑌) = 𝜎(𝑌) × 𝑇/√3where𝜎(𝑋𝑌)isthevolatilityofatimeseriesofreturnsoftheforwardcontract,and𝜎(𝑌)thevolatilityoftheinterestrate.AproofisgiveninAppendix2.ThenthevolatilityofthereturnsontheforwardcontractisdirectlyproportionaltoT.TotalForwardVolatilityFinally, we might consider the effect of all four risk factors (Index, IR,𝑞andAR)intheforwardrate(9.12)togivewhatwemightcallthetotalforwardvolatility.WecandosobyestimatingacorrelationmatrixbetweenthefourriskfactorsasshowninTable9.4:

Table9.4:CorrelationMatrixfortheFourMainRiskFactorsIndex IR 𝑞 AR

Index 1.00 0.00 -0.82 0.00IR 0.00 1.00 0.00 0.00𝑞 -0.82 0.00 1.00 0.00AR 0.00 0.00 0.00 1.00

Table9.5showsthevolatilitiesforthecomponentriskfactors:

Table9.5:VolatilitiesofComponentRiskFactorsComponentVolatility Value

𝜎34567 13%𝜎89 8.5%𝜎39 0.58%𝜎* 0.17%

WenextcombinethecorrelationsinTable9.4withthecomponentvolatilitiesinTable9.5 and thenapply (9.14) toobtain the termstructure for the total forwardvolatilityshowninTable9.6:

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Table9.6:TermStructureofTotalForwardVolatility𝑇 TotalForwardVolatility1 15.66%5 16.39%10 17.73%15 19.45%20 21.45%25 23.67%30 26.05%

If we worked with these results, we would apply a 15.66% volatility to the put fordecrement𝑡 = 1,a16.39%volatilitytotheputfordecrement5,andsoon,anda26.05%volatilitytotheputfordecrement30.49Figure9.3showsaplotofthetotalforwardvolatilityoverahorizonofupto40years.

Figure9.3:TermStructureofTotalForwardVolatility

Theblueplotshowstotalforwardvol,whichstartsatalittleover15%foramaturityof𝑇 = 1andrisestoalittleover31%for𝑇 = 40.Theredlineshowsthecontributiontothattotalforwardvolmadebythe𝑞andinterestrateriskfactors.Thesecontributionsstart(for𝑇 = 1)atclosetozero,butriselinearly(seeequation(9.14))with𝑇,reachingover24%for𝑇 = 40.Thus,as𝑇 increases,the𝑞andinterestrateriskfactorsbecomeincreasinglyimportantdriversoftotalforwardvolatility.ObtainingaSingleVolatilityEstimateforUseinAllPutDecrementsHavingestablishedthat,inprinciple,eachputdecrementrequiresitsown(potentially)differentvolestimate,itisstillpossibletouseasinglevolatilityforallputs,providedone

49 Amorecomprehensiveapproachwouldtakeaccountofcorrelationunderlessfrequenthedgingandgiveamoredetailedconsiderationofthecorrelationbetweentheindexandtheachievementrate.

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usesasinglevolatilitycalibrationthatisappropriatetotheborrower’sageandgender.Toobtainsuchacalibration,onecouldusetheexpectedvolatilityobtainedbyweightingeachvolatilityby itsexitprobordecrementprobability.Theformulafortheexpectedvolatility𝜎G isthen(9.15)𝜎G = ∑ [𝑒𝑥𝑖𝑡𝑝𝑟𝑜𝑏! × 𝜎!]! Table9.7showstheseexpectedvolatilitiesagainstborrowerage:formales26.1%forage55,27.3%forage70(andhenceourearlierbaselinesinglevolatilityrecommendationformalesaged70)andsoon.Theseexpectedvolatilitiesgiveresultsthatareveryclosetotheresultsthatonewouldhaveobtainedhadoneusedthefullvolatilitytermstructure.

Table9.7:ExpectedVolatilitiesforDifferentAges

BorrowerAge ExpectedVolatility(Males)

ExpectedVolatility(Females)

55 26.1% 27.3%60 23.8% 24.9%65 21.7% 22.7%70 20.0% 20.8%75 18.6% 19.2%80 17.6% 18.0%85 16.9% 17.1%90 16.4% 16.5%

Notes:ExitprobabilitiesasperFigure3.1.Table9.8showstheresultingNNEGandERMvaluations:

Table9.8:ERMandNNEGValuations:MaleAge70ValuationApproach 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

Black’76(usingvoltermstructure) £38.2 £43.5Black’76(usingexpectedvol) £38.2 £43.6

Notes:AsperTable3.1.Table9.9givesthesamevaluationsforage70aspercentagesoftheinitialloanamount,£40:

Table9.9:ERMandNNEGValuationsasPercentagesofLoanAmount:MaleAge70

ValuationApproach 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴Black’76(usingvoltermstructure) 95.6 108.8Black’76(usingexpectedvol) 95.5 109.0

Notes:AsperTable3.1.Figures9.4and9.5giveplotsofNNEG/loanvaluesagainstborrowerageandERM/loanvaluesagainstborrowerage.

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Figure9.4:NNEG/LoanRatiosVsBorrowerAge

Notes:Asindicated,otherwiseasperTable3.1.

Figure9.5:ERM/LoanRatiosVsBorrowerAge

Notes:Asindicated,otherwiseasperTable3.1.

Onenoticesthattheratiosof𝑁𝑁𝐸𝐺toloanamountfallsharplywithage.However,whatismostsignificantisthelowratiosof𝐸𝑅𝑀toloanamount.ConclusionsThecalibrationofthevolatilityparameterisamoreinvolvedsubjectthaniscommonlysupposed.Thecommonifnotuniversalpracticeistoestimatethevolatilityofthespotreturn,perhaps(ifyouareclever)adjustitforautocorrelationinthespotpriceprocess,and then input that volatility estimate in all the nneglet puts. This approach is quitemistaken,however.Sotooistheunnecessaryadjustmentforautocorrelation.Thatonlymakesabadapproachworse.

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Instead,weshouldtakeadifferentapproach.Weshouldtakeaccountoffourriskfactorsthataffectvolatility–thehousepriceindex,theachievementratesaroundthatindex,theinterestrateandthedefermentrate.Wethenendupwithavolatilitytermstructureinwhichdifferentnnegletputsorequivalentlydecrementshavedifferentvolatilities.Sincewealsohavetomapexitprobstothosedifferingvolatilities,wefindthattheappropriatevolatilitiesarealsodependentontheborrower’sageandgender.Thesepointsmade,ifwewish to use a single volatility estimate for all the decrements (and it is certainlyconvenienttodoso),wecanuseanexitprobweightedaverageofthosetermstructurevols,butthat‘single’volinputwillbebothageandgenderspecific.

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AppendixOnetoChapterNine:ProofofApproximation(9.12)Theforwardrate𝐹atanytime𝑡andforanymaturity𝑇isasfollows:(9A1.1) 𝐹!,2 = 𝑆!𝑒('(!,2)#*(!,2))2 where𝑆!isthespotpriceattime𝑡,𝑟(𝑡, 𝑇)istheinterestrateofmaturity𝑇attime𝑡and𝑞(𝑡, 𝑇) is thedeferment rateofmaturity𝑇 at time 𝑡.With thepassageof time∆𝑡, theforward ratewill change as a result of changes in𝑆,𝑟 and𝑞, and of coursewith thepassageoftimeitself.Thus(9A1.2)𝐹!0∆!,2#∆! = 𝑆!0∆!𝑒I'(!0∆!,2#∆!)#*(!0∆!,2#∆!)J(2#∆!) Thisexpressionisfairlycomplex,butwecanmakeanumberofsimplifyingassumptionsasfollows.First,wecanassumethatthetermstructureofboth𝑟and𝑞iscontinuous.Wehave assumed throughout a flat term structure 𝑞, so it follows that 𝑞(𝑡 + ∆𝑡, 𝑇 − ∆𝑡)equals𝑞(𝑡 + ∆𝑡, 𝑇).Wecannotassumethat the termstructureof interest rates is flat,because it will usually slope upwards or downwards at any time. However, we canreasonably assume that changes in the term structure will make no significantcontributiontovolatility.Thatis,achangeover1monthtothe10yearinterestratewillnotbesignificantlydifferentfromthechangeinthe9year11monthinterestrate.Thus𝑟(𝑡 + ∆𝑡, 𝑇 − ∆𝑡)willbeapproximatelyequalto𝑟(𝑡 + ∆𝑡, 𝑇),forsmall∆𝑡.Hence(9A1.3)𝐹!0∆!,2#∆! ≈ 𝑆!0∆!𝑒I'(!0∆!,2)#*(!0∆!,2)J(2#∆!)

Theouterterm(𝑇 − ∆𝑡)canalsobeeliminated,asitrepresentsaconstantcarrythroughtime.Astimepasses,if𝑟isgreaterthan𝑞,theforwardpricewillgraduallyfall,orif𝑟islessthan𝑞,theforwardpricewillgraduallyrise.Butvolatilitycorrespondstothemeandifference fromtheaverage,whereasthecarrytermwillbeclosetotheaverageitself.Hence(9A1.4)𝐹!0∆!,2#∆! ≈ 𝑆!0∆!𝑒I'(!0∆!,2)#*(!0∆!,2)J2 Weassumethatthedeterminantsofforwardvolatilityarethechangesinspot,interestrate and deferment rates alone, and that the passage of time is an insignificantcontributiontovolatility.Todeterminethevolatility,wemustfirstdeterminetheforwardpricereturn:(9A1.5)Forwardpricereturn=ln(𝐹!0∆!,2#∆!/𝐹!,2)Substitutingfromtheequationabove:(9A1.6)ln(𝐹!0∆!,2#∆!/𝐹!,2) =

lnr𝑆!0∆!𝑒I'(!0∆!,2)#*(!0∆!,2)J2s − 𝑙𝑛[𝑆!𝑒I'(!,2)#*(!,2)J2] =

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𝑙𝑛[𝑆!0∆!/𝑆!] + [𝑟(𝑡 + ∆𝑡, 𝑇) − 𝑟(𝑡, 𝑇) + 𝑞(𝑡, 𝑇) − 𝑞(𝑡 + ∆𝑡, 𝑇)] × 𝑇Now make the simplifying assumptions that 𝑟(𝑡 + ∆𝑡, 𝑇) − 𝑟(𝑡, 𝑇) = ∆𝑟! and 𝑞(𝑡 +∆𝑡, 𝑇) − 𝑞(𝑡, 𝑇) = ∆𝑞! .Wethenobtain:(9A1.7)forwardreturn≈D𝐻𝑃! + (D𝑟! − D𝑞!) × 𝑇whichwastobeproved,whereD𝐻𝑃 = ln((𝑆 + Δ𝑆)/𝑆).

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AppendixTwotoChapterNine:ProofofEquation(9.14)Weneedtodeterminethevolatilityofatimeseriesofpricereturnsforaforwardcontract,giventhatthematurity𝑇ofthecontractisconstantlydecreasing.Assumethefollowingstandardresultfortwoindependentvariables𝑋and𝑌:50(9A2.1) 𝑉𝑎𝑟(𝑋𝑌) = 𝑉𝑎𝑟(𝑋)𝑉𝑎𝑟(𝑌) + 𝑉𝑎𝑟(𝑌)𝐸[𝑋]) + 𝑉𝑎𝑟(𝑋)𝐸[𝑌])Let𝑋betheseriesofmaturities,and𝑌bethechangesininterestrate∆𝑟!(ordefermentrate∆𝑞!).Assumethattheaverageinterestrateordefermentratechangeiszero,i.e.that𝐸[𝑌] = 0. (9A2.2) 𝑉𝑎𝑟(𝑋𝑌) = 𝑉𝑎𝑟(𝑋)𝑉𝑎𝑟(𝑌) + 𝑉𝑎𝑟(𝑌)𝐸[𝑋]) + 𝑉𝑎𝑟(𝑋)𝐸[𝑌]) =

𝑉𝑎𝑟(𝑌)[𝑉𝑎𝑟(𝑋) + 𝐸[𝑋])]Thenwecan treat theseriesofmaturitiesasauniformdistribution fromthestartingmaturity 𝑇 down to zero. The variance 𝑉𝑎𝑟(𝑋) and the average 𝐸[𝑋]of a uniformdistributionovertheinterval(𝑥, 𝑦)areasfollows.(9A2.3)𝑉𝑎𝑟(𝑋) = (𝑦 − 𝑥))/12 = 𝑇)/12(9A2.4)𝐸[𝑋] = 𝑇/2Substituting:(9A2.5)𝑉𝑎𝑟(𝑋𝑌) = 𝑉𝑎𝑟(𝑌)[𝑉𝑎𝑟(𝑋) + 𝐸[𝑋])] = 𝑉𝑎𝑟(𝑌)[𝑇)/12 + 𝑇)/4] =

𝑉𝑎𝑟(𝑌) × 𝑇)/3(9A2.6)𝜎(𝑋𝑌) = t𝑉𝑎𝑟(𝑋𝑌) = 𝜎(𝑌) × 𝑇/√3whichwastobeproven.

50 L.A.Goodman(December1960).“OntheExactVarianceofProducts,”JournaloftheAmericanStatisticalAssociation55(292):708.

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ChapterTen:ModellingMortality1.IntroductionThischapterprovidesanintroductiontothemortalitymodellingissuesthatarisewithequityreleasevaluation.2.RealisedMortalityRatesConsiderthefollowingFigureshowingtherealisedmortalityratesofEngland&Walesmalesforages70andabove.

Figure10.1:RealisedMortalityRatesforMales70andAbove

Notes:AsperFigure3.1.

Byrealisedwemeanthemortalityratesexperienced for thesemalesover thesampleyears,1971:2017.51Weseethatthemortalityrateisalittleunder2%forage70andthenriseswithage,aswe would expect. However, there are two problems with the mortality rates in thisFigure.Thefirstisthattheyonlygouptoage89,whichisthemaximumageinoursampleagerange,andforNNEGvaluationwewantmortalityratesforhigheragesaswell.Thesecondisthattheserealisedmortalityratestakenoaccountofanyanticipatedlongevityimprovements.

51Let𝐷bethenumberofpeopleofacertainagewhodieinacertainyearandlet𝐸bethenumberofcorrespondingexposuresorpeopleofthatageatriskofdyingthatyear.Thedeathrate𝑚 = 𝐷/𝐸andthemortalityrateis𝑞 = 1 − 𝑒&'(see,e.g.,Cairnsetalia,2009,p.3).Themortalityrateisamoreconvenientratetousethanthedeathratebecauseitismathematicallymoretractableandbecauseitisguaranteedtobelessthan100%,whereasdeathratescanexceed100%due,e.g.,toerrorsintheexposuresdata,whichmightbeduetomisreportedbirthdatesortheoccasionalunreportedhomicide.

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3.ProjectedMortalityRatesAtraditionalsolutiontothislatterproblemistousesomespeciallypreparedmortalitytables,whichareinessenceexperteducatedguessesaboutfuturemortalityrates.SuchtablescanbeobtainedfromtheContinuousMortalityInvestigation,forexample.52Thisapproachiseasiertoapplybutissubjectiveandlackstransparency.Amorescientificallygroundedandmoretransparentandthereforebetterapproach(althoughthereisabitofalearningcurve)istousemortalityrateprojectionsfromastochasticmortalitymodel.53OnesuchmodelistheCBD-M5modelmentionedearlier,whichwasdesignedspecificallyforoldagemortalityprojections.Figure10.2givestherealisedandprojectedfuturemortalityratesformalesjustturned70:54

Figure10.2:ProspectiveMortalityRatesforMalesAged70

Notes:AsperFigure3.1.

Wehaveshowntheprojectedmortalityratesouttoage100,butthemodelallowsustoprojectthemtoanyagewewish.55Weseethattheprojectedmortalityratesgrowatalowerratethantherealisedrates.Thisdifferencebetweenthetwoplotsillustratestheimpactofprojectedimprovementsonfuturelongevity.

52 See https://www.actuaries.org.uk/learn-and-develop/continuous-mortality-investigation/cmi-mortality-and-morbidity-tables.53Thesemodelsnotonlyallowuserstoobtainlikelyprojectedmortalityrates,butalsoallowthemtoobtainstatisticallygroundedpredictionintervalsandscenarioanalyses,andtotakeaccountofrefinementslikeparameteruncertainty,individualdeathriskandtheimpactofBayesianpriorbeliefs.54Thesemortalityratesareknownascohortqrates,becausetheyfollowthecohortcurrentlyaged70astheyageovertimeandtheirmortalityratesreflecttheirincreasingage.55ForNNEGvaluation,weactuallyuseprojectedmortalityratesouttoage120,i.e.,formodellingpurposesweassume thatany individualswhoreach their121stbirthdayare thenautomaticallydispatched. It isimportanttotakeaccountoftheextremeoldage‘toxictail’whendealingwithlifetimefinancialproducts.

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4.ExpectedSurvivorRatesThenextstepintheanalysisistoobtainthecorrespondingexpectedsurvivorrates,𝑆! ,i.e.,theprobabilitythatanindividualalivenowwillsurvivetoyear𝑡.Ifwelet𝑞!bethemortalityrateforyear𝑡,thenthefollowingholdsfor𝑆!:(10.1)𝑆" = 1(10.2)𝑆$ = (1 − 𝑞$)(10.3)𝑆) = 𝑆$ × (1 − 𝑞$) = (1 − 𝑞$) × (1 − 𝑞))etc.Figure10.3showstheexpectedsurvivorratesforthepeopleontheir70thbirthday:

Figure10.3:ExpectedSurvivorRatesforMalesAged70

Notes:AsperFigure3.1.

Theexpectedsurvivorratesfallfrom100%onday1downeventuallytowardszero.5.ExpectedMortalityRatesWetaketheexpectedmortalityrate(ofacohortofgivencurrentage)forfutureperiod𝑡tobetheproductofthemortalityratefor𝑡andtheexpectedsurvivorratefor𝑡,i.e.,theexpectedmortalityrateequals𝑞! × 𝑆! .Theexpectedmortalityratesforacohortofmalescurrentlyaged70areshowninthenextFigure:

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Figure10.4:Figure3.1ExitProbabilitiesforMalesCurrentlyAged70

Notes:AsperFigure3.1.

whichisessentiallythesameasFigure3.1,ifweignorethepossibilityofhouseexitbygoingintoanursinghome.Weseethattheexpectedmortalityratesrisetopeakinthemidtolate80s,thenfallandeventuallygotozero.Now remember that these expectedmortality rates are theweights that apply to theNNEGputoptions.Thisweightingscheduletellsusthatthelowmaturityputsarenotsoimportant, because of the low probability of exercise, but the puts become moreimportantastheirmaturityrisesandtheonesthatareofmostsignificanceofthosewithmaturitiesofmaybe10toa littleover20years.Theputswith longermaturitieshavedecliningandeventuallyinsignificantimportance.6.ModelRiskinMortalityProjectionsAproblemwiththeseprojections isthattheyaredependentonanassumedmortalitymodel,M5.Theyarethereforeexposedtomortalitymodelrisk,i.e.,theriskoferrorfromtheuseoftheM5mortalitymodel.ThebestwaytoaddressthisissueistoconsideralternativemodelsandonethatwouldbesuitableisM7.ThismodelisanextensionofM5.Model5hastwoperiodeffects,butM7addsathirdperiodeffectandacohortoryearofbirtheffecttoM5.Furtherdetailscanbefoundin,e.g.,Cairnsetalia,2009.56Figure10.5showstheexpectedmortalityratesfor70yearoldmalesbasedonprojectionsfrombothmortalitymodels:

56Thosefamiliarwiththesemodelsmightbewonderingwhywedon’talsoincludemodelM6.Theansweristhatforreasonswedon’tquiteunderstand,M6givespotentiallyunreliableresultsforEngland&Walesfemales.Wethereforedropitfromconsideration.

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Figure10.5:ExpectedMortalityRatesforMalesCurrentlyAged70:ModelsM5

andM7

Notes:CalibrationsasperFigure3.1.

Theplotsaredistinctbuthavemuch thesameshape.Consequently,wewouldexpectthesemodelstogivefairlycloseNNEGvaluations.7.ModelRiskinMortalityProjectionsTable10.1showsthevaluationscorrespondingtoeachofthemortalitymodels:

Table10.1:BaselineNNEGandERMValuationsforMalesAged70CurrentHouse

PriceLoan

AmountModel 𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

£100 £40 M5 £81.72 £41.15 £40.63£100 £40 M7 £81.72 £40.50 £40.71

Notes:CalibrationsasperTable3.1WeseeacertainamountofvariationintheloanvaluesandNNEGvaluations,butmuchlessvariationintheERMvaluationsduetotheoffsettingimpactsoftheloanvaluesandtheNNEGvaluationsontheERMvaluations.8.FemalesTable10.2givesthecorrespondingvaluationresultsforfemales:

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Table10.2:BaselineNNEGandERMValuationsforFemalesAged70CurrentHouse

PriceLoan

AmountModel 𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

£100 £40 M5 £89.50 £51.12 £38.38£100 £40 M7 £90.06 £51.82 £38.23

Notes:AsperTable10.1butforfemales.Theappropriatevolatilityforfemalesaged70is20.8%.ThefemaleloanvalueandNNEGvaluationsarehigherthanforthemales,aswewouldexpectfromhigherfemalelifeexpectancy.Thevaluationsfromthetwomodelsarealsoveryclose. 9.JointLivesManyERMloansaretocouplesratherthanindividuals.Intheory,oneshouldmodeltheexitprobabilitiesassociatedwithsuchloansinawaythattakesaccountofthelongevityprospectsofbothpartnersandthepointthatexitwilloccurwhenthelongestsurvivingpartnerexitsthehouse.Suchananalysisisalittleinvolved,however,andthestandardapproachistotreatsuchanERMloanasifitwerealoantotheyoungestpartner.Atypicalcasewouldbeacoupleinwhichthemaleis70andthefemale66,i.e.,sowehaveayoungerfemalebutbearinmindthatfemaleshavelongerlifeexpectancy.SomeresultsaregiveninTable10.3:

Table10.3:NNEGValuationsandLifeExpectanciesforaTypicalCoupleCurrentHouse

PriceLoan

AmountPartners 𝑵𝑵𝑬𝑮 Lifeexpectancy

£100 £40 Maleaged70 £41.15 15.0£100 £40 Femaleaged66 £60.86 20.6

Notes:AsperTable10.2ThetableshowsNNEGvaluationsandassociatedlifeexpectancies,wheretheNNEGisthatofanERMloantoeachpartnerconsideredontheirown.TheNNEGvaluationsare£41.15fortheloantothe70yearoldmaleand£60.86fortheloantothe66yearoldfemale.BearinmindthattheNNEGforaloantoacouple,withexitdeemedtooccurwhenthelastremainingmemberexits,willalwaysbelargerthantheNNEGfromaloantoeitherindividualmemberalone.ThereasonisthatthelatterNNEGvaluationsdonotaccountfortherisk(tothelender)oftheotherpartnerexitinglaterthanthepartnertowhomtotheindividualloanwasmade.Thus,thetrueNNEGvaluationwouldbebiggerthaneitheroftheNNEGvaluationsshownintheTableanditisimmediatelyapparentthatonecouldgetamajorunder-estimationofthe𝑁𝑁𝐸𝐺ifonehadtreatedtheloanforNNEGvaluationpurposesasifitjustaloantoamaleaged70.Eventhehigher(female66)NNEGof£60.86willunder-estimatethetrueNNEG,butifonelooksatthelifeexpectancies,itisalsoclearthatthefemaleisexpectedtolive5.5yearslongerthanhermalepartner.Itisthereforeunlikelythatthemalewilloutliveher,

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andonecanmakeaplausiblehandwavingargumenttotheeffectthattheerrorintheNNEGvaluationwillnottooimportant.57However,anysuchargumentreliesonthespecificsofthecaseathand–andinparticular,ontheyoungermemberhavinganotablylongerlifeexpectancy–andthatconsiderationwillnotalwaysapply.Forexample,ifthemaleandfemalewerereversedinage,thenthe66yearoldmalewouldhavealifeexpectancyof17.1yearsandhis70yearoldfemalepartnerwouldhavealifeexpectancyof18.4years.Thereisthenamuchbiggerriskoftheoldermemberofthecoupleoutlivingtheyoungerone,andtheNNEGvaluationerror,associatedwith treating the loan as if itwere a loan to the youngermember,will begreaterthaninthepreviouscase. 10.ImpairedLives ERMcompanieswillsometimesoffermoreattractivetermstoborrowerswithimpairedlives, i.e., thosewith reduced longevity prospects. A first pass atmodelling fair-valueNNEGguaranteesforimpairedlivesborrowersistoestablishtheimpactoftheirhealthconditionon theirexpected longevity, thenoffer themanLTVbasedon thatexpectedlongevity.Togiveasimpleexample,ifaborroweraged70hasthelifeexpectancyofan80yearold,thenthelendermightofferthemtheLTVi.e.loantermsofan80yearold,e.g.,sotheborrowermightgetaloanbasedonanLTVof40%insteadofthestandard30%.ForNNEGvaluationpurposes,then,wemighttreatthe70yearoldasifhewerean80yearold.Asafurtherrefinement,wemightalsotakeaccountofwhetherandifsohow,theirhealthconditionmightaffecttheprospectsforthelengthoftheirendoflifeperiodincare.

57TherearealsotwooffsettingeffectsonanyNNEGunder-valuation.First,thereisthepossibilitythatthesurvivingmemberofthecouplewillmoveoutafterfirstonehasdied.Second,thereissomeevidencefora‘heartbreak syndrome’ bywhich the death of the firstmember raises themortality rate of the other,althoughanecdotalevidencesuggeststhattheoppositecansometimesoccurtoo.

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ChapterEleven:Long-TermCareWehavehithertoassumedthatexitoccurswithdeath.Inreality,manyolderpeopleleavetheirhomestomoveintosomeformoflong-termcare(LTC)andthetimespentincarecanbesubstantial.LengthofTimeinLong-TermCareWeunderstandthataruleofthumbinthisareaisthatpeopleareexpectedtospendtwoyears in long-termcare.Orperhapsa little longer:“Onaverage,olderpeoplestay inaresidentialcarehomefor30months,”statesarecentIndependentAgearticlecitinganearlier(2009)reportbyLangBuisson.58ThesenumbersarecomparabletothosefromtheU.S.Forexample,ThomasDay(2010)statesthat,the“averagestayforelderlypatientswhodieinanursinghomeisjustshyof2years,”59whilstBradBreedingcitesU.S.reportsof2009and2010thatreportaveragelengths of stay in assisted living facilities of about 28 months and 29 monthsrespectively.60Butwhateveryoneagreesonisthatthelengthoftimeincare,andwhetheronewillneedtospendanytimeincareatall,arehighlyuncertain.TakingAccountofExpectedTimeinCareThedifferencebetweentheexpectedtimetodeath(i.e.,lifeexpectancy)andthelengthoftimetohouseexitcouldthenmakeamaterialdifferencetoequityreleasevaluations.Theproblemishowtotakeaccountofthisdifference.ThestandardactuarialpracticeintheUKistoobtainexitprobabilitiesfrom(conditional)mortalityprobabilitiesbyimposingloadingfactorsonthelatter.AloadingfactorinvolvesmultiplyingthosemortalityratesbyafactorthatreflectssomeLTC‘addon’.Forexample,for thepurposesofobtainingexitprobabilities frommortalityrates,wemightaddanadditional 2% to the latter. Our exit probabilities would then be 102% times the(conditional)mortalityrates.Hostyetalia(2007)offerascheduleofsuchloadingfactors:

58 See “Cost of average length of stay in a residential home is equivalent to 26 years’worth of familyholidays.” IndependentAge26Oct2017; LaingBuissonCareofOlderPeopleMarketReport, 27 edition,London:LaingBuisson.59T.Day(2010)“AboutNursingHomes.”https://www.longtermcarelink.net/eldercare/nursing_home.htm 60 B.Breeding(2016)“SoI’llProbablyNeedLong-TermCare,ButforHowLong?myLifeSite(6July).https://www.mylifesite.net/blog/post/so-ill-probably-need-long-term-care-but-for-how-long/

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Table11.1:Hostyetalia(2007)MortalityLoadingFactors

Age Male(%) Female(%)≤ 70 2 3(70,80] 4 12(80,90] 5 13(90,100] 4 8

Source:Hostyetalia(2007,Table13).Theseloadingfactorsrisewithageandthenfallagainandreflectanintuitionabouttheexpectedlengthoftimespentincareoutsidethehome,dependingontheagewhenonegoesintocare.Thereareproblemswiththisapproach,however.OnewassetoutbyTunaru(2019,p.67):

Formulti-statemodellingconsidering the interactionbetween long-termcareentryandmortalityisparamountbecausethereissignificantlyhighermortalityexperiencedbylong-termcareresidentscomparedto“athome”mortalitymeansthattomaintainthesameaggregateassumptionformortalitybyagelighterthanaveragemortalityshouldbeassumedfor“athome”lives.

People incarewillbe lesswell thanpeopleof thesameagestillathomeandsotheirmortalityrateswillbehigher.Sincethepopulationmortalityratesareaveragesofthemortalityratesofpeople incareandpeopleoutsidecare, thenanyadjustmentfortheformerrequiresanoffsetting(butnotnecessarilyequallyoffsetting)adjustmentforthelatter. The problem here is that without further data on the numbers in care vs thenumbersoutsidecare,thenanysuchadjustmentsaredifficulttocarryout.Thereisalsoadeeperissue.Evenifwewereconfidentinourprojections,wehavenoreasontothinkthattheseloadingfactorsareanygoodinthefirstplace.Tocalculatethem‘properly’fromfirstprinciples,wewouldneed(a)reliableprojectionsforthemortalityratesofpeopleincare,(b)reliableprojectionsforpeopleoutsidecareand(c)reliableprojectionsoftherelativesizesofthesetwopopulations.Theproblemisthatwedon’thave any of these. Instead, we have are loading factors pulled out of thin air, i.e.,guesstimates,admittedlybylifeactuarieswithsomeintuitionforsuchissues,andthatisallwehave.Soweshouldn’tplacemuchrelianceontheHostyetalialoadingfactors.Fortunately,wedon’thaveto,becausethereisabetterwayanditissimpletoo.Supposewebelievethattheexpectedtimeinlong-termcareis2years.Ifwehavea70yearoldmale,thenhislifeexpectancyisabout15years,andweexpecthimtospendhislast2yearsincare,i.e.,weexpecthimtoexithishomein13years.Sowecanapproximatehistimetoexitbygivinghimthelifeexpectancyofa72yearold,andwecandothatbygivinghim theprojectedmortality ratesofa72yearold.Thus,we takeourNNEGandERMfunctions,andinputagesof72insteadof70.Table11.2givessomeillustrativevaluations.

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Table11.2:NNEGValuationsandLifeExpectanciesforaTypicalMale

CurrentAge 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴70 £41.15 £40.6372 £34.19 £41.94

Notes:AsperTable3.1WeseethatthisageadjustmentmakesasubstantialdifferencetotheNNEGvaluationbutamuchsmalleronetotheERMvaluation.If we believe that the expected time in care is 2.5 years, then we can obtain ourapproximationsbytakingourNNEGtobeaverageoftheNNEGsforages72and73,andsoforth.Ideally,wemightwanttobuildsomefancymodel,butintheabsenceofdata,wecan’t.Inthemeantimethesimplisticapproachjustsuggestedcouldbethebestwecando.

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ChapterTwelve:DelayedPossessionWehavehithertoimplicitlyassumedthattheloanisimmediatelyrepaidatthetimeofhouseexit.Inpractice,however,thereislikelytobesomedelay.Forexample,intheirNNEGstudy,Lietalia(2010)define𝛿astheaveragedelayintimefromthepointofexitto thesaleof thehouse.Theyuseabaselinecalibrationof𝛿=halfayear (p.16)andprovidesomesensitivityresults,buttheydonotprovideanyempiricaljustificationfortheir𝛿calibrations.Intheory,wetakeaccountoftheimpactofdelaysonourvaluationsifwehavegooddatathatallowsustoestimatetheexpecteddelay.Let’ssuppose,forexample,thatweexpecttheretobeahalfyeardelaybetweenhouseexitandhousesale.Wecanthenhandlethisdelaybytreatingtheborrowerasifheorshewerehalfayearyounger.Soiftheborrowerisactually70whentheloanistakenout,wetreattheborrowerasifheorshewere69andthenobtaintheNNEGorERMasequaltotheaverageoftheNNEGsorERMsfor69yearholdand70yearold.Theactuaryormodelercanpresumablyusetheirfirm’saccountdatabasetoobtainsomesenseofthedelayedexperiencedonpastERMloans.However,itmaynotalwaysbesostraightforward,andfortworeasons.Firsttherewillbedispersioninthe‘timetosale’statistics,andsomeofobservedtimestosalecanbelong.Inonecasewehaveseen,thetimetosalewasthreeyears.Asecondissueiswhethertheloanfixesatexitordeathandhasdelayedsettlementorkeepsrollinguntilsettlement.In the latter case, there is the potential for some serious abuse. One of ourcorrespondents,anexpertinERMpropertymanagementpractices,informsusthat

abankorbuildingsocietycanalsouselongtermassets(likeERMs)tohideitserrorsofvaluationatthebeginningofalongdatedbook...IhaveseensomenaughtyinstitutionsfiltertheharvestofERMassetsastheyrolloffbypostingthecashresultsofthosethatexpireatLTV<100%andjustallowingthe>100%onestorollonandon,withoutaddressingorcrystallisingthem.Sofroma reports and accounts perspective (where little of this level of analysis isexposedtothepublic)itlooksasthoughthecashisrollinginnicelyasplannedatinception,butthehorrors(theNNEGsthatshouldeatuptheaccumulatedcash accounting value of the assets) are also piling up unnoticed, becauseuncrystallised–dealwiththemlaterasitwere.Sothebooklookslikeit’sthebestcashgeneratingassetonthebalancesheet–cashflowsoaringandotherloansstillrollingup.Becausethere isnoexplicit“credit”effectonanERM,thereisnoobvious“default”ifno-onedecidestorecogniseit.Don’t forget thateven if someonedies, the recognitionof that factand themovementtosaleoftheunderlyingcollateralcanbedelayedforagestooasyoucanclaimtobenegotiatingwiththeheirsetcetc,andall thewhiletheassetisstillrollingupatthecompoundrate(eventhougheconomicallytheassetshouldhavebeencrystallisedmuchclosertoactualdeath/LTC)wheninfactitisgettingevermoreabove100%LTV.Butyoucanarguethatitisstillalegalclaimontheestate,evenifitsunlikelyevertobefullyrealised–youcan

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saythatyouwereenmeshedinnegotiationswiththeestateandhopingforasuddendoublingof thehousingmarket. Ihaveseenactualcases in theUKwhere the time from both mortgagors final death date to my portfolioappraisaldateasputativebuyerhasbeengreaterthan7years–andSTILLtheassetshowsascurrent,nicelyrollingupetcetc,whenitpassed100%BEFOREtheydied,andwhenpresumablyitsnowfullofsquattersandfallingdown!Liftthecornerofthismatandallthelittlebugscomerushingout……….

Evenso,hesuggests,“abitoflaggingissmallbeercomparedtothewide[achievementrate]dispersionnomatterwhat.”

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ChapterThirteen:CreditSpreadsWehavesofarassumedthatwearediscountingatrisk-free,i.e.,withnocreditspread.Noteveryoneagreeswiththisapproach,however.Therearetworeasonswhysomeonemightwanttoaddacreditspreadtothediscountrate. The first is because theywant to get lowerNNEG valuations, and the second isbecausetheybelievethatthereshouldbeacreditspread.Bothofthesearewrongandthefirstself-evidentlyso:thereisneveragoodreasonfortweakingcalibrationstogetdesired outcomes. But even if well-intended, it is still amistake to add in the creditspread.Agoodexampleof thecredit spreadargument isprovidedbyDavidLandat the IFoAsessionaleventonERMson11December2018.HeworksforRothesayLife,whoareamajorproviderofannuitiesandamajoruseroftheMatchingAdjustment.61TheyalsojustboughtawhackinggreatportfolioofEquityReleaseMortgages fromthetaxpayer.62 Itmakes sense that Rothesay would be interested in an approach that lowers NNEGvaluations.WethenwonderifRothesayareusingahigher-than-LIBORfundingratetocheapenthecostoftheirguarantees.Iftheyare,theyshouldn’tbe.In thediscussionat theDecember11event,Mr.Landaskedapointedquestion. If theworkingpartyhasn’tyet fixedtherightmethodofcalculatingtheforward, isn’t thataprettymajorsourceofpossibleerror?Nocoherentansweremerged,butLandraisedaninterestingpoint.Sowhatisgoingon?Well,ifwecan’tlowerthevalueofthenonegativeequityguaranteebyputtinginanoptimisticgrowthforecast,perhapswecantweakthefunding rate instead. He drops a hintwhen he suggests that there’s a large range ofpossiblefundingratesthatwemightconsiderandgoesontostatethat“ThePRAthinksthatyoucouldpossibly fundahouseatLIBORflat,whichseemsremarkablydifficult.”Theimplicationisthatweshouldbeaddinginaspread.Land’s argument is a seemingly plausible one and provides a much better case forloweringNNEGvaluationsthantheso-called‘realworld’approachthattheindustryaresoenamouredof.Itisstillwrong,butitiswronginamoreinterestingway.Nowweagreewithhimthat itwouldbeanunusual lenderwhomadearisky loanat‘LIBORflat’.Iftheyaretolendatall,theywouldchargeaspreadtocompensatefortheriskoflossfromdefault.Afterall,theborrowermightdefaultandleavethelenderwithalossbecauseinsufficientcollateralhasbeenposted.

61Seetheir2017SolvencyandFinancialConditionsReportathttps://www.rothesaylife.com/media/1183/rothesay-life-sfcr-2017.pdf62 See http://eumaeus.org/wordp/index.php/2018/09/28/doing-gods-work/ andhttp://eumaeus.org/wordp/index.php/2018/10/05/casting-magic-upon-daylight.The substance is thatin September 2018 Rothesay Life bought an £860m portfolio of equity release loans from UK AssetResolutionLimited(UKAR),agovernmentagencyestablishedon1October2010to‘facilitatetheorderlymanagement’oftheclosedmortgagebooksofbothBradford&BingleyandNRAM.(Source:OliverRalph,FT,27September2018,UKARwebsite).ThepreciseamountpaidbyRothesaywasnotdisclosed

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Butconsiderthefollowing.Supposewedidaddaspreadandsuppose(becauseitistrueand easily verified) that adding the spreadproduced a lowerNNEG. You thenhave asituationwheretheworsethecreditqualityoftheborrower,thehigherthespread;andthehigherthespread,thecheapertheNNEG,becausetheNNEGisbasedonasetofputoptionspricedofftheforward.Sothespreadargumentdoesn’tpassthesnifftest.Let’strytodrawoutwhatLandmayhavehadinmind.Startwith:(13.1)𝑅 = 𝑆 × 𝑒#*!(13.2)𝐹 = 𝑆 × 𝑒#('0K#*)!where𝑅 is thepriceof a reversion akadeferment contract,𝑆 is the spotpriceof theincome producing asset e.g., a property, 𝑞 is the discount rate aka deferment raterepresentingthepresentvalueoflostincomeoverthetermof𝑡years.𝐹isthepriceofaforwardcontract,𝑟theriskfreeand𝑠thefundingspreadoverLIBOR.IfweuseBlack76thentheunderlyingistheforwardprice𝐹ratherthanthedefermentprice𝑅.SincetheNNEGforanydecrement𝑡isaputoption,thehighertheforwardprice𝐹,themoretheoptionisoutofthemoney,andthelowerthecostoftheNNEGguarantee.SowecanmaketheNNEGatleastappearcheaperbyassuminganon-zerospread𝑠.Supposethenweareafirmholdingadefermentcontractforpossessionattandwewanttohedgeourriskbysellingthecorrespondingforwardcontract.Let’salsotakethecasemostfavourabletothecreditspreadargument,whichiswhereourcounterpartyhasnocollateralatall.Ifwepricethecontractperequation(13.2)above,itiscertainlytruethatwemaywanttochargeaspreadovertherisk-freerate𝑟tocompensatefortheriskthatthepurchaseroftheforwardcontractwilldefaultat𝑡.Sohowdowerepresentthevalueofourhedgedpositioninourbooks?Assumingnoriskofdefault,thepresentvalueofthepositionis(13.3)𝑅L = 𝑅 × 𝑒K! .Butthereclearlyisariskofdefault,becausewejustassumedonewhenweappliedthespread𝑠!Wemustthereforereserveagainstthatrisk,butbyhowmuch?Wellifsisaspreadrepresentingthecostofdefault,i.e.,ifthedifferenceinpresentvalueattributableto𝑠 ispreciselythecostofdefault, thenthatsamedifferenceistheamountwewouldreserve.Consequently(13.4)Reserve= 𝑅 × 𝑒K! − 𝑅Wemustthensubtractthereservefromourportfoliovalue:(13.5)Portfoliovalue= 𝑅L −reserve= 𝑅 × 𝑒K! − (𝑅 × 𝑒K! − 𝑅) = 𝑅Wearethenbackwherewestartedandthecorrectspreadiszero!Youcannotcheapenthevalueofadefermentcontractbyhedgingitinthederivativesmarket.Whateverincreasein

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valueisattributabletoapplyingaspreadoverLIBORrepresentsacompensationforrisk,whichmustbesubtractedagainasareserve.Remembertoothatthiscaseistheonemostfavourabletothecreditspreadargument,that inwhich the counterparty provides no collateral. If the forward contract is fullycollateralised,on theotherhand, then therewouldbenocredit spreadbecause therewouldbeno credit risk. So either the spread goes in and then comesout again, or itdoesn’tgointostartwith.Andintheintermediatecasewherethecontractispartiallycollateralised,wehaveaweightedaverageofthesetwocasesandagainnospread.Wehaveonlyconsideredthecaseofaforward,butthesameappliestoanoption.Wehavejustshownthatwecannotraisetheforwardratebytweakingthefundingrate.Thefundingrateshouldbetheriskfreerate,notthehigherratethatmightbeofferedbyabank to, say, a buy-to-let investor. But if we cannot change the forward rate, then itfollowsthattheNNEGvaluationmustremainunchangedaswell.

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ChapterFourteen:DrawdownDrawdownERMsdifferfromthe‘conventional’ERMs(oftencalledLifetimeMortgages,LTMs) that we have considered so far in that the borrower contracts to receive adrawdownfacilityratherthanaloantakenoutatanyonetime.Thisfacilitywouldgivetheborrower the rightbutnot theobligation todrawdownon the facility athis/herdiscretion,uptothemaximumamountofthedrawdownfacility.Typically,theborrowerwoulddrawdownan initialamountwhenthe facility issetupwithaviewtomakingsubsequentdrawdownslater.Theloancontractwouldspecifyacurrentdrawdownlendingrate,whichweunderstandwouldtypicallybealittlehigher(maybe5-6basispointshigher)thanthelendingrateonanLTMmortgage.Onceadrawdownismade, the lendingrateonthat loantranche isfixed.Thedrawdownlendingrate,liketheLTMloanrate,willchangeovertimeasthelenderperiodicallyadjuststherateinresponsetochangesinmarketconditionsanditsownlendingpolicy.TheexistenceofthedrawdownfacilitymakesDrawdownERMsmoreinvolvedthanLTMERMs,buttherehasbeenlittlediscussionofhowtohandlethedrawdownfacilityfromtheERMvaluationperspective.63One way tomodel valuations on drawdown ERMs and NNEGs would be to use pasthistoricaldata–whichthefirmwouldhave–onitsdrawdownexperience.Itsvaluationspecialistscouldthenusethesedatatoprojectthetimingoffuturedrawdownsandtheamountsdrawndowneachtime.Forexample,theymightanticipatethatafemaleaged70woulddrawdown50%oftheremainingfacilityatage74anddrawdowntherestofthefacilityatage78.Theycouldalsoforecastthedrawdownlendingratesatthesefuturetimes.ForERMandNNEGvaluationpurposestheycouldthentreatthedrawdownERMasequivalent toaportfolioof threeseparateLTMloansandwealreadyknowhowtovaluetheERMsandNNEGsofLTMloans:therewouldbeaninitialLTMloantakenoutnowwhentheborroweris70,asecondLTMloanthatisexpectedtobetakenoutin4years’timebythesamefemalewhensheis74andathirdLTMloanthatisexpectedtobetakenoutin8years’timebythesamefemalewhensheis78.Analternativewaytomodelvaluationsisjusttoassumethatthewholeofthedrawdownfacilityisdrawndowninonegowhenthecontractismade.Inthatcase,thedrawdownERMloanwouldbetreatedasequivalenttoasingleLTMloan.

63InitsCP7/19(section2.15to2.18),thePRAhasanextensivediscussionofdrawdownissuesthatis,nonetheless,quitesolution-lite.SeePRAConsultationPaper7/19“SolvencyII:EquityReleaseMortgages–Part2,”April2019.

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ChapterFifteen:EarlyRepaymentManyERMcontractsallowtheborrowertorepayearlierundercertainconditions.Nowearlyrepaymentofmortgagesisanotoriouslyfiendishproblem.64Itiswellknown,forexample,thatprepaymentratesonmortgage-backedsecuritiesaresensitivetointerestratesandMBSmodellershavehadgreatdifficultiesgetting thismodelling ‘right’.Onemight however suppose (or at least hope) that the prospects of future repaymentsinducedbylowerinterestrateswouldnotbesuchaproblematcurrentlowinterestrates.TheusualapproachsuggestedintheNNEGliteratureistoworkwithassumedrepaymentrates, whichmight be either pulled out of thin air or based on historical repaymentbehaviour. Tunaru (2019, section 10.9) has a gooddiscussion of this subject, but notmuchisknown.65Historicalprepaymentratestendtobehighintheearlyyearsbutthentailoff.Suchratesmakesenseintuitively:intheearlyyear,borrowersmayregrettheloanandstillbeabletopayitback;inthelateryears,itbecomesincreasinglydifficulttopayoffearlyastheloanamountrollsupandtheirfinancialcircumstancesdeteriorate.Thebestwecandoisprobablytousehistoricalrepaymentratesasaproxyforfuturerepaymentprobabilities.HowtoHandletoEarlyRepaymentOnewaytomakeuseoftheseforNNEGandERMvaluationgoesasfollows.ImagineforthemomentthatanERMcontracthadnofacilityforearlyrepayment.ThenthevalueoftheNNEG,𝑁𝑁𝐸𝐺,andthevalueoftheERM,𝐸𝑅𝑀,wouldbewhattheywereinpreviouschapters,andwecanmoveon.Putitthisway:if𝑁𝑁𝐸𝐺/M'G,NOisthevalueoftheNNEGwithoutthepossibilityofearlyrepaymentandsoforth,then(15.1)𝑁𝑁𝐸𝐺 = 𝑁𝑁𝐸𝐺/M'G,NO(15.2)𝐸𝑅𝑀 = 𝐸𝑅𝑀/M'G,NOWenowintroducethepossibilityofearlyrepaymentandaskourselves:howdoesthepossibilityofearlyrepaymentaffectNNEGandERMvaluations?Let’s startwith a pre-existing ERM loan, and let its value be𝐸𝑅𝑀M&P . If the old/pre-existingERMloanisrepaid,thenacertainamount–𝑋,say–ispaidtothelenderandtheERMisextinguished.𝑋wouldincludeanyearlyrepaymentchargesstipulatedbytheERM

64 Formore on these issues, see, e.g., JPMorganMBSPrimer, June 2006, or A. Davidson andA. LevinMortgageValuationModels:EmbeddedOptions,Risk,andUncertainty,OxfordUniversityPress,2014.65 Anotherproblem is that theratingsagencies takeaveryconservativeapproach to repayment.Theytypicallymodelmortgagesecuritiesusingayieldtomaturityapproach,andthenaccountforprepaymentas reinvestment at swapsminus 50 bps. This approachmakes prepayment appear very expensive forlenders.

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contract.The importantpoint for themodeller is thatwecaneasilyworkoutwhat𝑋mightbe.Weneednowtomakeanassumptionaboutthelenderdoeswiththemoneyitreceives.ItcouldinvestinT-bill,holditincashandsoon,butlet’ssupposethatthelenderusesthemoneythatisrepaidtomakeanewERMloan,𝐸𝑅𝑀/GQ .66Let’ssupposealsothatprocessofconvertingtheloanrepaymenttothepointwhereanew ERM loan is made costs the firm 𝑌, where 𝑌 would be the costs of marketing,distributionandsoon,netofanysuchchargestobeappliedtothenewcustomer.Again,themodellercanworkoutwhat𝑌mightbe.Consequently,theamountleftovertobeloanedtothenewcustomeris𝑋 − 𝑌.TomaketheloanamountexplicitinthevaluationofthenewERM,callthelatter𝐸𝑅𝑀/GQ|(𝑋 − 𝑌).Thus,𝐸𝑅𝑀/GQ|(𝑋 − 𝑌) is the value of thenewERM loan,where𝑋 − 𝑌 is the amountloaned at inception. Note that the input calibrations of the newERM loanmight andtypicallywouldbedifferentfromthoseoftheoldERMloan,e.g.,theloanratemighthavechanged.However,themodellerwouldhavetheinformationneededtoobtainthevalueofthenewERMloan.SowhenanERMloanispaidoffearly,theERMlenderlosesanassetworth𝐸𝑅𝑀M&P ,butacquires an assetworth𝐸𝑅𝑀/GQ|(𝑋 − 𝑌), andbyhypothesisweknowhow to obtain𝐸𝑅𝑀/GQ|(𝑋 − 𝑌).The‘transaction’mayormaynotbebeneficialtothelender,butthelenderhasnochoicebut to accept the loan repayment if the terms of original loan allow it, and we areassumingthatthebestthelendercandowiththemoneyrepaidistoinvestitintoanewERMloan.Exante,wedon’tknowwhethertheborrowerwillrepayearlyornot,butwecansupposethat the modeller has access to data on past loan repayment frequencies, e.g., if themodellerisanERMactuaryworkingforalargefirm,thenheorshewillhaveaccesstothefirm’sloanhistories,andcanthenestimatepastloanrepaymentfrequencies.Fromthese,themodellercanestimate𝜙,theprobabilitythatagivenERMloanwillberepaidoversomefutureperiod.ThemodellercanthenobtainthevalueoftheERMtakingintoaccountthepossibilityofearlyrepayment,e.g.:(15.3)𝐸𝑅𝑀 = 𝜙 × 𝐸𝑅𝑀/GQ|(𝑋 − 𝑌) + (1 − 𝜙) × 𝐸𝑅𝑀/M'G,NO

66Thisassumptionwillnotalwaysbeappropriate,however.Asoneofourreaderspointsout:“valuingoneERMassetontheassumptionthatyoucanoriginateanother,atwill,toorder,whenneeded,outofthinair,is going to struggle to pass IASB or SII orMAP constraints.Looks like falsely capitalising the value ofreinvestmentrisktome.”Hispointsarewell-taken.Ifre-investingtheproceedsfromanearlyrepaymentintoanewERMisanissue,thenoneneedstakeaviewonhowelsethoseproceedsshouldbeinvested.Forexample,wouldtheybeinvestedingilts,andsoon.Butthebasicapproachwesetoutherecaneasilybetweakedtohandlesuchalternatives.

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ChapterSixteen:Fees,ChargesandExpensesInprinciple,wemightwanttotakeaccountoffeesandcharges.Wesay“might”forgoodreason:iffees,chargesandexpensesarefairlyset,i.e.,theyareneitherexcessivenortoolow, then if those feesetc.arepaidbytheborrower, thentheyhaveno impactonthelenderandhencewouldbeirrelevantforNNEGvaluation.TheywouldalsobeirrelevantforERMvaluation,atleastfromthelender’sperspective.Whetherfeesandsofortharefairlysetisanothermatterandisbeyondthescopeofthisreport.Accordingly, in this chapterwesetoutwhat informationwehaveon feesandcharges,andleaveittopractitionerstodecideif,andifso,how,thesemightaffectNNEGvaluations.Forwhatitisworth,ourviewisthatthesearesecond-orderissuesrelativetothebigissueslikethecalibrationofthedefermentrateorvolatilities.Ontheotherhand,fees and chargeswouldbeof considerable interest toborrowersandwould certainlyrelevantwhendoingvalue-for-moneyanalyses.Thosehoweverarebeyondourremit.HostyetaliaonfeesandchargesHostyetalia(2007,p,29)reportthefollowingpolicyexpensessummary:

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Ourunderstandingisthatthe‘distributionandsales’and‘marketing’chargesbothapply.Thus,thetotalchargefordistribution,salesandmarketingwouldbe3.5%oftheloan.Hosty et al. (2007, p. 31) also report a ‘specimen product specification’ that includessimilaritemsbutinadditionincludes:

• provider’slegalfee(£300)andpropertyvaluationfee(£1permille)toaddedtotheloan;

• property sales expenses equal to 2% of final property value: these would bechargedtotheborrowerunlesstheNNEGisactivated;

• additionalcostsineventofnegativeequityclaim:£500tocovercostsofadditionalvaluationandadministration;and

• earlyrepaymentcharges:“Marktomarketwith25%cap”,whateverthatmeans.

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Thesechargesand feeswere in2007£s.According to theBankofEngland’s inflationcalculator the cumulative inflation since then has been 13.6%, so we would have tomultiplythesenumbersby1.136togettheirequivalentsattoday’spricelevel.67Hostyetalia(2008)providetheadditionalclarificationthat‘othercosts’arepassedtothecustomer,presumablywiththeexceptionofpropertysalesexpensesiftheNNEGisactivated,becausetodootherwisewouldviolatetheNNEG. ContemporaryFeesandChargesLoanrates:wehaveseencontemporary loanratesvarying from4.15%AERto6.78%AER.Itpaystoshoparound.Repaymentcharges:wehaveseencaseswhere thecharges torepay loanswereabout9.6%oftheamountowed.Foronelargefirmwehaveseenchargesandfeesthatinclude:

• Loanrate=5.35%AER• Startfee=£600• Advicefee=£995• Legalfee=£650• Endfee=£125.

Ourunderstandingisthatforthisfirm,allothercostsandexpenseswouldbepassedtotheborrower,unlessdoingsowouldviolatetheNNEG.Thereisalsotheissueofrepaymentcharges,andweareawareofonecontemporarycaseinwhichaborrowerfacedarepaymentchargeofabout16%oftherolled-uploanamountanditstruckusthatthischargewashigh.Therecouldalsobeotherfeesandcharges,butwehavebeenunabletodoasystematicsearch.A surprising example recently came to light, however. A recent Private Eye articlerevealedanicelittleschemeinvolvingAgeUK.68PotentialborrowerslookingattheAgeConcernwebsitewhoareinterestedinequityreleaseareencouragedto“dip[their]toeinthewaterwithhelpfromtheAgeCoUKEquityReleaseAdviceService,providedbyHubFinancialSolutionsLtd.”AstheEyearticlecontinues.“Hub,itturnsout,isownedby,er, Just Group. But its not just good business for Just. Age UK itself takes a handycommissionof “upto0.75percentof theamountadvancedundereachequityreleaseplansold,togetherwithacontributiontowardsmarketingsupport.””

67https://www.bankofengland.co.uk/monetary-policy/inflation/inflation-calculator68“JustReward.”PrivateEyeNo.149022February-7March2019,p.39.

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ChapterSeventeen:ScenarioAnalysisandStressTestingAscenarioanalysisisahypothetical‘whatif’exerciseinwhichweexaminewhatmighthappen to some variable of interest (e.g., a NNEG or ERM valuation) if some futurescenarioweretoplayout.Forexample,wemightexaminewhatwouldhappenaccordingtoourmodeliffuturehousepricesweretobehaveinaparticularway.Astresstestisascenarioanalysisinwhichthepositedscenarioisanadverseone(e.g.,alargedropinhouseprices).Type1HousePriceScenarioAnalysisorStressTestOnetypeofscenarioanalysis/stresstestistomodeltheimpactofanimmediateone-offhousepricefall.Weassumethathousepricesfall fiveminutesaftertheERMloanhasbeenmade.Thisexerciseiseasytocarryout.WefirstvaluetheNNEGorERMattheinitialhousepricevalue.Wethenre-valuethemimmediatelyafterthehousepricefallusingthenewLTV.Forexample,iftheinitialLTV=40%andhousepricesfallby50%,thenthenewLTVwillbe0.4 × 1/(1 − 0.5) = 80%. For this typeof stress testwedonotmakeanyprojectionsoffuturevariables,e.g.,futurehouseprices,otherthanthathousepricesfallshortlyaftertheERMloanismade.Table17.1givestheimpactofanimmediateone-offhousepricefallof50%.

Table17.1:ImpactofanImmediate50%FallinHousePrices 𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

Pre-stress £81.8 £41.1 £40.6Post-stress £81.8 £56.2 £25.6

∆𝑳 ∆𝑵𝑵𝑬𝑮 ∆𝑬𝑹𝑴Impactofstress £0 £15.0 -£15.0

%Impact 0 36.5 -36.5Notes:AsperTable3.1.

Forthegivensetofcalibrations,𝑁𝑁𝐸𝐺risesby£15.0and𝐸𝑅𝑀fallsbythesameamount.𝑁𝑁𝐸𝐺increasesby36.5%and𝐸𝑅𝑀fallsby36.5%.Figure17.1showstheimpactonNNEGandERMvaluationsofarangeofimmediateone-offhousepricefalls.

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Figure17.1:ImpactofaRangeofImmediateHousePriceFallsonNNEGandERM

Notes:AsperTable3.1.

Weseethatthegreaterthefallinhouseprices,thegreaterthefallinERMvaluation.Type2HousePriceScenarioAnalysisorStressTestInasecondtypeofexercise,weassumeahypotheticalrateofgrowthofℎ𝑝𝑖andshowhow,e.g., theERMvaluationwouldbehaveovertimeunderthepositedscenario.Thistypeofexerciseworksasfollows.WestartbyusingourERMvaluationmodeltoobtainthecurrentvalueoftheERM.ThenweobtaintheexpectedvalueoftheERMafteroneyear,whichwouldbe1yearexitprobabilitytimestheexpectedpayoffifexitoccursinyear1,plusthe1yearprobabilityofnoexittimestheERMvalueafteroneyear.ThislatterERMvalueisobtainedusingourERMvaluationmodel,buttakingaccountofthenewageafter1year(i.e.,1yearolder),thenewhousepriceafteroneyear(whichisequaltotheoldhousepricetimes𝑒+,-),andthenewLTVafteroneyear(whichisequaltotheoldLTVtimes𝑒&#+,-).WecarryoninlikemannerfortheexpectedvalueoftheERMaftertwoyearsandsoforth.Inpractice,wewouldoftenwanttocomparetwodifferentscenarios:abasescenario,whichmightbethescenarioweexpect,andastressoradversescenario.Forexample,wemightfollowJustGroupandassumeℎ𝑝𝑖 = 4.25%forourexpectedorbasescenario(seeChapter24below).Wethenpositsomestressscenario,e.g.,ℎ𝑝𝑖 = −1.7%,whichwastheaverageℎ𝑝𝑖inJapanover1990:2017.Figure17.2showsaplotofthesetwoscenarios:

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Figure17.2:ExpectedERMValuationsUnder4.25%vs.JapanHousePriceGrowthScenarios

Notes:AsperTable3.1.

WeseeamuchlowerexpectedERMvaluationprojectionunderthestressscenario.Thelessonhereisthatifwewererelyingonℎ𝑝𝑖 = 4.25%butactualℎ𝑝𝑖turnsoutto-1.7%,then the large𝐸𝑅𝑀 increaseswewereexpectingwillnot comepass, and futureERMvaluationswilldeclinetozeroconsiderablymorequicklythanwehadexpected.ExpectedCashflowsunderType2HousePriceStressTestAnothertypeofstresstestisprojectcashflowsunderanassumedhousepricescenario.Figure17.3showtheprojectedcashflowsunderthesametwoscenarios.Figure17.3:ExpectedERMCashflowsUnder4.25%vsJapanHousePriceGrowth

Scenarios

Notes:AsperTable3.1.

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Ifthe4.25%scenarioweretotranspirethensubsequentlyrealisedcashflowswouldbemuchlargerthaniftheJapanscenarioweretotranspire,andtheJapan-stylecashflowswouldbeevenlowerifwealsoexperienced,say,95%achievementrateinsteadofthe100%achievementrateassumedsofar:Figure17.4:ExpectedERMCashflowsUnder4.25%vsJapanHousePriceGrowth

Scenarios(II)

Notes:AsperTable3.1.

LongevityScenarioAnalysisorStressTestWecanalsocarryoutscenarioanalysesorstresstestsbasedonotherprojections.Animportantone in theequityreleasecontextwouldbea longevityscenarioanalysis, inwhichwepositsomechangetoexpectedlongevityandconsidertheimpactofthatchangeonNNEGorERMvaluations.Aneasywaytocarryoutsuchanexerciseistoassumeaparticularchangeinlongevity,say,weassumethatlongevitysuddenlyincreasesby3years.Wecanthenapproximatetheimpactofthisscenariobyreducingby3yearstheageoftheindividualinputtedintoourvaluationmodel,whilstkeepingotherparameters(andespeciallytheLTV)thesameastheywere.Table17.2givestheresultsofsuchanexercise.

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Table17.2:Impactofa3YearIncreaseinLongevity 𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

Current £81.8 £41.1 £40.6Underscenario £81.8 £53.3 £38.4

∆𝑵𝑵𝑬𝑮 ∆𝑵𝑵𝑬𝑮 ∆𝑬𝑹𝑴Impactofscenario £9.9 £12.1 -£2.2

Notes:AsperTable3.1.Forthescenario,weinputanageof67.Weseethatboth𝐿and𝑁𝑁𝐸𝐺risebycomparableamountsandlargelyoffsetintheirnetimpactonthevalueoftheERM.Itisalsointerestingtonotethatthesignoftheimpacton𝐸𝑅𝑀isnegative.AnincreaseinlongevitydecreasesthevalueoftheERM.ThisresultmeansthattheimpactofincreasedlongevityonthevalueoftheERMhasthewrongsigntofunctionashedgetoanannuitybook.Theliabilitiesofanannuitybookincreasewhenlongevityrises,sotofunctionasahedge,anassetmustalsoincreaseinvalue,butinthiscasetheERMdecreasesinvalueinstead.

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ChapterEighteen:ThePRA’sGoodPracticeERMValuationPrinciplesIn its Supervisory Statement SS 3/17 published in July 2017, the UK PrudentialRegulationAuthoritysetoutcertaingoodpracticeprinciplesrelatingtoERMportfolios.TheseprinciplesincludetwothatimposeupperboundsonERMvaluations.PrincipleIIPrincipleIIstates:

TheeconomicvalueofERMcashflowscannotbegreaterthaneitherthevalueof an equivalent loan without an NNEG or the present value of deferredpossessionofthepropertyprovidingcollateral.

i.e.,(18.1)𝐸𝑅𝑀 ≤ 𝐿and𝐸𝑅𝑀 ≤ 𝑃𝑉(𝐹)wherePV(.)isthepresentvalueofthetermin(.).Proofthat𝐸𝑅𝑀 ≤ 𝐿Startwith(3.1)𝐸𝑅𝑀 = 𝐿 − 𝑁𝑁𝐸𝐺.Weknowthat(18.2)𝑁𝑁𝐸𝐺 ≥ 0.If𝑁𝑁𝐸𝐺 > 0then(18.3)𝐸𝑅𝑀 = 𝐿 − 𝑁𝑁𝐸𝐺 < 𝐿.If𝑁𝑁𝐸𝐺 = 0then(18.4)𝐸𝑅𝑀 = 𝐿 − 𝑁𝑁𝐸𝐺 = 𝐿.Hence(18.5)𝐸𝑅𝑀 ≤ 𝐿whichwastobeproved.

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Proofthat𝐸𝑅𝑀 ≤ 𝑃𝑉(𝐹)The Present Value (PV) period 𝑡 payoff to 𝐸𝑅𝑀!ismin[𝐿! , 𝑃𝑉(𝐹!)], where 𝐿! is thepresentvalueoftheloanassumingitmaturesin𝑡yearsandassumingthatthereisnoNNEGinvolved,and𝑃𝑉(𝐹!)isthepresentvalueoftheperiod𝑡forwardcontract,forall𝑡.But(18.6)min[𝐿! , 𝑃𝑉(𝐹!)] ≤ 𝑃𝑉(𝐹!)forall𝑡.Therefore(18.7)𝐸𝑅𝑀! ≤ 𝑃𝑉(𝐹!)forall𝑡.Hence(18.8)𝐸𝑅𝑀 ≤ 𝑃𝑉(𝐹)whichwastobeproved.PrincipleIIIPrincipleIIIstates:

Thepresentvalueofdeferredpossessionofapropertyshouldbelessthanthevalueofimmediatepossession

i.e.,(18.9)Defermenthousevalue<spothousevalue.OntheValidityofPRAPrincipleIII,i.e.,WhyDefermentPropertyValuesareLowerthanCurrentPropertyValuesAt the risk of belabouring the obvious (because wemust!), we provide a number ofalternativedemonstrationsofthevalidityofPrincipleIII.Demonstration#1 Comparethevalueoftwocontracts,onegivingimmediatepossessionoftheproperty,theothergivingdeferredpossessionwhenexitoccurs.Theonlydifferencebetweenthesecontractsisthevalueofforegonerights(e.g.,torentalincomeortouseoftheproperty)duringthedefermentperiod,andthevalueoftheseforegonerightsshouldbepositivefortheresidentialpropertiesusedascollateralforERMs.Itthenfollowsthatthepresent

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valueofdeferredpossessionshouldbelessthanthevalueofimmediatepossession,i.e.,weobtainPrincipleIII.PrincipleIIIthusfollowsfromelementaryeconomics.Whywouldwenotpaylesstogetless?Demonstration#2 Asanalternativedemonstration,recall(3.10)𝑅! = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡ℎ𝑜𝑢𝑠𝑒𝑝𝑟𝑖𝑐𝑒 × 𝑒#*!where𝑞isthedefermentrateand𝑅!isthedefermentprice,andnotethatthespothousevalueandthecurrenthousepricewillbeequal.Weprovedearlier(seepp.35-36)that(18.10)𝑞 = 𝑛𝑒𝑡𝑟𝑒𝑛𝑡𝑎𝑙𝑟𝑎𝑡𝑒 = 𝑑/𝑆where𝑑isthenetnominalannualrental.Giventhatitisreasonabletopresumethat𝑑 >0,then(18.11)𝑅! = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡ℎ𝑜𝑢𝑠𝑒𝑝𝑟𝑖𝑐𝑒 × 𝑒#*! = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡ℎ𝑜𝑢𝑠𝑒𝑝𝑟𝑖𝑐𝑒 × 𝑒#*!whichimplies(18.12)𝑅! < 𝑐𝑢𝑟𝑟𝑒𝑛𝑡ℎ𝑜𝑢𝑠𝑒𝑝𝑟𝑖𝑐𝑒.Itisthenreasonabletosupposethatthedefermenthousevaluewillbeequalto𝑅!andPrincipleIIIfollows.Demonstration#3 Alongerandmorerigorousdemonstrationgoesasfollows:Let𝑞",𝑞$,𝑞),….bethesetofnetrentalyieldsforapropertyfromnow,period0,toforever.Thesenetrentalservicesaretheuse-benefitswegetfromlivinginaproperty(e.g.,thebenefitsofhavingaroofoverourheads)ortherentalincomeswecouldobtainbyrentingthepropertyout.Letusassumethattheseareallpositive.Afterall,zeroornegativerentalyieldsdonotmakemuchsense.LetAbethesetofthosenetrentalyields𝑞",….forperiods0toforever.LetBbethesetofnetrentalyields𝑞! ,𝑞!0$….fromperiodsttoforever,where𝑡 ≥ 1.

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LetCbethesetofnetrentalyields𝑞",...𝑞!#$,forperiods0tot-1.AssumeforthemomentthatthepricesofA,BandCallexist.Sincethesetsofrentalsarepositiveandhencevaluable, thenthepricesofA,BandCshouldeachbepositive.Bythelawofzeroarbitrage,thepriceofAshouldalsobeequaltothesumofthepricesofBandC.ButsincethepriceofCispositive,itmustfollowthatthepriceofB<thepriceofA,i.e.,thedefermentpricemustbelessthanthecurrentpriceandPrincipleIIIisestablished.Tochallengethisconclusion,itisnecessarytoarguethatsomeofthesepricesdonotexist.SincethepriceofAisthespotprice,thenthepriceofAclearlydoesexist,soonewouldhavetoarguethatthepricesofBand/orCdonotexist.Let’snotetobeginwiththattheempiricalbasisofanysuchclaimisarguable.Whilstitismanifestly obvious that the prices of B and/or C will rarely exist for some specificproperty, it isoftenpossible to inferproxyprices fordifferent typesofproperty fromcomparisonsoffreeholdandleaseholdpricesanditistheseproxypricesthatonewoulduseforvaluationpurposes.69Forexample,consideraleaseholdonaLondonflatwith99yearstorun.Thepriceofthisleaseholdwouldtypicallytradeatabout95%ofthepriceofavacantfreehold,andthecorrespondingfreehold,i.e.therighttoexclusivepossessionafter99years,wouldtradeat about 5%of the vacant value, and givesus theprice of possessiondeferredby99years.70Thefollowingchartshowsimplieddefermentprices–thedefermentpricesimpliedbyleaseholdprices,expressedasapercentageofthefreeholdvacantpossessionvalue71–forRICSprimecentralLondon2009andLeaseholdValuersLLP2017:

69Typically,onewouldobtaindefermentpricesforaparticularpropertybyapplyingrulesofthumbtothepricesfordifferentpropertytypes.Thesewouldtakeaccountofparticularfeaturesofapropertysuchaslocation,parkingavailability,thesizeofthegardenandsoon.70Seethe‘relativitygraphs’:http://www.graphsofrelativity.co.uk/.71Weassumethatthereisno‘marriagevalue’,i.e.thatthesumofthemarketvalueoftheleaseholdandfreeholdequalthevalueofvacantpossession.Foradiscussionofthisissue,seethenextfootnote.

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Figure18.1:EmpiricalImpliedDefermentPrices

Notes: RICS = Royal Institute of Chartered Surveyors. Sources:http://www.graphsofrelativity.co.ukandLeaseholdValuersLLP.

Thesefallasthedefermenthorizonlengthensandarealwayslessthanthespotprice.72Butsupposeforthesakeofargumentandcontrarytotheevidencejustpresentedthatsomeofthesepricesdonotexistanddonothavenearapproximationsorproxiesintermsofothermarketprices.Inthissituation,wesimplyswitchthemetricfrompricestovaluesandwecanestablish thevalidityofPrinciple III inmuchthesamewayasbefore.Forexample,ifweassumethateachnetrentalyieldhasapositivevalue,thenitimmediatelyfollowsthateachofA,BandChaspositivevalue,sothevalueofBmustbelessthanthatofAandPrincipleIIIfollows.Indeed,evenifweassumethatthecurrentnetrentalyieldispositiveandtheothersaremerelynon-negative,thenPrincipleIIIstillfollows.TochallengePrincipleIII,one is then lefthavingtoarguethatnetrentalyieldsorthevaluesofnetrentalyieldsarenegative.Let’sconsiderpossibleexamples.Oneiswherethepropertyandthelandwhichitstandsarepollutedbeyondanyfeasiblerepair.Chernobylcomestomind:evenifthelandcouldberestoredtoausablestate,thecostsofdoingsowouldbeprohibitive. In thiscase,all𝑞",𝑞$,….arenegativeandwillremain so. The property and the land itself would then be abandoned. In this casePrinciple IIIdoesnotholdbecausethespotanddeferredpricesarezero.This typeofsituationisrare,however.

72 Theseresultsareillustrationsonly,andotherfactorscomeintoplay.Forexample:(a)Currentleaseholdvaluesreflecttherighttoextendatamarketvalue,whereasERMborrowershavenosuchright(the‘lease’endswhen they exit into long term care or die, and the estate has no right of extension. (b)Wehaveassumednomarriagevalue–marriagevalue is theadditionalvaluean interest in landgainswhenthelandlord’s and the leaseholder’s separate interests are “married” into single ownership (see LawCommission,2018,p.23,n.62)-butaddingmarriagevaluewouldincreasetheimplied𝑞ratesfurther.and(c)ordinaryleaseholdstendnottoterminatewiththepropertyinruin,whereasthereisevidencethatveryoldERMborrowerstendtoneglecttheirproperty.

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Alessrarecaseiswherethepropertyisuninhabitableandrepairwouldbeuneconomic,butthelanditselfisvaluable.PartsofDetroitcometomind.Onemightthensaythatthe(currentornearcurrent)netrentalproceedswerenegative,butthissituationwouldnotlastbecausethelanditselfisvaluable.Thepropertywouldbedemolished,perhapsafterbeingsoldoff,andthesiteredevelopedtorestoreapositivenetrentalstream.A third and more common case is where the property needs repair and repair iseconomically feasible. The propertymight not generate any current net rental, but itwouldberepairedandapositiverentalstreamrestored.Thissituationisnotuncommon,butisstillrelativelyinfrequent,inthatitdoesnotapplytomostpropertiesmostofthetime.Thegeneralcaseisthatmostpropertiesmostofthetimegenerateapositivenetrentalstream.Therefore,whenlookingforageneralruletoassessdefermentvalue,theonlysensibleruleistoassumeapositivenetrentalstream–andapositivenetrentalstreamimpliesthatthedefermentvaluewillbelessthanthecurrentpropertyvalue.Inshort,ifthepricesofA,BandCallexistandarepositive,thenthevalidityofPrincipleIIIfollowsfromzeroarbitrage.IfanyofthepricesofA,Band/orCdonotexist,however,thenwecanstillobtainPrincipleIIIbyswitchingovertoarationalvaluationargument,inwhich it suffices to argue that thevaluesofA,B andC are all positivebecause theunderlyingrentalshavepositivevalue.Demonstration#4:Thefiduciaryprinciple Thereisalsoanormativeargumentthatonecancallthe‘fiduciaryprinciple’.Evenwheremarketpricesdonotexist, accountingprinciples say that theaccountantshouldvalueeconomically similar assets in the same way and imply that valuationshouldreflectrationalinvestorpreferences.Theword‘should’or‘ought’appears,e.g.,inIFRS13B14a:“Cashflowsanddiscountratesshouldreflecttheassumptionsthatmarketparticipantswould use when pricing the asset or liability.” The fiduciary principle says that anaccountantorauditororsomeotherperson,whohasanobligationoftrusttowardsalessknowledgeable investor, must value an asset or liability as a rational knowledgeableinvestor(ormarketparticipant,orknowledgeable,willingindependentperson)would.Thisprincipleprovidesasafeguardagainstinterestedpartiescomingbackalongthelinesof “no arbitrage doesn’t apply here, so we can make up any price that benefitsmanagement,othernon-fiduciariesoranyoneelsewechoose.”Applyingthisprinciple,theaccountant,actuaryetc.mustacknowledgethatrentalserviceshavepositivevalueandthisacknowledgementsufficestoestablishPrincipleIII.

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MisconceptionsAboutPrincipleIIIGuyThomasGuyThomastakesissuewithPrincipleIII inarecentposting(Thomas,2018).73 Inhispiece, he acknowledges that the loss of foregone rights (e.g., to income or use of theproperty) during the deferment period [i.e., the argument underlying Principle III]“appears a reasonable argument” but even so, adds that “there are also reasonablecounter-arguments.”Asheputit:

Housingtodayisownedmainlybyowner-occupiers.Theyhaveapreferencefora current interest to a deferred interest, because they need a roof over theirheads,theylikelong-termsecurityofoccupation,theylikebeingabletomaketheirownchoicesonextensionsandrepairs,etc.Inotherwords,theylikethepracticalandsentimentalbenefitsofhomeownership.Aminorityofownersarebuy-to-let landlords: they like understandable form of the investment, theunusual ability to finance it largely with borrowed money, and perhaps thedisengagementitfacilitatesfromthedistrustedpensionsandsavingsindustry.

Wewouldputitalittledifferently.Anyonewholivesinapropertygetsthe‘netrentalservices’ofthatproperty–theuse-valuebenefitsofaroofovertheirheadsandsoforth.Some people choose to obtain those benefits by buying their property and others byrentingthepropertytheylivein.Inthelattercase,thepropertyownergetsthebenefitofthe rent tenantspay, and inmostplausible situations, theownerwho rentsout theirpropertywillreceivearentthatmorethancoversthecostsofmaintainingtheirproperty.Thereareexceptionsaswehaveexplained,buttheseareunusual.

Foraninsurer,ontheotherhand,thesepracticalandsentimentalbenefitsofacurrent interest inahousehavenorelevance.Themainpotentialbenefitofacurrent(asopposedtodeferred)interestisthepotentialincomefromletting.

True,andthispointappliestoanyownerwhorentsouttheirproperty.But a current interest also has several disbenefits [sic]: tenants need to bemanaged, houses need to be maintained, from time to time there are costs(Including possibly PR costs) of evicting tenants in arrears, and there is apossibility(throughexistingornewlegislation)thattenantsmightacquirenewrights.

Yes,therearecostsandriskstohavingtenants.Ifontheotherhandhousesarekeptvacant,thisgivesanothersetofproblems:counciltax,securityandmaintenancecosts,andpossiblyveryconsiderablePRcostsofowningsubstantialamountsofemptyhousing.

73 G. Thomas, “No-Negative Equity Guarantees: Black-Scholes and its Discontents.” http://guythomas.org.uk/blog/?e=28

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Yes,therearealsocostsfromkeepingpropertiesinvacantpossession.

These disbenefits are not fanciful; theirmateriality can be inferred from theobservablefactthatdespitetheexcellentlong-termperformanceofhousingasaninvestment,neitherinsurersnoranyotherfinancialinstitutionshaveshownanyenthusiasmoverthepastseveraldecadesforhousingasanassetclass.

These passages are a roundabout way of saying that there are benefits and costs ofowningpropertybutifanownerregardsthecostsasoutweighingthebenefits,thenthesensiblechoicefortheowneristosell.Thepropertywillthenendupinthehandsofanownerwhodoesvaluethebenefitsasmorethanthecosts–otherwisetheywouldn’thaveboughtthepropertyandsomeoneelsewould.The lackofenthusiasm(orotherwise)of financial institutions forhousingasanassetclassisanotherquestion.Hecontinues:

Socurrentinterestsinhousesareevidentlynotattractivetoinsurersandotherinstitutional investors. Deferred interest might well be more attractive,particularly if in the form of cash-settled financial contracts, so that all theproblems of current interests are permanently avoided. Even if a deferredinterest is not strictly preferred, the relative valuation of a deferred interestcomparedtoacurrentinterestseemsverylikelytobemuchhigherforaninsurerthanatypicalindividualowner.(Ouremphasis)

Nowiftherewereasubstantialmarketfordeferredinterests,themoneyweightof individuals’preferenceforcurrent interestsversus insurers’preferencefordeferredinterestswoulddeterminetherelativemarketpricesforthetwotypesofinterest(i.e.whatthePRAcallsthe‘defermentrate’).Butwehavethesameproblemaswiththehedgingarguments:themarketfordeferredinterestsdoesnotexistonanymeaningfulscale.(Ouremphasis)

Leaving aside that amarket for deferred interests does exist (see above), Thomas iscomparingonehypotheticalnon-marketvaluation (i.e., insurers’valuationsof currentpossession)againstanother(i.e.,theirvaluationsofdeferredpossession).Acomparisonof the relative valuations of spot and deferred possessionmade by a party that is exhypothesi not amajor player in themarket does not establish (a) anything about themarketpricesorplausiblevaluesforcurrentpossessionorthemarketpricesorplausiblevaluesfordeferredpossessionoranyrelationshipbetweenthem.Inanycase,nosuchcomparisonestablishes(b)thatdeferred,forwardorfuture‘interests’havethenegativevaluenecessarytounderminethevalidityofPrincipleIII.Tomakepoint(a)inadifferentcontext,supposewevalueatypicalstatelyhomeasbeingworth2timesthevalueofatypicalcastle,butthemarketvaluesatypicalstatelyhomeasbeingworth3timesthevalueofatypicalcastle.Ourviewsmightbesincerelyheld,buttheyareofnorelevanceifwedon’thaveanyportfoliosofcastlesorstatelyhomesandarenotinthemarkettradingthem.Becausewearenotinthemarkettradingthesethings,ourviewsabouttheirrelativevaluationshavenorelevancetoanyonebutourselves.Theonlyvaluationsthatmatterarethoseofthemarket.

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Ortogiveanothercomparison.WemightbelievethatHollywoodmoviesarerubbishandBollywoodmovies are great, or the other way round. Makes no difference. The onlyinferenceswecouldreasonablydrawfromthatexpressionofrelativepreferencesisthatwewouldnevergotoseetherubbishmovies.Butalotofpeopledogotoseetherubbishmoviesbecausetheydon’tshareourgoodtaste,sotherubbishmovieshaveeconomicvalue.Thereforewemightrationallyinvestinthem,evenifwewouldneverwatchthem.Thepoint isnot toconfusesubjectivetastewitheconomicvalue,andwhat is thebestestimateoftheeconomicvalue?Itiswhatthemarketwillpay.Inanycase,nosuchcomparisonsestablishthatdeferred, forwardor future ‘interests’havethenegativevaluenecessarytounderminethevalidityofPrincipleIII.Thomas’sargumentalsorunsafoulofthefiduciaryprinciple:evenifanactuaryorauditorhasaprivateviewoftherelativevaluationsofdeferredandcurrentpossession,theyarestillrequiredtoreportthemarketvaluations,andonlythosevaluations.Inshort,thevalidityofPrincipleIIIcanbebuttressedbysoundeconomictheory,solidaccounting principles and abundant empirical evidence, but the counter argumentscannot.RaduTunaruonPrincipleIIITunaru also challengesPrinciple III. In his report he argues (p. 50) that the “prepaidforwardprice”,i.e.thedefermentprice,willalwaysbelowerthanthecurrenthouseprice(i.e.thefreeholdpriceofvacantpossession):

the idea that the prepaid forward price should always be lower than thecurrenthousepricecanbechallenged.Thisconditionwillworkobviouslyinnormal market conditions and for shorter maturities. However, in theaftermathoffinancialandeconomiccrises,conditionsmaybereversed.Forexample, in the aftermath of the subprime crisis house prices droppedsignificantly.Theusualquestionof“howmuchshouldyoupaytogetahouseinfiveortenyearstime?”shouldbereplacedwiththequestion“whatpricecanyougetonthemarkettosignnowforpossessionofahouseinfiveortenyearstime?”Evenifthehousepricemarketwasdepressedintheaftermathofthesubprimecrisis,theexpectationofthemarketwouldnaturallybethatthemarketwill recover after some time and the forward curvewill be incontango.Thus,itispossiblethatthemarketwillrequireaprepaidforwardthatishigherthanthecurrenthouseprice.

Thisargumentisnottheoreticallydefensibleandwearenotawareofanyevidenceforit.Froma theoretical point of view,whywould a rational investorwho expected housepricestorecoverpaymoreforacontractforpossessionin(e.g.)5yearstimethantheywould pay for a contract for vacant possession?The prices of both contracts will beidenticalafter5years,soinbothcasestheinvestorwillbenefitfromthedesiredrecovery.Howeverwithvacantpossession,theinvestorwillbeabletousethepropertyorrentitout,whereaswithdeferredpossessionthereisnosuchbenefit.Therationalpreference

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wouldbeforvacantpossession,andaswenotedabove,anaccountantwhowasvaluingthe contracts for the benefit of shareholders is professionally obliged to reflect thatrationalpreference.Froman empirical point of view,weknowof no evidence suggesting that longdatedfreeholdsintheLondonmarketweretradingabovethevacantpossessionpriceintheaftermathofthe2008collapse.A99yearleaseonaLondonapartmentwouldtypicallytradeat95-99%ofthevacantfreeholdvalue.Wehavenoevidencethatleaseholdvaluescollapsedtozero,orwentnegative,duringthecrisis.Perhapssuchevidenceexists,butgiventhetheoreticalunlikelihood,theonusshouldbeonTunarutoproduceit:strongclaims require strong evidence. He points (p.51) to the fact that short-dated EUREXfuturescontractstradedwithnegativedefermentratesforshortaperiodfollowingthecrisis,buttheserefertofuturesnotforwards,anddonotrefertocontractsforlong-dateddeferred possession such as in the freehold and leasehold markets, which are theforwardsrelevanttothecaseofequityrelease.ConsideralsoevidencesuchasthefollowingfromaDailyMailarticlepublishedon24July2009.74Thearticlementions‘therecentlargepricefallsinthehousingmarket’andthenpointsout:

“Recent price reductions of up to 25 per cent in most areas mean anequivalentdropin thecostofextendingyour leaseorbuying the freehold,”saysAngusFanshawe,ofLondonestateagencyDouglas&Gordon.

But if Tunaru were right, the freehold (or the extension) should be getting moreexpensive,notless,givenmarketexpectationsofpricerisesinthefuture,no?

At the very posh end of the market, an apartment in Chelsea's exclusiveCadoganGardenshasa43-year leaseand isonsale for£2.95million (02075813022,friendandfalcke.co.uk),butmaybeworthupto£4.5millioniftheleasewereextendedto99or125years.

Sothe43yearleaseholdhasapositivevalue,notthenegativeoneimpliedbyanegativedefermentrate.

…aflatinCentralLondonwithan88-yearleasewouldsellfor97percentofitsfullmarketpricebutthesameflatwith22yearsontheleasewouldsellforonly56percentandwithaten-yearleasefor32percent.

Soifthedefermentratewerenegative,wecouldbuyaflatwithvacantpossessionfor100%,carveoutan88yearleaseandsellfor97%,thenselltheremainingfreeholdforpossessionafter88yearsatover100%.Thatwouldbeanicetrade!All the relevant evidence indicates that deferment rates are positive as Principle IIImaintains.

74 https://www.dailymail.co.uk/property/article-1201766/Time-new-lease-life-Freehold-isnt-way-secure-future-profits.html

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IFoAMisconceptionsonPrincipleIIIIn June 2016, the Institute and Faculty of Actuaries issued “DP 1/16: Equity ReleaseMortgages:IFoAResponsetothePrudentialRegulationAuthority,”itsofficialresponsetothePRA’searlierDiscussionPaperDP1/16,whichhadhadaskedforindustryviewsonERMs.Toquotefromthisresponse:

33.Forthesecondrelationshipinparagraph4.9[i.e.,PrincipleIII]tohold,intheory,thereneedstobeadeepandliquidmarket.OtherwisetheimplicationisthattheaveragevalueoftheHPI[HousePriceInflation]assumptionislessthanorequaltothediscountrateassumedinthevaluationoftheNNEG.Inpractice,theapproachtosettingtheHPIassumptionvariessignificantlyfromfirmtofirm.

Thereareseveralhowlershere:

• Mistake#1isthatforPrincipleIII“tohold,intheory,thereneedstobeadeepandliquidmarket.”ThevalidityofPrincipleIIIhasnothingtodowithadeepandliquidmarketandwehavejustshownthatitsvalidityholdsundergeneralconditions.

• Mistake#2istosuggestthatthe“averagevalueoftheHPIassumptionislessthanor equal to the discount rate assumed in the valuation of the NNEG.” Thisstatementisjustplainwrong.ThecorrectstatementisthatwecanassumeanyHPIwewant to, but the assumed value of the HPI is always irrelevant to thevaluationoftheNNEG.

Para35thengivessome illustrationsofcircumstances inwhichPrinciple IIIallegedlymightnothold:

• One is the claim that Principle III “is a statement of ‘value’ and applies to anyindividual.Howeverthisisnotnecessarilytrueintermsoftheexchangevalue.”Thisstrangestatementisanimaginativeadditiontotheeconomictheoryofvaluebutisunfortunatelyalsowrong.TheclaimthatthePrincipleIII“isastatementofvalueandappliestoanyindividual”istrue,butthecorollaryisthatitalsoappliesto all individuals including (and not excluding!) when they engage in trade atmarketorexchangevalues.

• Another is the claim that “in a negative yield curve scenario, the relationship(PrincipleIII)wouldfailasthepremisethatdeferralcouldleadtoalowerpresentvaluenolongerholds.”Thisstatementisaheadscratcherbutonecanseethatitmustbewrongbecausethedefermentprice(orvalue,makesnodifferencehere)isequalto𝑆"𝑒#*!andthisexpressiondoesnotincludeanyinterestrateoryield,negativeornot.Torepeat,PrincipleIIIdependsonlyonthe𝑞ratesbeingpositive(ormostlypositive)anditisdifficulttoimagineplausiblesituationswherethatwouldnotbethecase.

So how come the distinguished actuaries of the IFoA could make such mistakes? Apossibleclueisthatthecoveringletteropenswiththefollowingstatement:

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The IFoA’sEquityReleaseMembers InterestGroup(ERMIG)andLifeBoardhavebeeninvolvedinthedraftingofthisresponse.Thecontributorstothisresponseincludememberswhoareactivelyengagedwithuseofequityreleaseassetsbylifeinsurers.(Myitalics)

TheIFoAhadalloweditselftobeusedasamouthpieceforERMindustrypractitionerstobroadcasttheirmisunderstandingoftheirproducts.But the authors of the IFoA official response to DP 1/16 are not alone inmisunderstanding these principles. Consider these passages from a recent DeloittecommuniquéonERMs:

Inourview,thethirdprinciple(thatfuturepossessionofapropertycannotbemorevaluablethancurrentpossession)islikelytoattractthemostfuturedebate.

ButPrincipleIIIisjustelementaryeconomics!

Very importantly, this principle implies that assumed future house pricegrowthcannotexceedthediscountrateappliedinthevaluation.…

Noitdoesnot.

ThePRAexpectstheretobeapositivevalueassociatedwithpossessionofaproperty.

Yes,obviously.The practical implication of this is that the assumed house price growthwithintheNNEGoptionpricingcalculationcannotexceedthediscountrate,asthiswouldimplythatfuturepossessionismorevaluable.Thisprinciple thereforeeffectivelysetsacapon firms’housepricegrowthassumptions.

Thesestatementsarenonsense.PrincipleIIIhasnoimplicationsaboutassumedfuturehousepricegrowth.YoucanmakeanyassumptionsaboutfuturehousepricegrowththatyoulikeandPrincipleIIIwouldbestillbevalid.

WewouldexpectfirmsinvestinginERMsandotherdirectinvestmentstoseeanincreasedlevelofscrutinyandquestioningfromthePRA,withthebarsetveryhighformanagement’sunderstandingofthevaluationofsuchinvestments.(Bulleyetalia,2017,ouritalics)

Theyareclearlyofftoaflyingstartonthatone.Theleadauthor,AndrewBulley,isapartnerinDeloitte’sCentreforRegulatoryStrategy.PriortojoiningDeloitte,Mr.BulleywasDirectorofInsuranceSupervisionattheBankofEngland.

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To challenge Principle III is thus to make an egregious intellectual error and it isremarkable that the IFoA has not only failed to condemn any such challenge but hasexplicitlygivenititsimprimatur.ThissituationisanalogoustotheUK’stopmathematicalinstitute,theInstituteofMathematicsanditsApplications,takingtheofficialviewthatthevalidityof2+2=4isanopinion.Yousee,somemathematiciansareoftheopinionthat2+2=4butothershaveadifferentview,includingsomewhospeakfortheInstitute.Weappeartohavehereanothercaseof‘actuarialjudgment’goneawry.WeareremindedofsomecommentsmadeonthissubjectbyTimGordonalmosttwodecadesago(Gordon,1999).Hewrote(p.4)abouttheactuarialconvictionthat“actuarialjudgmentistheonlytechniqueforvaluinglong-termliabilities”but‘actuarialjudgement’producesananswerthat“variesenormouslydependingonwhichactuarycarriesoutthecalculation.”Hecontinued:

actuaries assume that judgmentalmethods are the onlymethods availablewhich give sensible answers. What is more, the judgement involved issomethingwhichapparentlyonlycomeswithyearsofexperience. Inotherwords,weclaimtoknowtheanswerbutcannottellanyoneelsehowtoderiveitinadvance.

Theexperiencedactuaryknows itwhen theysee it.Romanaugurshad thesameskillreadingchickenentrails.Ashecontinuedfurther:

The problem is that the difference that actuarial judgement can make tovaluationsusingthetraditionalapproachisenormous.Itmeansthat:•weareexposedtopressurefromclientsseekingtomoveanswersinthedirectionwhichfavoursthem,and•we losecredibilitybecauseweareunabletoexplainpreciselyhowwearriveatananswer.

Actuarial judgment can also lose credibility when it produces answers that aredemonstrablywrong.IFoAERMWorkingPartyonPrincipleIIIInFebruary2020,theIFoAtookanotherstabatthePrincipleIII issueina ‘discussionnote’issuedbyitsERMworkingparty.75Inessence,theirview(p.4)wasthat‘wewouldexpectthedefermentrateofaresidentialpropertytousuallybepositive,’but‘thereisnologicalnecessityforthistoalwaysbethecase.’ThislatterphraseistobeunderstoodassuggestingthatitistheoreticallypossiblethatPrincipleIIImightnotholdforsomecases.Weagree.

75 “ADiscussionNoteontheEconomicValueofEquityReleaseMortgagesasPartofthePRA’sEffectiveValueTest,”InstituteandFacultyofActuariesEquityReleaseWorkingParty,17February2020.

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WegavetheexampleearlierofapropertyinChernobyl.Inthiscase,PrincipleIIIdoesnothold because the forward and spot values are identical, i.e., zero. However no equityreleaselenderwouldlendtopropertiesthatarenuclearwaste.ThatiswhyChernobylisnotanERMassetclass.Consequently,anyexceptionstoPrincipleIIIareirrelevant,atleasttoequityrelease,andwe should conclude that Principle III makes perfectly good sense for the kinds ofproperties that equity release lendersmight lend against. TheWP’s emphasis on thetheoreticalpossibilityofanegativedefermentrateisthenmisdirected,becausethereisnotmuchpointdwellingon theoreticalpossibilities thatdon’tapply toany realworldcasesthatequityreleasefirmsmightencounter.Butwearetalkingaboutactuarieshere.Theworkingpartyhasadeusexmachinaupitssleeve,however:anilliquiditypremium(ILP).Unfortunately,thistrickdoesn’tworkeither.Asthediscussionnoteexplains:

There isa technicalargument,presented inrecentactuarialERMvaluationresearch,76 that the presence of an illiquidity premium in the underlyinghouse price should reduce the cost of the NNEG (note that the illiquiditypremiumoftheresidentialpropertyisdistinctfromtheilliquiditypremiumofthemortgage).Specifically,thepresentvalueofthehousepriceilliquiditypremiumthatwillbeearnedoverthelifeoftheoptionshouldbeaddedtothehousepricethatisusedintheNNEGvaluation.ThisisequivalenttodeductingthehousepriceilliquiditypremiumfromthedefermentrateusedintheNNEGvaluation,implying:

Defermentrate=Netrentalyield–housepriceilliquiditypremium

Let’splayalongandcalibratethisequation.Ourballparkestimateofthenetrentalyieldis4.2%.Ifweacceptthisestimateforthesakeofargument,thenweneedanilliquiditypremiumofmorethan420basispointsperannumtogettoanegativedefermentratethatisn’trelevantanyway.Thatissomeilliquiditypremiumespeciallybearinginmindthemoreplausibleestimatesof15to20basispointsorsothatonesometimessees.There is however a deeper problem. ‘Technical argument’ or no, theworking party’sequation(18.13) 𝑞 = 𝑛𝑒𝑡𝑟𝑒𝑛𝑡𝑎𝑙𝑦𝑖𝑒𝑙𝑑 − 𝐼𝐿𝑃ispulledoutofthinairlikearabbitfromahat.Whereisthebodyofcommonlyacceptedtheoryunderlying it?Theworkingpartydoesn’t sayandonehas thesuspicion that itdoesn’texist,becausetheworkingpartywouldpresumablyhavedrawnonitifitdid.Theworkingpartyismakingitupasitgoesalong.77

76 ‘See Section 5.3.5 and 5.3.6, Jeffery and Smith (2019).’ 77 Alternatively, theymight have been trying to rewrite one of the fundamental equations of financialmathematicswhichwouldbeakintorewritingoneofthelawsofthermodynamics.Brave.

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Bethatasitmay,thedeeperproblemisthat(8.13)iswrong.Thecorrectequationis(7.8) 𝑞 = 𝑑/𝑆 = 𝑛𝑒𝑡𝑟𝑒𝑛𝑡𝑎𝑙𝑦𝑖𝑒𝑙𝑑Sowhyis(8.13)wrongand(7.8)right?ThereasonisbecausetheILP(or𝜆),assumingitexists,shouldappearintheformulaforthediscountfactor1/(1 + 𝑟 + 𝜋 + 𝜆).AccordingtobedrockfinancetheoryintheformofthemathematicsoftheDiscountDividendModel,𝜆canappearnowhereelse.Ifweacceptthisdiscountfactor,thentheanalysisfrom(7.2)onwards(seepp.34-35above)provestheresult(7.8),fromwhichitisclearthattheILPhasnoeffectonthedefermentrate.Therefore(7.8)iscorrect,andif(7.8)iscorrectthenunlesstheILPiszero,(8.13)mustbeincorrect.Toavoidanymisunderstanding,wearenotsayingthattheILPdoesnotexistordoesexistbuthasazerovalue.Wearenotarguing(much)aboutitssize.Asfaraswecare,youcansetitaslargeasyouwant.WearesayingthatiftheILPdoesexist,whateveritssize,itsimpactonthedefermentrategetscancelledout(alongwiththeimpactsofthegrowthrate𝑔,therisk-freerate𝑟andanyriskpremium𝜋),because(7.8)isalwaystrue.Inshort,theactuarial‘technicalargument’underlying(8.13)isfallacious.TheactuariesplayingpinDonkey’stailhavestucktheILPinthewrongendofthedonkey.Incredibly,itgetsevenbetter!Ifwegobacktothefundamentalvaluationequations(7.2)and(7.3),itispossibletoshow(18.14)𝑞 = 𝑟 + 𝜋 + 𝜆 − 𝑔TheproofisgivenintheAppendixtothischapter.78Compare(18.14)withtheactuaries’equation(18.13)anditisimmediatelyobviousthattheirequationishopelesslywrong.Thegoodnewsfortheactuariesisthatthisnewequationdoesincludethe𝜆term,butthebadnews is that that termenterswithapositive sign–howbigdidyouwant the ILPagain?–andomitsthe𝜋and𝑔parameters.Anyonewhousedtheactuaries’equationtoestimatethedefermentratewouldthenobtainsomemagnificentlyflawedresults.79Andthatistheendofthematter.

78Atfirstsight,(18.14)mightbereadassuggestingthat𝑞doesdependon𝜆afterall.Itdoesnot,because(18.13) shows that𝑞 does not depend on𝜆. There are therefore implicit constraints involved. Putting(18.13)and(18.14)together,theseconstraintsare𝑟 + 𝜋 + 𝜆 − 𝑔equals𝑞,where𝑞istreatedasagivenhereandwhereare𝑞 > 0and𝜋and𝜆 ≥ 0.Theimplicationisthatwecannotjustinsertanyvalueswewantfor𝑟,𝜋,𝜆and𝑔 into(18.14),because(18.13)mustalwayshold.Thissaid,therewouldbenoreasontoestimate𝑞from(18.14),because(18.14)hasthreeunobservablevariables(𝜋,𝜆and𝑔),whereas𝑞ismuchmoreeasily(andreliably)calibratedusing(18.13).79 Imaginethatsomeoneweretostretchlambdato4.2%togenerateazerodefermentrateusingtheirformula(18.13).Let’stake𝑟=1%,𝜋=3%and𝑔=2%asillustrativevalues,andtheyareallplausible.Thenacceptingtheir𝜆value,thetruedefermentratewouldbe1%+3%+4.2%-2%=6.2%.

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BoundsonERMandNNEGValuationsToreturntothemainstoryline,theimpactofthesetwoPrinciplesisillustratedinFigure18.2:

Figure18.2:IllustrationofPrinciplesIIandIII

Notes:AsperTable3.1.

PrincipleIIimpliesthattheblue(𝐸𝑅𝑀!)linemustbebelowboththegreen(𝐿!)lineandthe red (deferred possession) line, and Principle III implies that the red (deferredpossession)lineshouldslopedownwards.ThereissomeinterestingintuitionunderlyingtheFigure:

• Forverylowhorizons,𝑁𝑁𝐸𝐺!isveryoutofthemoneyandprobabilityofexerciseis very low.Hence the value of the optionwill be negligible and𝐸𝑅𝑀!will beindistinguishablyclosetothevalueoftheloan𝐿! .

• Forlonghorizonsorhigh𝑡,theoptioniswellintothemoneyandtheprobabilityofexerciseishighandapproaching1.Therefore,the𝐸𝑅𝑀!lineconvergestothedeferredhousevaluelineforperiod𝑡.

Underlyingthesegraphsaresomeelegantmathematics.𝐸𝑅𝑀!isgivenby(18.15)𝐸𝑅𝑀! = 𝑒#'!𝐿! − 𝑒#'![𝐿!𝑁(−𝑑)) − 𝐹!𝑁(−𝑑$)]

= 𝑒#'![1 − 𝑁(−𝑑))]𝐿! + 𝑁(−𝑑$)𝑒#'!𝐹!

where we have set the deferment price𝐷! = 𝑒#'!𝐹! . From the standard equivalence𝑁(−𝑥) = 1 − 𝑁(𝑥),wethenget(18.16)𝐸𝑅𝑀! = 𝑁(𝑑))𝑒#'!𝐿! + 𝑁(−𝑑$)𝑒#'!𝐹!This expression is simpler and reflects the shapes of the curves clearly. As 𝑑) getspositive, −𝑑$gets negative, so 𝑁(𝑑)) goes to 1,𝑁(−𝑑$) goes to zero and 𝐸𝑅𝑀!

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approachesthepresentvalueoftheloan.As𝑑)goesnegative,it’stheotherwayround,sothetermontheleftdisappearsandthetermontherightapproachesthedefermentprice𝑒#'!𝐹! .OneseestheseboundsatplayinFigure18.2. Besidestheirmathematicalelegance,theseboundsimpliedbyPrinciplesIIandIIIhaveahelpfulpracticaluse:theyareaneasilycalculatedcross-checkonanyproposedERMorNNEGvaluation.ConsiderFigure18.3,which shows theupperbound for𝐸𝑅𝑀!madeexplicitandhighlightedinblue.

Figure18.3:ERMUpperBound

Notes:AsperTable3.1.

Asanaside,ifwestartwithafigurelikeFigure18.2andletthevolatilitygetsmall,thenit iseasy toshowthatFigure turns intoFigure18.3and themessage is that theERMvaluationapproachesthePrincipleIIupperbound.ButiftheERMvaluationapproachesitsupperbound,thenthecorrespondingBlack76’optionvaluationmustapproachthePrinciple IINNEGvaluation lowerbound, i.e., as𝜎 → 0, theBlack ‘76NNEGvaluationapproachesthePrincipleIINNEGlowerbound.Wecanobtainthe𝐸𝑅𝑀!upperboundastheminimumof𝑒#'!𝐿!and𝑒#'!𝐹! .NotethatthisupperboundcanbeestimatedusingonlyinformationaboutthecurrenthousepriceandLTV (which together giveus the current amount loaned), the risk-free rate𝑟, thenetrental𝑞,theloanrate𝑙andtheexitprobabilities.Forexample,inthebaselinecase,weestimate the𝐸𝑅𝑀 upper bound to be £47.3, which compares to our earlier baselineestimateof𝐸𝑅𝑀as£40.6(seeTable3).Soevenwithoutestimating𝐸𝑅𝑀orits𝑁𝑁𝐸𝐺orestimatinganyunderlyingoptionmodelorcalibratinganyadditionalparameters,suchasthevolatility,weimmediatelyknowthatanyproposedvalueofERMthatexceeds£47.3mustbewrong.But if we can estimate an upper bound for ERMwithout requiring an option-pricingmodelorrelyingonanyvolatilityparameters,thenby(3.2)𝐸𝑅𝑀 = 𝐿 − 𝑁𝑁𝐸𝐺

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wecanalsoestimatealowerboundfor𝑁𝑁𝐸𝐺onthesamebasis.Giventhat𝐿=£81.7inourbaselinecase,theupperbound𝐸𝑅𝑀estimateof£47.3impliesa𝑁𝑁𝐸𝐺lowerboundequalto£34.4.This lowerboundcomparestoourearlier𝑁𝑁𝐸𝐺estimateof£41.2.Soeven without estimating the 𝑁𝑁𝐸𝐺 or relying on any NNEG valuation model or anyvolatilityestimatethatmightgointoanysuchmodel,weknowthatanyproposedNNEGvaluebelow£34.4mustbewrong.Tocuttothechase,giventhesevariousinputs–theassumedageandgender,theassumedhousepriceandLTV,theassumed𝑟,𝑞,and𝑙rates,andtheinputtedexitprobabilities–itisimpossibletogetaNNEGvalueanylowerthan£34.4whateveroptionpricingmodelonemightuseandregardlessofhowitmightotherwisebecalibrated.Attheriskofrepeatingourselves,wewouldstressthatthislowerboundNNEGvalueisnotdependentonBlack’76.TherecentInstitutereplytoCP13/18releasedon28Sep2018made a great deal of noise about how autocorrelation,mean reversion, lack ofGeometricBrownianMotion and so forthundermined the validity ofBlack ’76, and anumberofparticipantsatourLSEseminaron1October2018madesimilarpoints.Wewoulddisputethevalidityoftheseclaims–notleastbecausetheyconfusesufficientwithnecessaryconditionsforBlack-Scholestypevaluationstobevalid,andwehavemoretosayontheseissuesinChapter19–buteveniftheseclaimswereallvalid,theydonotapply to the bounds-based valuation offered here, because that argument is notdependentonanyoptionpricingatall,Black’76orotherwise.Wealsohavehereahandycross-checkofanyproposedNNEGvaluation:ifanyproposedmodelbasedonthesameinputcalibrationsgivesalowerNNEGvaluationthaneitheroftheNNEGlowerbounds,thenitmustbewrong.

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AppendixtoChapter18:Proofof(18.14)Recall(7.3)𝑆 = 𝑑(1 + 𝑔)/(1 + 𝑦) + 𝑑(1 + 𝑔))/(1 + 𝑦)) + 𝑑(1 + 𝑔)./(1 + 𝑦). +⋯= 𝑑(1 + 𝑔)/(1 + 𝑦)[1 + (1 + 𝑔)/(1 + 𝑦) + 𝑑(1 + 𝑔))/(1 + 𝑦)) +⋯]where𝑦 = 𝑟 + 𝜋 + 𝜆.Set𝑏 = (1 + 𝑔)/(1 + 𝑦)and𝑎 = 𝑑𝑏.Then(18A.1)𝑆 = 𝑎[1 + 𝑏 + 𝑏) +⋯ ].Multiply(18A.1)by𝑏(18A.2)𝑏𝑆 = 𝑏𝑎[1 + 𝑏 + 𝑏) +⋯]= 𝑎[𝑏 + 𝑏) + 𝑏.… ].Subtracting(18A.2)from(18A.1)(18A.3)𝑆 − 𝑏𝑆 = (1 − 𝑏)𝑆 = 𝑎⟹ 𝑆 = 𝑎/(1 − 𝑏).Substituteout𝑎and𝑏(18A.4)𝑆 = 𝑑(1 + 𝑔)/[1 + 𝑦 − (1 + 𝑔)]= 𝑑(1 + 𝑔)/(𝑦 − 𝑔).Substituteout𝑦(18A.5)𝑆 = 𝑑(1 + 𝑔)/(𝑟 + 𝜋 + 𝜆 − 𝑔).Rearrange(18A.6)𝑑(1 + 𝑔)/𝑆 = 𝑟 + 𝜋 + 𝜆 − 𝑔.So(18A.7)𝑑1/𝑆 = 𝑟 + 𝜋 + 𝜆 − 𝑔.whichwastobeproven,where𝑑1 = 𝑑(1 + 𝑔)istheendofyeardividend,notthestartofyeardividend𝑑.

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ChapterNineteen:TheMarketConsistentApproachDefinitionsA‘marketconsistent’(MC)approachtovaluationisonewhichgives‘marketconsistent’valuations,anda‘marketconsistent’valuationcanbedefinedasa‘fairvalue’valuationbasedontheIFRSdefinitionofafairvalueprice,namely:

Thepricethatwouldbereceivedtosellanassetorpaidtotransferaliabilityinanorderlytransactionbetweenmarketparticipantsatthemeasurementdate.80

AlthoughitisoftenidentifiedwithBlack’76,theMCapproachdoesnotmandatetheuseof anyparticularoptionpricingmodel.Theessenceof theapproach is simplymarketconsistency,notadherencetoanyparticularmodel.Therefore,anymodelisallowable,provideditmeetsthestandardsformarketconsistency.TheMCapproachissometimescalleda‘riskneutral’approachintheactuarialliterature.Thisalternative labelreflects theunderlyingderivativepricingmethodologybasedonrisk-neutralprobabilities.However,thislattertermisunfortunate,inthatitsuggeststothosewhodonotunderstandthismethodologythatitmustbeassumingthatinvestorsare actually neutral about risk and who would want to use a method that made aquestionableassumptionlikethat?Indeed,theargumentthatthe‘risk-neutral’approachis flawed because investors are not risk-neutral is probably the most commonmisconception one encounters against the approach, as if to suggestwe shouldwantsomealternative approach thatdoesnotmake this assumption.Butwe cannot stressoftenenoughthatthemarketconsistentor‘riskneutral’approachDOESNOTassumethatinvestors are risk-neutral. The truth of the matter is that the methodology worksregardlessofwhetherinvestorsarerisk-neutralornot;theirattitudetowardsriskisanentirely separate matter. The term ‘risk neutral’ has an unwarranted negativeconnotationwhenusedinbroadercirclesandisbestavoided.ActuarialMisconceptionsoftheMarketConsistentApproachWeoftenencountercriticismsofBlack’76madebyactuarieswhothinkthatiftheycan,forwantofabetterwayofputtingit,knockthemodeloffitsperchthentheycantherebyjustify lower NNEG valuations as if by default to an alternative that produces lowervaluations. This view is seriouslymistaken. If Black ’76 does not hold, then the validalternativeisnotthediscountedprojectionorTunaruapproaches(andmoreontheseinlater chapters) but some other MC-consistent model that will deliver higher NNEGvaluationsthanBlack‘76.WesuspectthatmanyactuarialcriticsofBlack’76wouldbealotlesscriticaliftheyunderstoodthispoint.

80 See,e.g.,https://www.iasplus.com/en/standards/ifrs/ifrs13.

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This point is a recurring theme in the literature that addresses option pricing in thepresenceoftransactionscosts.Thegeneralresultfromthatliteratureisthattransactionscosts increase option valuations above BS valuations or, in some cases, increase thevolatilitythatneedstobeinputintotheBSmodel,whichamountstothesamething.Thatliteraturealsoteachesusthatinsuchcircumstanceswemusttakeaccountofutilityorriskpreferenceswhich, inturn,makeforarangeofcorrectvaluations, i.e., thereisnouniquelycorrectsolution.81Intruth,weshouldn’tputanymodelonapedestal,whetherthatmodelbeBlack’76orsomesuperioralternative,letaloneaninferiorone.The most common actuarial misconceptions about BS relate to the validity of BlackScholesmethodology.Onefrequentlyencountersclaimsthatyoucan’tuseBlack’76orBlack-Scholesbecause:therearenoforwardcontractsoryoucan’tdoforwardcontracts,youcan’tshorttheunderlying,housepricesareautocorrelatedratherthanGBM,marketsare illiquid or incomplete and you cannot use no-arbitrage pricing in an incompletemarket,etc.Moregenerally,theyarguethattheassumptionsunderlyingthederivationofBlack-Scholes familymodelsdonothold empirically.The logical structureof theseargumentsisthat(a)BSmakessomesetofassumptions,(b)theseassumptionsdonotholdempirically,so(c)wecan’tuseBS.This argument is logically flawed, however. Yes, it is it true that in its usualclassical/textbookderivations,BSorBlack’76makeawholebunchofassumptions,someifnotallofwhichareempiricallyfalse.Evenso,itdoesnotfollowthatBS/B76givesthewrongresults.Therearetwologicalerrorshere.Let’sconsidereachinturn.IfpropositionAimpliespropositionB,thenestablishingthatAisfalsedoesnotestablishthatBisfalse.TheargumentthatAimpliesBsonot-Aimpliesnot-Bisthelogicalfallacyofdenyingtheantecedent.Let’ssaythatweclaimtohaveprovedthattheMoonismadeofgreencheese.Youthendisproveourproof.Fairenough,butyoustillhaven’testablishedthattheMoonisn’tmadeofgreencheese.To give an example in the BS context, BS assume that the underlying is lognormallydistributed.Empiricalevidenceshowsthathousesarenotlognormallydistributed,butitstilldoesnotfollowthatabsenceoflognormalitymakesBSvaluationsincorrect.Thesecondlogicalerrorinvolvesconfusionbetweensufficientandnecessaryconditions.Forastart,therearemultiplewaystoobtainBlack-Scholes–amongstothers,therearethemartingaleapproach,thebinomialapproach,CapitalAssetPricingModel(CAPM)and

81 See,e.g.,H.E.Leland(1983)“OptionPricingandReplicationwithTransactionsCosts,”JournalofFinance40:1283-1301;S.D.HodgesandA.Neuberger(1993)“OptimalReplicationofContingentClaimsunderTransactionsCosts,”ReviewofFuturesMarkets8:222-239;orG.BarlesandH.M.Soner(1998)“OptionPricingwithTransactionsCostsandaNonlinearBlack-ScholesEquation,”FinanceandStochastics2:369-397.

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utility-based approaches82 – and these will be based on different sets of sufficientconditions.Toshowthatthemodelis ‘wrong,’onewouldthenhavetoestablishwhichparticularassumptionswerenecessary.SowhichconditionsmadebyBSintheirderivationoftheformulawerenecessary?Theansweris:none.Notasingleassumptionmadeintheirderivationisnecessaryobtaintheformula.WeprovethisclaiminarecentpaperbyshowingthattheBSformulacanbeobtainedusingasingleassumption(thatofaconstantQuadraticVariationRate83)thatisnotmadebyBS.84TheimplicationisthatevenifeverysingleassumptionmadebyBSisempiricallyinvalid,BSisstillvalid.Notethatwesay‘is’andnot‘mightbe’.TheempiricalvalidityorotherwiseoftheassumptionsmadebyBSissimplyirrelevant.We now consider some examples where the BS formula is still valid in the face ofdeparturesfromtheclassicalBSassumptions.Black-ScholesisRobusttoNon-RandomnessBSassumeGeometricBrownianMotion.Let’stakeasimulatedGaussiandistribution,butsortitinorderofmagnitude,sothatwhileitisstillGaussian,itisnolongerrandom,sothe‘Brownian’bitof‘GeometricBrownianMotion’nolongerapplies.WethengettheplotsshowninFigure19.1:

82Formoreonthese,see,e.g.,J.Andreasen,B.JensenandR.Poulson(1998)“EightValuationMethodsinFinancialMathematics:TheBlack-ScholesFormulaasanExample,”MathematicalScientist23:18-40.83By‘cumulativequadraticvariation’wemeansimplythesumofthesquaredvariationsofthereturn,by‘quadraticvariationrate’wemeanthesumofthesquaredvariationsofthereturndividedbytime.TheQVRissimilartothevarianceofthereturn,butdoesnotentailtheuseofanexpectationsoperator.Itcanbeestimated in a similar way to the variance, however. Formore details see, e.g., S. Dunbar “StochasticProcessesandAdvancedMathematicalFinance:QuadraticVariationoftheWienerProcess,”orD.BucknerandK.Dowd“BlackScholeswithFewerAssumptions,”EumaeusDiscussionPaper(28February2020).84BucknerandDowd(2020,op.cit.).

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Figure19.1:SyntheticvsBlack-ScholesPutOption:GeometricNon-BrownianMotion

AswecanseefromthegreenlineintheFigure,theunderlyingpricefallsthroughoutthefirsthalfoftheseriesandthenrises.Theblue line shows theprice of a put optionprice struck at 90, computedusing thestandardBlack76optionformula.Asyouwouldexpect,thepricerisesastheunderlyingfallsbelow95,thenfallsbackastheunderlyingincreases,endingatzeroastheoptionexpires out of the money. The red line shows the value ofthesyntheticorreplicatingoptionconstructedusingthedeltaoftheput.WeseethattheBSvalue(blue)closelyfollowsthevalue(red)ofthesynthetic,i.e.,BSworkswell.Theimplicationisthat,whiletheassumptionofrandomnessmaybeoneofanumberofsufficientconditionsforBlack-Scholestohold,itisbynomeansnecessary.Black-ScholesRobusttoMeanReversionInitsreplytoCP13/18,theIFoAstates(p.10)that

Using theBlack-Scholes formula inpricingNNEGwill affect thecostof theguarantee, sinceallowance isnotmade for the featuresofmeanreversion,momentumandjumpsdescribedabove.UndergeometricBrownianmotionthevolatilityincreaseswiththesquarerootoftimewhileforothermodelsitdoesnot;thevalueforlongtermderivativessuchasNNEGcouldmateriallydifferfromthatassumedundertheBlack-Scholesmodel.85(Ouremphasis)

Let’shavealookatthismeanreversionclaim.Figure19.2plotsBlack-Scholesagainstaputvalueunderamean-revertingprocess:

85“CP13/18:SolvencyII–EquityReleaseMortgages:IFoAResponsetoPrudentialRegulationAuthority.”28September2018.

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Figure19.2:SyntheticvsBlack-ScholesPutOption:GBMvsMean-ReversionProcess

Thechartaboveshows(greenline)anunderlyingassetthatfollowsthepriceseries95,96,95,96…i.e.theserieshasameanof95.5towhichitcontinuallyreverts.Thebluelineshowsthepriceofaputoptionstruckat90,modelledbythestandardBSoptionformula.The red line shows the value of a hedge, constructedfrom a derivative of the sameformula(‘delta’)andtheyendupinapproximatelythesameplace.WeseethatBSworkswell,eventhoughtheseriesismeanrevertingandisnotdrawnfromanythingthatevenresemblesanormaldistribution.SoitistruethatunderaGBMprocessthevolatilityincreaseswiththesquarerootoftimewhilstforamean-revertingprocessitdoesnot,butthisdifferencedoesn’tmatterhere.Thelongerthesamplingperiodforthestrangedistributionabove,thelowerthesampledvolatility: the prices series is going exactly nowhere. Yet the standard option pricingmodelstillworkswell.Doadifferentsimulationoftheunderlyingpathandyouwouldgetsimilarresults.Black’76isUnderminedbyAutocorrelationinHousePricesAnotherexampleisautocorrelationinhouseprices.Onp.14ofhisreport,Tunaruwrites:

GBMas adatageneratingprocess forhouseprices is totally inappropriatebecause it ignores serial correlation and stickiness of prices, as well asclusteredvolatilityanddownwardjumps.

Thisargument ismistaken.Wehavealreadyseenthat itdoesnotmatterwhetherWeassumeGBMornot,anditdoesnotmatter(seep.50,Note42above)whetherthehousepriceprocessisseriallycorrelated.OurpaperonthederivationoftheBSformulaalsoimpliesthatitdoesnotmatterwhetherpricesarestickyorwhetherthereisclusteredvolorjumpsintheprice.ThesedeparturesfromGMBaresimplyirrelevant.

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‘NoShort-Selling’ArgumentThe ‘noshorting’argument isoftenwheeledout toclaimthat theusualarbitrage-freeargumentsunderpinningBlack’76don’tapply.CriticsclaimthatthevalidityofBlack’76dependsonano-arbitrageargumentthatitselfdependsonbeingabletoselltheforwardcontract,butthenclaimthatno-arbitragedoesnotapplybecause,e.g.,onecannotselltheforward.AnexampleisgivenbyTunaru:

In our opinion, the formula (10) [i.e. Black ‘76] simply does not apply forhousepricesand this isunrelated to theGaussiandistributionassumptionbehind the GBMmodel. The forward contract on a house price cannot becalculatedasin(11)[i.e.accordingtoourequation(3.9):forwardprice=spotprice× 𝑒('#*)!],simplyimitatingtheno-arbitrageformulaforastockpayingdividend,wherethedividendyield isreplacedbythenetrentalyield.Thatformula cannot work because currently we cannot shortsell the value of ahouse.Hence, theno-arbitrageprincipledoesnot applyhere to lock in theforward price as in the case of corporate stock. (Tunaru, 2019, p. 14, ouremphasis)

TheshortresponseisthatthevalidityoftheBSformuladoesnotdependonassumptionsabout the absence of short-selling constraints or the existence of a no-arbitrageequilibrium.Inanycase,thepremiseofhisargumentiswrong.Ofcourseyoucansellahouse!‘Can’justmeans ‘it is possible’ and there are no legal restrictions against doing so. Asweexplained in the previous chapter, anyone can enter into an agreementwhereby onepartyagreestodeliveranindividualpropertyofspecifiedsize,locationandstandardtotheotherparty,atanagreeddateandforanagreedprice.TheOver-the-Counter(OTC)marketmakesthesetradesallthetime.Ifnoindividualpropertyisavailable,thepartiescouldcontract tobuyor sell thevalueof, say, theHalifax index, times somecurrencymultiplier,atsomeagreeddateinthefuture.Partieswhowishtomakesuchtradescanusuallyfindinvestmentbankerswillingtobetheircounterparties. ‘RentalYieldisIrrelevantBecauseaHouseisaConsumptionAsset’A final argument is that the standard forward valuation (e.g., equation (3.9)) isinappropriatebecausetherentalyieldisirrelevanttothosewhopurchasehousestoliveinthem.ToquoteTunaruagain(p.30):

Theconceptofrentalyieldhasbeenintroducedintorealestatevaluationbyanalogywiththelinkbetweendividendsandshareprices.However,itcanbearguedthatthebuyerofahouseisnottheequivalenttoaninvestorbuyingahouseasaninvestmentasset.Forthemajorityofbuyers,housesplaytherole

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of a consumption asset and not that of an investment asset.86 There is noevidence that rental yieldsaredriving futurehouseprices so theexpectedhouse prices at various future long horizons cannot be determined withgrowthmodels in thesamewayexpectedsharepricesmaybedeterminedwithgrowthmodelslinkedtodividends.

OneresponseisthattheERMvaluationquestionappliestoinstitutionalinvestors,namelyERMlenders,whowanttoacquireresidentialpropertyexposure,andwhoarenotusingthepropertyasaconsumptionasset,butratherasaninvestmentasset.Ourpointisthatit is the institutional investorswhomatterhere,because theyare theoneswho issueNNEGs,nottheretailinvestors.In any case, why would the ordinary buyer approach the valuation of a deferredpossessionanydifferentlyfromaninstitutionalbuyer?Supposewearecurrentlylockedintoaleaseholdbutarelookingtomoveintoapropertyin2yearstimeafterourtenancyagreementrunsout.Youarelookingtomoveoutofyourpropertyatthesamedate,butneedmoneynow.Soweagreetopayyoumoneynowinordertopossessthepropertyatthat future date, but the price we pay now would not be the price of immediatepossession.Whyshouldwepaythefullpriceofthepropertynowifwecannotmoveinorrent itout fora full twoyears?Andhowwewouldvalue thedeferredpossession?Answer:atsomediscounttothevalueofcurrentpossession,wherethediscountwouldbedrivenbytherentalyield.87Tunaru writes that “there is no evidence that rental yields are driving future houseprices”butthatclaimisirrelevantevenifitistrue.Itisnotfuturehousepricesweareconcernedwith,butratherthepricenowofacontractforpossessioninthefuture.Wearecomparingthecurrentpricesoftwodifferentcontracts,oneforimmediate,theotherforfuturepossession,andwearenotspeculatingaboutfuturehouseprices,whichareirrelevanthere.Ofcourse,anticipatedfuturepricesmayberelevantifitisthedecisionthatisdeferred.Dowebuynowordowewaitfortwoyears?Ifweexpectpricestorise,wewillpayforimmediateordeferredpossession.Ifweexpectthemtofall,wewilldeferourdecisiontobuy,whetherthatbeimmediatepossessionordeferredpossession.Decisionisdifferentfrompossession,however.Ifwepaynowfordeferredpossession,wearealreadyexposedtofuturehouseprices.Ifweexpectpricestofall,wewillpayneitherforimmediatenordeferredpossession.Rather,wewillwait,i.e.wewilldeferourdecision.Allthesepointsfollowfromelementarypricingeconomics.

86Arelatedargumentwehaveheardisthatwhiletheutilityofnothavingtopayrentalmayberelevantto1sttimebuyerstryingtoescaperentalintoowner-occupier,oncepeoplehaveahomeoftheirown,theydon’t think about rental yield when looking at moving house “up the ladder.”This type of argumentfocussesonhowpeople thinkof their ownhome, i.e. as somewhere to live, andnot as an investment.However,wewouldarguethatthepsychologyofindividualhomeownersisirrelevant,ifonlybecausetherelevantowneristheERMlender,whohasapileofpropertywithrightstodeferredpossessionontheirbooks.87Or still another versionof the ‘householder is different’ argument.AndrewRendell at theARCERMLaunchEventon28thFebruary2019arguedthatwemustconsidertheutilitytotheoccupieroflivinginthehouse,withoutsayinghowwemightmeasurethatutility.Butwhatifweaskyoutovacateyourhousefor2years,soyounolongerhavethatutility?Whatisthecostofit?Surelythecostofrentingfor2years.

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ChapterTwenty:TheDiscountedProjectionApproachThestandardapproachusedbyERMactuaries in theUK is theDiscountedProjection(DP)approach,sometimesalsocalledthe‘realworld’approach.ThisapproachisbasedontheuseofaprojectionoffuturehousepricegrowthtovaluetheNNEG.Inparticular,it replaces the forwardhouse price as the underlying in theMC approachwith some‘expected’ future house price. Equivalently, it replaces the forward rate 𝑓 in the MCapproachwithsomeassumedrateoffuturehousepricegrowthℎ𝑝𝑖.StandardoptiontheoryInstandardoptiontheory,thepriceofanyoptionisthesumofitsintrinsicvalueanditstimevalue.Theintrinsicvalueisthevalueoftheunderlyinginstrumentorassetminusthestrikeprice.The intrinsicvalue is thereforedrivenbythevalueof theunderlying,whichforaBlack76optionisthecurrentpriceofaforwardcontract.Aforwardcontractisfordeferredpossessionoftheunderlyingasset,withpaymentatdefermentdate.Thecurrentpriceoftheforwardcontractwillthereforereflectthecosttothebuyerofnotreceivingtheincome(e.g.,rentalincomeorusevalue–thebenefitsofhavingaroofoverone’shead)fromtheassetuntildefermentdate,andthecosttothesellerofdeferringpaymentuntilthesamedate,namelyimputedinterest.Thepriceof the forwardcontractapproaches thepriceof theunderlyingassetas thedefermentdateapproaches. Wecanthenexpressthepriceoftheforwardcontractintermsofadiscountrelativetothepriceofthespotcontract,wherethediscountisexpressedintermsofadiscountrate𝑞,andapremiumrelativetothepriceofthespot,wherethepremiumisexpressedintermsoftherisk-freeinterestrate𝑟.Hencethe(current)priceoftheforwardcontractis(20.1)𝐹 = 𝑆𝑒R! = 𝑆𝑒('#*)! with𝑡asthetimetodefermentdate.Notethatthereisnoelementofforecastinginvolvedindeterminingthecurrentpriceoftheforwardcontract.88We call 𝑞 the deferment rate, defined as the discount rate that when applied to thefreeholdpriceofvacantpossessionresultsinthepriceofdeferredpossession.Wehaveshownalready(seepp.36-37above) that𝑞 isequal to thenetrentalyield,whichwedefineas𝑑/𝑆where𝑑isthenetnominalannualrental.89 (20.1)implies(20.2)𝑓 = 𝑟 − 𝑞.

88Wecantrytoestimatethefuturepriceoftheforwardcontract,buteconomictheorysuggeststhatisverydifficult,andinanycaseitsfuturepriceisirrelevanttoitscurrentprice.89TheequalityofthedefermentrateandthenetrentalyieldsisproveninBucknerandDowd(2020a,pp.34-35).

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DiscountedProjectionBy contrast, the valuation approach that firms are using is known as the DiscountedProjection(DP)approach.TheDPapproachwasproposedasanalternativetotheMCapproach,whichwecanconsiderforpresentpurposesasusingtheBlack’76putoptionmodelwiththeunderlyingvariablesintheputoptionsbeingforwardhouseprices.TheDPapproachreplacesthecurrentpriceoftheforwardcontractastheunderlyingwithsomeprojectedfuturehouseprice:(20.3)𝑝𝑟𝑜𝑗𝑒𝑐𝑡𝑒𝑑𝑓𝑢𝑡𝑢𝑟𝑒ℎ𝑜𝑢𝑠𝑒𝑝𝑟𝑖𝑐𝑒 = 𝑆𝑒+,-.!whereℎ𝑝𝑖 istheprojectedrateofhousepriceinflation.Equivalently,theDPapproachreplacestheforwardrate𝑓in(20.1)withℎ𝑝𝑖.Since𝑓 = 𝑟 − 𝑞,replacing𝑓byℎ𝑝𝑖gives(20.4)ℎ𝑝𝑖 = 𝑟 − 𝑞.Givenalsothatℎ𝑝𝑖hasbeenspecifiedand𝑟canbeeasilycalibratedfromthespotratecurve,(4)impliesthatwecanbackoutthefollowingimplied𝑞:(20.5)𝑞 = 𝑟 − ℎ𝑝𝑖.Togiveanexample, ifweset𝑟 = 1%andusethe4.25%ℎ𝑝𝑖assumedrecentlybyJustGroup90thenwecanbackout𝑞as(20.6)𝑞 = 1%− 4.25% = −3.25%.Thenegativesigninfrontofthe‘3.25%’ontheright-handsideof(20.6)isnotatypo.Forthiscalibration,andindeedforanycalibrationinwhichℎ𝑝𝑖exceedstherisk-freerate,theDPapproachproducesanegative𝑞rate.Onemightnotethatanegative𝑞rateissomewhatpeculiar,becausethe𝑞rateisequaltothenetrentalratewhichinturnequals𝑑/𝑆,andwewouldexpect𝑑tobepositive.WehaveherearedflagregardingthereliabilityoftheDPapproach.3.UnderlyingProblemwiththeDiscountedProjectionApproachThereisasingleunderlyingproblemwiththeDPapproach.TheoriginalBlack(1976)article tells us that the underlying is a forward contract.91 Presuming, as seemsreasonable,thatBlack’sderivationiscorrect(it is)andthathismodel isapplicable(it

90Thefirmreportedusingthisnumberinbothits2016and2017AnnualReports(seepp.163and110respectively).Thesamenumberalsoappearsinits2018H1results(p.18).91Strictlyspeaking,Black’76focusesonfuturescontracts,butfuturesareastandardisedformofforwardcontract.

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is92), then there are no grounds to replace the underlyingwith some other, differentunderlying,suchastheprojectedfuturehousepriceorthepriceofgreencheese.Anyunderlying other than the current forward price is therefore incorrect. And if theunderlyingisincorrect,thenvaluationsbasedonthatunderlyingmustbeexpectedtobeincorrectaswell.Thus,itiswrongonprincipletoreplacethecurrentforwardpriceintheputpricingequationwiththeexpectedfuturehouseprice(orequivalently,toreplacetheforwardratewiththeexpectedfuturehousepriceinflationrate).Toillustratethemagnitudeofthisdifference,considertheplotsinFigure20.1:

Figure20.1:FuturevsDefermentHousePrices

Notes:Basedoncurrenthouseprice=£100,ℎ𝑝𝑖=4.25%and𝑞=4.2%.

Theblueplotgivesthecorrectpricetouseintheoptionpricingformula,i.e.,thepriceofdeferredpossession.Foramaturityof,say,15years,andourrecommendeddefermentrateof4.2%,thedefermenthouseprice(thepriceofaspotcontractforpossessionin15years)is£53.3.TheredplotgivesthepricethatwouldbeusedinDPapproach,i.e.,thefuturehouseprice,inthiscasebasedonanassumed4.25%ℎ𝑝𝑖rate.Theprojectedfuturepricein15yearsis£189.3.Basedonthesecalibrations,theDPapproachimpliesthatwecanuseanunderlyingvalueof£189.3intheoptionpricingequationwhenthecorrectvalueis£53.3.Ifyoubelievethat theDPapproach is sound, thenyouarebelieving that anasset, a forwardworth£53.3,isactuallyworth£189.3.Inshort,thecorrectapproachistostartwiththeforwardprice𝑆𝑒('#*)!(see(1))astheunderlyingassetprice.However,theDPapproachincorrectlytreatstheforwardpriceastheprojected futureprice𝑆𝑒+,-×! .TheDPapproach thengives thewronganswers ingeneralexceptinthespecialcasewhereℎ𝑝𝑖 = 𝑟 − 𝑞.

92Forafulltreatment,seeBucknerandDowd(2020a,b).

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Ithasbeenobjectedtous93thatwiththedefermentratecurrentlyhigherthantherisk-freeinterestrate,theforwardpriceislowerthanspot,withtheimplausibleresultthathousepriceswillfalloveralongperiod.Thisobjectionismistaken.Ourpricingformula(20.1)abovetellsusthatif𝑞isgreaterthan𝑟,thenthecurrentforwardprice𝐹willbelowerthanthespotprice.Theformulaalsotellsusthatastimetothedefermentdatedecreasesandapproacheszero,thespotandforwardpricewillconverge–acontracttopossessthehouseinthefutureeventuallyturnsintopossessionitself.Buttheformulaissilentaboutthepriceinthefutureatwhichtheywillconverge.Theforwardpricemayremainunchanged,inwhichcasethehousepricewill have to fall in order tomeet it. Alternatively, the house pricemay remainunchanged,inwhichcasetheforwardpricewillhavetorisetomeetit.Orbothmayrise.Or bothmay fall. The future house price (and hence the future price of the forwardcontract)isfreetodowhateveritwantsandcan’tpossiblybetieddownbyanypresentconditions.Thusequation(20.1)doesnothaveanyimplicationsatallaboutthebehaviouroffuturehouseprices.4.OriginandRationaleoftheDiscountedProjectionApproachSowheredoestheDPapproachcomefrom?2005IFoAreportonNNEGvaluationIn 2005, the IFoA published a report on NNEG valuation (IFoA, 2005). This reportconfirmed that itwas reasonable to use Black-Scholesmethodologywhen seeking toobtain a ‘market consistent’ NNEG valuation whilst noting that it “is not without itsdifficultiesandshortcomings.”Wehavenoargumentwiththatassessment.However,italsonotedthat,“Othersmayhowever,prefertoapproachtheassessmentoftheNNEGusingmoreofa“realworld”stochasticmodellingapproach,”whateverthatmightbe,andtheydidnotexplain.AndsowehavethejuxtapositionofBSasareasonableapproachto‘market consistent’ NNEG valuation, versus an alternative unspecified ‘real world’approachakaDPapproachthatispulledoutofthinairandgivesadifferentvaluation.TheHostyetalreportonNNEGvaluationTwoyearslater,theIFoAissuedanotherreportonNNEGvaluation,Hostyetalia(2007).ThisreportstartedwithsomeconcernsaboutthedeclineofprofitabilityanditsimpactonthedevelopmentoftheERMmarket:

the competitive environment that has driven product innovation has …resultedinlowerproductmargins.Thisisallgoodfortheconsumer,butitis

93 In conversation and by email with actuaries and consultants.

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increasinglydifficultforproviderstoreachtargetreturnsoncapital,andthisisdeterringsomeprospectivenewentrants.Oneofthepurposesofthispaperistoinvestigatetheprofitabilityoftypicalschemesinthemarketatpresent,andsotoaddressthequestionofwhethercompetitionhasforcedthemarkettofunctionatnon-profitablelevels.…Wewillaimtoprovidearationalpricingmethodologywhichcanbeadoptedbyanyorganisationactiveinthemarket,andwehopethatthiscansupportthemarketasitexpandsoverthecomingyears.There is now concern that providersmay not be able to offer a productprofitably at currentmargins. Some competitive pressure is clearly a goodthing,asitwillforceproviderstofindmoreefficientwaysofprovidingtheirproduct to consumers. In the equity releasemarket, toomuch competitivepressuremaybeabadthing.(pp.1-2,ouremphasis)

To cut to the chase: their main concern is that overly high NNEG valuations mightunderminetheabilityoffirmstomeettheirprofittargets.Wearesuretheyareright,butthequestionishowtoreconciletheseconcernswiththeirobligationsunderactuarial(see section 7 below)andaccountingstandards(seesection8below)toprovideunbiasedandfairvaluevaluations.If therewasn’t a conflict between these twoobjectives (i.e., profits andunbiased/fairvalue), then there would be no issue to discuss. The fact that ERM industry leadersemphasisetheconflictbetweenthetwoobjectivesandtheirpreference forcommercialconsiderationsoverunbiased/fairvaluationindicatesthattheseobjectivesareinconflict.Otherwisetheywouldargueforunbiased/fairvaluationsandwouldn’tneedtoarguefortheprimacyofcommercialconsiderationsoverfairvalues.Theythenexaminewhattheyunderstandthe“marketconsistent”approachtobe.Theydo not define the term “market consistent” however and the nearest we get to anexplanationisthatthisapproachisbasedonan

approximate market consistent basis similar to the pricing of options onstocks.…Themainchallengewithamarketconsistentbasisisthefactthatthereisnounderlyingmarkettospeakof.Accordinglywehavetriedtocreateaproxymarketconsistentbasisusingtechniquesthatarestandardinsimilarmarkets,specificallyBlackScholesstylemodelling.(p.26)

Thecounter-argumentisthatthereisalwaysanunderlyingmarket!Almostallpropertytransactions are forwards, admittedly, short-maturity forwardsbut there areno legalbarrierstolongermaturityforwardsandERMfirmscouldalwaysapproachinvestmentbanksforquotes.WhetherERMfirmswishtotradeatthoseratesisanothermatter.Hostyetaliathenexplainwhattheymeanbyproxyvaluation:

Usingariskneutralbasis,housepriceinflationshouldbelinkedtothereturnon long term risk free instruments (i.e. government stocks) less anassumptionforrentalincome(netofexpenses).(p.26,ouremphasis)

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i.e.,theydrawon(4)above,whichstatesthatℎ𝑝𝑖 = 𝑟 − 𝑞.TheHostyetaliauseoftheincorrect term ‘house price inflation’ instead the correct term ‘forward house price’suggeststhat theyconsiderthat the futurehouseprice(orℎ𝑝𝑖 rate,dependingontheformulationonewishestouse)shouldgointotheBlack’76model,butwehavealreadyexplainedintheprevioussectionthatneitherthe(expectedorotherwise)futurehousepricenortheexpectedℎ𝑝𝑖ratebelonginthosemodels.Youcaninputthemifyouinsist,butyoushouldn’t,becausethemodelgivesyounoleavetodoso.Theuseofterm‘houseprice inflation’ in this context indicates a seriousmisunderstandingofhowBSoptionpricingworksandleadstoamajorunder-estimationoftheNNEGvalue.ButHostyetaliagoontomakeplainthattheydonotliketheMCapproach:

Inrealitytheabsenceofanunderlyingmarketmeansthatthisproxymarketconsistentapproachisonlyoflimitedacademicvalue…(p.27,ouritalics)

By“absenceofanunderlyingmarket”theymeantheabsenceofaliquidmarketinwhichthe option can be hedged using e.g. a zero-arbitrage trading strategy. The counter-argument is that it is perfectly feasible to apply theMCapproach in theUKpropertymarketcontext(seeBucknerandDowd,2020b,chapter10)andHostyetaliahavenolicencewhatevertoreplacetheforwardhousepriceintheoptionpricingequationwithsomeguess-estimateofthefuturehouseprice.The“onlyoflimitedacademicvalue”jibeis presumablymeant to suggest that theMC approach – or “proxymarket consistentapproach” as theyput it – is of nopractical ‘realworld’ use andperhaps tohint thatpractitionersshouldbe looking foramore ‘realworld’-friendlyalternative?Again,wedisagree.TheMCapproachisnotonlyfeasiblebuthasnofeasiblealternative.ThentheymakeafurthercriticismoftheMCapproach:

For providers attempting to price the NNEG on amarket consistent basisthereisinsufficientproductmargininordertoprovideacompetitiveproductunlesstheyhavestrongcompetitiveadvantagesinoneormoreoftheothercostareas.(p.30)

Whetherornotthisclaimistrue,thisstatementbegsthecentralissue,i.e.,whethertheMC-based valuations are reliable, and Hosty et al. provide no convincing scientificgroundstoquestionthem.SoHostyetalia’smainobjectiontoMCvaluationboilsdowntoitgivingvaluationsthattheydon’tlike.ButremembertheproblemsthatEquitableLifegotinto20yearsagowhenitwasdiscoveredtohavebeenundervaluingitslong-termguarantees!Section7.3.2oftheirreportexaminestheirpreferredalternative,an“insurancepricingbasisusing “realworld'' assumptions.”What theseassumptionsmightbe theydonotexplain;nor,dotheyexplainhowthis“realworld”approachmightbeconsistentwithaveryun-realworldnegativenetrentalrate. Infact, theydon’texplainwhattheir“realworld”approachevenis.Section7.3.2consistsofonly143wordsandisherereproducedalmostinfull:

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7.3.2“Realworld:assumptions”Thealternativemethodwehaveusedistocalculatetheoptioncostusing“realworld”basis.Themethodologywehaveusedisasfollows:- Usethelognormalmodelasbefore(withsamevolatility).- Abestestimateof4.5%p.a.forHPIinthefuture(seeSection4.4).Thisis

thenthemeanreturnunderthemodel.- Wehaveassumedthatarealworlddiscountrateof4.75%perannum.- Wehavenotassumeda“meanreversion'' so that therandomwalk in

each future period is applied independently of the position is [sic]preceding periods. The authors acknowledge that use of a “meanreversion”approachisequallyvalid.…

Ascanbeseen[fromTableshown],theresultingcostsaresignificantlybelowthoseassessedusingourproxymarketconsistentbasis.

Thereisnotawordofexplanationastowhyweshouldregardthis‘realworld’approachasreliable,butthephrasethatjumpsoutis“Abestestimateof4.5%forHPIinthefuture,”i.e.,theRWapproachisbasedonaguessaboutfutureHPI!Wenowseetheseedgerminate.The2005IFoAreportintroducedtheTrojanHorseofhouse price inflation, but at least did the calculations correctly. This error could beforgivenasaninnocuousterminologicalone,exceptthatthepassagequotedopensthedoortofull-scalemisuseandseemstoconfirmthattheHostyetalia2007‘realworld’valuationapproachisbasedonexactlythaterror.TheinclusionofHPIisnolongerameremislabeling,butabedrockprincipleoftheRW/DPapproach.Tospellitout,HPIisnowakeyinputinitsownright.Whichpointsconfirmthatthisapproachisinconsistentwithestablishedoptionpricingtheoryandthereforewrong.Section 7.3.3 clarifies the authors’ views on which approach is to be preferred. Wereproducepartofithere:

7.3.3Marketconsistentorrealworld?On our proxymarket consistent approachwe have derived a cost for theNNEG which would render the product non-profitable, whilst real worldmodellinghasproducedasignificantlylowercost.

Thisstatementhasmajorrepercussions.IfNNEGvaluationsonanMCbasiswouldmakeERMsunprofitableandifthereisnojustifiablealternativetoNNEGvaluationsonanMCbasis,thendoesn’tthatmaketheERMsectorunprofitable?Andifthesectoronlyappearstobeprofitablebecausethe‘realworld’NNEGvaluationsmakeitappearso,thendoesn’tthatmeanthattheprofitsthatthefirmshavebeenmakingmayhavebeenmoreapparentthanreal?However,theissueisnotwhetherthefirms’NNEGvaluationswouldgoupiftheyusedtheMCapproach.The issue iswhether firmsareusing the rightapproach in the firstplace.IffirmsareusingavaluationapproachthatgreatlyundervaluestheirNNEGs,thentheyhavegreatlyunder-estimatedtheircostsandthosecostsarealreadybeingborneby

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firms and their investors, regardless of whether firms acknowledge that fact. Firmsshouldbefacinguptothisprobleminsteadofdenyingit.Fortheirpart,analystsshouldbewonderinghowbigthisproblemmightbeandaskingthemselvesaboutthepotentialimpactonfirms’financialconditions.Under-valuedcostsmeanhiddenlossesandover-estimatedcapital,potentiallyonalargescale.Imagine if Bosch are under-valuing the guarantees they issue with their washingmachines.Theirmanagementthendiscoverthatthecostsofreplacingorrepairingtheirwashingmachinesaregoingtobehigherthantheyhadexpected,buttheydon’tyetknowhowmuchhigher.Theproblemmightonlybeasmallproblembutthenagainitmightnot. Sowhat is themost appropriate response from themanagementwhen they areinformed of it? Should they deny it on the grounds that theywouldn’t like it if theirguaranteesturnedouttobemorecostlythantheyhadthoughtorshouldtheylookintothe issuewithaview to fixing theproblembefore it getsanyworse?Wewouldhavethoughtthattheanswertothatquestionwasobvious,butthenwhywouldtheanswerbeanydifferentifitwasERMsratherthanwashingmachineswhoseguaranteeswerebeingunder-valued? And if the losses involved might potentially be on a large scale, thendoesn’tthatreinforcetheneedtoaddresstheproblemasamatterofsomeurgency,lestapotentiallylargeproblemgrowintoalargerproblemdowntheroadifnothingisdoneaboutit? 5.IllustrativeValuationsWenowprovidesome illustrativevaluationsbasedonourbaselineparametervalues,alongwith𝑞 = −3.25%fortheDPapproach.Table20.1showstheresultingvaluationsfromtheMCandDPapproaches:Table20.1:BaselineERMandNNEGValuations:MarketConsistentvs.Discounted

ProjectionApproachesApproach 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

Marketconsistent £42.1 £40.6Discountedprojection £10.5 £71.3

Notes:AsperTable3.1.DPprojectionbasedon𝑞 = −3.25%. In this case, theDP approach givesNNEG valuations that are less than 30%of thoseproducedbytheMCapproach.TheresultisaconsiderableovervaluationoftheERM,inthiscaseby72%.Butaskyourself:dotheDPvaluationsevenlookright?Ifyoubelievethem,thenyouhavetobelievethatthe‘true’𝑁𝑁𝐸𝐺isonly10.5/81.8=12.8%of𝐿.ThisNNEG/𝐿ratiolooksawfullylowwhenyouconsiderthespreadbetweentheloanrateandtherisk-freerate,whichis5.25%-1%=4.25%.IftheloanhaslittleriskbecausetheNNEGissolow,thenwhyistheriskspreadsohigh?TheDPapproachfailstheintra-oculartraumatest.

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It is also interesting to compare the DP valuations with the PRA’s two sets of‘impossibility bounds’ (i.e., thePrinciple II andPrinciple III bounds) calibrated to thebaselineparameterinputs.TheseresultsareshowninTable20.2:Table20.2:BaselineERMandNNEGValuations:DiscountedProjectionValuations

vsPRAImpossibilityBoundsApproach 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

Discountedprojection £10.5 £71.3 𝑵𝑵𝑬𝑮lowerbound 𝑬𝑹𝑴upperbound

PRAPrincipleIIbounds £34.4 £47.3PRAPrincipleIIIbounds £23.0 £58.8

Notes:AsperTable20.1.TheDPvaluationsviolateallthesebounds.TheDPNNEGvaluation(£10.5)fallsbelowtheNNEGlowerbounds(£34.4and£23.0)andtheDPERMvaluation(£71.3)exceedstheupperbounds(£47.3and£58.8).Thus,ineverycaseconsidered,theDPapproachproducesresultsthatareknowntobeimpossible.6.IsthePRA’sMinimumRequired𝒒RateHighEnough?The PRA has sought to counter the NNEG undervaluations and corresponding ERMovervaluationsproducedby theDPapproachby imposingboundson theminimum𝑞ratesthatfirmsareallowedtouseintheirvaluationmodels,andthecurrentminimumrequired𝑞rateis1%.94Thequestionisthenwhetherthisminimumrequired𝑞rateishighenoughtoruleoutimpossiblevaluations.Theshortanswerisno.Considerthefollowingresultsbasedonaminimumrequired𝑞rateof1%.

Table20.3:DiscountedProjectionValuationsvsPRABoundsApproach 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

Discountedprojection(𝑞=1%) £24.4 £54.4 𝑵𝑵𝑬𝑮lowerbound 𝑬𝑹𝑴upperbound

PRAPrincipleIIbounds £34.4 £47.3PRAPrincipleIIIbounds £23.0 £58.8

Notes:AsperTable20.1butwithfirstlinebasedon𝑞 = 1%.

94SeePRAPolicyStatement31/18,December2018,p.11.

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TheDPvaluationfortheNNEGisnow£24.41butthePrincipleIIlowerboundis£34.4,so𝑁𝑁𝐸𝐺 is still below the Principle II lower bound. Correspondingly, the DP ERMvaluation(£54.4)isabovethePrincipleIIupperbound(£47.3).AsforthePrincipleIIIbounds,itistrivial,since𝑞 = 1% > 𝑞 = 0%,thatthePrincipleIIIboundswillbesatisfied.However,toavoidgeneratingimpossibleresultsitisessentialthatbothsetsofboundsbesatisfied,andthePrincipleIIboundsarenot.Thus,thePRA’sminimumrequirementof𝑞 = 1%isnothighenoughtoruleoutimpossiblevaluations.Thenextquestionis,howhighmusttheminimumrequired𝑞ratebeinordertoensurethatthePrincipleIIboundsaresatisfied?AnindicationoftheanswerisgiveninTable20.4:

Table20.4:DiscountedProjectionValuationsvsPRABounds(II)Approach 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

Discountedprojection(𝑞 = 2.5%)

£34.1 £47.7

Discountedprojection(𝑞 = 3%)

£36.2 £45.6

𝑵𝑵𝑬𝑮lowerbound 𝑬𝑹𝑴upperboundPRAPrincipleIIbounds £34.4 £47.3

Notes:AsperTable20.1butwithfirstlinebasedon𝑞 = 2.5%andsecondlinebasedon𝑞 = 3%.Thistableshowsthatweneedminimum𝑞 ratesofcloseto3%inordertosatisfythePrincipleIIbounds.Twofinalpoints.First,theonlywaytheseresultscanbechallengedisbychallengingthecalibrations,butthecalibrationsareplausibleandwehaveexplainedthematlengthinearlierchapters.Second,theresultsshowninTable20.4arespecifictotheillustrativecaseofamale70yearold.Itisentirelypossiblethatevenaminimumrequired𝑞rateof3%willnotbehighenoughinothercases.Webelievewehavemadeourpoint,however:thecurrentminimum𝑞raterequiredbythePRAisnowherenearhighenoughtoruleoutimpossiblevaluations.8.ConclusionsTheDiscountedProjectionapproach:(a)hasneverbeenexplainedletaloneconvincinglyjustified by those who advocate it; (b) is based on an elementary error about theunderlyingprice,namely,theconfusionbetweenthefuturepriceofspotpossessionandthe current price of deferred position; (c) is being promoted by practitionerswith avested commercial interest who are promoting it for commercial reasons and aredismissiveoftheonlyapproachthatisscientificallyrespectablebecausetheydonotlikethevaluationsitproduces;(d)hasnotbeenendorsedbyasinglerecognisedindependent

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expert;(e)doesnotappearinthecorpusofrecognisedscientificresearchjournalsthataresubjecttorigorouspeer-review;95(f)iscontradictedbyalternativeapproachessuchasBlack’76thatareusedandtaughtallovertheworldandhavebeenpublishedintoptierrefereedacademic journals; (g)entailssignificantunder-valuationsofnonegativeequityguaranteesandover-valuationsofequityreleasemortgages, (h) fails the intra-oculartraumatestand(i)canproducevaluationsthatareknowntobeimpossible.Thereare also the issues discussed below whether the DP approach is inconsistent withactuarialandaccountingprofessionalstandards.TheuseoftheDPapproachhasinterestingprudentialimplicationsandtheregulatoryresponsetodatehasbeenlessthanfullyidealifonlybecauseitstillallowsfirmstoreportimpossibleNNEGandERMvaluations.

95 Admittedly,Hostyetalia(2007)waslaterpublishedintheBritishActuarialJournal,butitisnotclearwhetherBAJarticles(orforthatmatter,anyarticlesandreportspublishedbytheIFoA,e.g.,suchastheTunaru report) are subject to “rigorouspeer review”and theonly thing that is clearabout the reviewprocess,whateverthatmightbe,isthatitisunclear.ToquotefromtheBAJ’swebsite,“BritishActuarialJournalcontains thesessional researchprogrammeof the InstituteandFacultyofActuariesalongwithtranscriptsofthediscussionsanddebates.ItalsocontainsPresidentialaddresses;memoirsandpapersofinteresttopractitioners.”Thereisnothingaboutrigorousscientificpeerreview.

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ChapterTwenty-One:TheTunaruReport On19February2019,theActuarialResearchCentrepublishedProfessorRaduTunaru’smuchawaited reportonNNEGvaluation.96His reporthadbeencommissionedby theAssociationofBritishInsurersandtheIFoAinthehopethatitwouldresolvesomeoftheNNEGcontroversy.TheIFoApressreleaseaccompanyingtheTunarureportreassuredthepublicthatthe“valuations arising from insurers’ current models and bases are sufficient,” as if tosuggestthattheTunarureportputsanyconcernsaboutthesemodelstorest.Tunarudoesnosuchthing.Instead, Tunaru offers an approach that is free of the DP approach’s most glaringweaknesses – the reliance for option pricing on forecasts and on absurdly low andtypicallynegativedefermentrates–butgeneratesNNEGvaluationsthatareofthesameorderofmagnitudeasthatapproach.Thereforeittoofailsthesnifftest,inthatitproducesNNEG/𝐿ratiosthataretoolowtojustifythespreadbetweenlendingandrisk-freerates.Nonetheless,wecanseewhytheindustrywouldwelcomeit.Fromtheirpointofview,Tunaruoffersanotherwaynot to skin thecatwithoutgetting the flakaboutnegativedefermentrates.ToquotefromthetranscriptoftheStapleInnlauncheventon28February2019:

GarethMee(EY):ExcellentnewresearchpublishedtodayonEquityReleaseMortgages.Reallymovestheindustryforwardsintermsofunderstandingandriskmanagement.

Itseemstoushoweverthatthecelebrationmightbeabitprematureandwearenottheonly ones to have doubts. Similar concerns have been expressed byTony Jeffery andAndrewSmithintheirequityreleasereporttotheSocietyofActuariesinIrelandon28March2019,andbyanumberofactuarieswhohavecorrespondedwithusprivately.ExcessiveParameterisation:theARMA-EGARCHModelProfessorTunaruopenshisreportwithanoft-citedquotefromGeorgeBox:“Allmodelsarewrongbutsomeareuseful.”97Heusesthisquoteasawaytointroducehispreferredmodel,theARMA-EGARCHmodel–ortogiveititsfullname,theAutoregressiveMovingAverageExponentialGeneralisedAutoregressiveConditionalHeteroskedasticmodel.98

96 R.S.TunaruandE.Quaye“UKEquityReleaseMortgages:areviewoftheNoNegativeEquityGuarantee.”ActuarialResearchCouncilandInstituteandFacultyofActuaries. 97G.E.P.Box(1976)“ScienceandStatistics,”JournaloftheAmericanStatisticalAssociation71(356):791-799.98See,e.g.,R.F.Engle(1982)“AutoregressiveConditionalHeteroscedasticitywithEstimatesoftheVarianceof United Kingdom Inflation,”Econometrica50(4): 987–1007; T. Bollerslev (1986) “Generalized

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This model has been around for a long time and is well known to time-serieseconometricians and applied economists, who use it principally to model short-termvolatilitydynamicsinassetmarkets.Itisnotobvioushoweverwhyonewouldwanttoapplyamodelofshort-termvolatilitydynamicstoalong-termproblemsuchasNNEGvaluation.Justas‘is’doesnotimply‘ought’,‘can’doesnotimply‘should’.99IntroducingtheARMA-GARCHmodelmerelyaddsneedlesscomplexitytoasetofproblemsthatthatarealreadymuchmisunderstood.The ARMA-EGARCH has a lotmore parameters than Black ’76, and on the subject ofchoosingbetweendifferentmodelsProfessorBoxwentontostate

Sinceallmodelsarewrongthescientistmustbealerttowhatisimportantlywrong.Itisinappropriatetobeconcernedaboutmicewhentherearetigersabroad.(Box,1976,p.792)

Thekeyphraseis“importantlywrong”,asopposed,e.g.,totriviallywrong.Hispointwasthenwhenchoosingbetweenmodels,itisadvisabletogowiththemoreparsimoniousoneunlessthereisgoodreasontothecontrary.ToquoteBoxagain(1976,p.792):

Since all models are wrong the scientist cannot obtain a "correct" one byexcessiveelaboration.OnthecontraryfollowingWilliamofOccamheshouldseekaneconomicaldescriptionofnaturalphenomena.Justastheabilitytodevisesimplebutevocativemodelsisthesignatureofthegreatscientistsooverelaborationandoverparameterizationis

nottoberecommended.Sowhenconsideringaddingnewparameterstothemodel,oneshouldestablishaclearpurposeindoingso.Ataminimum,onewouldhavetoshowthattheexistingsimplemodelwasinadequateinsomerespectandthatfixingthisinadequacyrequiresaddingatleastonemoreparameter.Tunaruoffersnosuchdemonstration.TheARMA-EGARCHmodelfailstheBoxtest.InconsistentonMarketConsistencyTunaru repeatedly criticizes the ‘risk-neutral’ pricing methodology that underpinsstandardoptionpricingformulas,e.g.,

“Intheabsenceofanunderlyingmarket,liquidandfreeofcounterpartyrisk,itisnotpossibletohaveadirectrisk-neutralapproach.”(p.11)

AutoregressiveConditionalHeteroskedasticity,”JournalofEconometrics31(3):307–327;andD.B.Nelson(1991)“ConditionalHeteroskedasticityinAssetReturns:ANewApproach,”Econometrica59(2):347–370.99ThismodelisnotnewtotheNNEGliterature.ItsfirstappearanceinanacademicjournalisJ.S.-H.Li,M.R.HardyandK.S.Tan(2010)“OnPricingandHedgingtheNo-NegativeEquityGuaranteeinEquityReleaseMechanisms,”JournalofRiskandInsurance77(2):499–522.

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Unfortunately,Black(1976)modelcannotbeappliedinthecurrentcontextfortheNNEGmarketsincethereisnofutureshousepricecontractcurrentlytradedintheUK.…In ouropinion, the [Black ‘76] formula… simplydoesnot apply forhousepricesandthisisunrelatedtotheGaussiandistributionassumptionbehindtheGBMmodel.Theforwardcontractonahousepricecannotbecalculatedasin[Black‘76]simplyimitatingtheno-arbitrageformulaforastockpayingdividend,where thedividendyield is replacedby thenet rental rate.Thatformula cannotworkbecause currentlywe cannot shortsell the value of ahouse.Hence, theno-arbitrageprincipledoesnotapplyhere to lock in theforwardpriceasinthecaseofcorporatestock.(p.14)

WehavedealtwiththeseandothercriticismsofBlack’76inChapter19.Asanalternative,heproposes(pp.16-17)analternativerisk-neutralpricingrulebasedonanEsschertransform.100There is an entertaining inconsistency here. Having repudiated option pricingtheorybecause of incomplete markets, inability to take short positions etc., he thenrecommendstheEsschertransformrisk-neutralpricingruleasasolution.Howeverthearguments for the existence of any risk-neutral pricing rule hang onmuch the samecompletemarketorcostlessshortingassumptionshehasjustrejected.Tospellitout,TunarudismissesBlack’76becauseitreliesonthestandard‘risk-neutral’pricingmethodology,whichreliesonabunchof‘unrealistic’assumptionsthathedoesnotlike.HetheninvokeshispreferredEsschertransformpricingmethodology,butthatpricingmethodologyreliesonmuchthesamesetofassumptionshehasjustrejected.The problem is that if those underlying assumptions can be invoked to justify hispreferred model, they can also be invoked to justify Black ’76 and in that case hisargumentsagainstBlack’76fallaway.Ontheotherhand,ifthoseargumentscannotbeinvokedbecausemarketsareincompleteetc.,thenTunarucan’tinvokethoseargumentsfortheARMA-GARCHmodeleither.Inthatcasehehasnopricingmethodology.Hecan’thaveitbothways.Soeitheryoubelieve in theapplicabilityofa ‘risk-neutral’pricingapproachbasedonmarketcompletenessoryoudon’t.Ifyoudo,youcanuseitandifyoudon’t,thenyoucan’tuseit.Butyoucannotrejectonepricingapproachbecauseitreliesonassumptionsyoudon’tlikeandthenproposeanalternativepricingapproachthatalsoreliesonthosesameassumptions.

100 See F. Esscher (1932) “On the Probability Function in the Collective Theory of Risk,”SkandinaviskAktuarietidskrift15(3):175–195;orH.U.GerberandE.S.W.Shiu,“OptionPricingbyEsscherTransforms,”TransactionsoftheSocietyofActuaries46:99–191.

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InsufficientVolatilityTherearealsoissuesaboutTunaru’scalibrationandlet’sstartwiththevolatility.TunarubeginsbysettingoutsomeMaximumLikelihood(MLE),GeneralisedMethodofMoments(GMM)andMethodofMoments(MM)volatilities.ThesehereportsinhisTable1:

Notes:Tunaru(2019,Table1).

We would describe these estimates as referring to annualized spot volatilities, asopposed,e.g.,toanyforwardvolatilities.Onthebasisoftheseestimates,heproposesabaselinevolatilityof3.9%.He then presents a set of comparable results for alternative sample periods and UKregionsbasedontheNationwidequarterlyhousepriceindex,overafullsampleperiodof1971to2018.Theseresultsindicatethat“arangeofvaluesbetween3.85%to6.5%seemsrepresentativeforGBMvolatilityparameter”(p.1,referringtohisTable2).Oneisimmediatelystruckbythefactthathisbaselinevolatilityisjustnexttotheminimumofthe volatility range: one usually puts baseline parameters somewhere in themiddle,becauseotherwisetherangebecomesredundant.Itisthenstrange,inourview,thathedoesnotrevisehisbaselinevolatilityestimateupwardsinlightoftheresultsinhisownTable2.Hegoesontoargueagainstde-smoothingforhouseprices.De-smoothingisanapproachfordealingwithautocorrelatedhouseprices.101Heexplains,“Thispoint[whyoneshouldnotde-smoothforhouseprices]isimportantsinceintheNNEGliterature10%volatilityistakenasindicativefortheUK.Basedontheresults[his]inTable2wecanseethatavalue of 10% is already a very conservative stressed upwards estimate.” (p. 21)Elsewhere(p.1)heelaboratesbystatingthat“10%or13%isthenmoreofastressedscenariovalue.”SoTunarudoestheright thing in thesensethathedoesn’tapplyanydesmoothing, but for questionable reasons, i.e., because he regards 10% to 13%as astressedvalue.However,theupperendofhisrangeisalsothePRA’scentralestimate.RecallCP13/18(p.9):

ThePRA’scentralestimateisofa13%volatilityassumptionfortypicalholdingperiods forERMs,althoughuseofalternativedatachoicesgivesarangeof13%-16%,andmakinganallowanceforparameteruncertaintygivesarangeof11%-18%.EstimatesforpropertyvolatilityprovidedtothePRAbyfirmsaregenerallyintherange10%-15%.(Ouremphasis)

101 Formoreonde-smoothing,seeP.BoothandG.Marcato(2004)“TheMeasurementandModellingofCommercialRealEstatePerformance,”BritishActuarialJournal10(1):5–61.

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SoTunaru’svolestimatesarewayoutoflinewiththePRA’s.Whatmakesthatallthemoreodd is that he acknowledges that his baseline calibrations are “selected based ondiscussionswithexpertsworkingonERMsandusingpublicavailabletablesfromLegal&General,JustGroupandEquityReleaseCouncil”(p.27).Ourpoint is thedisconnectbetween the firms reporting10% to15% to thePRAandTunarugoing for3.9%basedonadvice fromexpertswhowerepresumablymuchthesamepeoplewhoreportedthose10%to15%(orwhatever)estimatestothePRAinthefirstplace.In short, he goes for the spot volatility of 3.9% and recommends that this volatilityestimatebeusedasthesinglevolatilityinputforallNNEGputs.ImplementingthislowvolatilityintohisNNEGmodelthenhelpsproducethe(very)lowNNEGvaluationsthathereports.ComparedtoourrecommendedapproachinChapter9,Tunarugetsas farasthespotreturnvol(i.e.,(9.4),getsthatwrong,andtakesnoaccountoftheadditionalriskfactors(achievementrate,𝑟and𝑞)thatalsoimpactthevol.Thenetresultisthathegoesfor3.9%forallagesandgenders,whereaswerecommend(seeTable9.7)age-dependentvolsthatstartat26.1%formalesaged55andgodownto16.4%formalesaged90,andhigherforfemales.TheRentalYieldandtheTunaruMultiplierAfinal issueisTunaru’sunusualwayofestimatingrentalyields.Tunarustarts(p.31)withanestimateofthemeangrossrentalyieldofjustunder5.2%.Hethennotesthatlessthan20%ofpropertiesarerentedoutandconcludes:

Thismeansthataroughcalculationwouldgiveatotalrentalyield,weightedby the 20% representing the actual rentingmarket, of 1.03% (5.1776%×20%)perannum.(Tunaru,2019,p.32)

Thisconclusiondoesnotfollow.AsAndrewSmithobservesinprivatecorrespondence:

Th[is]rentalyieldanalysisjustseemswrongtome-arguing,asfarasIcantell,thatonly20%ofthemarketisrented,sotherentalyieldonthemarketasawholeisafifthoftheyieldonrentedproperties.Thatimpliesthatowner-occupiersplacenovalueontheirrighttooccupytheirownproperty.Ican'tseeanyperspectivefromwhichthismakessense.(Ouremphasis)

ItisalsoapparentfromthediscussionatStapleInnthattherewaslittlesupportfortheTunarumultiplierthereeither.Tobefair,Tunaruacknowledgesthepoint.Inafootnotehewrites:

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Ithasbeendebatedwithotheracademicsandmarketpractitionerswhoarenotentirelyconvincedabouttheweightingbeingapplied.(Tunaru,2019,p.32,note16)

Wearenotentirelyconvincedeither.Oneofthesewashisowncolleague,DanAlai,whoraisedtheissueataKentUseminaron28January2019:

I was wondering why you multiply the rental yield by the proportion ofpropertiesthatarerentedout.Inotherwords,whyis5.1776%dividedby5.Ijustdonotseehowitisrelevantwhetherotherpropertiesarebeingrentedoutornotindeterminingtheappropriaterentalyieldforacertainproperty.(QuotedinTunaru,2019,p.74)

Dr.Alaiiscorrect.Theeconomicrental–thatis,theusevalue,thevalueofthe‘roofoverone’s head,’ etc. – is still enjoyed by someone (or potentially enjoyable even if thepropertyisvoid)regardlessofwhetherthepropertyisrentedoutornot.Whatmattersisthattheeconomicrentalisvaluable,notwhetherthepropertyisactuallyrentedout.Evenifthereisanowneroccupier,thepropertystillhasusevalueandthebestminimum102estimate for the market value of that use value comes from the rents for similarpropertiescurrentlyprevailinginthepropertyrentalmarket,notthoserentsmultipliedby20%.103Rememberthatwhatwearetryingtodohereisusetherentalyieldonrentedpropertiestocomeupwithanapproximatecalibrationforthe impliedrentalyieldonERMedproperties.Theproportionofpropertiesthatarerentedoutisirrelevanttothedefermentrate.ImaginethatIhavejustERMedmyhouse,buttheidenticalhouse(plusgardenetc.)nextdoorhadjustbeenrentedout.ThenIcanestimatethevalueoftherentalservicesonmyhousefromtherentalonthehousenextdoor,andmyestimateofthevalueofthoserentalserviceswillbeatleast100%ofthevalueofnextdoor’srental.Itisassimpleasthat.

102 Wesay‘minimum’becauseiftheowneroccupiervaluedrentingoutmorethanoccupation,hewouldn’tbeanowneroccupier. 103Othershavesaidmuchthesamething.ToquoteDavidRule’s‘DearCEO’letterof3April2019:“Oneofthe most financially significant parameters is the deferment rate, which the research estimates byconsidering rental yields. Several commentators have already challenged the research’s judgement tomultiplytherentalyieldbyafactor(currently20%)representingtheproportionofpropertiesrentedout–toitscredit,theresearchhighlightsachallengetothisjudgementmadebyanacademicreviewer.ThePRA’s own view is that the challenges are well-founded, the justification for the 20% factor isnotpersuasive,andthatitisnecessarytoconsiderthebenefitsofowner-occupationonpropertiesthatarenotrentedout(suchasthoseonwhichERMsarewritten).”SeealsoTurnbull(2019),whosuggeststhattheTunurumultiplier“isbestleftbehind.”Finally,theFebruary2020IFoAERMworkingpartydiscussionnotehadthistosay:“Contrarytothe[Tunarumultipliersuggestion]the lossofpotentialrental incomethatarisesduetoowner-occupationshouldnotbedeductedfromtheestimateofrentalincomeforthepurposesofassessmentofthedefermentrateforuseintheNNEGvaluation.Owner-occupierschoosetooccupytheirproperty because they are willing to bear the opportunity cost of the income foregone by owner-occupation.Thisdoesnotimplytheopportunitycostiszero.Itseemsself-evidentlyreasonabletoassumetheowner-occupierwouldprefertoownthepropertytodayratherthandeferownershipofthepropertytosomefuturedateandthusberequiredtopayrenttotheownerinthemeantime.Thissuggestsowner-occupation does not imply a deferment rate of zero, and that the deferment rate of owner-occupiedproperties is a function of the rental income that the property could generate if it were not owner-occupied.”SotheTunarumultiplierwaseventuallyrepudiatedevenbytheIFoA,whichhadcommissionedtheresearchinthefirstplace.

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Inanycase,thereportfailstoengagewiththerationalegivenbyCP13/18(para3.16,p.19),thattheonlydifferencebetweenacontractforimmediatepossessionandonefordeferredpossession, isthevalueof foregonerights (e.g. torental incomeoruseof theproperty)duringthedefermentperiod.Youwillpaylessfordeferredpossessionbecauseyouwilllosetheincomethatyoucouldgetbyrentingthepropertyoutor,alternatively,youwilllosetheusebenefitthatyoucouldgetbylivingintheproperty.Whywouldyourobyourselfbypretendingthatyouhaveonlylost20%ofthatincomeoruse?Youhavelost100%ofwhatyouhavelost.Ifthisargumentisnotclearthenconsiderthefollowingreductioargument:byTunaru’slogic,iftheproportionofrentedpropertiesweretofalltosay0.001%,thentherentalyieldtobeusedinERMcalculationswouldbe5.1776%×0.001%=0.005%,effectively0%.Arentalyieldofscarcelyoverzerocannotbecorrect.Why?Well,ifit’sisnotobvious,itisbecausetherentalonmyhousecannotbe0%oftherentalontheidenticalhouserentedoutnextdoor.Summary:TunaruDoesNotWorkTheTunaruapproachproducesNNEGvaluationssimilar to thoseproducedbytheDPapproach,butwithouttherelianceonforecastsor incredibledefermentratesthatarebelowzeropercent.Butithasmajorweaknesses,inorderofmagnitude:(1)Itusesanoverly parameterized model, when a simpler model such as Black ‘76 would havesufficed.(2)HisargumentsagainstBlack’76underminehisownARMA-GARCHmodelaswell.(3)Hisrecommendedvolatilityiswaytoolowand(4)hisrecommendeddefermentrateisbasedonanelementaryerrorandisonefifthofwhatitshouldbe.StripawaythefirsttwoerrorsandhemayaswellhaveusedBlack’76.StripawaythesecondtwoerrorsandhewouldhaveendedupwithBlack ’76basedonmuchthesameassumptionswemade.

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ChapterTwenty-Two:JustGroup’sDefermentRateCalibrationsInthepreviouschapterwestatedthatJustGrouphaduseda4.25%hpiassumptionwhenvaluingitsNNEG.WhenwemadeapresentationonequityreleasetotheLondonSchoolofEconomicson1October2018anequityreleaseanalystintheaudiencesuggestedthatwehadmadethenumberup.104Thefactsareeasilyverified:thefirmreportedusingthisnumberinbothits2016and2017AnnualReports(seepp.163and110respectively)andtherelevanceofthisnumberinthesereportsisalsoclear,becauseitistheIFRSreportsthatshareholderswouldbeinterestedin.Thesamenumberalsoappearsinits2018H1results(p.18).Howeverintheir2017SFCR,thefirmreportsanexplicit𝑞rateof0.5%(p.54).Thelatteraccompaniesanalmost£1billionhittotheirbalancesheetthatisoffsetbehindtransitionals.Page 83 of the firm’s 2017 Solvency and Financial Condition report reconciles thestatutory with the regulatory balance sheet. The almost £1bn figure appears as thechangein‘othervaluationdifferences’fromend2016toend2017.However,thisalmost£1billionlossdoesnotmakeasignificantimpactoncapitalbecauseitislargelyoffsetbyanincreaseinthePRAtransitionalarrangement,whichisanentryontheassetsideoftheregulatorybalancesheetthatcanbeusedtocreateextraregulatorycapital. It isapuzzlewhythislatteritem(whichismeanttobeslow-movinganddecliningovertime)shouldhave increasedsomuchover justoneyear. Ithasbeenput tous that the firmincreasedthisitemmerelytohidethehittoitscapital,butwefinditdifficulttobelievethatareputablefirmlikeJustwouldhaveresortedtosuchdissimulation,sotheremustbesomeinnocentexplanationthatweareunawareof.Thesituationfor2016isrelativelystraightforward.Toquoteits2016AnnualReport:

Whencalculatingthevalueoftheno-negativeequityguaranteeonthelifetimemortgages,certaineconomicassumptionsarerequiredwithinthevariantoftheBlack-Scholesformula.…In the absence of a reliable long-term forward curve for UK residentialproperty price inflation, the Group has made an assumption about futureresidentialpropertypriceinflation.Thishasbeenderivedbyreferencetothelong-termexpectationoftheUKretailpriceinflation,“RPI”,(consistentwiththeBankofEnglandinflationtarget)plusanallowancefortheexpectationofhousepricegrowthaboveRPI(propertyriskpremium) lessamargin foracombination of risks including property dilapidation and basis risk. Thisresultsinasinglerateoffuturehousepricegrowthof4.25%.(p.163)

Thenatural reading is that theyareusing theBlack76 formula (which takes forwardprices,notspot)usinganhpiof4.25%.ThisreadingsuggeststhattheyaretakingtheBlackforwardrate,whichshouldbeequaltoriskfreeminusthedefermentrate:(22.1)𝑓 = 𝑟 − 𝑞

104K.Dowd,“TheMarcusBarnardShow,”TheEumaeusProject,11October2018.

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andreplacingit(incorrectly)withtheforecastℎ𝑝𝑖,i.e.,(22.2)ℎ𝑝𝑖 = 𝑟 − 𝑞as per the bad old ‘discounted projection’ approach that no-one should be using.Rearranging,wegetanimplied𝑞:(22.3)𝑞 = 𝑟 − ℎ𝑝𝑖Ifweassumethat𝑟 = 1%thenweget(22.4)implied𝑞 = 1%− 4.25% = −3.75%.Wecanthencriticisethisimplied𝑞valueasmakingnosense.Thesituationfor2017ismoreinvolved,however.Their2017AnnualReportstates

Thereturnonequityreleaseassetsisadjustedtoallowfortherisksassociatedwiththeseassets–namely,thepotentialshortfallresultingfromtheNo-NegativeEquityGuarantee(“NNEG”).TheGroupcalculatestheshortfallinrespectoftheNNEGusingavariantoftheBlack-Scholesoptionpricingmodel.Inputsrequired(e.g.currenthouseprices,futurehousepricegrowthandhousepricevolatility)arederivedfromavailablemarketdata.(p.51)Intheabsenceofareliablelong-termforwardcurveforUKresidentialpropertyprice inflation, the Group has made an assumption about future residentialpropertypriceinflationbaseduponavailablemarketandindustrydata.Theseassumptionshavebeenderivedwithreferencetothelong-termexpectationoftheUKretailpriceinflation,“RPI”,(consistentwiththeBankofEnglandinflationtarget)plusanallowancefortheexpectationofhousepricegrowthaboveRPI(property risk premium) less a margin for a combination of risks includingpropertydilapidation andbasis risk.An additional allowance ismade for thevolatilityof futurepropertyprices.Thisresults inasinglerateof futurehouseprice growth of 4.25%, with a volatility assumption of 12% per annum.(ibid,p.110)

Thispassage is consistentwith thepreviouslyquotedpassed from their2016AnnualReport.Goingthroughthesamecalculationsasbefore,wethengetthesameimplied𝑞 =1.5% − 4.25% = −2.75%.Sofar,sosame.Thenthefunstarts.Intheir2017SFCR,thefirmclaimedtobeusingadefermentrateof0.5%:

Asat31December2017,theBoardconsiderstheMatchingAdjustmentintheGroup’sbalancesheetinrespectofLTMnotessatisfiestheprinciplesofSS3/17givingrisetoanimpliedpropertyvolatilityof12%andapositivedefermentrateof0.5%onariskneutralbasis.(2017SFCR,p.54)

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Theissuethenishowtoreconcilethisimplied𝑞 = −2.75%withtheexplicit𝑞 = 0.5%thattheyalsoclaimtobeusing.Thedifferencebetweenthetwoisenormous.Furthermore,theaverage𝑞ratewearediscussinghereisaslow-movingvariable,whichcannotmovemuchfromoneperiodtoanother.Ajumpof325basispointsfrom-2.75%inoneyearto0.5%inthenextyearisimplausible,evenleavingasidethefactthatboth𝑞ratesarewayoutoflinewiththeempiricalevidencesetoutinChapter7.Now for the awkward bit. In its 2018H1 results (p. 18) the firm offers the followingtreatmentofanimpliedHPIvsan‘actual’orexplicitHPI:

Implied HPI=actual HPI – volatility/dilapidation – effect of capitalrequirement–effectofsecuritisation=4.25%-3%-1.5%-1.4%=-1.65%

whichtheyroundto-1.65%.So the firm has gone from an explicit ℎ𝑝𝑖 = 4.25% to an implicit ℎ𝑝𝑖 = −1.65%, adifferenceofcloseto600basispoints!There is another problem. The adjustment for ‘volatility’ is odd, given that the Blackformulaalreadyincludesanexplicit treatmentofvolatility,namelythedirect input forvolatility(whichthefirmtellsusis12%).Ifthefirmisusinga4.25%hpiinputfortheforward rate calculation and a volatility input for the put valuation, then itwould bewholly incorrect to include an additional volatility ‘adjustment’ aswell. Likewise it isspurioustoincludethecapitalrequirement,becausethecalculationisfortheamountofcapitalavailable,notthecapitalrequired.TheNNEGcalculationisaninputdeterminingthe amount of capital available only. So there appears to be some obvious doublecounting.The‘effectofsecuritisation’itemisalsostrangeandwehavenoideawhatitisorwhyitisthere.Wenowrearrange(22.5)as(22.6)𝑟 = 𝑞 + ℎ𝑝𝑖andsubstitute𝑞 = 0.5%andℎ𝑝𝑖=-1.7%into(22.6)toobtain(22.7)𝑟 = 0.5% − 1.7% = −1.2%!Soinmakinganexplicitassumptionof𝑞 = 0.5%andgoingfromexplicitℎ𝑝𝑖 = 4.25%toan implicitℎ𝑝𝑖 = −1.7%, the firmisalso implyinganastonishing𝑟 = −1.2%,but thatcan’tberighteither.What seems tohavehappened is this: In2016, the firmusedanexpectedhpi rateof4.25%tomodelitsNNEG,equivalenttousinganimplicitqrateof-2.75%orthereabouts.Aswe have repeatedly stated, this approach ismanifestlywrong, because the q rateshouldbemuchhigher.In2017,thefirmagainusedthe4.25%expectedhpirateof4.25%tomodelitsNNEG,butthistimeit introducedaseriesof(mostlyinappropriate)extraitemsdrivingawedgebetweenthishpirateandanimpliedhpiratethatisonlyconsistentwith the firm’s assumed𝑞 rate of 0.5% if 𝑟 = −1.2%, which confirms that the firm’s

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reconciliationofitsexplicitandimpliedHPIratesmakesnosense.Thederivationoftheimpliedexpectedhpirateandthecorresponding0.5%𝑞rateisthustotallyhalf-baked.Table22.1summarisesthefirm’sNNEGvaluationapproachesfor2016and2017intermsoftheir(a)explicitparameterassumptions,(b)their impliedparametersand(c)theirerrors.

Table22.1KeyParametersofJust’sNNEGValuation:2016vs2017 2016 2017Explicitparam ℎ𝑝𝑖 = 4.25% ℎ𝑝𝑖 = 4.25%

𝑞 = 0.5%Implicitparam 𝑞 = −2.75% ℎ𝑝𝑖 = −1.7%Error 𝑞 ≪ 0 𝑟 = −1.2%

Inshort,thefirmhasahighlyoriginalapproachtoitsNNEGmodellingthatdefeatsoureffortstomakesenseofit.

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ChapterTwenty-Three:ActuarialStandardsTheIFoAhaveafairamountofmaterialontheirwebsiteaboutactuarialstandardsandregulation.Herearesomequotesfromtheirwebsite(buttheitalicsareours):https://www.actuaries.org.uk/about-us“UnderourRoyalCharterwehaveadutytoputthepublicinterestfirst”https://www.actuaries.org.uk/about-us/our-brand“Our vision is for the Institute and Faculty of Actuaries (IFoA) to serve the public byensuringthatwherethereisuncertaintyof futureoutcomes,actuariesaretrustedandsoughtafterfortheirvaluedanalysisandauthority”“Integrity

• Weare:Doingtherightthing fortheorganisation,ourmembers,theprofessionandthepublicinterest

• Bybeing:o Honesto Accountable,ando Professional.”

https://www.actuaries.org.uk/upholding-standards/standards-and-guidance/actuaries-code“TheActuaries’CodeistheethicalCodeofConductthatallMembersoftheIFoAmustadhereto.Itcameintoforceon18May2019.TheCodehassixprinciples.1. Integrity–Membersmustacthonestlyandwithintegrity.2. Competenceandcare–Membersmustcarryoutworkcompetentlyandwithcare.3. Impartiality– Members must ensure that their professional judgement is not

compromised,andcannotreasonablybeseentobecompromised,bybias,conflictofinterest,ortheundueinfluenceofothers.

4. Compliance– Members must comply with all relevant legal, regulatory andprofessionalrequirements.

5. Speakingup-Membersshouldspeakupiftheybelieve,orhavereasonablecausetobelieve,thatacourseofactionisunethicalorisunlawful.

6. Communication–Membersmustcommunicateappropriately.” On point 2, given the problems entailed by the DP approach (including, e.g., that itproduces not inconsiderableNNEGunder-valuations) the question iswhether/how itmeetstherequirementof“competentlyandwithcare.”Onpoint3,thereistheissueofwhetherit isappropriatetopreferoneapproachoveranotherbecauseofcommercialconsiderations.Theexamquestionisthis.ExplainhowtheDPapproach:(a)iscompliantwiththe‘dutytoput thepublic interest first;’ (b) is ‘doing the right thing;’ (c) involveswork that is

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carried out ‘competently andwith care;’ and (d) is compliantwith ‘all relevant legal,regulatoryandprofessionalrequirements.’Answerstobesentto:PuttingThingsRight,OfficeoftheChiefExecutive,InstituteandFacultyofActuaries,HolbornGate-7thFloor,326-330HighHolborn,LondonWC1V7PP.ThefulltextoftheActuaries’Codeishere.IntegrityOn the question of honesty, there is also a legal standard namely,what any ordinarypersonwouldreasonablyregardasdishonest.LordLaneCJsetoutthewellknown‘GhoshTest’ortwolimbapproachtotheissueofdishonestyinRvGhosh1982.Thetestis(i)whetheraccordingtotheordinarystandardsofreasonableandhonestpeoplewhatwasdonewasdishonest,and(ii)Ifso,didthedefendantrealisethatwhatwasdonewasbythosestandardsdishonest.JusticeCookeappliedthetestinthewell-knowncaseagainstTomHayes,thetraderwhowas convicted for the manipulation of LIBOR.105 Cooke ruled that the standard fordishonestyisabsolute,andcannotchangebyreferencetomarketstandardsormarketethos, standard practice in an industry or any common understanding amongstemployees.Thereisnoauthorityforthepropositionthatobjectivestandardsofhonestyaretobesetbyamarket.Quitethecontrary:

Thehistoryofthemarketshaveshownthat,fromtimetotime,marketsadoptpatternsof behaviourwhich aredishonestby the standardsof honest andreasonablepeople;insuchcases,themarkethassimplyabandonedordinarystandardsofhonesty.Eachofthemembersofthiscourthasseensuchcasesand the damage caused when a market determines its own standards ofhonesty in this way. Therefore to depart from the view that standards ofhonestyaredeterminedbythestandardsofordinaryreasonableandhonestpeople is not only unsupported by authority, but would undermine themaintenanceofordinarystandardsofhonestyandintegritythatareessentialtotheconductofbusinessandmarkets.

Impartiality“3.Membersmustensurethattheirprofessionaljudgementisnotcompromised,andcannotreasonablybeseentobecompromised,bybias,conflictofinterest,ortheundueinfluenceofothers.

105https://www.judiciary.gov.uk/wp-content/uploads/2015/12/r_-v_tom_alexander_william_hayes_redacted_approved.pdf

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3.1Membersmusttakereasonablestepstoensurethattheyareawareofanyrelevantintereststhatmightcreateaconflict.3.2Membersmustnotactwherethereisanunreconciledconflictofinterest.”Onpoint3,thereisaninterestingquoteintheindustrymanualonNNEGvaluation,Hostyetalia(2007):“ForprovidersattemptingtopricetheNNEGonamarketconsistentbasisthere is insufficient productmargin in order to provide a competitive product…” (p.30.).Section7.3.3ofthatmanualalsoexplainsthatunderamarketconsistentapproachtheproductwouldnotbeprofitable,whilstthediscountedprojection(akarealworld)model“hasproducedasignificantlylowercost”andisthereforetobepreferred. Speakingup“5.Members should speakup if theybelieve,orhavereasonablecause tobelieve, thatacourseofactionisunethicalorisunlawful.5.1 Members should challenge others on their non-compliance with relevant legal,regulatoryandprofessionalrequirements.5.2MembersmustreporttotheInstituteandFacultyofActuaries,assoonasreasonablypossible, anymatterwhich appears to constituteMisconduct for the purposes of theDisciplinary and Capacity for Membership Schemes of the Institute and Faculty ofActuaries and/or a material breach of any relevant legal, regulatory or professionalrequirementsbyoneofitsMembers.5.3Inadditiontocomplyingwithanylegalrequirementstoreportmatterstorelevantregulators or other authorities, Members should also report to those bodies anybehaviourthattheyhavereasonablecausetobelieveisunethicalorunlawful,andcarriessignificantriskofmateriallyaffectingoutcomes.5.4Membersmusttakereasonablestepstoensureusersareawareofanysubstantialissueswithapieceofwork forwhich theyareresponsibleor inwhich theyhavehadsignificantinvolvement,ifthoseissuesmightreasonablyinfluencethedecision-makingorjudgementofusers.”Actuarial standards are summarised in the following fourdocuments, fromwhichwereproducekeypassages.ThefirstisAPSX1:

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APSX1:ApplyingStandardstoActuarialWork106“8.2.Membersmustbeabletojustifythestandardsapplied(and/ornotapplied)totheirActuarialWork,ifreasonablycalledupontodoso.9.1.AfailuretocomplywiththisAPSmayresultinafindingofmisconductintermsoftheIFoA’sDisciplinaryScheme.”APSX2:ReviewofActuarialWork107ThesecondisAPSX2,whichappliestothereviewofactuarialwork,andwhichisrelevanttotheIFoA/ABIworkingparty’sworkonNNEGvaluation:“1.3.In considering for the purposes of paragraphs 1.1 and 1.2whether and towhatextentWorkReviewshouldbeappliedtoapieceofwork(includingwhetherandtowhatextentWorkReviewshouldbeintheformofIndependentPeerReview),Membersshouldhaveregardtoalloftherelevantcircumstances,includingthefollowing:1.3.1.thedegreeofdifficultyofthepieceofworkanditscomplexity1.3.2.thesignificanceofthepieceofwork,includinganyfinancial,reputationalorotherconsequencesfortheperson(s)forwhomtheworkisproduced”Thewords“reputationalorotherconsequences”jumpoutabit.“1.3.3.whetherthecircumstancesof thepieceofworkmakeitmore likelythaterrorscouldbemade”Yes,itispossiblethatanapproachthatcanproduceimpossiblevaluationscouldproduceerrors.“1.3.4.thereasonableexpectationsoftheperson(s)forwhomtheworkisproduced;1.3.5.theextenttowhichjudgementand/oranalysisisrequired1.3.6.theapplicationofotherqualityassurancecontrolstothepieceofwork;”Wehavetriedandtriedtoelicitinformation–hardinformation,asopposedtoboilerplatewaffle–aboutthequalityassuranceprocessesusedbytheIFoA/ABIworkingparty’sonNNEGvaluation,butno-oneinanypositionofresponsibilitywilltakeresponsibility,eveninprivate,letaloneinpublic.Whatwastheindependentscrutinyprocess,whoweretheseniorfiguresintheIFoAorActuarialResearchCouncilwhosignedoff,etc?Soasregards

106https://www.actuaries.org.uk/documents/aps-x1-applying-standards-actuarial-work107https://www.actuaries.org.uk/documents/aps-x2-review-actuarial-work

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thequalityassuranceinthiscase,wecan’tworkoutwhatitwasandno-oneintheknowwilltellus.“1.3.7.thedesirabilityofassuringpublicconfidenceinthequalityoftheworkinquestion.”Thequestionishowfailingtoanswerconcretequestionsaboutthequalityassuranceprocesshelpstoassurepublicconfidenceinthequalityoftheworkinquestion.Technical Actuarial Standard 100: Principles for Technical Actuarial Work,FinancialReportingCouncilDecember2016.108“TechnicalActuarialStandard100:Principles forTechnicalActuarialWork (TAS100)promoteshighqualitytechnicalactuarialwork.ItsupportstheReliabilityObjectivethatusersforwhomactuarialinformationiscreatedshouldbeabletoplaceahighdegreeofrelianceonthat information’srelevance,transparencyofassumptions,completenessandcomprehensibility, including the communication of any uncertainty inherent in theinformation.”Thequestionishowuserscanplace“ahighdegreeofreliance”onvaluationsproducedbyanapproachthatcanproduceimpossiblevaluations.“1.Judgementshallbeexercisedinareasonedandjustifiablemanner;materialjudgementsshall be communicated to users so that they are able to make informed decisionsunderstandingthemattersrelevanttotheactuarialinformation.”Thetaskistoexplainwhatis“reasonedandjustifiable”abouttheDPapproachandtoexplaininwhatsensearedecisionsbasedonanapproachthatcanproduceimpossiblevaluationstobeconsidered“informed,”e.g.,asopposedtotheopposite.“2.Datausedintechnicalactuarialworkshallbeappropriateforthepurposeofthatworksothatuserscanrelyontheresultingactuarialinformation.”“appropriate”,“rely”…“2.1Datashallberelevantforthepurposeofthetechnicalactuarialwork.”Actuariesareusingassumptionsaboutℎ𝑝𝑖topricetheforwardintheirNNEGvaluationmodels,butwehaveexplainedearlier thatℎ𝑝𝑖 is irrelevant.109 So inwhatsense isanirrelevantvariablerelevant?

108https://www.frc.org.uk/getattachment/b8d05ac7-2953-4248-90ae-685f9bcd95bd/TAS-100-Principles-for-Technical-Actuarial-Work-Dec-2016.pdf 109 Seealso,e.g.,PRASS3/17(p13,para3.17)whichstates:“Itisimportanttonotethatviewsonfuturepropertygrowthplaynoroleinpreferringonecontractovertheother.Investorsinbothcontractswillreceivethebenefitoffuturepropertygrowth(orsufferanypropertydepreciation)becausetheywillownthe property at the end of the deferment period. Hence expectations of future property growth areirrelevant...”(ouremphasis)

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“3.Assumptionsused,orproposedforuse,intechnicalactuarialworkshallbeappropriateforthepurposeofthatworksothatuserscanrelyontheresultingactuarialinformation.”Howcanitbeappropriatetouseanincorrectapproachthatdependsonanirrelevantvariable?Andhowareresultsbasedonaninappropriateassumptionaboutanirrelevantvariablereliableforusers?“Technical Actuarial Standard 200: Insurance,” Financial Reporting CouncilDecember2016110ThefourthisTAS200,whichappliestoinsurance.“8.Measures,assumptionsandjudgementsusedtoderiveanyestimatesdescribedas“bestestimate”, “central estimate” or other similar terms shall be neither optimistic norpessimisticandshallnotcontainadjustmentstoreflectadesiredoutcome.”HeavenforbidthatanyonemightwanttopushdefermentratesdowntoobtainlowerreportedNNEGS.Honisoitquimalypense. Finally,someadviceonwhattheIFoAmightdo:

Ifwemakemistakeswewanttoputthingsright.Bymonitoringanyconcernsraised, including any formal complaints, and by taking prompt correctiveactionwherenecessary,weseektolearnfromwherethingshavegonewrongandimprovethestandardofourserviceforfutureusers.111

ThatlastquotecomesfromtheIFoA’sdocument“Puttingthingsright.”

110 https://www.frc.org.uk/getattachment/c866b1f4-688d-4d0a-9527-64cb8b1e8624/TAS-200-Insurance-Dec-2016.pdf111https://www.actuaries.org.uk/documents/putting-things-right.

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ChapterTwenty-Four:AccountingStandardsThischaptersetsouttheaccountingbasicsastheyapplytoequityreleasevaluation.FairValueForUK-recognisedinsurancefirms,thevaluationoftheregulatorybalancesheetissetoutintheSolvencyIIregulationsastransposedintoUKlaw.Thisprocessisstrictlyrulesbased. The valuation of the statutory balance sheet, by contrast, is governed byaccountingstandardssuchasIFRS(‘InternationalFinancialReportingStandards’)whichtend to bemore principles-based. Undermodern accounting standards such as IFRS,valuationsmustbebasedontheprincipleof‘fairvalue’.IFRSdefinesa‘fairvalue’priceas:

Thepricethatwouldbereceivedtosellanassetorpaidtotransferaliabilityinanorderlytransactionbetweenmarketparticipantsatthemeasurementdate.112

IFRSdoesnotdefine“fair,”buttheassumptionisthatamarketparticipant,i.e.someonewhoisindependent,knowledgeable,ableandwillingtoenterintothetransaction,wouldnot be duped into anunfair transaction. So currentmarket pricesmust deemed fair,becauseamarketparticipantwouldnotbedupedintobuyingatgreaterthanthemarketprice, or be duped into selling at less than themarket price. Consequently fair valueequalsmarketprice,wherethemarketpriceexists.Butwhatisfairvalueifthemarketpricedoesnotexist?TheanswercomesfromtheLevel1/Level2/Level3fairvaluehierarchy.Level1fairvalueisthemarketprice,wherethemarketpriceexists.WherenoLevel1fairvaluesexist,i.e.wheretherearenomarketprices,IFRSusesLevel2fairvalues:thesearethepricesofrelatedinstrumentsthatcanbeusedasproxiesforunobservable values. An example in the equity release context would be the use ofleaseholdandfreeholdmarketpricesasproxiesforthevaluesofthenotional“leasehold”grantedtotheequityreleaseborrowerwhenanERMistakenout. WherenoLevel2pricesareavailable,IFRSusesLevel3ormark-to-modelfairvalues,i.e.,Level3 involves theuseof amodel toobtain fair values.However, themodel and itscalibrations should still reflect “the assumptions that market participants would usewhen pricing the asset or liability, including assumptions about risk.” In the equityreleasecontext,thenaturalexampleisaNNEGmodel.Suchamodel,whichisbydefinitionmark tomodel,would require under Level 3 to be calibrated using assumptions thatmarket participantswouldmake. One such assumptionwould be Principle II that no

112See,e.g.,https://www.iasplus.com/en/standards/ifrs/ifrs13.

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valueoftheERMcanexceedthevalueofforwardcontract(see(19.1)above);anotherisPrincipleIII, thatthedefermenthousevaluemustbe lessthanthecurrentspothousevalue,reflectingthepoint thatamarketparticipantwouldwantcompensation for theincomeorusethatwaslostthroughdeferment.Whateverlevelisused,theunderlyingprincipleisalwaysthesame.ToquoteFinancialReportingStandard102:

2.2Theobjectiveoffinancialstatementsistoprovideinformationaboutthefinancialposition,performanceandcashflowsofanentitythatisusefulforeconomicdecision-makingbyabroadrangeofuserswhoarenotinapositiontodemandreportstailoredtomeettheirparticularinformationneeds.113

Thisinformationshouldenableuserstotakeaneutralandobjectiveviewofthecompanyandensurethattheyarenotbeingcheated.Qualitative of this information include understandability, substance over form,completeness,comparabilityandtimeliness,aswellas(quotingFRS102):

• Relevance:2.5Theinformationprovidedinfinancialstatementsmustberelevanttothedecision-makingneedsofusers. Informationhasthequalityofrelevancewhenitiscapableofinfluencingtheeconomicdecisionsofusersbyhelpingthemevaluate past, present or future events or confirming, or correcting, their pastevaluations.

• Materiality: 2.6 Information is material—and therefore has relevance—if itsomissionormisstatement,individuallyorcollectively,couldinfluencetheeconomicdecisionsofuserstakenonthebasisofthefinancialstatements.

• Reliability:2.7Theinformationprovidedinfinancialstatementsmustbereliable.Informationisreliablewhenitisfreefrommaterialerrorandbiasandrepresentsfaithfullythatwhichiteitherpurportstorepresentorcouldreasonablybeexpectedtorepresent.Financialstatementsarenotfreefrombias(ienotneutral)if,bytheselection or presentation of information, they are intended to influence themakingofadecisionorjudgementinordertoachieveapredeterminedresultoroutcome.

• Prudence: 2.9 The uncertainties that inevitably surround many events andcircumstancesareacknowledgedbythedisclosureoftheirnatureandextentandby the exercise of prudence in the preparation of the financial statements.Prudenceistheinclusionofadegreeofcautionintheexerciseofthejudgementsneeded inmaking theestimatesrequiredunderconditionsofuncertainty,suchthat assets or income are not overstated and liabilities or expenses are notunderstated. However, the exercise of prudence does not allow the deliberateunderstatementofassetsorincome,orthedeliberateoverstatementofliabilitiesorexpenses.Inshort,prudencedoesnotpermitbias.

113FinancialReportingStandard102TheFinancialReportingStandardapplicableintheUKandRepublicofIreland.”FinancialReportingCouncil,September2015.https://www.frc.org.uk/getattachment/e1d6b167-6cdb-4550-bde3-f94484226fbd/FRS-102-WEB-Ready-2015.pdf

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Relevant,reliable,freefrommaterialerrorandbias,assetsnotoverstated,liabilitiesnotunderstated,…TheaccountantisthenhiredbythemanagementofthecompanytodrawupaccountsonthebasisoftheseprinciplesandinaccordancewithIFRSrulesandexistinglaw(e.g.,theCompaniesAct).Theseaccountswouldbeapprovedbythedirectors,whoaredeemedtohavepreparedtheaccountsandthenpresentedtotheauditorsforsign-off.“Anauditoris an independently qualified person who is appointed to give shareholders anindependent,professionalandinformedopiniononthefinancialstatementspreparedbythedirectors”114andthe

auditor’s objectives are to obtain reasonable assurance aboutwhether thefinancialstatementsasawholearefreefrommaterialmisstatement,whetherdue to fraud or error, and to issue an auditor’s report that includes theauditor’s opinion. Reasonable assurance is a high level of assurance ….Misstatements canarise from fraudorerrorandare consideredmaterial if,individuallyorintheaggregate,theycouldreasonablybeexpectedtoinfluencethe economic decisions of users taken on the basis of these financialstatements.115(Ouremphasis)

SufficetonotethatitisfarfromobviousthattheDPapproachisconsistentwiththesestandardseither.ImprovingonFairValue?Thereareanumberofcommonobjectionstofairvalue.Thefirstrelatestotheissueofwhetherpeopleshouldseekto‘improve’onthefairvalue/marketvalueprice.Theshortansweris“no.”Forexample,actuariessometimesclaimthatcurrentmarketvaluesshouldbeignoredbecausetheyarecurrentlytoolowortoohigh,relativetotheactuary’sjudgementofwhatthe“long-term”priceshouldbe.AsDavidWilkieonceputit,"Theactuaryis…sayingthatthemarkethastemporarilygotitwrong,butthat,induecourse,itwillgetitright.”116Sothe actuary is suggesting that themarket price should be replaced by a non-marketvaluationbasedonactuarialjudgementor,ifyouwanttoputitthatway,byanimplied(notevenexplicit!)“actuarialforecast.”Objection#1:Atleastiftheforecastwereexplicitwecouldscrutinisethemethodologyonwhichitisbasedandcometoaninformedviewofitsmerits.However,allwehaveto

114 http://www.corplaw.ie/blog/bid/337442/The-Responsibility-Of-Auditors.Deanlinkdoesn’twork. 115 FRC“DescriptionoftheAuditor’sResponsibilitiesfortheAuditoftheFinancialStatementsApplicablefor Audits of Financial Statements for Periods Commencing on or after 17 June 2016.”https://www.frc.org.uk/auditors/audit-assurance/auditor-s-responsibilities-for-the-audit-of-the-fi/description-of-the-auditor%E2%80%99s-responsibilities-for116DavidWilkieonp.549oftheDiscussionofA.C.L.DysonandC.J.Exley,(1995)“PensionFundValuationandInvestment.”BritishActuarialJournal1(5):965-977.

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go on here is a nebulous ‘actuarial judgement’. Basing a critique of market pricevaluationsonan“Iknowit’swrong”gutfeelingiswrongonprinciple.117 Objection#2: It isdoubtful thatareliable forecastexists.Wecanpredicteclipses, thereactionofhydrogenandoxygentoaflame,theaccelerationduetogravityandsoforth,butsciencehasn’tfoundawaytopredictthepathofmarketprices.Theproblemisthatthemarketpriceofanassetitselfinvolvesaforecast,bythemarket,offuturecashflows,sointryingtopredictwherethemarketwillbeinayear’stime,wearetryingtoforecastaforecast.Insteadoftryingtopredicttheresultofthenextelection,itisliketryingtopredictwhattheTimeswillpredictittobe.Goodluckonthat.Also, either themarket price is the best forecast, or it is not. If the former,we can’timproveonit.Ifthelatter,wehavetoforecastwhatthebadforecastwillbeinayear’stime.Butwhichbadforecastdowechooseandhowweselectit?Togototheheartofthematter,wecanbeprettyconfidentthatthemarketpricewillchangeall the time,but theproblem is thatwedon’tknowhow themarketpricewillchangefromoneperiodtothenext.Themarketvaluationmightnotbeverygood,butit’sthebestwehave. Objection#3:Evenifwehadperfectforesight,suchasGodmighthave,wewouldstillhavenoleavetomarkthevalueofanassettoanythingotherthanthecurrentmarketprice.Itmaybethatthemarketisinsomesense‘wrong’.Onemightarguethatthemarketpricemustbe‘wrong’mostofthetime,becauseitischangingallthetime.Evenifwegrantthatargument,ifwemarkanassetonafirm’sbooksathigherthanthemarketonthegroundsthatwehaveperfectforesight,orbetterjudgmentthanthemarket,thenwearedefraudingprospectiveshareholdersof the firm,because theywouldpaymore forsharesthantheywouldhavepaidhadwemarkedthesharestomarket.Ifwemarkthevalueatlowerthanthemarketprice,becauseourflawlessjudgmentvaluesitatless,thenwearedefraudingexistingshareholders,because their shareswouldbevaluedat lessthan they would have been had we marked the shares to market. If God were anaccountant,Hewouldnotvalueanassetdifferentlyfromitsmarketvalue,despitebeingomniscient, for God is also Perfectly Good, and so would not get involved in falseaccounting.Wellclearly,ifevenGodwouldnotsuperimposeHisJudgementoverthatofthemarket,thenthereisn’tmuchofacaseforanyoneelsetosuperimposehisorherjudgementoverthatofthemarketeither.

117 It lacks objectivity and is scientifically indefensible: “we claim to know the answer but cannot tellanyoneelsehowtoderiveitinadvance”asTimGordoneloquentlyputit.SeeT.Gordon(1999)“ThePriceofActuarialValues.”Paperpresented to the Staple Inn Actuarial Society (16 February).

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TheIlliquidityPremiumFallacyTheargumentisoftenmadethatifafirmholdsanilliquidasset,abond,say,andintendstoholdthatassettomaturity,thenthefirmisentitledtomarkuptheassetto‘capture’theilliquiditypremium.118Oneproblemwiththisargumentisthatallweknowinpracticeisthattheassethasaspread over the risk-free, but whether that spread is a risk spread or a illiquiditypremium,orsomuchofoneandsomuchoftheother,wedonotknow.Weonlyobservethespread,nottheilliquiditypremium,andanyclaimwemightmakeaboutthesizeoftheilliquiditypremiumismerelyahypothesis.Acounterargumentwehavesometimesencounteredisthat illiquiditypremiumshavebeenreliablyestimatedinapaperbyWebberandChurmpublishedinthe2007Q4Bankof EnglandQuarterly Bulletin.119 This paper presents a chart purporting to show thatmodelledcorporatebondspreadsareabout50%oftheobservablespread,fromwhichtheyinferthattheresidualis‘illiquidityrisk’thatcouldbecapturedbyholdingtheassettomaturity.However,wehavedoneourown reconstructionandwe found that afterusing adifferent leverageparameter, almost all thedifferencebetweenmodelled andobservedspreaddisappears.120Soitappearsthattheyhadmisspecifiedtheequationandthenmisidentifiedtheresidualfromtheirmisspecifiedequationas‘liquidityrisk’.Butforthesakeofargument,let’streattheilliquiditypremiumasifWebberandChurmhadcorrectlycaptureditandlet’sevengranttheargumentthatthefirmisjustifiedinmarking up the asset to capture that premium. Therefore, when a prospectiveshareholder comes topurchase the shares,hepays thepremium.However,when thesameindividualbuysthesameassetinthemarket,hewillpayless,becausethemarketpriceoftheassetismarkeddownbythesamepremium.Sincethemarketpriceisthefairvalueoftheasset,heispayingtoomuchwhenhebuystheshareandisthereforebeingdefrauded.Anexampleof this ‘illiquiditypremiumargument’ is tobe found in remarksmadebyAndrewRendellatthe28FebruaryStapleInneventfortheTunarureport:

…ifyouhaveacorporatebond,istheeconomicworthtotheinsurerthesameasitistoeverybodyelse,arguablyitisn’t,andthereasonforthatbeingthatatypicalmarketparticipantwillputadiscounttothepricethattheywouldbepreparedtopayforit,becausethatcorporatebondhasrisksaroundliquidity,andithasrisksaroundpricevolatilityoverthedurationoftheasset.

118Notethatthisnotionofanilliquiditypremiumisquitedifferentfromtheonefamiliartoeconomists.Theeconomicnotionofanilliquiditypremiumrefersthepricedifferencebetweentwootherwisesimilarassets,whereoneassetisliquid(i.e.,easytosell)andtheotherisnotliquidormayprovetobedifficulttosellinthefuture.119L.WebberandR.Churm(2007)“DecomposingCorporateBondSpreads.”BankofEnglandQuarterlyBulletin2007(Q4):533-541.120SeeD.Buckner(2019)“FromthePostbag–MertonModelPartII.”TheEumaeusProject(19June).

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Theinsurersays,“WellIdon’tcareaboutthat,becauseI’mgoingtoholdmyassettomaturity,andthereforeIdon’tneedthatdiscount,sothecorporatebondisworthmoretomethanitistoatypicalparticipant.”…SothequestionthenishowdoesthatmapthroughtotheERM,inparticularthepropertysideofERMs,so…what’stheeconomicworthofthatproperty?…youarenotgoingtoseeanycashoutofthatassetforsometime,youcan’tsellit,sotomanyinvestorsthatwouldbequiteasignificantdisadvantage,butmaybelesssotoaninsurerthathaslongtermliabilitiesanditcanjustwaitforthatvaluetoemerge.121

Thereare threeproblemshere. [1]Mr.Rendell talksabout theeconomicworthto theinsurer,butwhoistheinsurer?Theinsurerisn’taperson.Theinsurerisacompanywhichhasshareholders,soweshouldbeaskingabouttheeconomicworthtotheshareholders.Sowhatistheeconomicvaluetotheshareholdersofanassetwhereyouhaveto“waitforthatvaluetoemerge”asheputsit?Well,theeconomicvaluetoshareholdersisjustthesameasiftheyweren’tshareholdersatall,butwerebuyingintheopenmarket.Ifthemanagement of the insurance company decides that a prospective shareholdermustpaynowforthevaluethatwillemergelater,sothattheycangetawhackinggreatbonus,then they have defrauded the prospective shareholder,whomust paymore than themarketprice.[2]Asfor“waitingforthevaluetoemerge”,themanagement(orexistingshareholders)haven’twaitedatall.Theyhavecrystallised thevaluenowandtherebyrobbedfutureshareholdersoftherewardstheywerewaitingpatientlyfor.Thosewhobelieve in “waiting for the value to emerge,” shouldwait for the value to emerge. [3]Finally,itisthecurrentorprospectiveshareholderorinvestor,notjustthecompany,towhomtheaccountanthasafiduciarydutytoreportafairvalue.Ifwestillhaven’tpersuadedyou,considerthefollowingargument.Webuytherighttopossessionofapropertywhoseestimatedvacantvalueis£1m,i.e.afreeholdencumberedbyaleasehold.Theleaseisfor99years.Basedonavailabledata(e.g.fromtheLandTribunal)thevalueofthedefermentwouldbeabout£30,000.Wesell thesamerighttoyoufor£1mbecauseyouthinkthere isamassive illiquiditypremiumthatcanbemarkedupnow.Wehavethenmadeanimmediatemassiveprofit(£970k)fromthisdeal.Wehavejustearnedanilliquiditypremiumthatwouldhaveotherwisehavetakenus99yearstoearn,and you have paid away the same thing! You will earn nothing from the illiquiditypremium,whichhasallgonetous.Whyshouldyoupaythepremiumtosomeoneelse?Youmay argue that it is differentwith an insurance company, because an insurancecompanyisthereforthelongtermandhasthepatiencetowait99years.Maybeso,butwhywould the insurance companymark upnow a illiquidity premium thatmust beearnedover99years?Youcanonlyobtaintheilliquiditypremiumafter99years,andifyou sell the asset, you only get the currentmarket pricewhich does not include any

121https://youtu.be/DdLRqcIvR20?t=3794

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illiquiditypremium.Butaboveall,whywouldanyonepay£1mwhentheycouldpay£30kinsteadforthesameassetinthemarket?The fallacy is not in asserting that there may be an illiquidity premium that can be‘captured’byholdingtheassettomaturity.Theexistenceandextentofanysuchpremiumisanempiricalmatterand(astheWebber-Churmexampleshows)isdifficulttopindown.Instead,thefallacyliesinthebeliefthatanilliquiditypremiumtobeearnedinthefutureshouldbemarkedupnow,abovethecurrentlyprevailingmarketpriceoftheasset.Orconsider this finalargument.Suppose thereexistsabond typeasset thatoffersanilliquiditypremium.Solet’ssetupacompanywhereweborrowlongdatedliabilitiesatriskfreeandinvesttheproceedsintheselong-datedilliquidassetswiththeir(certain)illiquiditypremium.Persuadeshareholders/PRAetcthatanilliquiditypremiumexiststhat‘can’tbearbitragedout’.Createapileofequitybydiscountingliabilitiesatriskfree+premium,payyourselfalotofdividendsorsellthecompany,andretiretothebeach.Congratulations!Youhavejustarbitragedouttheilliquiditypremium,whichnolongerexists.122 TheBuyn’HoldFallacyArelated(andcommon)argumentisthataninsurercanmarkuphigheryieldingassetsbecausetheinsureris(a)goingtoholdtheassetstomaturity,so(b)willnotbeexposedtochangesinmarketvalue,and(c)willrealisetheadditionalreturn.Thisargumentmightseemplausiblebutquicklyunravelsonexamination:First,howdoweknowthattheinsurerwillholdtheassetstomaturity?Wedon’t.Itmayintendtoholdthemtomaturity,butitcouldbeforcedtoselllater.Whoknows?Second,evenifwegrantthattheassetwillbeheldtomaturity,itis(usually)falsetoclaimthattheholdingentityisnotexposedtochangesinthemarketvalueoftheasset.Iftheassetisabond,thebondmightdefaultbeforematurity.Ifitisanequity,thematurity(oftheannuity)maycoincidewithadownturninthemarket,andifitisanERM,maturitycouldbeintheaftermathofaJapanstylehousingdecline.Theonlyassetsthatarenotsubject to riskof loss (innominal terms)wouldbegilts,butgiltsdon’toffera returnhigherthanrisk-free.Thustheargumentisinternallycontradictory:ifanassetoffersahigherreturnthanrisk-free,thenthatreturnmustberisky,sothereisriskoflossatmaturity;butifthedoesnotofferhigherthanrisk-free,becauseitisrisk-free,thenthereisnohigherreturntorealise.Soyoucanhave(b)oryoucanhave(c),butyoucan’thaveboth(b)and(c).

122SeealsoK.Dowd“JustOneMoreThing,”TheEumaeusProject4July2019.

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MarioWüthrich:AnIlliquidityPremiumisInconsistentwithMarketConsistencyProfessor Mario Wüthrich from ETHZ offers a trenchant criticism of the IlliquidityPremium argument in a 2011 article in the founding issue of theEuropean ActuarialJournal.123Hismessageisthatanilliquidityspreadisinconsistentwithmarketconsistentvaluation.Heoffersanumberofarguments:

[1]Valuationofinsuranceliabilitycashflowswithilliquidbondscontradictsmarket-consistent actuarial valuation because it corresponds to a hold-to-maturityview,butmarket-consistentvaluesarebasedeitheronatransferorproductioncostview(withtheappropriaterewardforriskbearing).Onemaydebate…theconceptofmarket-consistentvaluationinaccounting,butoncethisframeworkisgivenwearenotallowedtomeasuresomeinstrumentsbytransfervaluesandothersinahold-to-maturityview,becausethiswillleadto a rather inconsistent full balance sheet valuation that may allow foraccountingarbitrage.(p.95)

He is correct: market consistent valuation is not consistent with two inconsistentvaluationapproaches.

[2]Theargumentthat insurancecashflowsarepredictablecontradictsthevery notion of insurability. However, we could measure the degree ofpredictabilitychoosinganappropriateriskmeasure,butthenthis,ofcourse,alwaysdependsonthechoiceof theriskmeasure.Sincewehave infinitelymanyacceptableriskmeasurechoicesweareagaininanincompletemarketsetting which requires more information about preferences of financialagents.(loc.cit.)

Thefirstsentenceissomewhatparadoxical,butthesecondpointaboutneedingmoreinformationinanincompletemarketssettingisindubitablycorrect.

[3]Thereisnoclearconcept[about]howtheilliquidityspreadismeasuredanddistinguishedfromthecreditspread.Therefore,wecanneitherpricethissinglecomponent inamarket-consistentway,norcanwereplicate it inanappropriateway.(loc.cit.)

Absolutely correct. If there is no reliable methodology to estimate and separate theilliquidity spread from the credit spread, then one has no reliable estimates of theilliquidityspreadtoworkwith.

[4]Oftenthedegreeofilliquidityiscomparedtoadegreeofpredictabilityofinsuranceliabilitycashflows.Wearenotawareofatheorythatwouldallowforameaningfulanalysis[ofthisissue].(loc.cit.)

Neitherarewe.

123M.V.Wüthrich(2011)“Anacademicviewontheilliquiditypremiumandmarket-consistentvaluationininsurance.”EuropeanActuarialJournal1(1):93-105.

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[5]Thegeneralaiminmarket-consistentvaluationistoreplicateinsuranceliabilities with financial instruments that have reliable market prices.Therefore,we cannot choose illiquid instruments because their values areerratic and hence the market-consistent value of the insurance liabilitiesbecomesarbitrary.(loc.cit.)

Thekeyphrasehereis“themarket-consistentvalue…becomesarbitrary.”Lastbutnotleast:

[6]Theexistenceoftheilliquidityspreadcanbeobservedonthemarketandcanaswellbeincorporatedinatheoreticalmodel.However,underthecurrentaccountingandsolvencyrules(market-consistentone-yearview)itcanonlybeappliedontheassetsideofthebalancesheet(withanappropriateriskmargin).If regulationdecides thatalso the liability sideof thebalance sheet canbediscounted with the illiquidity spread then one needs to change theaccounting and solvency rules such that they allow for a hold-to-maturityview(withallpossibleconsequences),i.e.webelievethatitisnotthetaskoftheactuary to introduce fancyarguments for the illiquiditydiscounting thatcircumventsthecurrentregulation.(p.95,ouremphasis)

Amentothat.PrudentialRegulationandFairValueAnother class of objections to fair value is that fair value does not apply becauseprudentialregulatorshavedifferentobjectivesorareworkingtodifferentstandards.Asan example, consider the following letter of 19 March 2019 that we wrote to HansHoogervorst,thechairoftheInternationalAccountingStandardsBoards(IASB):“DearMrHoogervorst,Oneofus(Buckner)wrotetoyouon9January2019raisingtheconcernthatsomelifeinsurance firms are valuing embedded guarantees in a way radically different fromacceptedfinancialtheory,anddifferent(inourview)fromthewaytheywouldandshouldbevaluedunderbroadIFRSprinciples,i.e.howamarketparticipantwouldvaluethem.TheIASBresponseon4March2019tothisconcernwasapuzzlingone.Thereplyclaimsthat ‘differing valuations for prudential purposes are largely driven by the differentobjectivesofthetwomeasurementbases’.

Prudential regulators focus onmeasures of regulatory capital that absorblosses, whereas accounting standard setters are concernedwith “financialinformationaboutthereportingentitythatisusefultoexistingandpotentialinvestors, lendersandothercreditors inmakingdecisionsaboutprovidingresourcestotheentity”(IFRSConceptualFrameworkOB2).Thosedifferent

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objectives and approaches to measurement will sometimes give rise todifferentvaluationsforthesameinstrument.

Wehaveworkedformorethan40yearsbetweenusintheareasofcapitalmeasurementandcapitalmanagementandtheviewthattherearedifferingvaluationsforprudentialpurposes is news to us. To be sure, prudential regulators are concerned with theadequacyofanygivenamountofcapital.Ifoneinsurerhas£10bnoflowqualityassets,another insurer has the same amount of high quality assets, and both have £9bn ofpension liabilities at the same duration, then both have the same amount of capital,namely£1bn.124Buttheadequacyofthatamountofcapitalisadifferentmatter,andtheobjectiveoftheprudentialregulatoristoassesscapitaladequacybymeansofestablishedtechniques such as value at risk, stress testing etc. There is no equivalent of capitaladequacyorcapitalrequirementinIFRS,however.Theissueraisedintheletterof9Januarywasnotabouttheadequacyofcapital,butratherabouttheamountofcapital,andinparticular,abouthowthatamountismeasured.Inthiscasetheobjectivesofregulatorsandofaccountingstandardsettersareidentical,namelytovalueassetsandobligationsasamarketparticipantwouldvalue them,andweareastonishedthattheIASBwouldgoonrecordtoclaimanydifferent.Insurersevenprovideareconciliationofregulatoryandstatutorybalancesheetsintheirfinancialreports,fromwhich it is clear that, while there are minor differences, the purported objective ofmeasurementisthesame,i.e.arm’slengthvaluation,fairvaluemeasurementetc.…TheIASBresponsegoesontosaythataccountingstandardsettersareconcernedwith“financialinformationaboutthereportingentitythatisusefultoexistingandpotentialinvestors,lendersandothercreditorsinmakingdecisionsaboutprovidingresourcestotheentity.”Butcanyoupleaseexplainhowisitusefultomarketparticipantsiffirmsvalueassets andobligations inways that amarketparticipantwouldnot?How thenwouldinvestorsknowwhetherthereportedvalueswerefair/useful/reasonableorevenlegal,ornot?Ifvaluationsarehigherthanthosethatamarketparticipantwouldplaceonthem,thenprospectiveinvestorsarebeingunfairlydisadvantaged;ifvaluationsarelowerthanthosethatamarketparticipantwouldplaceonthem,thenexistinginvestorsarebeingunfairlydisadvantaged.Theonlyfairwaytoavoideitherpartybeingdisadvantagedistomakethesamevaluationsthatamarketparticipantwouldmake.Faithfully,etc”Wearestillawaitingareply.TheFiduciaryPrinciple Wethencomeback to the fiduciaryprinciple.Evenwheremarketpricesdonotexist,accountingprinciplessaythattheaccountantshouldvalueeconomicallysimilarassetsinthesamewayandimplythatvaluationshouldreflectrationalinvestorpreferences.To

124Weleaveoutdetailssuchasregulatorycapitalincludingsubordinateddebt.Suchdetailsareirrelevantinthepresentcase.

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quote,e.g.,IFRS13B14a:“Cashflowsanddiscountratesshouldreflecttheassumptionsthatmarketparticipantswouldusewhenpricingtheassetorliability.”Anaccountantorauditor or some other person, who has an obligation of trust towards a lessknowledgeable investor,must value an asset or liability as a rational knowledgeableinvestor(ormarketparticipant,orknowledgeable,willingindependentperson)would.This principle applies regardless of whether the accountant, auditor or whoever hasprivateviewsaboutvaluationthatdifferfromfairvaluevaluations.Italsoappliesevenifheorshehassuperiorknowledgeofthefuture:evenifonehadaperfectcrystalball,oneisnotallowedtouseittoprovidevaluationsthatdifferfromfairvalueones.FalseAccountingThose who report valuations are also required to desist from false accounting, thepracticeofwhichisacriminaloffence.Tobeprecise,theoffenceoffalseaccountingisasubclassoftheoffenceoftheft125andiscreatedbysection17oftheTheftActof1968whichstates:

17.-(1)Whereapersondishonestly,withaviewtogainforhimselforanotherorwithintenttocauselosstoanother,-• destroys, defaces, conceals or falsifies any account or any record or

documentmadeorrequiredforanyaccountingpurpose;or• infurnishinginformationforanypurposeproducesormakesuseofany

account, or any such record or document as aforesaid, which to hisknowledge is or may be misleading, false or deceptive in a materialparticular;

heshall,onconvictiononindictment,beliabletoimprisonmentforatermnotexceedingsevenyears.(2)Forpurposesofthissectionapersonwhomakesorconcursinmakinginanaccountorotherdocumentanentrywhichisormaybemisleading,falseordeceptive in a material particular, or who omits or concurs in omitting amaterial particular from an account or other document, is to be treated asfalsifyingtheaccountordocument.(Ouremphasis)

Weemphasisetwopointsaboutthisdefinition.Thefirstisthattobeguiltyoftheoffenceoffalseaccountingitisnotenoughtoreportinformationthatisormightbemisleading.One must also do so dishonestly, i.e., knowing that one’s behaviour is dishonest byreasonablestandards,andonemustdosowiththeintentofpersonalgainortodeprivesomeoneelseofwhatislawfullytheirs.Theotherpointisthatthephrase“concealsorfalsifiesany account orany record or documentmade or required forany accountingpurpose…”ishighlyencompassing,andwouldincludepublicaccounts,internalbooks

125TheoffenceoftheftisdefinedunderSection1oftheTheftActof1968:“Apersonisguiltyoftheftifhedishonestlyappropriatespropertybelongingtoanotherwiththeintentionofpermanentlydeprivingtheotherofit…”Seehttp://www.legislation.gov.uk/ukpga/1968/60/pdfs/ukpga_19680060_en.pdf.

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andotherdocumentssuchas,e.g.,spreadsheetsandinternalmemos.Italsopotentiallyincludesknowinglymakingon-the-recordstatementsthatareanythingbutthefulltruth.ConsideralsothefollowingNationalFraudandCybercrimeReportingCentrestatementonfalseaccountingfraud:

False accounting fraud happens when company assets are overstated orliabilities are understated in order to make a business appear financiallystrongerthanitreallyis.False accounting fraud involves an employee or an organisation altering,destroyingordefacinganyaccount;orpresentingaccountsfromanindividualor an organisation so they don’t reflect their true value or the financialactivitiesofthatcompany.…Someexamplesoffalseaccountingfraudinclude:• anemployeemakinginflatedexpensesclaims• acustomeroranemployeefalsifyingaccountsinordertostealmoney• anemployeeusing falseaccounting tocoverup lossesbuiltup through

tradingorfraudulentactivity.…• attheextremeendofthescale,thefraudmaymeanthatacompanyhas

incurredseriousfinanciallossesand/oristradingwhileinsolvent.126

Therearealsorelatedoffencessuchasconspiracy(wheretwoormorepeopleplantoengageinanotheroffence,e.g.,falseaccounting)127andaiding,abetting,counsellingandprocuringthecommissionofanotheroffence,e.g.,falseaccounting.128

126/www.actionfraud.police.uk/a-z-of-fraud/false-accounting-fraud127http://serious-crime-solicitors.co.uk/conspiracy.php128http://www.legislation.gov.uk/ukpga/Vict/24-25/94/section/8

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ChapterTwenty-Five:RecommendationsforGoodValuationPracticeAnyreasonableapproachtoNNEGandERMvaluationmustuseamodelthat is fit forpurposeandbebasedonreasonable(i.e.,plausibleanddefensible)calibrations.TheserequirementsnarrowthefieldtosomeformofMCapproach.ThealternativestoanMCapproacharetheDPandTunaruapproaches,butneitherofthesemeettherequirementsofbeingfitforpurposeorbeingreasonablycalibrated.Wecan implementanMCapproachusingBlack ’76oraversionofBSM inwhich theunderlyingbearsadividendyield.Thenthereistheissueofcalibration.Threeoftheinputparamsarestraightforwardtocalibrate:

• TheLTVcanbeobtainedempirically(e.g.,forsomeparticularERMloanorclassofloans)butifonewishestouseadefault,thenapplythe‘ageminus30’rule.

• Therisk-freeratecanbeobtainedfromtheimpliedyieldsongiltspublishedbytheBankofEngland,butareasonablevalueincurrentinterestrateconditionsis1%.

• TheERMloanratecanbeobtainedfromlookingattheratesthatfirmsareoffering.Bear inmindhoweverthat there isconsiderablediscrepancyamong loanratesofferedbydifferentfirmsandacrossdifferentERMproducts.

Thedefermentrateandvolaremoreinvolved.Ideallythedefermentrateshouldbeestimatedfrommarketdata(asperChapter8)butifonewishestouseasingledefaultvalue,wewouldsuggest4.2%.Thedefermentrateshould be set equal to the net rental yield without any adjustments for illiquiditypremiumsorotherjiggerypokery.Weshouldkeepinmindthatanyproposeddefermentrates that arewell away fromplausiblemarketnet rental yieldsarenot credible.Forexample,defermentratesof-2.75%(usedbyacertainfirm),0%(aboundemphasisedbythe IFoAERMworkingparty)or0.5%(thecurrentminimumsetby thePRA)arenotcredible.ThevolatilitycalibrationshouldnotbebasedonreturnsobtainedfromHPIdataalone,and thevolatility shouldnotbeadjusted for autocorrelation in theHIP.Thevolatilitycalibration volatility should take account of the impact of the contributions ofachievementraterisk,interestrateriskand𝑞rateriskonthevolatility.Inprinciple,oneshouldusedifferentvolinputsineachofthedecrementputs,butonecanuseacommonvolinputprovideditisbasedonthevoltermstructureandtheappropriateexitprobs.Anysuchvolwouldbespecifictotheborrowerageandgender,sooneshouldnotusethesamesinglevolinputforallpossibleborrowertypes.UndernocircumstancesshouldpractitionersuseamodelthatusesprojectedorexpectedfuturehousepricesasaninputintheirNNEGvaluationmodels.

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ChapterTwenty-Six:RecommendationsforGovernanceandDisclosure Poorvaluationpracticesareendemicintheequityreleasesector,buttheIFoAhasdonenothingtocondemnthem.Indeed,theInstituteitselfisonrecordashavingendorsedanumberoferrorsonNNEGvaluation.TheIFoAneedstoputthissituationright,andpromptly.Werecommendthatitissueacorrectivestatementalongthefollowinglines:

TheIFoAacknowledgesandregretsthatseriouserrorshavebeenmadebyanumber of equity release actuaries in the valuation of No-Negative EquityGuarantees.TheIFoAaffirmsthemarketconsistencyprinciple:anyvalidapproachmustbemarketconsistent.

TheIFoAregardsapproachesthatarenon-marketconsistentasnotmeetingtechnical actuarial standards. In particular, it regards the so-called ‘RealWorld’ or ‘Discounted Projection’ approach as inherently fallacious andadvisesthatpractitionersshouldrefrainfromusingit.

The PRA needs to take action too. We recommend that the PRA issue a new PolicyStatementstatingthatequityreleaseposesaprudentialproblemandconfirmingthatitregardsonlyMCapproachesasbeingconsistentwithgoodactuarialpractice.ThePRAshouldalsomakeclearwhatitregardsgoodpracticetobe.Besidesmarketconsistency,good practice should cover issues such as acceptable modelling practices, plausiblecalibrationandtheimportanceofdisclosureandtransparency.Inparticular,thePRAshouldmakeclearthatgoodpracticevaluationmethodsshouldnotbeswayedbynotionsofprofitability.Iffirmscan’tmaketheprofitstheywant,orcan’tmakeprofitsatall,thentheyarefreetoexittheindustry,butthatshouldn’tbethePRA’sconcern.ThePRA’sconcernshouldbetoensurethatthevaluationsaredoneproperly.The PRA would still face the problems associated with firms’ gaming the regulatorysystem,soitneedstotakeactiontominimisefirms’scopeforgaming.Itcoulddosointhefollowingways:

• ThePRAshouldimposeamarketconsistencyrequirementonthedefermentrate,withfirmstoprovideaplausibledefenceoftheirassumptions.Withthecurrentrateofgrossrentalyields(2019around5%)wewouldfindanyratemuchlessthansay,3.5%,difficulttojustify.

• Firmsshouldoffer thePRAaplausiblemarket-baseddefenceof theirvolatilityanalysis and assumptions, demonstrating consistency with observed values ofachievement dispersion, interest rate and deferment rate volatility, andcorrelation.

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Weunderstand that thePRAalreadyrequires firms to reportNNEGandERMboundsbasedonitsPrinciplesIIandIII,andwillnotacceptanyvaluationsthatviolateanyofthesebounds.Itshouldgofurtherhoweverandrefusetoacceptresultsbasedonmodelsthatcangiveresultsthatcouldviolatethesebounds.Thirdly,wehavesomeadviceforaccountingandauditpracticebasedonourevidencetotheBrydonReviewonthequalityandeffectivenessofaudit:129

• Auditors shouldbe encouraged to ensure, as far aspossible, that shareholdershavetheinformationtheyneedtodeterminetheadequacyofafirm’scapital.

• Firmsshouldberequiredtoreportallthemajorriskfactorsthatmightimpedethequalityofcapital.Somefirmsalreadydoso,butitisdoubtfulthattheycapturethe whole truth. In our experience firms will often choose to ignore or tounderstatethemostmaterialrisks.

• Auditorsshouldbeencouragedtodeterminewhetheranymaterialrisktocapitalhasbeenomittedfromthefinancialstatements.Thistaskshouldnotbedifficult.Giventhatitiseasytoidentifybookswithexcessorunusualreturns,theauditorshouldbydefaultdeclaresuchbookshighrisk,andcheckthattheriskhasbeencommunicatedclearlyinthestatements.

• Riskshidden in thematuritystructureshouldnotbeconcealed in the financialstatements. Many firms report the structure of debt up to 5 years, with anaggregate of debt longer than 5 years. Yet the sensitivity to debt is (roughly)proportionate to the term! Our research has uncovered some staggering risksensitivitiesconcealedinthisway.

• Material risks should not be concealed in some ‘other’ category. Carillion’saccountsrepresentedanearlypaymentfacilityas‘othercreditors’,meaningthatthat was not incorporated in a debt to earnings ratio presented to lenders.130Eumaeushasreviewedanotherfirmwhichreports‘othervaluationdifferences’ofnearly£1bn,morethanhalfofitsreportedcapital!131

Asfortheindustry,goodluck,butitisessentialtogetthevaluationsright.

129 https://www.gov.uk/government/publications/the-quality-and-effectiveness-of-audit-independent-review130‘Carillion’,Business,EnergyandIndustrialStrategyandWorkandPensionsCommittees2018,p.43131Seee.g.http://eumaeus.org/wordp/index.php/2018/10/19/past-and-present-tens.