a computer approach to ordinary life assurance policy valuation - a new look at concepts. -...

7/27/2019 A Computer Approach to Ordinary Life Assurance Policy Valuation - A New Look at Concepts. - Whittaker, G. p.169

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COMPUTER PPRO CH TO ORDIN RY LIFE SSUR NC E POLICY V LU TION NEW LOOK

T CONCEPTSby

GRAHAM WHITT KER1. Introduction This paper ha s been written as an ExpositoryArticle and, as such, is not meant to be complete. Its main objectis to describe useful concepts and methods that are not covered bythe standard reading for the Ins titute s examinations. As the titleof the paper implies not all of these concepts are new. Some, in fact,are first principles which are particularly useful when performing anactuarial valuation using a computer. The title also suggests that thepaper is confined to Ordinary Life business. However, much of whatis said could also be applied to Industrial or even Non-life business.2. What are the requirements of a system for valuing OrdinaryLife insurance policies ?

2.1.An economic system.2.2.A fast system.2.3. The results should be produced in reports which are clear anduseful to management and also in a form suitable for statutoryreturns.2.4. A simple system, as far as possible.2.5. The system should be capable of providing results by differentmethods of valuation such as the net premium method, the grosspremium bonus reserve method, or by methods where the netpremium is adjusted using Zillmer-type techniques.2.6. The system should allow revaluation on different bases.2.7. The results should be as accurate as the economic constraintswill allow.2.8.A desirable feature would be that the valuation could be m ade asat any day of the year.2.9. The system should be capable of producing additional itemsnecessary for an analysis of surplus.

169

JSS 19 (3) (1971) 169-180


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170 GR H M WHITT KER3. What are the features of traditional valuation systems in theU.K.?

3.1. Policy valuation data are grouped by (for example) currency,policy class, bonus group, an integral age, an integral duration, etc.In the early days, contracts were valued individually, referencebeing made to the original papers for the necessary information.Soon, however, the large number of contracts and the need toreduce work led to the practice of valuing in groups and the use of aseparate file of valuation da ta. 13.2. ctuarial Practice of Life ssuranceby H. F. Fisher and J. Young.The use of exact ages and durations would involve prohibitivework, and various assumptions are made. 13.4. ibid.3.2. Each group is valued.3.3. The results are combined for management purposes and statu-tory returns.3.4. The process is comparatively slow.3.5. The policy valuation data are continuously updated, in theirvaluation groups using the continuous m ethod , or the valuationfile is updated, and sorted into valuation groups for the valuation. In a small office using punched-card equipment the cards can besorted into valuation groups and tabulated at each valuation bu t in asmall office using hand-written cards or in a large office this is notpracticable. Most offices therefore obtain their data by a continuousmethod 16.10.Life and Other ontingenciesby P. F. Hookerand L. H. Longley-Cook.4. What are the alternative possible features of a valuation systemusing a computer ?

4.1. A traditional grouping method could be used or the valuationcould be made policy by policythe individual valuation m ethod isfeasible once again. In the case of the individual method it is neces-sary to accumulate valuation results into management and statutorygroupings. Figure 1 illustrates this concept. The individual methodof valuation is a practical method for a large file of policies for thefollowing reasons:4.1.1.Using processors which are extremely fast a valuation can beperformed in a time comparable with the time it takes to read thevaluation file into the computer s m ain storage. A technique termedoverlap may be used to process a policy, read the next policy and


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ORDINARY LIFE ASSURANCE POLICY VALUATION 171write the valuation results for the previous policy, all at the sametime;this is possible as each operation is independent. W hether theultimate is achievable depends upon whether the processing timeis less than the read time .

Valuation R esult for a single policy

Array of accumulatorsValuationitemValuationGrouping .

Sterling, E/A, WP Sterling, E/A, N P

Sterling, W/L , WPSterling, W/L, NPDollars, E/A, WPDollars, E/A, NPDollars, W/L, WPDollars, W/L, NPMarks, E/A, WPMarks, E/A, NPMarks, W/L, WPMarks, W/L, NP

NumberofPoliciesAnnualgrossPrems

AnnualNetPrems

E/A = Endowment AssuranceW/L = Whole of Life

SumsAssured P.V.ofFutureNetPrems

P.V. ofSumsAssuredReserve

WP = With profitsNP = Without profitsFIG. 1. Simplified example of individual policy valuation results accumulation

4.1.2. Because of new technologies the cost of computer equipmentis reducing at a very fast rate; labour costs continue to rise. Econo-mies of scale can be gained also, by using larger capacity hardware.It should be noted, however, tha t systems should be flexible andeasily modifiable because of increasing reprogramming costs.As the concepts of the traditional grouping method are wellknown, and can be applied also to a computer system, the remainderof this paper is confined to the individual method.4.2. The accumulated results are printed. Further combinations ofresults could be made during this process.4.3. It may be necessary to produce the valuation data file from alarger file.

] Sterling, E/A, N P | 1 120 | 103 | 1400 395 | 1295 | 900


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172 GR H M WHITT KER'Current developments in the use of electronic equipment mightmake it possible once again to assemble the data direct from the

primary file, even eliminating the intermediate use of cards. . . .' 13.3. Fisher Young.These developments, of what is termed a 'consolidated file', havebeen or are being made in some companies at the time of writing.Each policy record of the consolidated file holds such information asbasic policy information, money owed to or by the policy holder,loan information, accrued bonuses, valuation data, etc. Where sucha file is used it is usual to produce a smaller valuation file by meansof a program which assesses sequentially each record of the consoli-dated file, picks off the relevant information, formats this informa-tion into a valuation record, and produces a file of these records on amagnetic tape. This program would be run whenever a valuation wasrequired. For policies with multiple benefits this program couldproduce one or more valuation records.4.4. It may or may not be necessary to sort the valuation file beforethe valuation is performed. It depends on the valuation system itselfIn some cases the sorting overhead will not be justified by the reduc-tion in run time of the valuation program ; in other cases sorting is aprerequisite to the valuation.4.5. The following are alternative methods of producing actuarialvaluation factors:

(i) The computer program calculates the necessary actuarialfactors for each policy individually. This method is flexib le, butcan be slow, as the processing time for each policy can beexcessive.(ii) The computer program calculates a set of actuarial factorsnecessary to value a continuous stream of policies each timethere is a 'control break' (for example, a change in policy class).The actuarial factors would be held in a table in the computer'smain storage. It is evident that the policies would requiresorting. This method saves having to recalculate factors forevery policy, and is also flexible.(iii) The computer program reads into main storage a set of actuarialfactors necessary to value a continuous stream of policies eachtime there is a control break. The factors would have beencalculated previously and stored on an external storage mediumsuch as a magnetic tape, disk or drum.


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ORDIN RY LIFE SSUR NCE POLICY V LU TION 173This method is flexible and is very similar in concept to tha tdescribed in (ii) above. It is used extensively in the U.S.A. wherestatutory valuation bases are fixed. However, it is also highlysuitable for countries where flexibility of basis is required. Thepolicies normally require sorting.(iv) The actuarial factors are carried on the policy record, havingbeen updated previously. The updating would probably takeplace once a year for each policy. This m ethod is inflexible as itis equivalent to valuing on a fixed basis. No sort is required.(v) There are many possible variations of the above methods. Forexample, it is theoretically possible to hold all the valuationfactors in main storage. In practice main storage space is limitedand so it might be decided to hold core resident the factors forthe largest valuation classes only. The advantage of this type ofapproach is that it is not necessary to sort a large file of valuationrecords.

4.6. Alternative methods of accumulating the individual results areas follows:(i) The results are accumulated into groups in the computer s mainstorage as each result is produced. An array of accumulatorssimilar to th at shown in Figure 1 would hold the results. Theprogram would decide for a particular policy which line of thearray to add into. The program would print the accumulatorsafter processing the file of policy records.(ii) The results are accumulated into groups on an external directaccess device, such as a magnetic disk or drum , as each result isproduced. The results are printed from the storage device by a

subsequent program. This method is slow.(iii) The results are written individually onto an external device inthe same sequence as the valued policies, in preparation for asecond accumulation and prin t program. The device wouldprobably be a magnetic tape.4.7.(i) The actuarial factors could be produced initially as a t thevaluation date. If the factors are produced at integral durations,this method is imprecise and requires adjustment. If the factorsare produced at non-integral durations the technique is similarto that described in (ii) below.


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174 GR H M WHITT KER(ii) The actuarial factors could be produced initially as at the policyanniversaries straddling the valuation date and the factors as at

the valuation date found by interpolation. This method takesinto account the exact duration of the policy, and allows thevaluation to be made on any day of the year.5. The following section assumes that the less well-known methoddescribed in section 4.7(ii) is used. It elaborates on section 4.5 bydescribing the central valuation program .

5.1. actors calculated on a control change (see 4.5(ii))

The policies would have previously beensorted.This routine would also calculate netpremiums. It could be class-dependent,or mainly independent of class if it usesa table of benefits and premiums. Itcould use iterative methods.This routine is probably independent ofpolicy class as its main function is tointerpolate and multiply.What this routine does depends on thesystem required. If the first system isused, the program will finally print theaccumulators after valuing all thepolicies. If the second system is used, asubsequent program would read, accu-mulate and print the results.

ReadPolicyRecord

CalculateFactorsRoutineControlChange

ReserveRoutine

Ye s

N o

ccumulate ResultsOR Write Resultsonto SequentialDevice

5.2. Sets of factors held on an external storage med ium(see 4.5(iii))The policies would normally be sorted before the valuation pro-grams. The calculate factors routine would be replaced by aroutine which reads in a set of factors from the external storage intomain storage.A preceding program would produce the factors on the externalstorage, possibly by an iterative process.


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ORDIN RY LIFE SSUR NCE POLICY V LU TION 1755.3.Individual calculation of factors (see 4.5(i))

ReadPolicyRecordIt may be desirable to sort the policiesprior to the valuation program.

ReserveRoutine This routine could use sub-routines tocalculate net premiums and factors.

Accumulate ResultsOR Write Resultsonto SequentialDevice

5.4. actors held on policy record (see 4.5(iv))

ReadPolicyRecord No sort wiH be necessary.

ReserveRoutineThis routine is probably independent ofclass as its main function is to in terpo lateand multiply.

Accumulate ResultsOR Write Resultsonto SequentialDevice


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176 GR H M WHITT KER6. The ensuing section develops practical valuation formulae

suitable for the technique described in section 4.7 (ii), 'the policyanniversary factor method'.6 1 Definitions:

A t = benefits valuation factor at integral duration t years. Ingeneral this factor could also value future bonuses.a,= premiums valuation factor at integral duration tyears. Ingeneral this factor could also allow for future contractualincreases or decreases in premium.S = initial sum assured.P = premium valued, per unit sum assured.F = fraction of year from previous policy anniversary tovaluation date.k = fraction of year from previous premium-due date to valua-tion date.k = fraction of year from valuation date to next premium-duedate.F = fraction of year from next premium-due date to the nextpolicy anniversary.m = number of mode premiums in a policy year.

6.2. Annual policiesP.V. of sum assured = S [A,_t (1 - F) + A,F] (1)P.V. of future premiums = SP [(t_ 1- 1 ) (1 - F)+ t,F] (2)Reserve = (1) - (2)i.e.SV,_1+F = S [ ( V , _1 + P)(1-F)+V,F] (3)

Note that no reserve adjustments are required. Moreover, theformula is insensitive to age approximations and so an approximateintegral age could be used.

'The use of the age next birthday approximation is based on theimplicit assumption that the birthdays of the lives assured aredistributed uniformly over the calendar year. This assumption is areasonable one and in practice any error which it involves is of noconsequence.' 15.2 Hooker Longley-Cook.6 3 Non annual policiesThe formulae for non-annual policies are not quite so simple, anddepend upon whether premiums are true or instalment. Figure 2shows the growth of policy reserves for an annual policy and also for


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ORDIN RY LIFE SSUR NCE POLICY V LU TION 177a non-annual instalment premium policy. The figure is also approxi-mately accurate for non-annual true premiums.If equivalent annual net premiums are valued

4that is

(5)(6)

The formulae could also be used in practice for true premiumpolicies.6.4. Combining annual and non annual formulae.

and adjustment to Reserve = +SPK for all policiesAlternatively,

Adjustment to Reserve = SP F for all policies (including annual)where F = fraction of year from next premium-due date to nextpolicy anniversarynumber of modes outstanding

number of modes in year6.5.In passing it is interesting to look at this formula if the traditionalgrouping assumption is made that on average policies are half-waythrough the policy year on the valuation date.Substituting F = 1/2 for all policies

Reserve = SAdjustment to Reserve =

[number of modes in second half of year]number of modes in year

Reserve =

Reserve =

7

8

Value sum assured byand value future premiums by


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178 GRAHAM WHITTAKERSimilarly we could substitute F = F o wherepolicies. for all

V t 1

FIG.2. Reserve approximation for a quarterly instalment premium policyNote V, is reserve for an annual policy.The premiums shown are 1) The equivalent annual premium ; 2) One quarterof the above.

For true premiumsWhole of LifeEndowmentIt is clear that the approximation ,V x m)= tx is good for endowmentpolicies, but less accurate for whole of life policies. A suitable furtheradjustment to the reserve for whole of life policies is

9)6.6. terativeformulae for valuationfactorsThe following formulae could be used to calculate sets of factorswhich are stored on an external storage device section 4.5. ill)) or tocalculate sets of factors during the main valuation program section


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ORD IN RY LIFE SSUR NCE POLICY V LU TION 1794.5. (ii)). The importance of these formulae is, firstly, that they areiterative and therefore fast, and, secondly, that they are independentof policy class.Definitions:

t,V is the reserve value of a policy at the end of the tib policy yearPt, is the premium payable a t the beginning of the (t + l) h policyyearS, is the Sum assured payable at the end of the t +l ) t hpolicy yeart, is the annuity factor at the end of the f th policy yearA, is the assurance factor a t the end of the tth policy year

whereand

(10)(11)(12)(13)(14)

The program would either calculate or look up in tables values ofP,and S, for the more complicated classes of policy. Similar formulaemay be developed if it is assumed death claims are payable part waythrough the policy year ( immediate payment of claims ).7.Final thoughts

7.1. It would be useful to compare a traditional valuation not usinga computer, a traditional valuation using a computer and a valuationby computer using the policy anniversary factor method, by meansof the criteria set out in section 2 of the paper.7.2. What are the practical advantages of the various methods ofproducing factors described in section 4.5?7.3.Is the individual valuation method a practical proposition for alarge file of policies if a projection of income and outgo is a furtherrequirement?7.5. A mortality or exit investigation by the census method couldalso be carried ou t by the individual valuation method; the calcula-tions are fewer and simpler, but the number of groupings is probablylarger. Censuses could be produced more than once a year if animprovement in accuracy is required.


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180 GR H M WHITT KER7.6. The valuation of ONs and OFFs for an analysis of surpluscan be carried out on similar lines to those described in the paper.7.7. Year-end Accounting items may be produced using similarsystems.

B I B L I O G R A P H YFISHER H . F. and YOUNG J. Actuarial Practice of Life Assurance chapters 3 1318.HOOKER P. F. and LONGLEY-COOK L. H.Life and OtherCon tingencies chapters1 15 16. chapters

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