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NBER WORKING PAPER SERIES

THE MARKET PRICE OF CREDIT RISK:AN EMPIRICAL ANALYSIS OF INTEREST RATE SWAP SPREADS

Jun Liu

Francis A. Longstaff Ravit E. Mandell

Working Paper 8990http://www.nber.org/papers/w8990

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138June 2002

We are grateful for the many helpful comments and contributions of Don Chin, Qiang Dai, Robert Goldstein,Gary Gorton, Peter Hirsch, Antti Ilmanen, Deborah Lucas, Josh Mandell, Yoshihiro Mikami, Jun Pan,Monica Piazzesi, Walter Robinson, Pedro Santa-Clara, Janet Showers, Ken Singleton, Suresh Sundaresan,Abraham Thomas, Rossen Valkanov, Toshiki Yotsuzuka, and seminar participants at Barclays GlobalInvestors, Citigroup, Greenwich Capital Markets, Invesco, Mizuho Financial Group, Simplex AssetManagement, UCLA, the 2001 Western Finance Association meetings, the 2002 American FinanceAssociation meetings, and the Risk Conferences on Credit Risk in London and New York. We areparticularly grateful for the comments and suggestions of the editor William Schwert and an anonymousreferee. All errors are our responsibility. The views expressed herein are those of the authors and notnecessarily those of the National Bureau of Economic Research.

© 2002 by Jun Liu, Francis A. Longstaff and Ravit E. Mandell. All rights reserved. Short sections of text,not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including© notice, is given to the source.



The Market Price of Credit Risk: An Empirical Analysis of Interest Rate Swap SpreadsJun Liu, Francis A. Longstaff and Ravit E. Mandell

NBER Working Paper No. 8990June 2002JEL No. E4

ABSTRACT

This paper studies the market price of credit risk incorporated into one of the most important

credit spreads in the financial markets: interest rate swap spreads. Our approach consists of jointly

modeling the swap and Treasury term structures using a general five-factor affine credit framework and

estimating the parameters by maximum likelihood. We solve for the implied special financing rate for

Treasury bonds and find that the liquidity component of on-the-run bond prices can be significant. We

also find that credit premia in swap spreads are positive on average. These premia, however, vary

significantly over time and were actually negative for much of the 1990s.

Jun Liu Francis A. Longstaff Ravit E. MandellUCLA UCLA Salomon Smith BarneyAnderson School Anderson School

110 Westwood Plaza, Box 951481Los Angeles, CA 90095-1481and NBER [email protected]



the extreme liquidity of Treasury bills, their yields tend to underestimate the e ff ectiveriskless rate. Since the implied short-term riskless rate could also be interpreted asspecial repo rate for the on-the-run Treasury bonds in the sample, we contrast themwith repo rates for generic or general Treasury collateral. 1 We nd that these impliedspecial repo rates are slightly less than the general repo rates on average, suggesting

that the prices of the on-the-run Treasury bonds in the sample include premia for theirliquidity or specialness relative to o ff -the-run Treasury securities. These specialnesspremia can be large in economic terms. For example, the specialness premium for theten-year Treasury note can be as much as 0.57 percent of its notional amount. Theestimated specialness premia match closely those implied by a sample of market termspecial repo rates provided to us.

We then solve for the implied spread process. We nd that this spread variessigni cantly over time, but is nearly zero for an extended period during the mid tolatter 1990s. Interestingly, much of the variation in the implied credit spread is relatedto the di ff erence between the general and implied special repo rates, suggesting that

changes in the liquidity of Treasury bonds may be one of the major driving factors of swap spreads.

Finally, we examine the implications of the model for the market prices of interest-rate and credit-related risk. Consistent with previous research, we nd that there aresigni cant time-varying term premia embedded in Treasury bond prices. We also ndthat there are signi cant credit premia embedded in the swap curve. On average,these premia are positive, ranging from 0.1 basis points for a one-year horizon to 45basis points for a ten-year horizon. These credit premia also display substantial timevariation. Surprisingly, we nd evidence that credit premia were often negative duringan extended period in the 1990s. These results suggest that there have been majorchanges over time in the expected returns from bearing the default and liquidity riskinherent in interest rate swaps.

This paper complements and extends the recent paper by Du ffi e and Singleton(1997) who apply a reduced-form credit modeling approach to the swap curve andexamine the properties of swap spreads. Our results support their nding that bothdefault-risk and liquidity components are present in swap spreads. By modeling boththe swap and Treasury curves simultaneously, however, we are also able to addressthe issue of how credit risk is priced in the market, which is the primary focus of this paper. Another related paper is He (2000) who independently uses a multi-factoraffi ne term structure framework similar to ours in modeling swap spreads. While Hedoes not estimate the parameters of his model, our empirical results provide supportfor both swap spread modeling frameworks. Grinblatt (2001) models the swap spread

1 In a recent paper, Du ffi e (1996) studies the causes and e ff ects of special repo ratesin the Treasury repo market. See also Sundaresan (1994), Jordan and Jordan (1997),Buraschi and Menini (2001), and Krishnamurthy (2001).

2



C (t, T ) = E Q ^expX− 8 T

tr s + λ s ds~ , (2)

where λ is a credit-spread process. 3 They also show that this credit-spread process maybe viewed as the product of the time-varying Poisson intensity and the recovery-rateprocess. Furthermore, they argue that the credit-spread process could also include atime-varying liquidity component which may be either positive or negative. In thispaper, we simply refer to λ as the credit-spread process, keeping in mind, however,that λ may include both default-risk and liquidity components. Consequently, the termcredit risk is used in a general sense throughout this paper, re ecting that variationin credit or swap spreads may be due to changes in either default risk or liquidity.

In applying this credit model to swaps, we are implicitly making two assump-tions. First, we assume that there is no counterparty credit risk. This is consistentwith recent papers by Grinblatt (2001), Du ffi e and Singleton (1997), and He (2000)that argue that the e ff ects of counterparty credit risk on market swap rates shouldbe negligible because of the standard marking-to-market or posting-of-collateral andhaircut requirements almost universally applied in swap markets. 4 Second, we makethe relatively weak assumption that the credit risk inherent in the Libor rate (whichdetermines the swap rate) can be modeled as the credit risk of a single defaultableentity. In actuality, the Libor rate is a composite of rates quoted by 16 banks and,as such, need not represent the credit risk of any particular bank. 5 In this sense, thecredit risk implicit in the swap curve can be viewed essentially as the average creditrisk of the most representative banks providing quotations for Eurodollar balances. 6

3

In the Duffi

e and Singleton (1999) model, the recovery rate is linked to the valueof the bond immediately prior to the default event. While this assumption has beencriticized, the fact that Libor is computed from a set of banks that may change overtime if some banks experience a deterioration in their credit rating argues that thisassumption may be more defensible when applied to the swap curve.4Even in the absence of these requirements, the e ff ects of counterparty credit riskfor swaps between similar counterparties are very small relative to the size of theswap spread. For example, see Cooper and Mello (1991), Sun, Sundaresan, and Wang(1993), Bollier and Sorensen (1994), Longsta ff and Schwartz (1995), Du ffi e and Huang(1996), and Minton (1997).5The offi cial Libor rate is determined by eliminating the highest and lowest four bankquotes and then averaging the remaining eight. Furthermore, the set of 16 banks whosequotes are included in determining Libor may change over time. Thus, the credit riskinherent in Libor may be “refreshed” periodically as low credit banks are droppedfrom the sample and higher credit banks are added. The e ff ects of this “refreshing”phenomenon on the di ff erences between Libor rates and swap rates are discussed inColin-Dufresne and Solnik (2001).6For discussions about the economic role that interest-rate swaps play in nancial mar-

4



To model the discount bond prices D (t, T ) and C (t, T ), we next need to specifythe dynamics of r and λ . In doing this, we parallel the approach used by Du ffi eand Kan (1996), Du ffi e and Singleton (1997), Du ff ee (1999), Dai and Singleton (2000,2001), and others by working within a general a ffi ne framework. In particular, weassume that the dynamics of r and λ are driven by a vector X of ve state variables,

X = [ X 1 , X 2 , X 3 , X 4 , X 5]. Following Dai and Singleton (2000), we assume that

r = δ 0 + X 1 + X 2 + X 3 + X 4 , (3)

where δ 0 is a constant. Thus, the dynamics of the riskless term structure are drivenby the rst four state variables. This four-factor speci cation of the riskless termstructure is consistent with recent evidence by Dai and Singleton (2001) and Du ff ee(2002) about the number of signi cant factors a ff ecting Treasury yields. 7

In modeling the dynamics of the spread λ , we assume that

λ = δ 1 + γ r + X 5 , (4)

where δ 1 and γ are constants which may be positive or negative. Note that thisspeci cation allows the spread λ to depend on the state variables driving the risklessterm structure in both direct and indirect ways. Speci cally, λ depends directly onthe rst four state variables through the term γ r in Eq. (4). Indirectly, however,the spread λ may be correlated with the riskless term structure through correlationsbetween X 5 and the other state variables.

The advantage of allowing for both a direct and an indirect relation between the

spread λ and the factors driving the riskless term structure is that it enables us toexamine in more depth the determinants of swap spreads. For example, our approachallows us to examine whether the swap spread is an artifact of the di ff erence in thetax treatment given to Treasury securities and Eurodollar deposits. Speci cally, in-terest from Treasury securities is exempt from state income taxation while interestfrom Eurodollar deposits is not. Thus, if the spread λ were determined entirely by thediff erential tax treatment, the parameter γ would represent the marginal state tax rate

kets, see Bicksler and Chen (1986), Turnbull (1987), Smith, Smithson, and Wakeman(1988), Wall and Pringle (1989), Macfarlane, Ross, and Showers (1991), Sundare-san (1991), Litzenberger (1992), Sun, Sundaresan, and Wang (1993), Brown, Harlow,and Smith (1994), Minton (1997), Gupta and Subrahmanyam (2000), and Longsta ff ,Santa-Clara, and Schwartz (2001).7Also see the empirical evidence in Litterman and Scheinkman (1991), Knez, Lit-terman, and Scheinkman (1994), Piazzesi (1999), and Longsta ff , Santa-Clara, andSchwartz (2001) indicating the presence of at least three signi cant factors in termstructure dynamics.

5



To study how the market compensates investors over time for bearing credit risk,it is important to allow a fairly general speci cation of the market prices of risk inthis a ffi ne A0(5) framework. Accordingly, we assume that the dynamics of X underthe objective measure are given by

dX = − κ (X − θ)dt + Σ dB P , (6)

where κ is a diagonal matrix, θ is a vector, and B P is a vector of independent standardBrownian motions. This speci cation has the advantages of being both tractable andallowing for general time varying market prices of risk for each of state variables. 9

Given the risk-neutral dynamics of the state variables, closed-form solutions forthe prices of riskless zero-coupon bonds are given by

D (t, T ) = exp( a(t) + b (t)X ), (7)

where

a(t) = − δ 0(T − t) +12

L β − 1 ΣΣ β − 1L(T − t)

− L β − 1 ΣΣ β − 2(I − e− β (T − t ) )L + 3i,j1 − e− (β ii + β jj )( T − t )

2β ii β jj (β ii + β jj )(ΣΣ )ij L i L j ,

b(t) = β − 1 pe− β (T − t ) − I QL,

and where L = [1, 1, 1, 1, 0], and I is the identity matrix. Similarly, the prices of riskyzero-coupon bonds are given by

C (t, T ) = exp( c(t) + d (t)X ), (8)

9

It is important to note, however, that even more general speci

cations for the marketprices of risk are possible. In principle, for example, the diagonal matrix κ could begeneralized to allow nonzero o ff -diagonal terms. Our speci cation, however, alreadyrequires the estimation of ten market price of risk parameters which approaches thepractical limits of our computational techniques. Adding more market price of riskparameters also raises the risk of creating identi cation problems.

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where K is a diagonal matrix with i-th diagonal term e− κ ii ∆ t , and Ω is a matrix withij -th term given by

Ω ij =1 − e− (κ ii + κ jj ) ∆ t

κ ii + κ jj(ΣΣ )ij .

Let t + ∆ t denote the vector of di ff erences between the observed value of R2,t + ∆ t andthe value implied by the model. 10 Assuming that the terms are independently dis-tributed normal variables with zero means and variances η2

i , the log likelihood functionfor t + ∆ t is given by

− 12 t + ∆ t Σ − 1

t + ∆ t − 12

ln | Σ | , (14)

where Σ is a diagonal matrix with diagonal elements η2i , i = 1 , . . . , 4.

Since X t + ∆ t and t + ∆ t are assumed to be independent, the log likelihood function for[X t + ∆ t , t + ∆ t ] is simply the sum of Eqs. (13) and (14).

The nal step in specifying the likelihood function consists of changing variables fromthe vector [ X t , t ] of state variables and error terms to the vector [ R1,t , R 2,t ] of ratesactually observed. It is easily shown that the determinant of the Jacobian matrix isgiven by | J t |= | ∂ h (R 1 ,t )

∂ R I

1 ,t| . Summing over all observations gives the log likelihood

function for the data

− 1

2

T − 1

3t =1

(X t + ∆ t − θ − K (X t − θ)) Ω − 1 (X t + ∆ t − θ − K (X t − θ))

+ ln | Ω | + t + ∆ t Σ − 1t + ∆ t + ln | Σ | + 2 ln | J t |= (15)

Given this speci cation, likelihood function depends explicitly on 37 parameters.

From this log likelihood function, we now solve directly for the maximum like-lihood parameter estimates using a standard nonlinear optimization algorithm. Indoing this, we initiate the algorithm at a wide variety of starting values to insure thatthe global maximum is achieved. Furthermore, we check the results using an alter-native genetic algorithm which has the property of being less susceptible to nding

local minima. These diagnostic checks con rm that the algorithm converges to the10 We assume that the terms are independent. In actuality, the terms could becorrelated. As is shown later, however, the variances of the terms are very smalland the assumption of independence is unlikely to have much e ff ect on the estimatedmodel parameters.

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global maximum and that the parameter estimates are robust to perturbations of thestarting values.

Table 2 reports the maximum likelihood parameter estimates and their asymp-totic standard errors. As shown, there are clear di ff erences between the objective andrisk-neutral parameters. These di ff erences have major implications for the dynamicsof the key variables r and λ which we consider in the next two sections. The di ff er-ences themselves re ect the market prices of risk for the state variables and also haveimportant implications for the expected returns from bearing credit and liquidity risk.We note that the β and κ parameters are all estimated to be positive; the estimationprocedure does not constrain these parameters to be positive. One key result thatemerges from the maximum likelihood estimation is that the ve-factor model ts thedata well, at least in its cross-sectional dimension. For example, the standard devia-tions of the pricing errors for the CMS2, CMS3, CMS5, and CMT5 rates (given by η1 ,η2 , η3 , and η4 , respectively) are 12.2, 9.0, 6.4, and 6.0 basis points respectively. Theseerrors are fairly small and are on the same order of magnitude as those reported in

Dai and Singleton (1997) and Duff

ee (2002). Note, however, that we are estimatingboth the Treasury and swap curves simultaneously. In addition, with the exceptionof a few of the parameters of Σ , all of the parameters of the model are statisticallysigni cant based on their asymptotic standard errors.

Table 3 reports the correlation matrix for the state variables implied by the max-imum likelihood estimates of the parameters de ning Σ . As shown, a number of thecorrelations are negative. As pointed out by Du ff ee (2002) and Dai and Singleton(2002), it is not possible to capture the empirically-observed humped term structureof interest rate volatility within an a ffi ne model unless there are negative correlationsamong some of the factors. Thus, our results are consistent with these empiricalproperties.

5. THE IMPLIED FINANCING RATE

The instantaneous riskless rate r plays a central role in many continuous-time termstructure models. In addition to being the shortest-maturity rate, r can also beviewed as the cost of borrowing on short-term riskless loans such as those fully securedby riskless Treasury bond collateral. Traditionally, the cost of riskless borrowing isequated to the Treasury-bill rate since this is the rate at which the U.S. Treasurycan borrow short-term funds. Among practitioners, however, the Treasury-bill rate is

generally viewed as a noisy measure of the true riskless rate (see also recent academicwork by Du ff ee (1996)). The reason for this is the widespread belief that the extremeliquidity of Treasury bills makes them worth slightly more than the present value of their cash ows, and hence, that Treasury-bill rates can represent downward biasedestimates of the true cost of riskless borrowing. Some recent papers (for example,Longsta ff (2000)) suggest considering general collateral Treasury repo rates as an

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alternative measure of the riskless rate. The rationale for this measure is that repoloans that are overcollateralized by default-free Treasury bonds are essentially risklessshort-term loans. Because repo loans are nancial contracts rather than securities,however, they may be less a ff ected by liquidity events such as short squeezes (althoughthere are many examples of illiquid nancial contracts).

A useful feature of our approach is that we can solve for the value of r endoge-nously and then contrast it with market rates. This allows us to explore directly thequestion of whether the implied value of r more closely resembles Treasury-bill ratesor repo rates. In this model, the implied rate r represents the cost of carrying a posi-tion in the longer-term Treasury bonds de ning the CMT rates. If these longer-termbonds do not have special liquidity value, then r should represent the riskless interestrate for the market. On the other hand, if longer-term Treasury bonds have liquidityvalue, then the estimated value of r takes on the interpretation of a special repo ratein the sense of Duffi e (1996). Speci cally, since r is implied from the cross section of CMT rates (and from the swap rates), r represents the average or typical short-term

special repo rate for the on-the-run bonds in the sample.11

In this case, r may then beless than the true riskless rate. To re ect the unique role that r plays in this model,we designate r the implied nancing rate.

To make estimates of the implied nancing rate comparable with the three-monthgeneral collateral repo and Treasury-bill rates in the sample, we rede ne the implied nancing rate slightly to be the yield implied by a three-month riskless zero couponbond. 12 Using the maximum likelihood parameter values, the values of the state vari-ables are implied from the data as described previously. Given the values of the statevariables, the value of the riskless bond is obtained directly from the closed-form ex-pression in Eq. (7). Table 4 reports summary statistics for the three-month generalcollateral repo rates, implied nancing rates, and Treasury-bill rates along with thespreads between these rates. Fig. 2 graphs the di ff erence between the implied nanc-ing rate and the Treasury-bill rate, and the di ff erence between the repo rate and theimplied nancing rate.

As shown, the implied nancing rate generally lies between the general collateralrepo rate and the Treasury-bill rate. On average, the implied nancing rate is 8.0basis points below the repo rate, but 27.6 basis points above the Treasury-bill rate.The median implied nancing rate is 2.9 basis points below the median repo rate, but

11 We are using a slightly broader interpretation of the special repo rate since specialrepo rates are typically associated with a speci c Treasury bond. The advantage of

this approach, however, is that since these implied special repo rates are based on thedynamics of r , they re ect not only the current liquidity of the bonds, but also thepossibility of future increases in their liquidity. Thus, this interpretation lends itself well to comparisons with term special repo rates.12 See Chapman, Long, and Pearson (1999) for a discussion of the e ff ects of usingthree-month rates as a proxy for instantaneous rates.

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27.5 basis points above the median Treasury-bill rate. These results are consistentwith the view that the general collateral repo rate may be closer to the actual risklessrate than the Treasury-bill rate.

The spread between the implied nancing rate and the Treasury-bill rate canbe interpreted as a measure of the relative liquidity of Treasury bills and on-the-runTreasury bonds. As shown in Fig. 2, this spread is typically positive, suggesting thatTreasury bills tended to be more liquid than Treasury bonds during much of the sampleperiod. During the 1990-93 period, however, the liquidity of Treasury bonds and billsappears to converge. After the hedge fund crisis of 1998, the implied nancing rateactually dips below the Treasury-bill rate, which suggests that longer-term on-the-runTreasury bonds may have become more liquid that Treasury bills. This could possiblybe related to the fact that the U.S. Treasury no longer auctions one-year Treasurybills on a regular basis.

Note that our estimates of this liquidity spread are consistent with those estimatedby Amihud and Mendelson (1991) and Kamara (1994) from the di ff erences betweenthe yields on off -the-run Treasury notes and Treasury bills. For example, Amihudand Mendelson nd that the spread between o ff -the-run Treasury notes and T-billsaverages 42.8 basis points. Kamara reports an average di ff erence between Treasurynote and bill yields of 34 basis points. Since we use on-the-run bonds while Amihudand Mendelson and Kamara use o ff -the-run bonds in computing the liquidity pre-mium in Treasury bills, one would expect our estimate to be less than theirs by theamount of the liquidity di ff erence between on-the-run and o ff -the-run Treasury bonds.Subtracting our estimate of 27.6 basis points from their estimates implies a liquid-ity diff erence between off -the-run and on-the-run bonds of 6.4 to 15.2 basis points.While we acknowledge that that comparing spreads from di ff erent studies (calculatedusing diff erent data and time periods) is far from a rigorous analysis, it is intriguingthat these estimates are very consistent with the evidence from term repo rates to bepresented later in this section.

If one is willing to equate the general collateral repo rate with the riskless rate,then the di ff erence between the repo rate and the implied nancing rate could be givena simple interpretation of the average implied specialness of the on-the-run Treasurybonds used to compute CMT rates. Fig. 2 shows that this implied specialness variessigni cantly over time. During the rst part of the sample period, the implied special-ness is as high as 40 basis points, suggesting that the prices of Treasury bonds have alarge liquidity component. During the 1994-1998 period, the implied specialness of theTreasury bonds essentially disappears and the implied nancing rate approximates thegeneral collateral repo rate. After the hedge-fund crisis of 1998, however, the impliedspecialness of the bonds increases dramatically, reaching a high of more than 75 basispoints near the end of the sample period.

To provide a simple “back-of-the-envelope” estimate of the size of the liquidity orspecialness component in the prices of on-the-run Treasury bonds, we do the following.

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premia. The two-year, ve-year, and ten-year on-the-run bonds are denoted by anasterisk.

As shown in Table 6, a number of Treasury bonds trade special in the repo mar-ket. For many of these bonds, the di ff erence between the term special and generalcollateral repo rates is small, and the implied specialness premium is likewise small.For the on-the-run bonds, however, the value of the specialness premia is substantial.In particular, the specialness premia for the two-year, ve-year, and ten-year on-the-run bonds given in Table 6 are 8.0 cents, 50.5 cents, and 64.3 cents respectively. 13 Thisagrees well with the average implied specialness premia reported in Table 5. Further-more, as discussed earlier, these estimates of the liquidity premia in on-the-run specialbonds are also in broad agreement with the estimated liquidity premia in Treasurybills relative to on-the-run bonds shown in Table 5 and the average liquidity premiafor Treasury bills relative to o ff -the-run bonds reported by Amihud and Mendelson(1991) and Kamara (1994) (see also Boudoukh and Whitelaw (1993), Longsta ff (1995),Jordan and Jordan (1997), Buraschi and Menini (2001), and Krishnamurthy (2001)).

This provides additional evidence that the model is capturing key features of theTreasury and swap term structures. 14

6. THE SPREAD PROCESS

The spread process λ plays a particularly important role in the Du ffi e and Singleton(1997) credit modeling framework. Recall that in this framework, the spread λ mayconsist of both default-risk and liquidity components. Since the Libor rate is ttedexactly in the maximum likelihood estimation, the implied spread λ can be thought of as the di ff erence between the Libor rate and the implied nancing rate. From Eqs. (3)

and (4), λ is a function of all ve state variables. Table 2 reports that the maximumlikelihood estimate of the parameter γ is -.07126, which implies that there is a strongnegative relation between the level of λ and the level of the riskless rate r . Thisrelation is consistent with the negative relation between rates and spreads implied bya number of fundamental models of credit spreads including Merton (1974), Blackand Cox (1976), and Longsta ff and Schwartz (1995) and documented by Du ff ee (1998)and others. As discussed earlier, an additional in uence on the value of γ could betaxation, since Treasury bills are not taxable for state income tax purposes while Liborcash ows are. That γ is negative, however, argues against the hypothesis that swapspreads are primarily an artifact of di ff erential taxation.

13 There is no on-the-run three-year bond on June 30, 2000 because the Treasury hadpreviously stopped auctioning three-year bonds.14 Conversations with market practitioners indicate that there are occasional periodsduring which the specialness premia for some on-the-run Treasury issues implied bymarket term special repo rates are on the order of twice those shown in Table 6.

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r + b (t)(( β − κ)X t + κθ )) . (18)

The rst term in this expression is the riskless rate and the sum of the remaining termsis the instantaneous term premium for the bond. This term premium is time varying

since it depends explicitly on the state variables. To solve for the unconditional termpremium, we take the expectation over the objective measure of the state variableswhich gives b (t)βθ .

Now applying Ito’s Lemma to the closed-form expression for the price of therisky zero-coupon bond given in Eq. (8) leads to the following expression for theinstantaneous expected return

r + d (t)(( β − κ)X t + κθ )) . (19)

The rst term in this expression is again the instantaneous risky rate. The sum of

the remaining terms can be interpreted as the combined term premium and creditpremium. To solve for the credit premium separately, we simply take the di ff erencebetween the expected return of a risky zero-coupon bond and the expected return ona riskless zero-coupon bond with the same maturity. As before, the credit premiumis time varying through its dependence on the state variables. Taking the expectationwith respect to the objective measure for the state variables and subtracting theexpression for the unconditional term premium gives ( d(t) − b(t)) βθ .

Focusing rst on the unconditional premia, Table 7 reports the unconditionalterm premia for riskless zero-coupon bonds with maturities ranging from one to tenyears. Table 7 also reports the unconditional credit premia for risky zero-coupon

bonds with the same maturities. These unconditional premia are also graphed inFig. 5. As shown, the mean term premia are positive and monotonically increasingfunctions of time to maturity. Mean term premia range from about 97 basis points fora one-year horizon to about 321 basis points for a ten-year horizon. These estimatesof unconditional term premia are similar to those reported by Fama (1984), Fama andBliss (1987), and others.

Table 7 and Fig. 5 also show that unconditional credit premia are positive andincreasing functions of maturity. The mean credit premium for a one-year horizon isonly 0.1 basis points. Thus, there is very little compensation on average for bearingshort-term credit risk. At longer horizons, however, the mean credit premium is muchlarger. For example, the mean credit premium for a ten-year horizon is 45 basispoints. The convex shape of the unconditional credit premium curve indicates thatinvestors require sharply higher credit premia as the maturity of the bond increases.This pattern contrasts with that observed for the unconditional term premia.

To give some sense of the time variation in term and credit premia, Fig. 6 graphsthese premia for a ten-year maturity zero-coupon bond. As illustrated, the term

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Rate Swaps, Financial Management 17, 34-44.

Sun, T., S. M. Sundaresan, and C. Wang, 1993, Interest Rate Swaps: An EmpiricalInvestigation, Journal of Financial Economics 36, 77-99.

Sundaresan, S. M., 1991, Valuation of Swaps, in Sarkis J. Khoury, ed., Recent Devel-opments in International Banking and Finance, vol 5, New York: Elsevier SciencePublishers.

Sundaresan, S. M., 1994, An Empirical Analysis of U.S. Treasury Auctions: Implicationsfor Auction and Term Structure Theories, Journal of Fixed Income , 35-50.

Turnbull, S., 1987, Swaps: Zero-Sum Game?, Financial Management 16, 15-21.

Vasicek, O., 1977, An Equilibrium Characterization of the Term Structure, Journal of Financial Economics 5, 177-188.

Wall, L. D. and J. J. Pringle, 1989, Alternative Explanations of Interest Rate Swaps:A Theoretical and Empirical Analysis, Financial Management 16, 15-21



T a b l e 1

S u m m a r y S t a t i s t i c s f o r t h e D a t a . T h i s t a b l e r e p o r t s t

h e i n d i c a t e d s u m m a r y s t a t i s t i c s

f o r b o t h t h e l e v e l a n d

r s t

d i ff e r e n c e o

f t h e i n d i c a t e d d a t a

s e r i e s . T h e t e r m ρ

r e p r e s e n t s t h e

r s t - o r d e r s e r i a l c o r r e l a t i o n c o e ffi c i e n t . T h e d a t a c o n s i s t o f 7 3 4 w e e k l y o b s e r v a t i o n s

f r o m J a n u a r y 1 9 8 8 t o F e b r u a r y

2 0 0 2 . L i b o r

d e n o t e s t h e t h r e e - m o n t h L i b o r r a t e ,

C M S d e n o t e s t h e s w a p r a t e

f o r t h e i n d i c a t e d m a t u r i t y ,

C M T d e n o t e s t h e c o n s t a n t m a t u r i t y T r e a s u r y

r a t e f o r t h e i n d i c a t e d m a t u r i t y , a n d

S S d e n o t e s t h e s w a p s p r e a d

f o r t h e i n d i c a t e d m a t u r i t y w h e r e t h e s w a p s p r e a d i s d e n e d a s t h e

d i ff e r e n c e

b e t w e e n

t h e c o r r e s p o n d i n g

C M S a n d

C M T r a t e s .

L e v e l s

F i r s t D i ff e r e n c e s

S t a n d a r d

S t a n d a r d

M e a n

D e v i a t i o n

M i n i m u m

M e d i a n

M a x i m u m

ρ

M e a n

D e v i a t i o n

M i n i m u m

M e d i a n M a x i m u m

ρ

L i b o r

5 . 8 1 7

1 . 8 0 1

1 . 7 5 0

5 . 6 8 8

1 0 . 5 6 0

. 9 9 8

- . 0 0 7

. 1 1 3

- . 8 0 0

. 0 0 0

. 5 6 3

. 0 9 7

C M S 2

6 . 4 1 7

1 . 6 1 3

2 . 8 3 8

6 . 2 0 5

1 0 . 7 5 0

. 9 9 5

- . 0 0 7

. 1 5 6

- . 5 5 5

. 0 0 0

. 6 3 8

- . 0 3 1

C M S 3

6 . 6 6 8

1 . 5 1 1

3 . 4 6 6

6 . 3 9 5

1 0 . 5 6 0

. 9 9 5

- . 0 0 7

. 1 5 4

- . 5 1 0

- . 0 1 0

. 6 8 1

- . 0 5 4

C M S 5

6 . 9 9 1

1 . 3 9 5

4 . 2 0 8

6 . 6 9 2

1 0 . 3 3 0

. 9 9 4

- . 0 0 6

. 1 4 8

- . 4 6 4

- . 0 0 7

. 6 5 5

- . 0 7 0

C M S 1 0

7 . 3 6 9

1 . 3 0 5

4 . 9 0 8

7 . 0 5 3

1 0 . 1 3 0

. 9 9 4

- . 0 0 6

. 1 3 7

- . 4 2 0

- . 0 1 0

. 6 0 8

- . 0 8 1

C M T 2

6 . 0 1 9

1 . 5 2 0

2 . 4 0 0

5 . 8 9 5

9 . 8 4 0

. 9 9 6

- . 0 0 7

. 1 4 2

- . 5 0 0

- . 0 1 0

. 4 7 0

. 0 1 8

C M T 3

6 . 1 9 9

1 . 4 4 3

2 . 8 1 0

6 . 0 2 0

9 . 7 6 0

. 9 9 5

- . 0 0 6

. 1 4 3

- . 4 8 0

- . 0 1 0

. 4 8 0

. 0 0 9

C M T 5

6 . 4 7 1

1 . 3 3 0

3 . 5 8 0

6 . 2 4 0

9 . 6 5 0

. 9 9 4

- . 0 0 6

. 1 4 0

- . 4 6 0

- . 0 1 0

. 4 5 0

- . 0 2 1

C M T 1 0

6 . 7 6 7

1 . 2 7 6

4 . 3 0 0

6 . 5 3 0

9 . 4 9 0

. 9 9 5

- . 0 0 6

. 1 2 9

- . 3 8 0

. 0 0 0

. 4 6 0

- . 0 3 5

S S 2

. 3 9 9

. 1 9 8

. 0 4 0

. 3 9 8

. 9 5 1

. 9 1 6

. 0 0 0

. 0 8 1

- . 3 4 0

. 0 0 0

. 3 5 8

- . 4 5 0

S S 3

. 4 7 0

. 2 1 3

. 1 0 0

. 4 9 9

1 . 0 0 9

. 9 3 7

. 0 0 0

. 0 7 5

- . 3 0 0

. 0 0 0

. 3 8 0

- . 4 5 7

S S 5

. 5 2 0

. 2 4 0

. 1 2 0

. 5 2 3

1 . 1 2 1

. 9 5 4

. 0 0 0

. 0 7 3

- . 2 9 0

. 0 0 0

. 4 5 7

- . 4 2 4

S S 1 0

. 6 0 2

. 2 5 3

. 1 5 5

. 5 7 0

1 . 3 4 3

. 9 6 8

. 0 0 0

. 0 6 4

- . 2 8 0

. 0 0 0

. 4 3 7

- . 4 3 5



Table 2

Maximum Likelihood Estimates of the Model Parameters. This table reports the maximum likeli-hood estimates of the parameters of the ve-factor term structure model along with their asymptotic standarderrors. The asymptotic standard errors are based on the inverse of the information matrix computed fromthe Hessian matrix for the log likelihood function.

Parameter Value Std. Error

β 1 15.89001 1.12163β 2 1.03109 .01902β 3 .29298 .00669β 4 .00004 .00001β 5 .00487 .00028

κ 1 10.12189 .78817κ 2 11.67224 .88389κ 3 1.78303 .42228

κ 4 .46752 .14111κ 5 1.61688 .38056

θ1 -.00425 .00170θ2 .00523 .00103θ3 -.03555 .00458θ4 .31595 .07564θ5 -.14303 .00376

σ11 .08453 .00483σ21 -.00326 .00129σ22 .05613 .00125σ31 .00073 .00135σ32 -.00255 .00045σ33 .04297 .00108σ41 -.00354 .00025σ42 .00001 .00006σ43 -.00194 .00041σ44 .00926 .00019σ51 .00008 .00012σ52 .00253 .00024σ53 -.00011 .00022σ54 -.00157 .00019σ55 .00516 .00011

δ 0 -.23047 .07574δ 1 .15072 .00861γ -.07126 .00341

η1 .00122 .00025η2 .00090 .00018η3 .00064 .00014η4 .00060 .00011



T a b l e 4

S u m m a r y S t a t i s t i c s f o r t h e T h r e e - M o n t h G e n e r a l C o l l a t e r a l R e p o R a t e , t h e I m p l i e d T h r e e - M o n t h F i n a n c i n g R a t e , a n d t h e T h r e e -

M o n t h T r e a s u r y - B i l l R a t e . T h i s t a b l e r e p o r t s t h e i n d i c a t e d s u m m a r y s t a t i s t i c s

f o r t h e l e v e l a n d s p r e a d s o f t h e d a t a s e r i e s . T h e t e r m ρ

r e p r e s e n t s

t h e r s t - o r d e r s e r i a l c o r r e l a t i o n c o e ffi c i e n t . T h e d a t a c o n s i s t o f 7 3 4 w e e k l y o b s e r v a t i o n s

f r o m J a n u a r y 1 9 8 8 t o F e b r u a r y 2 0 0 2 .

S t a n d a r d

M e a n

D e v i a t i o n

M

i n i m u m

M e d i a n

M a x i m u m

ρ

G e n e r a l

C o l l a t e r a l R a t e

5 . 5 0 9

1 . 7 4 8

1 . 5 6 0

5 . 4 5 0

1 0 . 1 5 0

. 9 9 8

I m p l i e d F i n a n c i n g R a t e

5 . 4 2 8

1 . 7 7 3

1 . 0 0 4

5 . 4 7 9

1 0 . 2 9 7

. 9 9 7

T r e a s u r y - B i l l R a t e

5 . 1 5 2

1 . 5 5 3

1 . 5 8 0

5 . 0 5 0

9 . 0 3 0

. 9 9 8

G e n e r a l

C o l l a t e r a l M i n u s I m p l i e d F i n a n c i n g R a t e

. 0 8 0

. 2 3 7

- . 6 4 1

. 0 2 9

. 7 6 4

. 8 8 4

I m p l i e d F i n a n c i n g M i n u s T r e a s u r y - B i l l R a t e

. 2 7 6

. 3 5 8

- . 7 1 6

. 2 7 5

1 . 5 5 6

. 9 4 6

G e n e r a l

C o l l a t e r a l M i n u s T r e a s u r y - B i l l R a t e

. 3 5 6

. 2 6 4

- . 2 0 0

. 3 1 0

1 . 4 1 0

. 9 3 0



T a b l e 5

S u m m a r y S t a t i s t i c s f o r t h e L i q u i d i t y o r S p e c i a l n e s s P r e m i a i n O n - T

h e - R u n T r e a s u r y B o n d P r i c e s . T h i s t a b l e r e p o r t s t h e i n d i c a t e d

s u m m a r y s t a t i s t i c s

f o r t h e e s t i m a t e d

l i q u i d i t y o r s p e c i a l n e s s p r e m i u m i n c o r p o r a t e d i n t o t h e v a l u e s o f t h e T r e a s u r y

b o n d s . T h e p r e m i u m i s e s t i m a t e d

f r o m t h e s p r e a d

b e t w e e n t h e g e n e r a l c o l l a t e r a l r e p o r a t e a n d t h e i m p l i e d

n a n c i n g r a t e . T h e p r e m i a a r e r e p o r t e d i n u n i t s o f

d o l l a r s p e r

$ 1 0 0 d o l l a r

n o t i o n a l a m o u n t . T h e d a t a c o n s i s t o f 7 3 4 w e e k l y o b s e r v a t i o n s

f r o m J a n u a r y 1 9 8 8 t o J u n e 2 0 0 2 .

S t a n d a r d

M e a n

D e v i a t i o n

M i n i m u m

M e d i a n

M a x i m u m

ρ

T w o - Y e a r T r e a s u r y B o n d

. 1 4 9 0

. 0 3 6 5

. 0 3 9 5

. 1 4 2 2

. 2 5 8 6

. 8 8 9 7

T h r e e - Y e a r T r e a s u r y B o n d

. 2 0 9 9

. 0 3 6 0

. 1 0 4 1

. 2 0 2 9

. 3 2 2 9

. 8 9 9 6

F i v e - Y e a r T r e a s u r y B o n d

. 3 0 5 9

. 0 3 8 6

. 2 0 3 3

. 2 9 8 5

. 4 2 7 2

. 9 3 1 8

T e n - Y e a r T r e a s u r y B o n d

. 4 3 2 7

. 0 6 1 4

. 3 1 3 4

. 4 2 3 4

. 5 7 3 9

. 9 8 2 6



T a b l e 6

S p e c i a l a n d G e n e r a l C o l l a t e r a l T e r m R e p o R a t e s a n d t h e I m p l i e d S p e c i a l n e s s P r e m

i a f o r T r e a s u r y B o n d s a s o f J u n e 3 0 , 2 0 0 0 . T h i s

t a b l e r e p o r t s t h e l o n g e s t q u o t e d t e r m s p e c i a l r e p o r a t e s f o r t h e i n d i c a t e d T r e a s u r y

b o n d s a l o n g w i t h t h e c o r r e s p o n d i n g t e r m g e n e r a l c o l l a t e r a l r e p o

r a t e . T h e i m p l i e d s p e c i a l n e s s i s c o m p u t e d

b y m u l t i p l y i n g t h e d i ff e r e n c e

b e t w e e n t h e t w o r a t e s

b y t h e t e r m o f t h e r e p o r a t e s m e a s u r e d i n y e a r s a n d

r e p r e s e n t s t h e s p e c i a l n e s s p r e m i u m p e r

$ 1 0 0 v a l u e . T h e o n - t h e - r u n

i s s u e s a r e

d e n o t e d b y a n a s t e r i s k .

L o n g e s t R e p o

G e n e r a l

C o l l a t e r a l

S p e c i a l T e r m

I m p l i e d

C o u p o n

M

a t u r i t y

T e r m i n D a y s

T e r m R e p o R a t e

R e p o R a t e

D i ff e r e n c e

S p e c i a l n e s s

5 . 8 7 5

3 0 - N o v - 0 1

9 2

. 0 6 4 6 1

. 0 6 4 5 0

. 0 0 0 1 1

. 0 0 3

6 . 6 2 5

3 1 - M a y - 0 2

7 8

. 0 6 4 2 4

. 0 6 0 5 0

. 0 0 3 7 4

. 0 8 0

6 . 3 7 5

3 0 - J u n - 0 2

3 2

. 0 6 3 4 6

. 0 5 5 5 0

. 0 0 7 9 6

. 0 7 0

5 . 2 5 0

1 5 - A u g - 0 3

9 2

. 0 6 4 6 1

. 0 6 4 0 0

. 0 0 0 6 1

. 0 1 6

4 . 2 5 0

1 5 - N o v - 0 3

2 8 6

. 0 6 7 1 4

. 0 6 6 5 0

. 0 0 0 6 4

. 0 5 0

4 . 7 5 0

1 5 - F e b - 0 4

9 2

. 0 6 4 6 1

. 0 6 3 5 0

. 0 0 1 1 1

. 0 2 8

6 . 0 0 0

1 5 - A u g - 0 4

9 2

. 0 6 4 6 1

. 0 6 4 0 0

. 0 0 0 6 1

. 0 1 6

5 . 8 7 5

1 5 - N o v - 0 4

9 2

. 0 6 4 6 1

. 0 6 3 0 0

. 0 0 1 6 1

. 0 4 1

6 . 7 5 0

1 5 - M a y - 0 5

3 5 8

. 0 6 7 6 5

. 0 6 2 5 0

. 0 0 5 1 5

. 5 0 5

6 . 5 0 0

1 5 - M a y - 0 5

9 3

. 0 6 4 6 2

. 0 6 3 5 0

. 0 0 1 1 2

. 0 2 9

5 . 6 2 5

1 5 - F e b - 0 6

9 3

. 0 6 4 6 2

. 0 6 4 0 0

. 0 0 0 6 2

. 0 1 6

6 . 8 7 5

1 5 - M a y - 0 6

9 3

. 0 6 4 6 2

. 0 6 4 0 0

. 0 0 0 6 2

. 0 1 6

7 . 0 0 0

1 5 - J u l - 0 6

9 3

. 0 6 4 6 2

. 0 6 3 5 0

. 0 0 1 1 2

. 0 2 9

6 . 5 0 0

1 5 - O c t - 0 6

9 3

. 0 6 4 6 2

. 0 6 4 0 0

. 0 0 0 6 2

. 0 1 6

6 . 2 5 0

1 5 - F e b - 0 7

9 2

. 0 6 4 6 1

. 0 6 4 0 0

. 0 0 0 6 1

. 0 1 5

6 . 6 2 5

1 5 - M a y - 0 7

9 3

. 0 6 4 6 2

. 0 6 4 0 0

. 0 0 0 6 2

. 0 1 6

6 . 1 2 5

1 5 - A u g - 0 7

9 3

. 0 6 4 6 2

. 0 6 4 0 0

. 0 0 0 6 2

. 0 1 6

5 . 5 0 0

1 5 - F e b - 0 8

9 3

. 0 6 4 6 2

. 0 6 4 0 0

. 0 0 0 6 2

. 0 1 6

5 . 6 2 5

1 5 - M a y - 0 8

9 3

. 0 6 4 6 2

. 0 6 4 0 0

. 0 0 0 6 2

. 0 1 6

4 . 7 5 0

1 5 - N o v - 0 8

1 8 3

. 0 6 5 9 2

. 0 6 5 0 0

. 0 0 0 9 2

. 0 4 6

5 . 5 0 0

1 5 - M a y - 0 9

2 5 9

. 0 6 6 8 7

. 0 6 6 0 0

. 0 0 0 8 7

. 0 6 2

6 . 0 0 0

1 5 - A u g - 0 9

2 9 8

. 0 6 7 2 4

. 0 6 6 0 0

. 0 0 1 2 4

. 1 0 1

6 . 2 5 0

1 5 - F e b - 1 0

3 5 4

. 0 6 7 6 3

. 0 6 1 0 0

. 0 0 6 6 3

. 6 4 3



T a b l e 7

U n c o n d i t i o n a l T e r m a n d C r e d i t P r e m i a . T h i s t a b l e r e p o r t s u n c o n d i t i o n a l v a l u e s o f t h e t e r m p r e m i a i n z e r o - c o u p o n T r e a s u r y

b o n d p r i c e s i m p l i e d

b y t h e m o d e l .

A l s o r e p o r t e d a r e t h e u n c o n d i t i o n a l v a l u e s o

f t h e c r e d i t p r e m i a i n z e r o - c o u p o n r i s k y

b o n d s i m p l i e d

b y t h e m o d e l .

M a t u r i t y i n Y e a r s

1

2

3

4

5

6

7

8

9

1 0

T e r m P r e m i u m

. 9 6 6

1 . 5 0 9

1 . 9 6 3

2 . 3 1 9

2 . 5 9 1

2 . 7 9 5

2 . 9 4 8

3 . 0 6 1

3 . 1 4 6

3 . 2 0 8

C r e d i t P r e m i u m

. 0 0 1

. 0 3 1

. 0 6 8

. 1 1 1

. 1 5 9

. 2 1 3

. 2 6 9

. 3 2 9

. 3 8 9

. 4 5 2



88 90 92 94 96 98 00 020

50

100

150

88 90 92 94 96 98 00 020

50

100

150

88 90 92 94 96 98 00 020

50

100

150

88 90 92 94 96 98 00 020

50

100

150

Ì

Û

Ó

¹

Ö

Ë

Û

Ô

Ë

Ô

Ö

Ì

Ö

¹

Ö

Ë

Û

Ô

Ë

Ô

Ö

Ú

¹

Ö

Ë

Û

Ô

Ë

Ô

Ö

Ì

Ò

¹

Ö

Ë

Û

Ô

Ë

Ô

Ö

Ù Ö ½ º Ë Û Ô Ë Ô Ö × º Ï Ð Ý Ø Ñ × Ö × Ó × Û Ô × Ô Ö × Ñ × Ù Ö Ò

× × Ô Ó Ò Ø × º Ì × Ñ Ô Ð Ô Ö Ó × Ö Ó Ñ Â Ò Ù Ö Ý ½ Ø Ó Ö Ù Ö Ý ¾ ¼ ¼ ¾ º



88 90 92 94 96 98 00 02

−50

0

50

100

150

88 90 92 94 96 98 00 02

−50

0

50

100

150

Á

Ñ

Ô

Ð

Ö

¹

Ì

¹

Ð

Ð

Ê

Ø

¹

Á

Ñ

Ô

Ð

Ö

Ù Ö ¾ º Á Ñ Ô Ð Ò Ò Ò Ê Ø Ë Ô Ö × º Ï Ð Ý Ø Ñ × Ö × Ó Ø

× Ô Ö Ø Û Ò Ø Ñ Ô Ð ¬ Ò Ò Ò Ö Ø Ö Ò Ø Ì Ö × Ù Ö Ý ¹ Ð Ð Ö Ø ¸ Ò Ó

Ø × Ô Ö Ø Û Ò Ø Ò Ö Ð Ó Ð Ð Ø Ö Ð Ö Ô Ó Ö Ø Ò Ø Ñ Ô Ð ¬ Ò Ò Ò

Ö Ø º Ë Ô Ö × Ö Ñ × Ù Ö Ò × × Ô Ó Ò Ø × º Ì × Ñ Ô Ð Ô Ö Ó × Ö Ó Ñ Â Ò Ù Ö Ý

½ Ø Ó Ö Ù Ö Ý ¾ ¼ ¼ ¾ º



88 90 92 94 96 98 00 02

0

20

40

60

80

100

120

140

Á

Ñ

Ô

Ð

Ë

Ô

Ö

Ù Ö ¿ º Ì Á Ñ Ô Ð Ö Ø Ë Ô Ö º Ì Û Ð Ý Ø Ñ × Ö × Ó Ø

Ñ Ô Ð Ö Ø × Ô Ö º Ì Ö Ø × Ô Ö × Ñ × Ù Ö Ò × × Ô Ó Ò Ø × º Ì

× Ñ Ô Ð Ô Ö Ó × Ö Ó Ñ Â Ò Ù Ö Ý ½ Ø Ó Ö Ù Ö Ý ¾ ¼ ¼ ¾ º



0 20 40 60 80 100 120−50

0

50

100

150

−80 −60 −40 −20 0 20 40 60 80−50

0

50

100

150

Ù Ð Ø ¹ Ê × È Ö Ó Ü Ý

Ä Õ Ù Ø Ý È Ö Ó Ü Ý

Ö

Ø

Ë

Ô

Ö

Ö

Ø

Ë

Ô

Ö

Ù Ö º Ë Ø Ø Ö Ö Ñ × Ó Ø Ö Ø Ë Ô Ö Ò × Ø Ù Ð Ø Ê ×

Ò Ä Õ Ù Ø Ý È Ö Ó Ü × º Ì Ø Ó Ô Ö Ô Ô Ð Ó Ø × Ø Ñ Ô Ð Ö Ø × Ô Ö Ò × Ø

Ø Ù Ð Ø ¹ Ö × Ô Ö Ó Ü Ý º Ì Ó Ø Ø Ó Ñ Ö Ô Ô Ð Ó Ø × Ø Ñ Ô Ð Ö Ø × Ô Ö Ò × Ø

Ø Ð Õ Ù Ø Ý Ô Ö Ó Ü Ý º Ì Ö Ø × Ô Ö Ò Ø Ù Ð Ø ¹ Ö × Ò Ð Õ Ù Ø Ý Ô Ö Ó Ü ×

Ö Ñ × Ù Ö Ò × × Ô Ó Ò Ø × º



1 2 3 4 5 6 7 8 9 1050

100

150

200

250

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Å Ø Ù Ö Ø Ý

Å Ø Ù Ö Ø Ý

Ì

Ö

Ñ

È

Ö

Ñ

Ù

Ñ

Ö

Ø

È

Ö

Ñ

Ù

Ñ

Ù Ö º Í Ò Ó Ò Ø Ó Ò Ð È Ö Ñ º Ì Ø Ó Ô Ö Ô Ô Ð Ó Ø × Ø Ù Ò Ó Ò Ø Ó Ò Ð

Ø Ö Ñ Ô Ö Ñ Ù Ñ Ó Ö × Ð × × Þ Ö Ó ¹ Ó Ù Ô Ó Ò Ó Ò Ò × Ø Ø Ñ Ø Ù Ö Ø Ý Ó Ø Ó Ò Ò

Ý Ö × º Ì Ó Ø Ø Ó Ñ Ö Ô Ô Ð Ó Ø × Ø Ù Ò Ó Ò Ø Ó Ò Ð Ö Ø Ô Ö Ñ Ù Ñ Ó Ö × Ý Þ Ö Ó ¹

Ó Ù Ô Ó Ò Ó Ò Ò × Ø Ø Ñ Ø Ù Ö Ø Ý Ó Ø Ó Ò Ò Ý Ö × º È Ö Ñ Ö Ñ × Ù Ö

Ò × × Ô Ó Ò Ø × º



88 90 92 94 96 98 00 02

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

88 90 92 94 96 98 00 02

−0.05

0

0.05

0.1

Ì

Ö

Ñ

È

Ö

Ñ

Ù

Ñ

Ö

Ø

È

Ö

Ñ

Ù

Ñ

Ù Ö º Ó Ò Ø Ó Ò Ð È Ö Ñ º Ì Ø Ó Ô Ö Ô Ô Ð Ó Ø × Ø Ó Ò Ø Ó Ò Ð Ø Ö Ñ

Ô Ö Ñ Ù Ñ Ó Ö Ö × Ð × × Ø Ò ¹ Ý Ö Þ Ö Ó ¹ Ó Ù Ô Ó Ò Ó Ò º Ì Ó Ø Ø Ó Ñ Ö Ô Ô Ð Ó Ø × Ø

Ó Ò Ø Ó Ò Ð Ö Ø Ô Ö Ñ Ù Ñ Ó Ö Ö × Ý Ø Ò ¹ Ý Ö Þ Ö Ó ¹ Ó Ù Ô Ó Ò Ó Ò º Ì × Ñ Ô Ð

Ô Ö Ó × Ö Ó Ñ Â Ò Ù Ö Ý ½ Ø Ó Ö Ù Ö Ý ¾ ¼ ¼ ¾ º

the market price of credit risk

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