the whys of the lois: credit skew and funding spread ... · volatility of the...

Equilibrium ModelLOIS Formula

Empirical corroboration (EUR market 15/08/2007 � 16/04/2012)

The Whys of the LOIS:

Credit Skew and Funding Spread Volatility

Stéphane CrépeyUniversité d'Evry Val-d'Essonne, Laboratoire Analyse & Probabilités

Séminaire du CEFRA13 juin 2013

The research presented in these slides bene�ted from the support ofthe �Chaire Risque de crédit� and the �Chaire Marchés en mutation�,

Fédération Bancaire Française

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LIBOR

Most interest-rate derivatives have LIBOR-indexed cash-�ows (LIBOR�xings)

What is LIBOR?

LIBOR stands for London InterBank O�ered Rate. It is produced for10 currencies with 15 maturities quoted for each, ranging fromovernight to 12 Months producing 150 rates each business day.LIBOR is computed as a trimmed average of the interbankborrowing rates assembled from the LIBOR contributing banks.

More precisely, every contributing bank has to submit an answer tothe following question: "At what rate could you borrow funds, wereyou to do so by asking for and then accepting inter-bank o�ers in areasonable market size just prior to 11 am?"

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OIS

In most currencies there is also an interbank market of overnight loans,at a rate referred to OIS (spot) rate henceforth

In some currencies the OIS rate (like the EONIA rate for the euro)can be viewed as a short-tenor limit of LIBOR

In others (like US dollar) this view is simplistic since the panel of theLIBOR and of the OIS rate is not the same, and the OIS rate re�ectsactual transaction rates (as opposed to a purely collected LIBOR)

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LOIS

Divergence Euribor (�L�) / EONIA-swap (�R�) ratesLeft: Sudden divergence between the 3m Euribor and the 3mEONIA-swap rate that occurred on Aug 6 2007Right: Term structure of Euribor vs EONIA-swap rates, Aug 14 2008

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Drying-up of the interbank market

Segmentation between LIBOR markets of various tenors since thecrises

LIBOR-OIS swap spreads, basis swap spreadsOIS (overnight) market versus LIBOR 3m market, LIBOR 6mmarket,..

Credit risk and liquidity risk fundamentals of these spreads

Deterioration of the average credit quality of the panelists of aLIBOR during the length of the tenor

Credit spread skew constant component

D. Filipovic and A. Trolle: The term structure of interbank risk.Journal of Financial Economics (forthcoming).

MORE OPTIONALITY WHEN LENDING SHORTER

Volatility of the cost-of-capital (including volatility of credit spreads)√T/2-component

Crépey, S. and Douady, R.: LOIS: credit and liquidity Risk

Magazine (June 2013 issue, forthcoming).

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Some other references

Bean, C. (2007). An indicative decomposition of Libor spreads.Quaterly Bulletin of the The Bank of England 47 (4), 498-99.

Brunnermeier, M. and L. Pedersen (2009). Market Liquidity andFunding Liquidity. The Review of Financial Studies 22 (6),2201-2238.

Crépey, S., Z. Grbac, and H. N. Nguyen (2012). A multiple-curveHJM model of interbank risk. Mathematics and Financial Economics6 (3), 155-190.

Morini, M. (2009). Solving the puzzle in the interest rate market.SSRN eLibrary.

Smith, J. (2010). The term structure of money market spreadsduring the �nancial crisis. Preprint.

Wheatley, M. (2012). The Wheatley Review of LIBOR: �nal report.Technical report.

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Outline

1 Equilibrium Model

2 LOIS Formula

3 Empirical corroboration (EUR market 15/08/2007 � 16/04/2012)

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Dt Time-t short term debt of the bank

ρt(D) Funding rate of a lending bank with short term debt D

A complex output possibly impacted by, depending on the treasurymanagement of the bank, beyond the level of the short term debt D:

the O/N rt

the bank's CDS spreadother macro-economic global variables

αt = ρt(Dt) Interest rate charged to the bank �on the last euro thatit borrowed`� for funding its loan

�current� re�nancing or funding rate of the bank

ρt(Dt + x) Interest rate charged to the bank on the last euro of itsre�nancing debt e(Dt + x)

x Putative extra amount of re�nancing capital borrowed by the bankOIS or Libor

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Liquidity issue → ρt(D) increasing in D (or ρt(Dt + x) increasing inx)

�the next euro borrowed costs more than the previous one�

Marginal cost ρt(Dt + x) integrated over x from 0 to the cumulativeamount N → global re�nancing cost convex in N (as the integral ofan increasing function)

Optionality feature of the funding liquidity issue

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Lending OIS

TU((rt); (Nt)) = E( ∫ T

0

Ntrtdt −∫ T

0

∫ Nt

0

ρt(Dt + x)dxdt)←− max (Nt) (1)

Nt Amount of notional that the bank is willing to lend at the O/Nrate rt between t and t + dt

Lending Libor

TV(L;N) = E(NL(T ∧ τ)−

∫ T∧τ

0

∫ N

0

ρ(Dt + x)dxdt − 1τ<TN)←− max N (2)

N Amount of notional that the bank is willing to lend at the Liborrate L over the period [0,T ]

τ stylized default time of the borrower re�ecting the deterioration ofthe average credit quality of the Libor contributors during the lengthof the tenor

Default events that could have been avoided, had the loan be madeas rolling overnightSurvivorship bias that overnight loans bene�t from 14 / 29



U((rt)) = max(Nt) U((rt); (Nt)), V (L) = maxN V(L;N)

Best utilities a bank can achieve by lending OIS or Libor

O/N process (rt) assumed to be given depending on thesupply/demand of liquidities and on base rates from the central bank

LOIS equation

V (L∗) = U((rt)) (3)

Indi�erence value at the optimal amounts prescribed by the solution toboth optimization problems

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Linear funding costs model

ρt(Dt + x) = αt + βtx (4)

Eg αt = 2%, βt = 50bp

The last euro borrowed by the bank was at an annualized interestcharge of 2 cents

If the bank would be indebted by e100 more, the next euro to beborrowed by the bank would be at an annualized interest charge of2.5 cents

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−→ TU((rt); (Nt)) = E∫ T

0

((rt − αt)Nt −

1

2βtNt

2

)dt (5)

Denoting by λt the (assumed) intensity of τ and letting γt = αt + λt and

`t = e−∫

t

0 λsds , we also have:

TV(L;N) = E∫ T

0

((L− γt)N −

1

2βtN

2

)`tdt (6)

where a standard credit risk computation was used to get rid of thedefault indicator

Merton 74 A high-quality credit-name has a decreasing CDS curvere�ecting the expected deterioration of his credit

→ intensity λt of τ proxyed by the slope of the credit curve of a Liborrepresentative (and therefore high-quality) borrower

Di�erential between the borrower's 1y CDS spread and the spread ofher short term certi�cate depositsCurrently 10 to a few tens of bp for major banksCredit skew of a Libor representative borrower

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Outline

1 Equilibrium Model

2 LOIS Formula


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ct := αt − rt Funding spread

OIS problem (1), (5) resolved independently at each date t

according to

ut(rt ;Nt) = ctNt −1

2βtN

2

t ←− max Nt

−→ U((rt)) = U((rt); (N∗t )) = E

(1

T

∫ T

0

c2t2βt

dt

)Libor problem (2), (6) solved as

TV(L;N) = NE∫ T

0

(L− γt)`tdt −1

2N2E

∫ T

0

βt`tdt ←− max N

−→ V (L) = V(L;N∗) =

(E 1

T

∫ T0(L− γt)`tdt

)22E 1

T

∫ T0βt`tdt

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Let R := E 1

T

∫ T0

rtdt

Stylized LOIS de�ned as (L∗ − R), where L∗ is, given the process(rt), the solution to (3)

Assumed to exist, then unique and ≥ R

LOIS FORMULA

L∗ − R ≈ λ∗ + σ∗√T/2 (7)

λ∗ Reference credit skew of the borrower

intrinsic value component of the LOISa borrower's credit skew component

σ∗ Reference volatility of the instantaneous funding spread processct = αt − rt

assumed di�usive

factor 1/2 under the square-root due to the fact that we deal withtime-space variances

time-value of the LOISa lender's liquidity (credit volatility) component

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Outline

1 Equilibrium Model

2 LOIS Formula


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Time series of the �red� intercepts (in %; credit component of the LOIS),�blue� and �purple� slopes (in %; liquidity component of the 3m- and 6m-LOIS) and �green� R2 coe�cients of the regressions of the 1m to 1yrLOIS against

√T/3m or

√T/6m

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Three market regimes

Until Q1 of 2009, the market seems to �try to understand� whathappens, with an R2 becoming signi�cant together with very largeand volatile credit (�red intercept�) and liquidity (�blue or purpleslope�) LOIS components

spike of both components at the turn of the credit crisis followingLehman's default in Sept 2008, during which the interbank drop oftrust created both a credit and a liquidity crisis

Between Q2 of 2009 and mid-2011, the situation seems �stabilized�with an R2 close to 1, a liquidity LOIS component of the order of30bps on the 3m or 45bps on the 6m and a much smaller creditLOIS component.

The ongoing Eurozone crisis, prompted by the US downgrademid-2011, reveals a third pattern with a much higher liquidity LOIScomponent, of the order of 60bps on the 3m or 90bps on the 6m,revealing increased funding liquidity concern of banks, due to harderregulatory constraints (e.g. government bonds no longerrepositable).

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14 Aug 2008

Left Euribor / EONIA-swap ratesRight Square root �t of the LOIS(T = 1m to 12m)

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28 Apr 2010


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1 Dec 2011


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Confrontation with the economical determinants of λt andαt

Intercepts of e.g. 10 bp appear as reasonable for a �credit skew�,that is, the di�erential between the one year CDS spread and theshort term certi�cate deposit credit spread of a major bank

Coe�cients of√T/2, ranging between 100 and 200 bp/yr

(corresponding to magnifying on Fig. 19 by a factor 2 the purplecurve, or slope coe�cients of the regression against

√T/6m), is

quite in line with recent orders of magnitude of the volatility ofmajor banks' one year CDS spreads.

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CONCLUSION

Theoretical developments corroborated by empirical evidence on theEUR market studied in this paper on the period half-2007 half-2012

LOIS explained in a balanced way by credit and liquidity until thebeginning of 2009 and dominantly explained by liquidity since 2009

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APPLICATIONS

Implying the value σ∗ �priced� by the market from an observed LOIS(L∗ − R) and a borrower's CDS slope taken as a proxy for λ∗

σ∗ can then be compared by a bank to an internal estimate of itsrealized funding spread volatility, so that the bank can decidewhether it should rather lend Libor or OIS

much like with going long or short an equity option depending on therelative position of the implied and realized volatilities of theunderlying stock

σ∗ can also be used for the calibration of the volatility σ∗ of thefunding spread process ct in a stochastic model for the latter

e.g. in the context of multiple-curve CVA computations

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