Credit and Basket Default Swaps - CiteSeerX

Credit and Basket Default Swaps Dilip B. Madan, Robert H. Smith School of Business and Bloomberg LP. Michael Konikov, Mircea Marinescu Bloomberg LP 499 Park Avenue New York, NY 10022 September 26, 2004 Abstract We evaluate extreme value distribution models for the time to default distribution embedded in market credit default swap quotes. Two distribution classes, the Weibull and Frechet are considered and it is observed that though both are adequate, the Weibull model has a better ¯t to the data. We also provide details for the pricing of N th to default contracts using the one factor Gaussian and the Clayton copulas. An illustration of the ¯rst, second and third to default contracts on a basket of six ¯rms in the ¯nancial sector is conducted. Gaussian ¯rst to default prices are seen to be closer to the independence boundary relative to the Clayton copula prices. The Weibull is shown to possibly also be adequate for the distribution of the ¯rst, second and third to default times.

1

Introduction

The pricing of default risk has been recognized in the corporate bond markets and models for pricing corporate bonds have priced default in a variety of ways. Early work by Merton (1974) modeled default on a bond as a put option on the ¯rm value held by the equity holders. This was extended to permit early default and hence have nondegenerate default time density in Longsta® and Schwartz (1995). Jarrow and Turnbull (1995), among others introduced the reduced form approach that focused attention directly on modeling the instantaneous likelihood of default or what is now also referred to as the hazard rate of default. These perspectives have been built into the pricing of corporate bonds (Du±e and Singleton (1997), Du®ee (1999)) or sovereign debt (Du±e, Pedersen and Singleton (2003)). Additionally we also have the issues of strategic default developed in Leland (1994), Leland and Toft (1996). The ¯nancial markets have recently developed securities that price directly the credit event called credit default swaps. The ¯rst such security was issued 1

by Deutsche Bank in 2000. These securities payout only on the occurrence of a credit event, like a bond default and they have prices quoted in basis points as a percentage of a notional. We refer the reader to SchÄonbucher (2003) for a detailed description of the contract. The data base of price quotes on these contracts is expanding but there has been relatively little work done on directly modeling these prices, with the exception of Houweling and Vorst (2001) and Cossin and Hricko (2001). The general theory for pricing such contracts is set out in Arvanitis, Gregory and Laurent (2001) and SchÄonbucher (2003). This paper formulates a class of parametric closed form models for pricing a variety of credit derivative contracts and reports on the results of estimation, using market quotes. Two contracts are studied, these are the credit default swap, and the nth to default in a basket of m > n obligors, with special attention on the ¯rst to default. Critical to the modeling is the speci¯cation of the joint density of default times and their correlation or dependence. A number of approaches are possible. One could specify a variety of stochastic processes for interest rates and the underlying ¯rm values, de¯ne the default event, and infer the distribution of default times or the joint densities as a consequence. This is in keeping with what is now called the structural approach Bielecki and Rutkowski (2001). For a recent application to credit spreads we refer to Giesecke (2004). Alternatively, one may directly model hazard rates as adapted to economic information and then infer the default time densities by computing the implied survival probabilities. This corresponds to the reduced form approach referred to above in the context of bond pricing. For an application to credit default swaps we note the recent paper by Yu (2003). From a modeling perspective both of these methods are indirect as they specify or directly model ¯rm value processes or hazard rates and infer the densities of default times. We take a more direct approach and model the default time densities themselves. For the marginal densities of ¯rm default times we employ probabilities from the class of extreme value distributions. Speci¯cally, we use the Weibull and Frechet families of densities. Joint densities are constructed using the method of copulas. Two copulas are studied, these are the one factor Gaussian copula and the Clayton copula. The remainder of the paper is organized as follows. Section 2 presents the model for the marginal default time densities. The credit default swap is priced in section 3. Section 4 summarizes the data, estimation methodology and empirical results on evaluating the Weibull and Frechet models for the CDS rates. In section 5 we present the copulas to be considered in pricing the j th to default and we price this contract in section 6. Section 7 presents an illustrative study of the ¯rst, second and third to default contracts for a basket of ¯nancials. Section 8 concludes.

2

2

The Marginal Default Time Density

We suppose the probability space for the economy is (-; F ; P ) where - is the event space and P is the true probability measure de¯ned on the ¾ ¡ f ield F of subsets of -: The information ¯ltration for the economy is given by the right continuous, increasing and complete family of ¾ ¡ f ields Ft µ F for 0 < t · 1 with F1 = F : Consider an obligor in the economy with some outstanding debt exposed to the risk of default. We view the obligor as exposed to unanticipated losses due to unforeseen circumstances. In a simple view of the world we may take the magnitude of such losses as given and the ¯nancial standing of the obligor is sound enough to withstand a large number of these events. However, for some large value of N of the mumber of losses, the obligor will have to default if there are N such losses. Let T i be the time of the i th unanticipated loss. We suppose that these loss times are identically distributed and default occurs at ¿ where the default time is ¿ = M ax fT i j1 · i · N g : The default time is then seen as the maximum of a large number of draws from some ¯xed distribution. The distribution of such a time ¿ then belongs in the family of extreme value distributions. It is well known (Embrechts, Kluppellberg and Mikosch (1997)) that these distributions are of three types, Frechet, Gumbell of Weibull. By way of a distributional model on the positive half line, we focus attention on the Frechet and the Weibull. We report results using both distributional families but our evidence supports the Weibull model in favor of the Frechet. We also note that the Weibull has the additional °exibility of accomodating increasing as well as decreasing hazard rates and this is useful given the concentration of our interest in a range of maturities between one to possibly ten years. We shall conduct the discussion primarily with respect to the Weibull family. The Weibull density has two parameters, a scale parameter c; and a shape parameter a: The density is best described by its cumulative distribution function F (t) that de¯ned by µ µ ¶a ¶ t F (t) = 1 ¡ exp ¡ c The density function is µ µ ¶a¶ a a¡ 1 t p(t) = a t exp ¡ c c and the hazard rate is given by h(t) =

a a¡1 t ca

and we have increasing hazard rates for a > 1: The moments of the Weibull 3

distribution are also easily computed and given by µ ¶ £ k¤ k E T =¡ + 1 ck ; a

where T denotes the random variable with the Weibull distribution. The Weibull model is a popular model and it is widely used in the study of a variety of lifetimes. For a comprehensive and recent reference we cite Murthy, Xie and Jiang (2003). In general the pricing of claims is done under a risk neutral measure Q that is equivalent to P and de¯ned on the same ¾ ¡ f ield F T : Such a change of measure induces a change of probability on the random variables in the economy and the distribution of the time of default under Q may not be the same as that under P: Since our interest is in the market prices of credit default swaps we model directly the risk neutral probability as belonging to the Weibull family. The default time, risk neutrally, is also the maximum of a large number of random variables and this is suggestive of a continued use of the extreme value distributions, under the new probability. When considering the j th to default contract for a basket of names we shall take the marginal risk neutral distributions of the default time for each name to be in the Weibull class with parameters as estimated by the credit default swap market for each name separately. These marginal distributions are put together to construct joint densities using the method of copulas.

3

Pricing Credit Default Swaps

The credit default swap contract is best viewed in two parts, one describing the receipt side of the cash °ows and the other the payment side. There is a notional amount M associated with the receipt side and the actual level of cash °ows received is M times one minus the recovery rate R on the defaulting asset. These funds are received on the default date ¿ : Against this stream of receipts the holder makes regular coupon payments, typically until the default date or the maturity whichever is smaller. At the default date one o®sets against the receipt the accrued coupons from the last coupon date before ¿ to this date. We denote time measured in days by n and in years by t: For each day n we de¯ne the function ³ (n) that gives the number days between the end of day n and the last day on which a coupon was paid. We also denote by N the maturity of the contract in days while T is the maturity in years. By convention we take (N; n) = 360(T ; t). Let ¢(n) be the default indicator function that takes the value 1 if default has occurred on or before day n and is zero otherwise. Further, let kT denote the annual coupon rate quoted on the contract for maturity T :The cash °ow receipts on day n; Rn are then written as µ ¶ M kT (n ¡ ³(n)) Rn = (¢(n) ¡ ¢(n ¡ 1)) M (1 ¡ R) ¡ 360 4

Let N P denote the number of coupon payment dates and let np j be the day number of the the j th payment date. The j th payment P j occurring on day np j is then given by M kT (npj ¡ np j¡1 ) (1 ¡ ¢(np j )) ; 360 where the multiplication by the complementary default indicator function recognizes that no coupon payments are made after a default date. For the valuation of claims we employ a discount function Bn that gives the present value of a dollar promised on day n: There are a variety of approaches to constructing such discount functions. For example, one may ¯t to data on Treasury swap rates a yield curve in the Nelson-Siegel parametric form with Pj =

rt = a + (b + ct)e ¡dt and then de¯ne Bn

= exp (¡rt t) n t = : 365 The random present value of cash °ows to the credit default swap contract, V (T ) is then given by V (T ) =

N X

n=1

Rn Bn ¡

N P X

P j Bnpj

j=1

We are supposing here that conditional on the outcome of the default or the realization of the default indicator function the expected values of future dollars are the same. This is equivalent to supposing that the defaults of single names in the economy do not contain any information about macro movements in interest rates or that interest rate evolutions are independent of the default process. We leave for future research the modeling of joint evolutions of interest rates and defaults. The credit default swap quotes in markets are set at levels consistent with a zero price at the initiation of the swap contract. Hence we have that under the risk neutral measure E Q [V (T )] = 0 From our risk neutral Weibull distribution for the default time we infer that E Q [(1 ¡ ¢(n))] = F n (c; a) Def ³ ³ n ´a ´ = exp ¡ 365c The probability of a default on a particular day n may be approximated by the density of default times the length of the day in years. Hence we have that E Q [(¢(n) ¡ ¢(n ¡ 1))] = ¼ n (c; a) Def

³ ³ n ´a ´ a ³ n ´a¡1 1 = exp ¡ 365c ca 365 365 5

We may then compute the value of the credit default swap as ¸ N · X M kT (n ¡ ³ (n)) n ¼ (c; a) ¡ E Q [V (T )] = M (1 ¡ R) ¡ B 365 n 360 n=1 NP X M kT (np j ¡ npj¡ 1 ) B npj F npj (c; a) 365 365 360 j=1

Setting this value to zero and solving for the kT we obtain the Weibull credit default swap pricing model PN n ¼ (c; a) (1 ¡ R) n=1 B 365 n kT (c; a) = PNP (np ¡np ) (1) PN n¡ ³(n) j j¡1 n ¼ n (c; a) B npj F npj (c; a) + n=1 360 B 365 j=1 360 365

365

Equation (1) provides us with a two parameter credit default swap pricing model and it may be estimated from market prices quoting coupon rates on default swaps in basis points. An alternative model is based on the Frechet extreme value distribution G(t) that also has two parameters that we denote by c; a with the speci¯cation ³ ³ c ´a ´ G(t) = exp ¡ : t The scale parameter is c and the shape parameter is a: The probability density is given by aca g(t) = G(t) 1+a t For large t we note that G(t) approaches unity and the density has a power law tail that is associated with a hazard rate that decreases like t¡ 1 : We may replace the probability of no default by time n by

=

E Q [(1 ¡ ¢(n))] = G n (c; a) Def µ µ ¶a ¶ 365c exp ¡ n

and the probability of a default on a particular day n may be approximated by the Frechet density of default times the length of the day in years to get E Q [(¢(n) ¡ ¢(n ¡ 1))] = ¼ G n (c; a) Def

µ µ ¶a ¶ µ ¶ ¡(1+a) 365c 365 1 = exp ¡ aca n n 365 in the formula (1) to build the Frechet default swap model. This gives the Frechet based credit default swap rate as PN G n ¼ (c; a) (1 ¡ R) n=1 B 365 n G kT (c; a) = PN P (np ¡ np ) : (2) PN n¡ ³(n) j j¡1 np np n ¼ G (c; a) B j G j (c; a) + n=1 360 B 365 n j= 1 360 365

365

6

We report on the results of estimating these two models and testing them against each other using non-nested tests based on non-linear least squares analysis.

4

Data and Estimation Methodology

We obtained data on Credit Default Swap (CDS) rates on 21 companies over the period July 16 2003 to February 20 2004 on a daily basis for the three maturities of 1, 3 and 5 years. We partitioned this period into 11 subperiods of 20 days each. For each name and each subperiod we estimated by non-linear least squares the parameters of the Weibull and Frechet probability models for the distribution of the default time as embedded in the CDS rate models given by equations (1) and (2). The discount curves employed were obtained from Bloomberg. We report in Table 1 the average values for the Weibull and Frechet parameters over the 11 subperiods for each of the 21 names along with the average root mean square error in basis points for these estimations. TABLE 1 Average Weibull and Frechet Parameter Values Weibull Frechet Name a c RMSE a c BAC 1.9740 41.0441 0.6086 0.3756 235.2088 BSC 1.8435 37.0942 1.5622 0.3791 164.7747 CGI 1.9423 43.0353 0.9308 0.3643 270.7991 GS 1.7768 39.7373 1.2279 0.3633 193.1117 JPM 1.7966 42.0495 1.5243 0.3561 236.8073 LEH 1.5733 51.5518 1.3472 0.3244 307.0647 MER 1.5651 50.0127 1.3529 0.3251 276.7 MWD 1.8416 36.0546 1.5729 0.3868 148.6257 BACC 1.6724 37.1682 3.6513 0.3678 157.4412 CAT 1.7786 53.6282 1.9928 0.3315 421.4911 GE 2.0281 33.7199 0.9893 0.4256 164.6839 EDS 1.3156 21.1828 12.5504 0.4303 25.5902 IBM 1.6127 52.2856 8.3461 0.3272 313.3166 CA 1.3131 40.9251 6.7404 0.3403 108.9091 AOL 1.5318 36.82 3.6275 0.3679 106.5789 DIS 1.3868 53.9697 3.3481 0.3180 259.0787 VIA 1.5274 67.4905 1.9109 0.2980 542.6367 BCS 1.5938 51.3989 1.7365 0.3297 276.8461 SBC 1.5570 47.1867 2.0354 0.3309 230.7631 ATT 1.6039 24.6365 6.7028 0.4321 49.9827 VZGF 1.4786 49.2160 4.8730 0.3269 236.2542

RMSE 0.7309 1.9018 1.3184 1.8854 2.0903 2.0746 2.0864 1.1568 4.8354 2.1867 1.4157 15.80 8.4644 7.5482 3.9647 4.1930 2.1803 1.3216 2.8524 7.6651 5.6433

We observe from Table 1 that the Weibull model generally provides a better ¯t to the CDS data than the Frechet model. It is of interest to ascertain whether 7

this improvement is signi¯cant in a statistical sense or whether both models are equally acceptable as models for the risk neutral distribution of default times for single names implied by the CDS rate curves. In order to assess this question we performed non-nested tests based on the parameter encompassing principle. We take for the null hypothesis for example the Weibull model and ask if the Frechet parameters estimated from Weibull predicted CDS rates di®er signi¯cantly from those estimated using the original data. This produces an F test statistic for the test of equality of estimated parameters to the values obtained from the original Frechet estimation. We term this statistic F-Weibull for the null hypothesis of Weibull and evaluate its P value from the F density. This P value is the statistic P-Weibull. If we fail to reject equality and observe a P-Weibull value above :05 values then we accept the Weibull as an adequate model, otherwise we have evidence in favor of the Frechet over the Weibull. We then reverse the procedue and ask if the Weibull estimates from the data di®er signi¯cantly from the estimates obtained from using Frechet predictions of the CDS rates. This produces the F-Frechet statistic and the associated PFrechet P values. Table 2 reports the average P-Weibull and P-Frechet statistics over the eleven subperiods for the 21 names studied. TABLE 2 Weibull and Frechet Name P-Weibull AOL 0.8920 ATT 0.8201 BAC 0.9505 BACC 0.3334 BLS 0.6573 BSC 0.9583 CA 0.5590 CAT 0.8372 CGI 0.9282 DIS 0.6967 EDS 0.9243 GE 0.8567 GS 0.8409 IBM 0.9159 JPM 0.7355 LEH 0.6060 MER 0.7149 MWD 0.7504 SBC 0.6911 VIA 0.9814 VZGF 0.7516

P-Values P-Frechet 0.7434 0.4919 0.8118 0.1048 0.9523 0.7089 0.4345 0.6945 0.6011 0.2670 0.2987 0.7077 0.4669 0.7584 0.3757 0.2558 0.2843 0.8939 0.2291 0.5697 0.2999

We observe from Table 2 that both the Weibull and Frechet models are adequate in that neither is rejected in its ability to explain the estimation of 8

the alternative. However, the Weibull P values are signi¯cantly larger, with two exceptions (BLS and MWD). We conclude that of the two models the Weibull provides a better ¯t to the data on CDS rates in markets.

5

Joint Densities for Default Times

For the pricing of the ¯rst or m th to default among a set of M names one needs the joint density of the default times. For example, if ¿ i is the default time of the ith company in a basket of M companies then the time of the ¯rst default ¿ (1;M ) is ¿ (1;M ) = M in(¿ i; 1 · i · M ) and ¿ (1;M ) > a just if ¿ i > a for all i: This requires a speci¯cation of the joint densities of each of the default times. In the previous section we described how to employ market data on credit default swap quotes to identify marginal default time densities from market quotes. We now wish to construct joint densities consistent with these already identi¯ed marginals. The general approach to the solution of such a problem is to employ a copula, as they describe all the joint densities consistent with given marginals. A copula function is a joint distribution function de¯ned on the unit M ¡ cube; that is consistent with uniform marginal distributions. An arbitrary joint density of default times F (t1 ; ¢ ¢ ¢ tM ) consistent with the estimated Weibull marginals F i(ti); 1 · i · M has the form F (t1 ; ¢ ¢ ¢ tM ) = C (F 1 (t1 ); ¢ ¢ ¢ ; FM (tM )) for some copula C (u1 ; ¢ ¢ ¢ ; u M ). This is Sklar's theorem. We may also de¯ne a copula from any joint density F on recognizing that ¡ ¢ ¡1 C (u1 ; ¢ ¢ ¢ ; u M ) = F F 1¡ 1 (u1 ); ¢ ¢ ¢ ; F M (u M )

is always a copula when we take for the Fi0 s the marginals of F: There are many choices of copulas available in the literature that permit various structural approaches to modeling dependency among the given variables. We consider here two copulas, the one-factor Gaussian copula and the Clayton copula. The one-factor Gaussian copula is widely used and well understood. However, it fails to incorporate tail dependence and correlates the variables near the mean and not in the tails. The Clayton copula parameterizes dependence in the tail and has recently been shown to be a limiting copula for a wide class of joint distributional models. We present results on modeling the nth to default using these two copulas. For completeness we brie°y describe in two short subsections these two copulas and some of their properties.

5.1

The one factor Gaussian copula

The one factor Gaussian copula is the copula associated with multivariate normal random variables that display the correlation structure induced by linear 9

dependence on a single normally distributed factor. We will relate our time changes to these multivariate normal vectors by nonlinear transformations and hence we may restrict attention in the discussion of the copula to standard normal variates. Suppose then that X1 ; ¢ ¢ ¢ ; XM are marginally standard normal variates. We allow for dependence through linear dependence on a single factor V that also has a standard normal distribution. Speci¯cally we suppose that q Xi = ½i V + 1 ¡ ½2i Wi; i = 1; ¢ ¢ ¢ ; M

where the variables W i; i = 1; ¢ ¢ ¢ ; M are independent standard normal variates. The correlation structure induced by this model has the correlation matrix ½½T where ½ = (½1 ; ¢ ¢ ¢ ; ½M )T : Let ©(x) denote the standard normal cumulative distribution function. The copula de¯ned by the above model is denoted C ½ (u 1 ; ¢ ¢ ¢ ; u M ) for uniform marginal variates u1 ; ¢ ¢ ¢ ; u M , and is de¯ned as the probability P of the following event ¡ ¢ C ½ (u 1; ¢ ¢ ¢ ; u M ) = P X1 < ©¡ 1 (u1 ); ¢ ¢ ¢ ; © ¡1 (u M ) : This copula may be explicitly computed by conditioning on the factor V and then integrating the result with respect to the density of V; Á(v): The result is à ! Z 1 Y M © ¡1 (u i) ¡ ½i v C ½ (u 1; ¢ ¢ ¢ ; u M ) = © p Á(v)dv 1 ¡ ½2i ¡1 i= 1 With this copula the joint density of the M default times consistent with the prespeci¯ed Weibull marginals Fi (ti; ci; ai) is F (t1 ; ¢ ¢ ¢ tM )

= C ½ (F 1 (t1 ); ¢ ¢ ¢ ; F M (tM )) Ã ! Z 1 Y M ©¡ 1 (Fi (ti)) ¡ ½iv = © p Á(v)dv 1 ¡ ½2i ¡1 i=1

where we have suppressed the dependence on the parameters for notational convenience. One may also easily evaluate the joint survival function or the joint probability that each default time ¿ i exceeds ti as S(t1 ; ¢ ¢ ¢ ; tM ) and à ! Z 1 Y M ½iv ¡ © ¡1 (F i(ti)) p S(t1 ; ¢ ¢ ¢ ; tM ) = © Á(v)dv: 1 ¡ ½2i ¡1 i=1 The coe±cient of pairwise lower and upper tail dependence in a copula ij ¸ ij L ; ¸ U is de¯ned as ¸ ij L

=

lim P (u i < uju j < u)

ij ¸U

=

lim P (u i > uju j > u)

u#0 u"1

10

ij

ij

For the one factor Gaussian copula it may be shown that ¸L ; ¸U equal zero for all pairs i 6= j independent of the levels of correlation in the vector ½: Hence this copula, though widely understood, has the unsatisfactory feature that when we come to extreme events, there is in fact a lack of correlation in the tails. The model builds correlation near the mean and not in the tail of the distribution. These considerations lead us to consider other copulas that do in fact parameterize the level of tail dependence as de¯ned above.

5.2

The Clayton copula

The Clayton copula builds dependence between prespeci¯ed marginals on noting that if we take any positive random variable V then one may de¯ne conditional distribution functions as conditionally independent given V and of the from P (Xi < xjV = v) = Gi (x)v for some distribution functions Gi (x): Let fV (v) be the marginal density of the positive random variable V then the joint density of variables Xi will be Z 1Y M F (x1 ; ¢ ¢ ¢ ; x M ) = Gi(xi )v f V (v)dv (3) Z

=

0

1

0

= Ã

Ã

i=1 PM v i=1 ln(G i (xi ))

e

¡

M X

f V (v)dv !

ln(G i(xi ))

i=1

(4)

where Ã(s) is the Laplace transform of V and Z 1 Ã(s) = e ¡sv f V (v)dv: 0

From the expression (3) we also see that the implied marginal densities must be F i(x i ) =

Z

0

=

Z

1 1

G i(x i) v fV (v)dv e v ln(Gi( xi)) fV (v)dv

0

=

Ã(¡ ln(Gi (xi ))

and hence it follows that Gi (xi ) = e¡Ã

¡1

(F i(xi) )

:

Substituting (5) back into (4) we observe that ÃM ! X ¡1 F (x 1 ; ¢ ¢ ¢ ; x M ) = Ã Ã (Fi (xi )) i=1

11

(5)

In the special case of uniform marginals we get the Archimedean copula ÃM ! X ¡1 C (u1 ; ¢ ¢ ¢ ; u M ) = Ã Ã (u i) i=1

that one may construct from the Laplace transform of any positive random variable V . The speci¯c Clayton copula results when we choose for V a gamma random variable with density 1 1 f V (v) = ¡ 1 ¢ v µ ¡1 e¡ v ¡ µ for which we have that

Ã(s) Ã ¡ 1 (u) and hence C(u 1 ; ¢ ¢ ¢ ; u M ) =

¡1 µ

= (1 + s)

= u ¡µ ¡ 1 ÃM X i=1

µ u¡ ¡M +1 i

! ¡ µ1

With this copula and the Weibull marginals Fi (ti) we have that ¡ ¡ ¢¢ Gi (ti) = exp ¡ F i(ti )¡µ ¡ 1

and from (3) we see that F (t1 ; ¢ ¢ ¢ ; tM ) = S (t1 ; ¢ ¢ ¢ ; tM ) =

¡

¡

1 ¡1¢ µ

1 ¡1¢ µ

Z

0

Z

0

M 1 Y i=1

¡ ¡ ¢¢ 1 exp ¡v F i(ti) ¡µ ¡ 1 v µ ¡1 e¡ v dv

M 1 Y ¡ i=1

¡ ¡ ¢¢¢ 1 ¡1 ¡ v 1 ¡ exp ¡v Fi (ti)¡ µ ¡ 1 v µ e dv

For this copula we may evaluate that ij ¸U

=

¸ij L

=

¡ ¢¡ 1 1 ¡ 2u + u ¡µ + u¡µ ¡ 1 µ lim =0 u "1 1¡u ¡ ¡µ ¢ u + u ¡µ ¡ 1 1 lim = 2¡ µ u #0 u

and we have lower tail dependence parameterized by µ:

6

Pricing the jth to default

For much of this section we follow the methods described in Laurent and Gregory (2003) and SchÄonbucher (2003). The j th to default contract pays the notional 12

less recovery rate at the time of the j th default in a basket of m > j names. The time of the j th default exceeds t just if the number of defaults to time t is less than j: These considerations suggest a study of the process N (t) that counts the number of defaults to time t: Given a total of m names with default times ¿ i; 1 · i · m we de¯ne m X N (t) = 1 ¿ i·t i=1

the default counting process. We are interested in the probabilities p k (t) = P ( N (t) = k); k = 0; ¢ ¢ ¢ ; m: Associate with these probabilities their generating function à N(t) (u) de¯ned as à N(t) (u) = =

h i E uN (t) m X

p k (t)uk :

k=0

Modeling dependence as conditional on the factor V ; either using a one factor Gaussian copula or a Clayton copula, we also de¯ne the V conditional probability generating function for N by à NjV (u) where à N(t)j V (u) = =

h i E uN (t) jV m X

p kjV (t)u k

k=0

and p kjV (t) is the conditional probability that N (t) = k given V: Conditional on V the default times are independent by construction and hence one may obtain the conditional probability generating function in terms of the conditional probabilities p ijV (t) = P (¿ i · tjV ) as à N jV (u) =

m ³ Y i=1

1 ¡ p ijV (t) + p ijV (t)u

It follows that à N (t) (u) = E V [à N (t)jV (u)]: For the one factor Gaussian copula we have à ! ©¡ 1 (Fi (ti)) ¡ ½i v ijV p (t) = © p 1 ¡ ½2i

while for the Clayton copula we get

¡µ pijV (t) = e v(1¡F i(t) ) :

13

´

We therefore obtain " à ! à ! # Z 1Y m ½iv ¡ ©¡ 1(Fi (ti)) ©¡1 (Fi (ti)) ¡ ½i v à N (t)(u) = © p +© p u Á(v)dv 1 ¡ ½2i 1 ¡ ½2i 0 i=1 for the one factor Gaussian copula and Z 1Y m h i ¡µ ¡µ à N (t)(u) = 1 ¡ e v(1¡F i(t) ) + ev (1¡F i( t) ) u 0

k

i=1

¡

1 1 ¡ 1 ¢ v µ ¡1 e¡ v dv µ

for the Clayton copula. We may now obtain the V conditional complementary distribution function of the time of the j th default, ¿ (j) as ³ ´ P ¿ ( j) > tjV = P (N (t) < jjV ) j¡ 1

=

X

pkjV (t)

k=0

For the valuation of the payments on the j th to default contract we just need the risk neutral probabilities of ¿ ( j) exceeding t: For the receipt side we need to evaluate the probability density that the j th default occurs on a particular day. We may approximate this probability by di®erencing the complementary distribution function. Somewhat more generally we allow for di®erential recovery rates on di®erent names and in this case we need to separate out the probability that we have the j th default and that this default is that of name i: For this calculation we introduce the default counting processes N (i) (t) that count the number of defaults excluding name i from the basket. We then have analogously the entities ¡ ¢ (i) (i) p k (t) = P N (i) (t) = k ; à N (i)(t) (u); p kj V (t), and à N (i)(t) jV (u) and we may evaluate the conditional complementary distribution function ³ ´ P ¿ (i;n) > tjV = P (N (i) (t) < njV ) =

n¡1 X

(i)

pkjV (t)

k= 0 ( j)

th

The receipt side for the j to def ault with coupon on maturity T ; kT may then be evaluated by conditioning on V as · ¸ 2 3 (j) Pm PN M kT (n¡ ³(n)) n £ M (1 ¡ R ) ¡ B i i=1 n=1 360 365 5 EV 4 ¡ (i;j¡1) ¢ ¡ ijV ¢ ijV P ¿ > njV p (n) ¡ p (n ¡ 1) While the payment side is similarly evaluated as 2 3 NP ³ ´ X M k (np ¡ np ) T j j¡ 1 EV 4 B npj P ¿ (;j) > njV 5 365 360 j=1

14

The j th to default credit default swap rate is then obtained on equating these two expectations.

7

A First, Second and Third to Default Study

We report in this section on an illustrative study of the ¯rst, second and third to default contracts on a basket of names. The basket selected for the study is from the ¯nancial sector and is a subset of the ¯rms used in section 4 to study the individual credit default swap rates for the Weibull and Frechet models. Speci¯cally we consider the basket of six ¯rms BAC; GS; J P M ; LEH; M E R; and M W D: For the marginal densities we used the Weibull model with parameters at the average values reported in Table 1. The copulas used were the one factor Gaussian copula and the Clayton copula. The ¯rst task in implementing a ¯rst to default pricing is the determination of correlations for the Gaussian copula and the determination of lower tail dependence for the Clayton copula. We estimate these correlations from data on the equity returns. For the one factor Gaussian copula we estimate a null hypothesis of a one factor Gaussian model describing equity returns using factor analysis. If we let R denote the 6 by 1 vector of standardized daily returns on the six names then our hypothesis is that R = ¹ + ¤f + u where ¤ is a 6 by 1 vector of exposures to a Gaussian unit variance latent factor f and u is a vector of orthogonal Gaussian idiosyncratic disturbances. Our data on daily returns for the six names covered the period May 4 1999 to June 10 2003. Implementing a factor analysis on this data yielded the factor exposures and residual variances reported in Table 3. TABLE 3 Results of Factor Analysis Name Exposure Residual Variance BAC 0.6341 0.5979 GS 0.8762 0.2322 JPM 0.7434 0.4474 LEH 0.8901 0.2078 MER 0.8845 0.2176 MWD 0.9253 0.1437 The correlations employed in our ¯rst to default one factor Gaussian copula are given by the vector of factor exposures. For the Clayton copula we employed the relationship between Kendall's tau and the degree of tail dependence. In the bivariate case we have that µ=

2¿ 1¡¿ 15

where ¿ is the Kendall tau measure of correlation. Kendall's ¿ evaluates the proportion of time two variables move in the same direction less the proportion of time they move in opposite directions. Unlike the standard correlation measure no attention is paid to the size of the move, but just the direction. We report in the matrix ¨ below the pairwise correlations as estimated by Kendall's ¿ for this return data. 1 :3697 :5006 ¨= :4002 :3908 :4191

:3697 1 :4519 :5965 :5851 :6219

:5006 :4519 1 :4625 :4713 :5033

:4002 :5965 :4625 1 :6028 :6040

:3908 :5851 :4713 :6028 1 :6319

:4191 :6219 :5033 :6040 :6319 1

The average value for Kendall's tau is :5074 that yields a value for µ of 2:0605 and a lower tail probability of :7143: Alternatively, one could consider for the multivariate tail dependence the weakest link given by the minimum value for Kendall's tau of :3697 that has µ of 1:1732 and tail dependence of :5539: We also consider half this tail dependence of :2769 with an associated µ = :5398 and ¿ = :2125: For the three values of tail dependence of :7143; :5539 and :2769 we present in Figure (1) the contour plot of the Clayton pairwise joint densities with standard Gaussian marginals in the interval [¡3; 3]: We observe that a tail dependence of :7143 is quite large and makes the second default very likely given the ¯rst. We present in Figure (2) ¯rst to default Clayton curves for the levels of tail dependence of :5539 and :2769. Also shown are the ¯rst to default swap rates for the Gaussian copula, the maximum of the individual swap rates, and the sum of the individual swap rates. The marginals are the reported Weibull models, the maturities extend from one year to 10 years in steps of a quarter year, and recovery is assumed at 50%. We note Clayton dependencies associated with Kendall's tau take the ¯rst to default swap rates closer to the maximum of the individual swap rates relative to the one factor Gaussian copula. For half the tail dependence of the weakest lenk in Kendall's ¿ we get swap rates closer to the one factor Gaussian copula. We also understand that the Gaussian copulas have zero tail depedence and so low levels of tail dependence in Clayton approach the Gaussian swap rate curve. Finally, we ask if the Weibull model is capable of synthesizing the distribution of time to ¯rst, second and third default as well as the individual asset swap quotes. For an assessment of this property we treat the swap quotes on the ¯rst to default as if they were quotes on the default of a single name. We have a lot more data here and ¯t our Weibull model for the CDS rates to this data on the ¯rst to default quotes. Figure (3) presents of this estimation for the ¯rst to default on both models and Figures (4,5) present the results for the second and third to default on the Gaussian and Clayton copulas respectively. We observe that the hazard parameters of the ¯rst to default distribution fall just slightly to 1:5593 for the Gaussian copula and 1:7281 for the Clayton when compared with those for the individual names. Most of the adjustment towards 16

2 0

Tail Dependence .7143

-2 -3

2 0

-2

-1

0

1

2

3

-1

0

1

2

3

-1

0

1

2

3

Tail Dependence .5539

-2 -3

2 0

-2

Tail Dependence .2769

-2 -3

-2

Figure 1: Graphs of Clayton Contours for di®erent levels of tail dependence and Gaussian marginals

17

First to Default Swap Rates 250 Sum of Individual Swap Rates

200

Clayton Tail Prob .2769 Maximum of Individual Swap Rates Gaussian One Factor

Swap Rate

150

Clayton Tail Prob .5539 100

50

0 1

2

3

4

5

6

7

8

9

10

Maturity

Figure 2: First to Default Swap Rates with Weibull Marginals on a Basket of Six Financials for Maturities up to 10 years

18

Weibull Fit to First to Default Quotes 110

100 Weibull to Gauss FTD c=25.48 a=1.5593

90

80

Swap Quote

70

60

50 Weibull to Clayton FTD c=26.73 a=1.7281

40

30

20

10

1

2

3

4

5

6

7

8

9

10

Maturity

Figure 3: Weibull model ¯ts to ¯rst to default quotes for the Clayton and Gaussian Copulas. Quotes are quarterly and are indicated by circles. The solid lines represent the Weibull ¯ts.

19

Weibull Fit to Gauss 2nd and 3rd to default 60

50 2nd to Default c=31.9749, a=1.8257

Swap Rate

40

30

20

10 3rd to Default c=36.8103, a=2.0111

0

1

2

3

4

5

6

7

8

9

10

Maturity

Figure 4: The Second and Third to Default on the Gaussian one Factor copula

20

Weibull Fit to Clayton 2nd and 3rd to default 55

50

45

2nd to Default c=33.8872, a=1.7548

40

Swap Rate

35

30

25

20

15 3rd to Default c=41.6159, a=1.7633 10

5 1

2

3

4

5

6

7

8

9

Maturity

Figure 5: The Second and Third to Default on the Clayton copula

21

10

the ¯rst to default quotes is made by a reduction of the scale parameters to 25:48 for the Gaussian copula and 26:73 for the Clayton copula. This pattern is maintained for the second and third to default contracts on both copulas with the main adjustment being in the scale parameters. The Gaussian scale parameters for the second and third to default are respectively, 31:9749 and 36:8103: The corresponding shape parameters are 1:8257 and 2:0111: The scale parameter values for the second and third to default with the Clayton copula are 33:8872 and 41:6159 respectively while the corresponding shape parameters are 1:7548 and 1:7633:

8

Conclusion

This paper proposes an extreme value distribution for the time to default distribution embedded in market credit default swap quotes. Two distribution classes, the Weibull and Frechet are empirically evaluated. We observe that in the Weibull family, though the hazard rates of survival may be either falling or rising, they are typically estimated to be rising. This suggests that the Frechet model, where hazard rates are necessarily falling is not appropriate. At a statistical level both models are found to be adequate though the Weibull model provides a better ¯t to the data and also has a superior performance in non nested tests against the Frechet alternative. We then describe the details for the pricing of N th to default contracts using the one factor Gaussian copula and the Clayton copula. This pricing is illustrated on a basket of six ¯rms in the ¯nancial sector. We employ Gaussian correlations derived from a factor analysis of equity returns. For the Clayton copula we employ the relationship between tail dependence and Kendall's tau. Gaussian ¯rst to default prices are seen to be closer to the independence boundary relative to the Clayton copula prices. Finally we assess the ability of the Weibull model to synthesize the ¯rst, second and third to default time distributions as re°ected in the respective swap quotes. We ¯nd that the for the cases studied, it appears that the Weibull model is also adequate for the distribution of time to ¯rst, second or third default.

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