1 The Whys of the LOIS: Credit Risk and Refinancing Rate Voatiity Stéphane Crépey 1, and Raphaë Douady 2 1 Laboratoire Anayse et Probabiités Université d Évry Va d Essonne 9137 Évry, France 2 Centre d économie de a Sorbonne CNRS/University Paris 1 Pantheon Sorbonne August 6, 212 Abstract The 27 subprime crisis has induced a persistent disconnection between the LIBOR derivative markets of different tenors and the OIS swap market. Commony proposed expanations for the corresponding spreads are a combination of credit risk and iquidity risk. However in these expanations the meaning of iquidity in particuar is either not precisey stated, or it is simpy defined as a residua spread after remova of a credit component. In this paper we propose a styized equiibrium mode in which a LIBOR- OIS spread (LOIS) emerges as a consequence of a credit component determined by the skewness of the CDS curve of a representative borrower, and a iquidity component corresponding to the voatiity of the cost-of-capita of a representative ender, where this cost-of-capita invoves the ender s CDS spread. The credit component is thus in fact a credit skewness component, and the reevant notion of iquidity appears as the optionaity, vaued by the aforementioned voatiity, of dynamicay adjusting through time the amount of a roing OIS oan, as opposed to ending a fixed amount up to the tenor horizon on LIBOR. In diffusive modes this resuts in a square root term structure of the LOIS, with a square root coefficient reated to the CDS spread voatiity of a representative ender. Consistenty with this theoretica anaysis, empirica observations revea a square root term structure of the LOIS with a square root coefficient much in ine with the voatiity of major banks one year CDS. Keywords: LIBOR, OIS, LOIS, Mutipe-Curve, Credit, Liquidity, Interest Rate Spread, Equiibrium, Fixed-income Modeing, Funding Cost, Treasury Management, CVA. The research of S. Crépey benefited from the support of the Chaire Risque de crédit, Fédération Bancaire Française.
2 2 1 Introduction The interbank oan market has been severey impacted since the 27 subprime crisis and the ensuing iquidity squeeze. However the reference interbank rates, the LIBOR, coected daiy from major banks, are sti of primary infuence as underyings to most vania interestrate derivatives ike FRA, IRS, cap/foor and swaptions. The resuting situation where an underying to financia derivatives has become in a sense arbitrariy fixed by a pane of key payers in the derivatives market poses insider issues, as iustrated by the recent mock LIBOR affairs. But first of a, it poses a crucia funding issue as, on the one hand, in parae to the drying up of the interbank oan market, LIBOR got disconnected from OIS rates (see Fig. 1); whist on the other hand, as more and more trades are coateraized, their effective funding rate is the corresponding coatera repo rate which is typicay indexed on OIS. This creates a situation where the price of an interest-rate product, even the simpest fow instrument ike a FRA, invoves (at east) two curves, an OIS and a LIBOR curve, and the reated convexity adjustment (which for some products is found significant, see (Mercurio 21)). Via the interreations between CVA and funding this aso has some important CVA impications (see (Crépey 212a; Crépey 212b; Paavicini, Perini, and Brigo 211)). Figure 1: Divergence LIBOR-OIS. Left: Sudden divergence that occurred on Aug 6 27; Right: Term structure of LIBOR vs OIS swap rates, Aug (see Formua (62) in Subsection 4.3 of (Crépey, Grbac, and Nguyen 211) for the definition of OIS swap rates which is used in the right pane). Commony advanced expanations for the LIBOR-OIS spreads (often caed LOIS in the market) are a combination of credit risk and/or iquidity risk. See (Morini 29; Fiipović and Troe 211; Crépey, Grbac, and Nguyen 211; Mercurio 21; Bianchetti 21; Fujii, Shimada, and Takahashi 211; Moreni and Paavicini 21) for various mutipe-curve modes that have been introduced in the iterature. A cear tribute to credit risk and iquidity fundamentas is expicit in the first three references. However in these expanations the meaning of iquidity in particuar is either not precisey stated, or it is simpy defined as a residua spread after remova of a credit component. In this paper we propose a styized equiibrium mode to evauate at which rate does a bank find it interesting to end at a given tenor horizon, as opposed to roing an overnight oan which it can cance at any moment. In this setup LOIS emerges as a consequence of the skewness of the credit curve of a representative borrower, and of the voatiity of the cost-of-capita of a representative ender, where this cost-of-capita incudes in particuar the CDS spread of the ender.
3 3 2 Equiibrium Mode We assume that the funding rate of a ending bank with a short term debt D is given by a random function ρ t (D). This funding rate mainy depends on the eve of the short term debt of the bank as compared to its immediatey repositabe capita C t, and on the bank s CDS spread rate S t. For instance, in a first approximation, we may have ρ t (D t + x) = ϕ(t, S t, (D t + x)/c t ) (1) in which the time dependence in ϕ refects macro-economic goba variabes such as the evoution of interest rates. Let P and E denote the actuaria probabiity measure and the reated expectation. The probem of the bank ending overnight at rate K is modeed as ( Nt ) T U(K; (N t )) = E K N t dt ρ t (D t + x)dxdt max (N t ) (2) where a stochastic process N t represents the amount of notiona that the bank is wiing to end at the OIS rate K between t and t + dt. As opposed to the situation of ending overnight a revisabe notiona N t, when ending at the LIBOR of maturity T, a bank cannot modify a notiona N which is ocked between and T. Moreover, as the composition of the LIBOR pane is updated at reguar time intervas, there is more defaut risk in a LIBOR oan than in an OIS oan (even if the pane of contributors is the same at a given time, as is the case for EONIA with respect to EURIBOR; in case of the Fed rate for US doar the pane of contributing banks is in fact arger than for the reated LIBOR). The reated optimization probem of the bank can be modeed as ( τ N ) T V(L; N) = E LN(T τ) ρ(d t + x)dxdt 1 τ<t N max N (3) Here a constant N represents the amount of notiona that the bank is wiing to end on the LIBOR of maturity T at rate L between and T, and a styized defaut time τ of the borrower refects the deterioration of the average credit quaity of the LIBOR contributors during the ength of the tenor, since this deterioration ony affects the ender when ending LIBOR (see (Fiipović and Troe 211) for a detaied anaysis, and see the comments foowing (8) beow). As we are deaing with short-term debt, we assume no recovery in case of defaut. Incusion of a positive recovery rate woud not change the end-resuts of Subsection 3.1 though since the fina inputs therein are reformuated in terms of CDS spreads, which incorporate (1 rec y ) factors anyway. The utiity functions of the bank which are impicit in (2)-(3) are taken in a standard economic equiibrium formaism of Legendre transforms U or V of the OIS and LIBOR cost functions, represented by the integras over x in the right-hand side of (2)-(3). These utiity functions are inear to refect the genera risk-neutra behavior of banks when ending, in which gains and osses are assessed in terms of actuaria expectations. In other word, the choice of banks to end is ess driven by preferences than by an optimization of the cost of capita and credit protection (even if it is not bought, as its cost determines capita adequacy). One can incorporate a concavey distorted utiity function to account for risk aversion. Such a distortion woud appear in our mode as an increased voatiity of capita needs and of the corresponding borrowing rate. However, we beieve that short term ending decisions are more driven by the estimated cost of capita than by a trade-off between interest
4 4 returns and defaut risk. Defaut risk obviousy matters in the ending decision, but as a yes/no answer, based on avaiabe information on the borrower at the time of ending. Once a borrower is deemed acceptabe, the amount ent and the granted rate depend on the impact on the bank capita and its estimated cost, as we as on the borrower s defaut risk measured through the cost of protection (credit spread, CDS rate), which either is bought or is accounted for in the capita affected to the oan. This justifies the use of purey inear utiity functions. We aso did not introduce a discount factor, as we are deaing with short term debt. Extending this mode to onger term debt woud require this correction. We stick to the styized formuation (2)-(3) for tractabiity issues, and aso in order to emphasize that the voatiity terms which appear in the end-resuts (11), (13) and (14), are convexity adjustments refecting an inherent optionaity of LOIS, even without any risk premia. Letting U(K) = max (Nt) U(K; (N t )) and V (L) = max N V(L; N), our approach for expaining the LOIS consists, given a vaue of K, in soving the equation V (L ) = U(K) (4) which expresses an equiibrium reation between the utiity of ending overnight versus LIBOR for a bank invoved in both markets (indifference vaue at the optima amounts prescribed by the soution of the corresponding optimization probems). A second optimization probem, eft aside in this artice, woud be in the eve of the overnight rate K, depending on the suppy/demand of iquidities and on base rates from the centra bank. 2.1 Linear Case For tractabiity we assume henceforth that the funding rate ρ is inear in D, i.e. ρ t (D t + x) = α t + t x (5) where α t = ρ t (D t ) is the cost of capita of the ending bank at time t, and the coefficient t (positive in spirit) represents the margina additiona cost of borrowing one more unit of notiona for the bank aready indebted at the eve D t. For instance α t = 2%, t = 5bp means that the ast 1 euros that were borrowed by the bank cost an annuaized interest charge of 2 euros, whereas the next 1 euros to be borrowed by the bank woud cost an annuaized interest charge of 2.5 euros. E.g., in a specification aso consistent with (1), etting R t = D t /C t for constants A and B, so in (5). One then has T U(K; (N t )) = E ρ t (D t + x) = S t + AR t + BS t x (6) α t = S t + AR t, t = BS t ( (K α t )N t 1 ) 2 2 tn t dt (7) Denoting by λ t the intensity of τ, one aso has, etting γ t = α t + λ t and t = e t λsds T V(L; N) = E ((L γ t )N 12 ) tn 2 t dt (8)
5 5 where a standard credit risk computation was used to get rid of the defaut indicator functions in (8) (see for instance (Bieecki and Rutkowski 22)). As expained after (3), the styized defaut time τ refects the deterioration of the average credit quaity of the LIBOR contributors during the ength of the tenor. The intensity λ t of τ can thus be proxyed by the differentia between the borrower s CDS spread of maturity T and the spread of her Certificate Deposit. Accordingy we ca λ t the credit skewness of the borrower. Formay, λ t dt is the difference between the borrower s defaut probabiity over the interva [t, t + dt] measured with information known at t = and that known at time t, assuming that the bank appies a ending poicy preventing ending to counterparties that are considered too risky. Assuming that the borrower s defaut intensity is a random process with i.i.d. increments and that the bank ending poicy prevents ending when the defaut intensity is above a given threshod, we can thus see the difference λ t dt as the expectation of the defaut intensity when it exceeds the threshod (i.e. the conditiona expectation conditionay to exceeding the threshod, mutipied by the probabiity of exceeding it). Note that a possibe mock effect, or incentive for a LIBOR contributor to bias its borrowing rate estimate in order to appear in a better condition than it is in reaity, can be incuded as a spread in the borrower s credit risk skewness component λ. 3 Derivation of the Spreads Probems (2), (7) and (3), (8) are respectivey soved, for given K and L, as foows. The OIS probem (2), (7) is resoved independenty at each date t according to u t (K; N t ) = (K α t ) N t 1 2 tn 2 t max N t hence N t (K) = K α t t and u t (K; N t ) = (K α t) 2 2 t The expected profit of the bank over the period [, T ] is ( 1 T U(K) = U(K; (Nt (K α t ) 2 ) )) = E dt T 2 t hence In the Libor probem (3), (8), we must sove T V(L; N) = NE (L γ t ) t dt 1 2 N 2 E t t dt max N N = E 1 T (L γ t) t dt dt and V (L) = V(L; N ) = t t E 1 T ( E 1 T (L γ t) t dt t t dt Note that in case λ =, one necessariy has U(K) V (K), since the constant N soving the LIBOR maximization probem (3) is a particuar strategy (constant process N t = N ) of the OIS maximization probem (2). As U and V are increasing functions, the indifference pricing equation (4) in turn yieds that L K. Let V (K; N) be the utiity of ending LIBOR in case λ =. When λ >, for each given amount N, one has via λ which is present in γ in (8) that V(K; N) V (K; N) (up 2E 1 T ) 2 (9)
6 6 to the second order impact of the discount factor ), hence the maximum of V(K; ) in N is beow that of V (K; ), i.e. V (K) V (K) U(K), the atter inequaity resuting from the fact that τ doesn t appear in U(K; (N t )). One concudes as in case λ = that L K hods again. The seque of this paper is devoted to the computation of the spread L K where L is, given K, the soution of (4) (assumed to exist; note that V is continuous and increasing in L, so that a soution L to (4) can ony be uniquey defined). For notationa convenience et us introduce the time-space probabiity measure P product of P times dt T over Ω [, T ], as we as another time-space probabiity measure such that d P. For a process X = X dp t(ω) we denote the corresponding time-space averages and variances by X = EX = E 1 X t dt, σ 2 X = E(X T X) 2 [ X = ÊX = E X ], σ X 2 = Ê(X X) 2. As [ ] (K α) 2 U(K) = E and V (L) = 2 (L γ) 2 2 2E equaizing V (L ) = U(K) yieds [ ] 2 (L γ) 2 (K α) 2 = EE (1) In which So [ ] (K α) 2 EE = E [ (K α) 2 ] Cov [, 2 (L γ) 2 = Ê [ (K α) 2] Cov [, ] (K α)2 ] (K α)2 In particuar, at-the-money when K = α = γ λ, one has ] (L K λ) 2 = σ α 2 (α α)2 Cov [, A reasonabe guess is that the covariance is negigibe in the right-hand-side (in particuar this covariance vanishes when λ is zero and is constant). Our first spread formua foows as 1 L K λ + σ α. (11) Aternativey, Equation (1) in L can be written in terms of a third time-space probabiity measure such that d P dp 1 as 2 (L γ) 2 = E  1Ẽ [ (K α) 2] 1 Admitting that L K λ, a natura assumption as LIBOR ending shoud at east compensate for the credit risk over the tenor horizon period, cf. the comments foowing Equation (9).
7 7 or (L γ) 2 = 1Ẽ [ (K α) 2] (12) Another case of interest is then when K = α = Ẽα, in which case (12) bois down to (L γ) 2 = 1 σ 2 α. yieding our second spread formua 2 L K = λ + α α + 1 σ α λ + 1 σ α (13) as α α is negigibe with respect to other terms. Aso note that in case λ = and P = P, one has in the ast expression that = 1 and 1 1 by the Jensen s inequaity, with equaity in the constant case. 3.1 LOIS Formua In (11) as in (13), the two key drivers of the LOIS are on the one hand a suitabe timespace average of the borrower s credit skewness λ (which can be seen as the intrinsic vaue component of the LOIS, and is a borrower s credit component), and on the other hand a suitabe time-space standard deviation ( time-vaue component of the LOIS, a ender s iquidity component) of the ender s cost of capita α t = ρ t (D t ). Under the specification (6) and assuming here independence of the ender s credit spread S t and debt-ratio R t, as we as diffusive properties of the input processes λ t, S t and R t, with corresponding reference voatiities denoted by σ λ, σ S and σ R, and with a reference credit skewness λ of the borrower denoted by λ, one can approximate in (11) and (13) One then has in (11) λ T, σ α σ α (σ S + Aσ R ) T L K λ + σ α σ2 λ T λ + ( λ T )(σ S + Aσ R ) T and in (13) L K λ + 1 σ α λ + ( σ2 λ T λ T )(σ S + Aσ R ) T 2 Admitting non-negativity of L γ = L α λ = L K + α α λv, cf. the comments foowing Equation (9).
8 8 In both cases L K = λ + (σ S + Aσ R ) T + O(T 3 2 ) (14) which we ca the LOIS formua. The square root term structure in this theoretica formua has a very good fit with the market. Fig. 2 thus shows the fit between a theoretica square root term structure and the empirica LOIS term structure corresponding to the data of the right pane in Fig. 1. The sope (coefficient of T in the regression) and intercept (constant coefficient) are.99 and.9. A sope of.99 is quite in ine with current order of magnitudes of one year CDS spread voatiities on major banks, whereas an intercept of.9 is aso quite reasonabe for the differentia between the one year CDS spread and the Certificate Deposit spread of a major bank. The downward bump at one month coud be due to the credit-skewness component, which woud typicay dispay this kind of shape if banks appy a threshod-based ending poicy (see end of Subsection 2.1). Formua (14) can be used for impying the quantity Aσ R priced by the market from an observed LOIS L K, the borrower s CDS sope, taken as a proxy for λ, and the ender s CDS spreads voatiity, taken as a proxy for σ S. The quantity Aσ R thus impied through (14) can be compared by a bank to an interna estimate of its reaized Aσ R, so that the bank can decide whether it shoud rather end LIBOR or OIS, much ike with going ong or short an equity option depending on the reative position of the impied and reaized voatiities of the underying stock. Figure 2: Square root fit of the LOIS corresponding to the data of the right pane in Fig. 1 Concusion Since the 27 subprime crisis, OIS swaps and LIBOR swaps, or equivaenty in Euro, EONIA and EURIBOR swaps, diverged suddeny. We show in this paper that, by optimizing their ending, banks are ed to appy a spread over the OIS rate when ending at LIBOR. Theory impies that the corresponding LIBOR-OIS spread (LOIS) has two components: One corresponding to the credit skewness of a representative borrower, and one corresponding
9 9 to the voatiity of the cost-of-capita of a representative ender. In diffusive modes this resuts in a square root term structure of the LOIS, with a square root coefficient reated to the CDS spread voatiity of a representative ender. This is corroborated by empirica evidence, with an empirica coefficient of a LOIS square root term structure much in ine with the voatiity of major banks one year CDS. References Bianchetti, M. (21). Two curves, one price. Risk Magazine, August Bieecki, T. R. and M. Rutkowski (22). Credit Risk: Modeing, Vauation and Hedging. Springer. Crépey, S. (212a). Biatera Counterparty risk under funding constraints Part I: Pricing. Mathematica Finance. Forthcoming. Crépey, S. (212b). Biatera Counterparty risk under funding constraints Part II: CVA. Mathematica Finance. Forthcoming. Crépey, S., Z. Grbac, and H. N. Nguyen (211). A mutipe-curve HJM mode of interbank risk. Forthcoming in Mathematics and Financia Economics. Fiipović, D. and A. B. Troe (211). The term structure of interbank risk. SSRN elibrary. Fujii, M., Y. Shimada, and A. Takahashi (211). A market mode of interest rates with dynamic basis spreads in the presence of coatera and mutipe currencies. Wimott Mag., Juy Mercurio, F. (21). A LIBOR market mode with a stochastic basis. Risk Magazine (December), Moreni, N. and A. Paavicini (21). Parsimonious HJM modeing for mutipe yiedcurve dynamics. Preprint, arxiv: v1. Morini, M. (29). Soving the puzze in the interest rate market. SSRN elibrary. Paavicini, A., D. Perini, and D. Brigo (211). Funding vauation adjustment: a consistent framework incuding cva, dva, coatera,netting rues and re-hypothecation. Preprint, arxiv:1521.