00a exploring statistical arbitrage opps ye-xiaoxia

7/31/2019 00A Exploring Statistical Arbitrage Opps YE-XIAOXIA

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Exploring Statistical Arbitrage Opportunities in the TermStructure of CDS Spreads

Robert Jarrow, Haitao Li, and Xiaoxia Ye

October 2011

Jarrow is at the Johnson Graduate School of Management, Cornell University, Ithaca, NY 14850. Li is at the Stephen M.

Ross School of Business, University of Michigan, Ann Arbor, MI 48109. Ye is at the Risk Management Institute, National

University of Singapore, Singapore, 119613. We thank seminar participants at Xiamen University, Conference on Advances

in the Analysis of Hedge Fund Strategies at Imperial College, National University of Singapore, Workshop on Financial

Econometrics at University of Toronto; especially Peter Carr, Peter Jackel, Andrea Vedolin, Hai Lin, Linlin Niu, Yu Ren,

Jin-Chuan Duan, Oliver Chen, Ying Chen, and Fermin Aldabe for helpful comments. We are responsible for any remaining

errors.


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1 Introduction

The credit derivatives markets have experienced tremendous growth in the last decade. According to

the Bank for International Settlements (BIS), the notional value of outstanding credit derivatives by the

end of 2007 was 58 trillion dollars. The single-name credit default swaps (CDS) are the most liquid

and popular product, as they account for more than two thirds of all outstanding credit derivatives.

Though some exotic credit derivatives, such as subprime CDOs have caused tremendous problems in the

current nancial crisis, the vanilla CDS contracts do play important economic roles. The newly proposed

regulations, such as the establishment of central clearing house for CDS, would help to reduce systemic

risk and improve transparency in the CDS markets. Therefore, the CDS contracts are likely to remain to

be the preferred vehicle for investing, speculating, and managing single name credit risk.

The rapid growth of the CDS market makes it possible to speculate on the relative pricing of the

credit risk of a company across a wide range of maturities. Though ve-year CDS has been the most

liquid contracts until recently, nowadays a complete credit curve (CDS spreads over dierent maturities)is available for many companies. As a result, it is possible to buy and sell protections on a given rm at

dierent maturities.

A natural question arises in this market is whether the credit risk of a rm is consistently priced

across maturities. This is an interesting question to both academics and practitioners. From an academic

perspective, one interesting issue is whether existing credit risk models, either structural or reduced-

form, can capture the rich term structure behaviors of credit spreads. From a practical perspective, one

challenging issue is whether one can design trading strategies to exploit potential mispricings along the

credit curve.

In this paper, we take an applied approach to this problem. Based on a reduced-form model of credit

risk, we explore potential statistical arbitrage opportunities in the term structure of CDS spreads of a

large number of companies in North America. Specically, we consider 297 rms with continuous daily

observations of CDS spreads with maturities of 1, 2, 3, 5, 7, 10, 15, 20, and 30 years between January

4, 2005 and December 31, 2008. We estimate an ane model of credit risk for each company based on

its term structure of CDS spreads and identify mis-valued CDS contracts relative to the model. Based

on the estimated model parameters, we construct a portfolio of CDS contracts that are both delta- andgamma-neutral to potential changes in credit spread. Then we would long (short) the portfolio if it is

under (over) valued relative to our model and liquidate the portfolio a week later.

We conduct both in-sample and out-of-sample analysis on the protability of the above statistical ar-

bitrage strategy. In the in-sample analysis, we estimate model parameters, construct arbitrage portfolios,

and calculate trading prots using all the data. In the out-of-sample analysis, we estimate model para-

meters using the rst half of the sample, based on which we construct arbitrage portfolios and calculate

trading prots using the second half of the sample.

We nd that our arbitrage strategy can be quite protable both in sample and out of sample. For

most rms, the Sharpe ratio of the weekly returns of this strategy is above one. For more than half of

the rms, the Sharpe ratio can be well above two! Obviously the CDS contracts at dierent maturities

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might have dierent levels of liquidity, which are not reected in their quoted spreads. There could also

be counterparty risks in CDS contracts. Our analysis does not explicitly take into account of bid-ask

spreads and various transaction costs either. Despite these concerns, the high Sharpe ratios we document

do suggest that there could be interesting statistical arbitrage opportunities in the term structure of

CDS spreads that could potentially be exploited.

Though a huge literature has been developed on credit risk in the past decade, empirical studies onCDS that involves the modeling of the entire credit curve is still pretty rare. One main reason is that

until recently we do not have the data on the CDS spreads for a wide range of maturities. Two studies

that are closely related to ours are Zhang (2008) and Pan and Singleton (2008), who estimate default risk

models using the entire credit curve of sovereign CDS spreads. One important contribution of our paper,

however, is that we are probably one of the rst to focus on exploring potential statistical arbitrage

opportunities in the term structure of CDS spreads.

The rest of this paper is organized as follows. In Section 2, we discuss the ane term structure model

for credit risk. Section 3 discuses the econometric methods for estimating the model. Section 4 discusses

empirical results on model estimation and trading performances. Section 5 concludes.

2 The Model

In this section, we develop a one-factor ane model for the term structure of CDS spreads. We use only

one factor to capture the dynamics of credit risk because our principal component analysis (PCA) on CDS

spreads shows that the rst principal component captures 96% of the variations of CDS spreads. Ourmodel is similar to that ofLongsta et al. (2005), Due and Singleton (1999), Due and Singleton (1997),

Due et al. (2003), and Zhang (2008). For the sake of robustness of model performances (especially out-

of-sample performances), we assume that credit spreads are independent of interest rates and thus avoid

estimating a model for the default-free term structure. We obtain similar results with a two-factor ane

model for the default-free term structure, in which the credit spread is correlated with the two interest

rate factors.

Specically, we assume that the state variable, i.e. the default intensity Zt; follows a square root

process (CIR process) as

dZt = Z (Z Zt)dt + Zp

ZtdwQZ (t) ; (1)

where wQZ (t) is a standard Brownian motion under the equivalent martingale measure Q.

While we only need the dynamics of the state variable under the Q measure for pricing purpose, we need

its dynamics under the P measure for econometric estimation. Given the completely ane specication

of market price of risk, we model the P measure dynamics of the state variable as

dZt =

ZZ

P

ZZtd

t + Zp

Ztd

wP

Z (t) :

To compute the CDS spread, we assume a constant recovery rate. Since both the buyer and the seller

of credit protection in a CDS are exposed to counterparty risk, the quoted recovery rates might dier

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from the real recovery rates implicit in the CDS spreads. Therefore, unlike the common practice in the

literature which sets the recovery rate to a certain number (see, e.g., Longsta et al. (2005), and Zhang

(2008)), we estimate the value of the constant recovery rate along with model parameters from the market

prices of CDS spreads. Under the fractional recovery of face value (RFV) framework, which has been

widely used for pricing CDS, the recovery rate and the default intensity can be easily identied jointly.

To this end, we set recovery rate as1 y = exp (0) ;

where 0 > 01.

Then the CDS spread at t for a protection between t and t + equals

St =[1 exp(0)]

Rt+t

P(t; u)E2 (t; u)duRt+t

P (t; u)E1 (t; u)du; (2)

where P (t; T) is the time-t price of a default-free zero coupon bond that matures at T, and

E1 (t; u) = EQ

exp

Zut

Zsds

Ft

;

E2 (t; u) = EQ

exp

Zut

Zsds

Zu

Ft

:

Following Due et al. (2000), we consider the Transform and the Extended Transform respectively

below,

(w; Zt; t ; u) = EQ

exp

Z

u

t

Zsds

ewZu

Ft

; (3)

(v; w ; Zt; t ; u) = EQ

exp

Zut

Zsds

vZue

wZu

Ft

: (4)

Proposition 1 of Due et al. (2000) indicates that (3) has the following form:

(w; Zt; t ; u) = exp fA (t; u) + B (t; u) Ztg ;

where A and B satisfy the ODEs

_B (t; u) = 1 + ZB (t; u) 12

B (t; u)2 2Z;

_A (t; u) = ZZB (t; u) ;1 The positiveness of parameter 0 ensures that y 2 (0; 1).

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with boundary conditions B (u; u) = w and A (u; u) = 0, and

B (t; u) =Z tanh

h12

(u t) 12

lnZ+w2Z+Zw

2

Z

i

2Z;

A (t; u) = ZZZu

t

B (s; u)ds;

=q

22Z + 2Z:

Similarly, (4) is given by

(v; w ; Zt; t ; u) =@ (v + w; Zt; t ; u)

@

=0

= (w; Zt; t ; u) [C(t; u) + D (t; u) Zt] ;

where C and D satisfy the ODEs

_D (t; u) = ZD (t; u) 12

D (t; u) B (t; u) 2Z;

_C(t; u) = ZZD (t; u) ;

with boundary conditions D (u; u) = v and C(u; u) = 0, and

D (t; u) = v

2

ntanh h12 (u t) 12 ln Z+w

2

Z

+Zw2

Zio2

v

2

2 (Z w2Z)2;

C(t; u) = ZZ

Zut

D (s; u)ds:

Then we have

E1 (t; u) = (0; Zt; t ; u) ;

E2 (t; u) = (1; 0; Zt; t ; u) :

In practice, following Longsta and Rajan (2008), we discretize (2) as

St =

[1 exp(0)]4Xi=1

P

t; t + i4

1; 0; Zt; t ; t +i4

4Xi=1

P

t; t + i4

0; Zt; t ; t +i4

:

3 Model Estimation

In this section, we discuss the econometric method for estimating our ane model using CDS spreads.

When implementing the model, we rst need to back out the zero yields from Libor and swap rates to

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compute the prices of default-free zero coupon bonds P (t; T). Then, we use these zero coupon bond prices

multiplied by estimated E1 (t; T) as discount factors to calculate the present value of the premium and

the protection leg2 of the CDS contracts.

There are dierent econometric methods one can use to estimate the ane model. Instead of using

maximum likelihood, which typically assumes that certain spreads are observed without errors and then

obtains the state variables by inversion, we use the unscented Kalman lter (UKF) in conjunction withQMLE to estimate the credit risk model because CDS spreads are highly nonlinear in the state variable

Zt.

3.1 State Space Model

To use the UKF in empirical estimation, we re-cast our model in the framework of state-space model.

Although the transition density of the state variable in our model is not Gaussian, by applying the UKF

with QMLE, we only need to consider the rst two moments of the transition density. Therefore, we writedown the transition equation as if the state variable is conditionally normally distributed, as long as the

rst two moments are intact. Duan and Simonato (1999) shows that this approximation is fairly ecient

and accurate for estimating models with CIR type of state variables. Based on this approach, we provide

the state-space representation of the defaultable term structure model below.

Let t be the sampling interval in our study, which is a week. Then the transition equation for the

default state variable Zt is given as

Ett [Z(t)] =ZZ

PZ

1 exp PZt + exp PZtZ(t t) ;

Vartt [Z(t)] =ZZ

2Z

2 (PZ)2

1 exp PZt2 + 2Z

exp

PZt exp 2PZtPZ

Z(t t) :

Let CDSt be the CDS spread for protection between t and t+ : Then the measurement equation becomes

CDSt = St (Zt) + "

t ;

where "

t v i:i:d:N(0; v2

) and = 1; 2; 3; 5; 7; 10; 15; 20;and 30 years.

3.2 Unscented Kalman Filter

In this section, we briey discuss the implementations of the unscented Kalman lter. More detailed

discussions can be found in Harvey (1991) and Haykin et al. (2001).

One challenge in applying the Kalman lter to estimate the credit risk model is that the CDS spread is a

nonlinear function of the state variable. One solution to this problem, the so called extended Kalman lter

2

Premium or protection leg of the time t CDS contract with time to maturity is given by

CDSt

4Xi=1

P

t; t+

i

4

0; Zt; t ; t+

i

4

= CDS

t

4Xi=1

P

t; t+

i

4

E1

t; t+

i

4

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(EKF), is to consider a rst order Taylor expansion of the measurement equation around the predicted

state Ztjt1: Unlike the EKF, the UKF uses the exact nonlinear function St (Zt) and does not linearize

the measurement equation. Instead, the UKF approximates the conditional distribution of Zt using a

scaled unscented transformation. The essence of the UKF (Chow et al. (2007)) used in this paper can be

summarized briey as follows.

For each measurement occasion t, a set of deterministically selected points, termed sigma points, areused to approximate the distribution of the current state3 estimates at time t using a normal distribution

with a mean vector Ztjt1, and a covariance matrix, which is a function in the state variance PZ;t1jt1 (for

notational clarity, we normalize the time interval to one) and conditional variance Vart1 [Z(t)]. Sigma

points are specically selected to capture the dispersion around Ztjt1. They are then projected using the

measurement function St (), weighted, and used to update the estimates in conjunction with the newlyobserved measurements at time t to obtain Ztjt and PZ;tjt .

We start the UKF by choosing the initial values of the state variable and its variance as their steady

state values:

Z0j0 =ZZ

PZ; PZ;0j0 =

ZZ

2 (PZ)2

2Z:

Given Zt1jt1 and PZ;t1jt1; the ex ante prediction of the state and its variance are given by

Ztjt1 =ZZ

PZ

1 exp PZt + exp PZtZt1jt1;

PZ;tjt1 = e2P

ZtPZ;t1jt1 +

ZZ2Z

2 (PZ)

2 1 eP

Zt

2

+2Z

e

P

Zt e2PZt

P

Z

Zt1jt1:

Given an ex ante prediction of state Ztjt1, a set of 3 sigma points are selected as

tjt1 =h

0;t1 +;t1 ;t1

i;

where

0;t1 = Ztjt1;

+;t1 = Ztjt1 +p

(1 + )

expPZtpPZ;t1jt1 + pVart1 [Z(t)] ;

;t1 = Ztjt1 p

(1 + )

expPZtpPZ;t1jt1 + pVart1 [Z(t)] :

The term is a scaling constant and given by

= 2 (1 + ) 1;

where and are userspecied constants. In this paper, we choose = 0:001 and = 2. Since the

3 In typical UKF setting, both transition and measurement equations are nonlinear. Hence, sigma points are needed toapproximate the distribution of previous state estimates, in order to compute the ex ante predictions of state variablesmean and variance. However, in our paper, the transition equations are linear, so we can directly compute the ex antepredictions in the way as that in Classic Kalman Filter, and do not need sigma points at this stage.

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values of these constants are not critical in our analysis, we do not provide more detailed discussions for

brevity. Curious readers are referred to Chow et al. (2007) or Chapter 7 in Haykin et al. (2001) for details.

tjt1 is propagated through the nonlinear measurement function St () (i.e. nonlinear transformation

of the sigma points through measurement function)

Stjt1 = S

t

tjt1

;

where the dimension ofStjt1 is 9 3. We dene the set of weights for covariance matrix estimates as

W(c) = diag

1++ 1 2 + 2 ; 1

2 (1 + );

1

2 (1 + )

;

and the weights for mean estimates as W(m) =h

1+

12(1+)

12(!+)

i|

:

Predicted measurements and the associated covariance matrix are computed as

Stjt1 = Stjt1W(m);

Pyt =Stjt1 113 Stjt1

W(c)

Stjt1 113 Stjt1

|

+ V;

PZt;yt =

tjt1 113 Ztjt1

W(c)Stjt1 113 Stjt1

|

;

where V = diag [v2; v2; ; v2]99 :Finally, the discrepancy between model prediction and actual observations is weighted by a Kalman

gain t function to yield ex post state and variance estimates as

Ztjt = Ztjt1 + t

St Stjt1

;

PZ;tjt = PZ;tjt1 tPyt|t ;

where t = PZt;ytP1yt

:

3.3 Likelihood Function

We assume that the measurement errors are normally distributed. Then the transition density of S (t) =hCDS1t CDS

2t CDS30t

i|

given information set Ft1 is a 9-dimensional normal distribution withmean Stjt1 and covariance matrix Pyt; which are outputs from the UKF. Thus, the transition density of

S (t) can be written as

ft1 (S (t)) =

p29q

jPytj1

exp

1

2

S (t) Stjt1

|

P1ytS (t) Stjt1

;

then the log-likelihood function is given by

lnL _ nXi=1

ln jPyij nXi=1

S (i) Siji1

|

P1yiS (i) Siji1

;

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where n is the sample size.

4 Empirical Results

4.1 Default-free Zero Coupon Bond Prices

We back out the prices of default-free zero coupon bonds from Libor rates and plain vanilla xed for

oating Libor-quality interest rate swap rates, similar to Dai and Singleton (2003), Due et al. (2003),

and Zhang (2008). Though Treasury yields have been widely used as benchmark for risk free term

structure estimation, there are concerns that they may dier from the true risk free rates due to repo

eects, liquidity dierences, and tax shields (see, e.g., Houweling and Vorst (2005)). Moreover, interest

rate swap data are widely available at a range of xed maturities, which simplies the estimation process.

Specically, we obtain daily observations of zero yields with maturities from 3 months to 30 years with

equal increment of 3 months. These zero yields are bootstrapped from the Libor/Swap term structures,which consist of Libor rates with maturities of 3, 6 and 9 month and Swap rates with maturities of 1, 2,

3, 4, 5, 7, 10, and 30 year. The sample period is from January 4, 2005 to December 31, 2008. Figure 1

shows the dynamics of the default-free zero yields, which allow us to compute the prices of zero coupon

bonds with maturities from 3 months to 30 years with an equal increment of 3 months.

4.2 Estimation of CDS Spreads

The CDS data used in our analysis are obtained from Markit. Based on quotes provided by dierentdealers, Markit creates the daily composite quotes for each CDS contract. It also provides average recovery

rates used by data contributors in pricing each CDS contract. Moreover, an average of Moodys and S&P

ratings is also provided. We focus on US dollar denominated CDS contracts on all US non-sovereign

entities. We only use CDS on senior unsecured issues with modied restructuring (MR) clauses, as they

are the most popular CDS contracts in the US market.

To obtain accurate estimates of model parameters and enough observations for out-of-sample analysis,

we require all rms included in our study to have at least four years of daily continuous observations of

CDS spreads. Since the liquidity of CDS contracts with a maturity dierent from 5 year might not be

good during early years of the market, we restrict our sample to the period between 2005 and 2008. In

total, we have 297 rms in our nal sample with continuous daily CDS spreads with maturities of 1, 2, 3,

5, 7, 10, 15, 20, and 30 years.

Table 1 provides summary statistics on the distribution of the 297 rms in dierent rating groups

between 2005 and 2008. In results not reported, we see that the total number of rms in the dataset has

increased dramatically from 309 in 2001 to 1,268 in 2009. During our sample period, the number of rms

are relatively stable. Among the 8 rating groups (from AAA to D), A, BBB, and BB rated rms account

for about 70-80% of all the rms with CDS trading. The table also contains the distribution of the rms

among ten dierent industry groups, which include basic materials, consumer goods, consumer services,

nancials, health care, industrials, oil & gas, technology, utilities, and telecommunication. Financials have

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most rms with CDS trading, followed by consumer services, industrials, consumer goods, and utilities.

Other industries have less rms with CDS trading.

Tables 2 and 3 provide the mean and standard deviation of CDS spreads for dierent rating and

industry groups. For most rating groups, we see an upward sloping credit curve, which is consistent with

the notion that on average default risk is higher over longer maturity. For D-rated rms, however, we

see a downward sloping CDS curve. This is consistent with the notion that for speculative grade bonds,the default risk can be high in the near future but if the rm survives long enough, then the default risk

actually goes down. The average credit curve for most industries also slopes upward. One prominent

exception is nancial companies, whose CDS spread tends to decline with maturity. In contrast, we nd

that the standard deviation of CDS spread generally declines with maturity. In general, lower rated rms

have higher and more volatile CDS spreads than higher rated rms. One main exception is that the

AAA-rated rms actually have higher spreads than some lower rated rms. We believe this is mainly

because of the small number of AAA-rated rms in our sample.

Based on the prices of default-free zero coupon bonds, we estimate the credit risk model using the

whole term structure of credit spreads for each of the 297 rms. Table 4 presents the minium, maximum,

median, mean, and interquartile range of the variance ratios of CDS spreads at dierent maturities for

the 297 rms. Variance ratio measures the percentage of variations of yields explained by the model. It

is clear that the model can explain the variations of most spreads very well. The average variance ratios

for most maturities and rms are close to 90%. This suggests that our model does a reasonably good job

in capturing the term structure of CDS spreads.

Panel (a) of Table 5 reports the minium, maximum, median, mean, and interquartile range of eachestimated parameter based on the data during the whole sample period. We nd that the rm-specic

default intensity, Zt, commands a negative risk premium, since estimated PZs are signicantly bigger than

Zs. As we set 0 as a free parameter, we can estimate the recovery rate for each individual rm. And we

nd that the average recovery rate exp(0) is around 50% and higher than the average quoted recoveryrate, which is around 40%.

Panel (a) of Table 6 presents the average estimates of ZPZ

Z across ratings and sectors. The quantityZPZ

Z represents the mean of the default state variable Zt under the P measure. The results clearly show

that the lower the rating, the higher the average estimates of ZPZ

Z, which is consistent with the fact

that default risks are higher for lower rated rms. From this panel, we also see that the Health Care

sector has the lowest mean default rate under the P measure and that the Basic Material sector has

the highest mean default rate under the P measure. The ranking is dierent, however, during the rst

sub-sample period due to probably dierent macroeconomic environments.

Panel (a) of Table 7 presents the average estimates of Z across ratings and sectors. Z can be viewed

as the mean of the default state variable Zt under the Q measure. The mean of Zt is much higher under

the Q measure than that under the P measure, again conrming that the default intensity, Zt, commands

a negative risk premium. In general, the mean of Zt under the Q measure increases when the rating

deteriorates. The only exception is with the AAA rating category, which we suspect is due to the small

number of AAA-rated rms in our sample. Finally, the Oil & Gas and Industrial are the sector with

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the lowest and highest mean default rate under the Q measure, respectively.

Panel (a) of Table 8 presents the average estimates of the recovery rates exp(0) across ratings andsectors. For ratings, the estimated recovery rates range from 40% (CCC) to 55% (A). However, we do not

nd a monotone relation between recovery rate and rating, maybe because the estimated recovery rate

incorporates some counterparty risks. The deviation of recovery rates across dierent sectors is smaller,

ranging from 49% (Health Care) to 56% (Telecommunication), with an average of 51%.

4.3 Design and Performance of Statistical Arbitrage Strategies

4.3.1 Design and Implementation

Based on the estimated ane model for CDS spreads, in this section we develop trading strategies to

exploit potential statistical arbitrage opportunities in the term structure of CDS spreads. Our basic

approach is to construct market-neutral portfolios of CDS contracts that are immune to both rst and

second order changes in the default state variable. Then we would long (short) under (over) valued hedged

portfolios. We briey describe the strategies here and refer readers to Appendix A for the details of the

market-neutral strategy and the computation of portfolio hedging weights.

We consider a second-order expansion of the CDS pricing function around the backed out state variable

Zt with the following rst and second order derivatives H1 (t) =

@St(Z)

@Z

Z=Zt

, and H2 (t) =@2S

t(Z)

@Z2

Z=Zt

;

where is the maturity of the CDS contract. We then can combine a CDS with maturity 0 with two

other CDSs with maturities 1 and 2 to form a hedged portfolio. By choosing the appropriate weights

of the CDS contracts, we can hedge away uctuations in the value of the portfolio due to changes in Zt

up to the second order. Specically, we choose the weights of the other two CDS contracts, m1 (t) and

m2 (t) ; according to the following equations,

m1 (t) =H02 (t) H

21 (t) H01 (t) H22 (t)

H11 (t) H22 (t) H12 (t) H21 (t)

; m2 (t) =H02 (t) H

11 (t) H01 (t) H12 (t)

H21 (t) H12 (t) H22 (t) H11 (t)

:

Consequently, the change in the value of the hedged portfolio should be both delta- and gamma-neutral

to changes in Zt. To achieve best hedging performance, in our setting, we choose 1 and 2 as the two

closest maturities to 0, e.g., if 0 = 1, then 1 = 2, and 2 = 3; if 0 = 7, then 1 = 5, and 2 = 10; andif 0 = 30, then 1 = 15, and 2 = 20.

If our model is correctly specied and all the CDS contracts are fairly priced, then the excess return on

the portfolio over the risk free rate should be close to zero. On the other hand, if some of the CDS contracts

are mis-valued, then we might be able to earn positive excess returns. Given the estimated parameters,

each week we re-estimate the state variables to ret the cross-sectional data as well as possible. Then

based on the estimated Zt; we construct the hedged portfolio at dierent maturities. We compare the

market and model price of the above hedged portfolio. If the market price is less than model price, we

would long the portfolio. But if the market price is higher than model price, we would short the portfolio.

We hold each investment for one week and then liquidate the position.4 Since at each t; we only observe

4 In results not reported here, we also hold the investment for two, three, and four weeks, but we nd that the one-week

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the refreshed CDS spreads, we interpolate the CDS curve when we exit our position to approximate the

payo on our portfolio.

For each company, we form arbitrage portfolios of CDS contracts at each maturity. Then we decide

which of the nine portfolios to long and short. Though each portfolio should have zero market value when

we initiate the position, we assume that we have to put down one dollar, the notional amount of each

CDS into each portfolio. Weekly return of each portfolio is the sum of discounted (by default adjustedrate) future cash ows generated by one week trading, i.e. the weighted average of premium (protection)

legs of CDS contracts constituting the portfolio. So in total, we have nine portfolios, and the return we

earn in each rm is the sum of returns of each individual portfolio.

4.3.2 Performance

In this section, we provide in-sample and out-of-sample analysis on the protability of our arbitrage

strategy. We divide our data into two parts, with the rst part covering 2005 to 2006 and the second

part covering 2007 to 2008. In the in-sample analysis, we estimate model parameters using data that

cover the entire sample period, and construct trading portfolios using the second part of the data. In the

out-of-sample analysis, we estimate model parameters using the rst part of the data and implement the

arbitrage strategy using the second part of the data.

We rst look at the in-sample performance of our trading strategy. Panel (a) of Table 9 presents

the summary information of the in sample performance. In particular, we report the minimum, median,

mean, the rst and third quartiles, and the maximum of three important performance measures: the

accumulative prot, the Sharpe Ratio, and the maximum drawdown. And the average values of the threeperformance measures across ratings and industry groups are presented in Panels (a) of Table 10, Table 11,

and Table 12. Accumulative prot is the net prot generated by the strategy, i.e., the time-t accumulative

prot is the sum of the prots and losses realized up to time t. To calculate Sharpe ratio, we convert the

weekly return and standard deviation to annual terms and subtract the (annualized) one-week Libor rate

from the mean return. The max drawdown measures the largest drop of the accumulative prot from the

beginning of the investment until now, and is formally dened as

max(

max(AP (i))ji=1 AP (j)

max(AP (i))ji=1 + 9

)nj=1

where AP(i) stands for the accumulative prot at t = i.

We nd that the in-sample accumulative prots are all positive for the 297 rms and range from 0.34

to 14.76, with an average of 2.51. This means that in general the strategy makes money. The CCC-rated

rms have the highest average accumulative prots of $6, the AAA-rated rms have the lowest average

accumulative prots of $1, and the average accumulative prots are higher for lower-rated rms. For

dierent industry sectors, Consumer Service and Technology have the highest average accumulativeprots of $3, while Utility has the lowest accumulative prot of $1.7. Looking at the intersections of

holding period has the best performance and the Sharpe ratios of the other three holding periods are unimpressive. Ourresults suggest that the mis-pricing we identify disappears within a week.

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ratings and sectors in Panel (a) of Table 10, we see that CCC-rated rms in the Basic Material sector

have the highest average accumulative prots of about $15, while AAA-rated rms in the Financials sector

have the lowest accumulative prots of less than $1.

The excellent performance of the strategy is most evident by the Sharpe ratios it produces. Panel

(a) of Table 9 shows that the in-sample Sharpe ratios for the 297 rms range from -4.62 to 4.46 with an

average of 2.09. Though a few rms exhibit negative Sharpe ratios, 75% of the rms have Sharpe ratioshigher than 1.6, 50% of the rms have Sharpe ratios higher than 2.16, and 25% of the rms even have

Sharpe ratios higher than 2.6! Panel (a) of Table 11 shows that the strategy tends to produce higher

Sharpe ratios for lower rated rms. For dierent industry sectors, Consumer Services and Industrial

have the highest average Sharpe ratios of about 2.40, while Health Care and Utility have the lowest

average Sharpe ratios of about 1.8. Moreover, we see that CCC-rated rms in the Industrial sector have

the highest Sharpe ratio of 3.25, while AAA-rated rms in the Financials sector have the lowest Sharpe

ratio of 0.3712.

In addition to the accumulative prot and Sharpe ratio, we also look at the max drawdown to make

sure that the strategy does not lead to dramatic decline in portfolio value. Panel (a) of Table 9 shows

that the in-sample max drawdowns of all rms range from 0.1% to 9%, with an average of 1.57%. This

suggests that the average loss of the strategy is very small compared to the initial investments. Panel

(a) of Table 12 shows that except for AAA- and AA-rated rms, the average max drawndowns tend to

be higher for lower rated rms. But the max drawdown of even the rating-sector group with the highest

drawdown is only 7%. The average max drawdowns for all industry sectors range from 1.24% to 1.80%,

which again are very small numbers.While the above results show that our strategy performs very well in sample, it could be due to

overtting of the data. To test the robustness of the trading performance, we provide summary information

on the out-of-sample performance of our trading strategy in Panel (b) of Table 9. The most remarkable

result is that, on average, the out-of-sample performance of our strategy is almost as good as the in-

sample one. For example, the mean and median of out-of-sample Sharpe ratios for the 297 rms is 1.88

and 2.06, respectively, which are only slightly lower than that of in-sample Sharpe ratios. The out-of-

sample accumulative prots are only slightly lower than the in-sample ones as well. Though the highest

out-of-sample max drawdown is almost twice of that of the in-sample max drawdown, the mean and

median of the out-of-sample max drawdown are actually slightly lower than the in-sample ones. The

out-of-sample average accumulative prots, Sharpe ratios, and max drawdowns for rating-industry sorted

groups in Panels (b) of Table 10, Table 11, and Table 12, respectively, are roughly consistent with the

in-sample results. Overall, the out-of-sample analysis shows that performance of our trading strategy is

robust and is unlikely due to overtting of the data.

To further investigate how well the strategy applies to individual rms, we present summary sta-

tistics of P&L for 10 randomly selected rms in Table 13. These 10 rms are Cytec Inds Inc (Basic

Material, BBB), Procter & Gamble Co. (Consumer Good, AA), JetBlue Awys Corp. (Consumer

Service, CCC), Gen Elec Cap Corp. (Financials, AAA), Boston Scientic Corp. (Health Care, BB),

Honeywell Intl Inc. (Industrials, A), Marathon Oil Corp. (Oil & Gas, BBB), Hewlett Packard Co.

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(Technology, A), Intelsat Ltd. (Telecommunication, B), and CMS Engy Corp. (Utilities, BB). In

the Table, for each rm, we report the maximum, minimum, mean, standard derivation, accumulative

prot, Sharpe ratio, and max drawdown of the in-sample and out-of-sample weekly returns.

We nd that our strategy can generate very high Sharpe ratios for most of these 10 randomly selected

rms at one week holding period, either in sample or out of sample. For example, for CMS Engy Corp

the accumulative prot is about $3 and the Sharpe ratio is almost 4.0 both in sample and out of sample!While for other rms, the Sharpe ratios are lower, but they are still much higher than that of a lot of

other strategies. For example, the in-sample and out-of-sample Sharpe ratios of JetBlue Awys Corp are

about 2.4, and Cytec Inds Inc has in-sample Sharpe ratio of 1.7 and out-of-sample Sharpe ratio of 1.4.

For most rms, the dierences between the in-sample and out-of-sample performances of our strategy are

negligible. The only exception is Procter & Gamble Co whose Sharpe ratio drops from 1.8 in sample to 0.6

out of sample. This conrms the robustness of our strategys performance. Furthermore, the small max

drawdowns both in sample and out of sample indicate that the performance of our strategy is very stable.

Figures 2 and 3 provide the in-sample time series plots of the accumulative prots and the weekly returns

of the strategy with one week holding period. Figures 5 and 6 provide the out-of-sample counterparts. We

can see clearly the steady growth of the accumulative prot both in sample and out of sample. Though

the strategies suer losses some time, the exceptional out-of-sample performance shows that on average

we make money.

4.3.3 Tuning for Actual Application

The main purpose of this paper is proposing a basic idea to exploit the arbitrage opportunities in theterm structure of CDS spreads, in the sense of forecasting the future directions of the market neutral

portfolio. There are caveats that we need to keep in mind when looking at these high Sharpe ratios.

For example, we have not explicitly accounted for transactions costs and liquidity concerns, which could

eat into our prots. Nonetheless, the impressive Sharpe ratios our strategy generates do point out great

potentials for statistical arbitrage in the term structure of CDS spreads.

As far as actual application is concerned, there are some attentions need to be taken into account, and

the strategy should be tuned accordingly as well. For example, in the real trading, we need to consider

the bid-ask spread. So when deciding which portfolio to long or short, we should look at the discrepancy

between model value and bid-ask value (consists of bid-ask values of certain CDS spreads) in stead of

market value (consists of mid values of certain CDS spreads). As long as market value of the portfolio is

strongly mean-reverting, the bid-ask value would also admits mean-reverting tendency, especially those

with larger discrepancy from model value. And we should be aware of that it might take more time for

bid-ask value converging to model value because of the larger discrepancy from model value, therefore

proper extension of portfolio holding period might be probably needed. Another common issue in the real

trading is that some of CDS contracts might not be tradable, though their values are observed on themarket. In this case, when implementing the strategy, those untradable CDS spreads should be excluded

from the data used to estimate model parameters. By doing so, we can focus more on those spreads that

are actually traded, and make sure the market neutral portfolio constructed using those spreads admits

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strong mean-revering tendency, or, in other words, is predictable to a certain extension.

5 Conclusion

The rapid growth of the CDS market makes it possible to speculate on the relative pricing of credit risk

of a company across a wide range of maturities. Based on a reduced-form model of credit risk, we explorestatistical arbitrage opportunities in the term structure of CDS spreads of a large number of companies

in North America. Specically, we estimate an ane model for the term structure of CDS spreads of a

given company and identify mis-valued CDS contracts along the credit curve. We trade market-neutral

portfolios of mis-valued CDS contracts relative to our model, betting that the mis-valuation will disappear

over time. Empirical analysis shows that our arbitrage strategy can be very protable. For most rms,

the Sharpe ratio are higher than one, and for some rms, the Sharpe ratio is even above two. The

evidence we document shows that there could be interesting statistical arbitrage opportunities in the

term structure of CDS spreads.

A Technical Details of Market Neutral Strategy

To understand mathematically the idea of our market neutral strategy, we rst expand the time t CDS

pricing function of ZT around the backed out state variable Zt up to second order as

S

t ^

ZT

= S

t ^

Zt

+ H

1 (t) ^

ZT ^

Zt

+

1

2 H

2 (t) ^

ZT ^

Zt2

+ O ^

ZT^

Zt3

(5)

where

H1 (t) =@St (Z)

@Z

Z=Zt

H2 (t) =@2St (Z)

@Z2

Z=Zt

:

We assume for a short period of time t = T t, S

t ^

ZT

can approximate St

T ^

ZT

well, i.e.,

StT

ZT

St

ZT

:

Then, by the above approximation and ignoring high order terms, (5) can be rewritten as

StT

ZT

St

Zt

+ H1 (t)

ZT Zt

+

1

2H2 (t)

ZT Zt

2:

Therefore, given a small t, at time t for 1 unit of CDS with maturity of 0, we could employ other two

CDSs with maturities of 1 and 2 to form a hedged portfolio, whose values are immune to the variation

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ofZt up to second order, by setting the weights such that

S0t

Zt

+ m1 (t) S

1t

Zt

+ m2 (t) S

2t

Zt

S0tT

ZT

+ m1 (t) S

1tT

ZT

+ m2 (t) S

2tT

ZT

i.e.

H01 (t)

ZT Zt

+ m1 (t) H11 (t)

ZT Zt

+ m1 (t) H

21 (t)

ZT Zt

= 0

H02 (t)

ZT Zt2

+ m2 (t) H12 (t)

ZT Zt

2+ m2 (t) H

22 (t)

ZT Zt

2= 0

therefore

m1 (t) =H02 (t) H

21 (t) H01 (t) H22 (t)

H1

1 (t) H2

2 (t) H1

2 (t) H2

1 (t)m2 (t) =

H02 (t) H11 (t) H01 (t) H12 (t)

H21 (t) H12 (t) H22 (t) H11 (t)

:

We can see that model value of this portfolio is relatively stable over t.

If the model is correct, then the model value of the portfolio can be viewed as a historical average

of the market value of the portfolio,

CDS0t + m1 (t) CDS1t + m2 (t) CDS

2t

S0t

Zt

+ m1 (t) S1t

Zt

+ m2 (t) S2t

Zt

+ 0t + m1 (t) 1t + m2 (t)

2t ;

where CDSt is the market value of the CDS spread5 with maturity of at time t, and t is the corre-

sponding discrepancy between the market and model value. Therefore any deviation from model value

suggests the existence of arbitrage opportunities and that we might be able to make prots from either

shorting or longing these portfolios.

5 The reason we use value of CDS spread as a proxy of conventional value of CDS contract (value of proctetion or premiumleg of CDS contract) is that the value of proctetion or premium leg is contaminated by interest rate risk, model price of

market neutral portfolio of protection (premium) legs is not as stable as that of CDS spreads. Therefore, it is more indicativeto use CDS speads as trading indicator than proctection (premium) legs.

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Table 1: Number of Firms in Dierent Rating and Industry Groups

This table provides the number of rms used in our analysis in terms of rating and industry. The ten industries

are basic materials, consumer goods, consumer services, nancials, health care, industrials, oil & gas, technology,

utilities, and telecommunication.

Rating Year AAA AA A BBB BB B CCC D

2005 2 10 68 119 48 43 8 0

# of rm 2006 2 11 68 119 48 43 8 0

2007 2 12 72 127 62 52 8 0

2008 2 13 74 133 66 55 17 2

Sector BM CG CS Fin HC Ind OG Tec Tel Uti

# of rm 27 48 55 44 13 37 24 14 9 26

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Table 2: Summary Statisitics of CDS Spreads for Dierent Rating Groups

This table provides the mean and standard deviation of the CDS spreads in dierent rating groups for all available m

Rating 1y 2y 3y 5y 7y 10y 15y 20y 30y

AAA Mean 0.0156 0.0139 0.0136 0.0126 0.0117 0.0112 0.0113 0.0112 0.0113Std. 0.0428 0.0349 0.0325 0.0275 0.0240 0.0213 0.0205 0.0199 0.0194

AA Mean 0.0061 0.0058 0.0059 0.0062 0.0063 0.0066 0.0069 0.0071 0.0072Std. 0.0385 0.0306 0.0278 0.0238 0.0213 0.0190 0.0172 0.0178 0.0157

A Mean 0.0045 0.0048 0.0052 0.0059 0.0064 0.0070 0.0076 0.0079 0.0082Std. 0.0191 0.0172 0.0161 0.0141 0.0127 0.0115 0.0111 0.0109 0.0108

BBB Mean 0.0059 0.0066 0.0074 0.0090 0.0098 0.0108 0.0115 0.0118 0.0120Std. 0.0240 0.0203 0.0184 0.0160 0.0142 0.0128 0.0123 0.0121 0.0121

BB Mean 0.0132 0.0161 0.0187 0.0230 0.0243 0.0257 0.0266 0.0268 0.0268Std. 0.0329 0.0326 0.0316 0.0309 0.0285 0.0269 0.0263 0.0256 0.0257

B Mean 0.0335 0.0406 0.0459 0.0527 0.0540 0.0548 0.0554 0.0555 0.0548Std. 0.0723 0.0693 0.0663 0.0608 0.0567 0.0526 0.0516 0.0508 0.0493

CCC Mean 0.1273 0.1293 0.1303 0.1324 0.1314 0.1295 0.1269 0.1265 0.1216Std. 0.2642 0.2325 0.2192 0.2023 0.1943 0.1867 0.1773 0.1772 0.1702

D Mean 0.6703 0.6179 0.6010 0.5944 0.5636 0.5322 0.4743 0.4784 0.4140Std. 0.5959 0.3744 0.3500 0.3804 0.3752 0.3695 0.3009 0.2914 0.2672

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Table 3: Summary Statisitics of CDS Spreads for Dierent Industry Sectors

This table provides the mean and standard deviation of CDS spreads for dierent industry sectors for all available m

Sector. 1y 2y 3y 5y 7y 10y 15y 20y 30y

BM Mean 0.0163 0.0181 0.0199 0.0229 0.0237 0.0246 0.0252 0.0254 0.0253Std. 0.0873 0.0762 0.0741 0.0681 0.0630 0.0609 0.0579 0.0588 0.0555

CG Mean 0.0205 0.0226 0.0242 0.0267 0.0272 0.0277 0.0284 0.0284 0.0286Std. 0.0742 0.0667 0.0619 0.0573 0.0532 0.0500 0.0489 0.0475 0.0465

CS Mean 0.0157 0.0198 0.0229 0.0268 0.0280 0.0290 0.0298 0.0300 0.0300Std. 0.0528 0.0552 0.0563 0.0541 0.0524 0.0493 0.0480 0.0471 0.0467

Fin Mean 0.0158 0.0152 0.0150 0.0152 0.0149 0.0149 0.0151 0.0151 0.0153Std. 0.0549 0.0467 0.0421 0.0375 0.0335 0.0307 0.0288 0.0280 0.0273

HC Mean 0.0044 0.0060 0.0076 0.0103 0.0114 0.0125 0.0132 0.0133 0.0135Std. 0.0097 0.0123 0.0143 0.0170 0.0175 0.0178 0.0179 0.0177 0.0177

Ind Mean 0.0133 0.0144 0.0155 0.0178 0.0190 0.0199 0.0203 0.0208 0.0204Std. 0.0632 0.0594 0.0565 0.0564 0.0574 0.0568 0.0546 0.0551 0.0522

OG Mean 0.0046 0.0060 0.0073 0.0099 0.0110 0.0122 0.0129 0.0131 0.0133Std. 0.0090 0.0101 0.0110 0.0127 0.0128 0.0129 0.0132 0.0132 0.0128

Tec Mean 0.0122 0.0154 0.0179 0.0217 0.0231 0.0244 0.0252 0.0258 0.0255Std. 0.0534 0.0513 0.0488 0.0457 0.0444 0.0424 0.0415 0.0411 0.0401

Tel Mean 0.0084 0.0116 0.0149 0.0191 0.0206 0.0220 0.0229 0.0233 0.0238Std. 0.0168 0.0199 0.0232 0.0256 0.0250 0.0243 0.0242 0.0241 0.0240

Uti Mean 0.0048 0.0062 0.0077 0.0103 0.0114 0.0126 0.0133 0.0136 0.0139Std. 0.0093 0.0100 0.0110 0.0123 0.0125 0.0125 0.0128 0.0128 0.0128

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Table 4: Variance Ratio Summary

This table provides distribution of variance ratio, the percentage of variations of CDS spreads explained by the

credit risk model, of the 297 rms used in our empirical analysis at 1, 2, 3, 5, 7, 10, 15, 20, and 30 year

maturities

1y 2y 3y 5y 7y 10y 15y 20y 30y

Min -55% 17% 43% 60% 64% 44% 48% 37% 11%

1stQuantile 80% 87% 89% 89% 89% 85% 80% 78% 73%

Median 89% 93% 94% 94% 93% 91% 88% 87% 82%

Mean 81% 89% 91% 92% 91% 88% 85% 83% 79%

3rdQuantile 94% 96% 96% 95% 95% 94% 92% 92% 89%

Max 98% 99% 99% 99% 100% 100% 99% 98% 99%

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Table 5: Estimation of CDS Spreads

This table reports the distribution of parameter estimates of the 297 rms used in our empirical analysis. Panel

(a) is the summary of parameter estimates using full sample period (2005-2008), Panel (b) is the summary of

parameter estimates using rst sub-sample period (2005-2006)

(a) Full Sample Summary

Z Z Z PZ exp(0) "

Min 0.0001 0.0080 0.0185 0.0084 0.3679 0.0005

1stQuantile 0.0022 0.4282 0.0455 0.3133 0.4461 0.0009

Median 0.0028 0.5432 0.0693 0.6534 0.5393 0.0017

Mean 0.0110 1.7572 0.1329 1.2472 0.5130 0.0056

3rdQuantile 0.0056 1.1189 0.1353 0.9507 0.5652 0.0037

Max 0.7371 123.6500 4.4288 11.3416 0.6873 0.0362

(b) First Sub-Sample Summary

Z Z Z PZ exp(0) "

Min 0.0001 0.0260 0.0188 0.0329 0.0000 0.0003

1stQuantile 0.0025 0.4839 0.0620 0.4576 0.1982 0.0006

Median 0.0038 0.9240 0.1046 1.2391 0.5665 0.0010

Mean 0.0077 2.3544 0.1127 1.9424 0.5241 0.0029

3rdQuantile 0.0075 1.9862 0.1493 2.3345 0.8267 0.0023

Max 0.2165 146.4283 0.4411 13.5166 0.9535 0.0280

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Table 6: Average Estimated ZZ=PZ across Ratings and Sectors

This table reports the average estimatedZ

PZ

Z which is the mean of default state variable Zt under P measure.

Panel (a) is the result from full sample period (2005-2008), Panel (b) is the result from rst sub-sample period

(2005-2006)

(a) Full Sample Result

AAA AA A BBB BB B CCC Aver.

BM 0.0016 0.0109 0.0053 0.3933 0.0596 0.0864CG 0.0006 0.0017 0.0385 0.0240 0.0055 0.0511 0.0218CS 0.0008 0.0020 0.0071 0.0734 0.0414 0.0379 0.0291Fin 0.0007 0.0016 0.0528 0.1319 0.0458 0.3867 0.0779HC 0.0023 0.0017 0.0030 0.0022 0.0034 0.0024Ind 0.0020 0.0052 0.0053 0.0036 0.0273 0.0044OG 0.0015 0.0039 0.0124 0.0155 0.0059Tec 0.0015 0.0178 0.0095 0.0539 0.0191Tel 0.0077 0.0073 0.0071 0.0218 0.0089Uti 0.0030 0.0033 0.0581 0.0601 0.0140Aver. 0.0007 0.0014 0.0135 0.0265 0.0322 0.0727 0.1001 0.0314

(b) First Sub-Sample Result


BM 0.0030 0.0079 0.0180 0.0045 0.0221 0.0078CG 0.0025 0.0285 0.0024 0.0385 0.0802 0.0076 0.0314CS 0.0026 0.0063 0.0122 0.0243 0.0090 0.1946 0.0196

Fin 0.0032 0.0047 0.0081 0.0345 0.0074 0.0033 0.0165HC 0.0080 0.0066 0.0107 0.0008 0.0051 0.0073Ind 0.0118 0.0057 0.0058 0.1270 0.0254 0.0152OG 0.0040 0.0060 0.0016 0.0042 0.0048Tec 0.0050 0.0086 0.0045 0.0738 0.0207Tel 0.0017 0.0021 0.0051 0.4307 0.0503Uti 0.0130 0.0090 0.0256 0.0015 0.0100Aver. 0.0032 0.0044 0.0094 0.0109 0.0201 0.0438 0.0746 0.0177

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Table 7: Average Estimated Z across Ratings and Sectors

This table reports the average estimated Z which is the mean of default state variable Zt under Q measure.

Panel (a) is the result from full sample period (2005-2008), Panel (b) is the result from rst sub-sample period

(2005-2006)






BM 1.5619 0.8183 1.7551 2.1264 2.7668 1.4487CG 0.2945 1.1902 0.5824 2.1501 5.6043 0.7684 2.0788CS 1.6681 1.6955 1.8087 2.3714 5.2998 1.6160 2.7373

Fin 2.0219 0.8488 1.1492 11.04601.0663 0.4579 4.5765HC 1.1106 1.3577 1.4630 0.3853 1.5924 1.3324Ind 1.9051 1.6657 2.1625 5.6109 0.5541 2.0470OG 0.7116 1.4033 0.7966 3.4444 1.4385Tec 2.1909 1.1734 0.4998 5.2130 2.1854Tel 0.5171 0.7780 1.0062 6.5438 1.4114Uti 0.8739 0.8446 1.0931 6.4131 1.5108Aver. 2.0219 0.8460 1.4002 2.4776 1.7712 4.7837 1.2966 2.3544

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Table 8: Average Estimated Recovery Rates exp(0) across Ratings and Sectors

This table reports the average estimated recovery rates exp(0) which are assumed to be constant over time.Panel (a) is the result from full sample period (2005-2008), Panel (b) is the result from rst sub-sample period

(2005-2006)






BM 0.7563 0.5113 0.6399 0.1207 0.7212 0.5158CG 0.6899 0.7737 0.3798 0.3353 0.3015 0.1077 0.4109CS 0.9246 0.8096 0.6131 0.3374 0.2793 0.4109 0.5040


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Table 9: Summary of Strategy Performance

This table reports Minimum, Median, Mean, 1st&3rd Quartiles, and Maximum of 3 important performance

measures: Accumulative Prot(Accum.), Sharpe Ratio, and Max Drawdown(MDD). Panel(a) is the in sampleresult, Panel(b) is the out of sample result

(a) In Sample Result

Accum. Sharpe MDD

Min 0.3429 -4.6183 0.09%

1stQ 1.4885 1.5768 0.84%

Median 2.0763 2.1575 1.16%

Mean 2.5093 2.0890 1.57%

3rdQ 2.8655 2.6424 1.86%

Max 14.7611 4.4609 8.99%

(b) Out of Sample Result

Accum. Sharpe MDD

Min 0.2516 -3.4593 0.10%

1stQ 1.1825 1.2944 0.61%

Median 1.7791 2.0630 0.98%

Mean 2.2467 1.8793 1.45%

3rdQ 2.7860 2.5670 1.55%

Max 14.0332 4.3198 17.49%

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Table 10: Average of Accumulative Prots Across Rating and Sectors

This table reports average of accumulative prots distribution across rating and sectors, Panel(a) is the in

sample result, Panel(b) is the out of sample result






BM 1.1039 1.5688 3.5429 5.6203 4.8224 2.6345CG 1.2007 1.4863 2.1653 2.1549 4.2058 6.0833 2.5163CS 0.5916 1.1331 1.9001 3.6959 4.3283 2.6993 2.7422


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Table 11: Average of Sharpe Ratios Across Rating and Sectors

This table reports average of Sharpe ratios distribution across rating and sectors, Panel(a) is the in sample

result, Panel(b) is the out of sample result






BM 0.8930 1.7946 1.8387 2.1581 2.4812 1.6837CG 1.0654 1.7556 2.1086 2.4238 2.5959 3.0154 2.2213CS -0.374 1.1108 2.2688 2.6648 2.2363 2.3778 2.1252


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Table 12: Average of Max Drawdown Across Rating and Sectors

This table reports average of Max Drawdown distribution across rating and sectors, Panel(a) is the in sample

result, Panel(b) is the out of sample result



BM 1.08% 1.52% 1.61% 2.21% 1.66% 1.57%CG 1.10% 1.28% 1.45% 1.14% 2.49% 2.95% 1.56%CS 0.43% 1.01% 1.34% 1.71% 2.31% 6.13% 1.76%Fin 3.12% 2.08% 1.69% 1.71% 1.61% 1.14% 1.80%HC 0.35% 1.15% 1.48% 4.42% 1.24% 1.46%Ind 1.47% 1.18% 1.43% 0.89% 7.03% 1.45%OG 0.83% 1.32% 0.96% 1.81% 1.24%Tec 1.39% 0.87% 1.84% 3.32% 1.75%Tel 0.87% 2.00% 1.12% 1.23% 1.47%Uti 1.54% 1.03% 0.85% 2.56% 1.27%Aver. 3.12% 1.55% 1.32% 1.36% 1.47% 2.24% 4.17% 1.57%



BM 0.82% 1.29% 1.01% 4.40% 1.11% 1.76%CG 0.99% 1.15% 1.46% 1.17% 1.80% 2.65% 1.41%CS 0.32% 0.70% 0.89% 2.68% 3.50% 5.05% 2.00%

Fin 0.80% 1.46% 1.37% 1.25% 2.28% 1.02% 1.39%HC 0.24% 0.37% 0.67% 3.86% 1.60% 0.91%Ind 1.22% 0.83% 1.05% 0.63% 7.92% 1.16%OG 0.70% 0.98% 1.11% 2.55% 1.13%Tec 0.86% 0.82% 2.21% 1.45% 1.26%Tel 0.47% 1.81% 1.61% 1.50% 1.43%Uti 0.91% 0.74% 0.84% 4.39% 1.19%Aver. 0.80% 1.13% 0.98% 1.05% 1.70% 2.80% 3.80% 1.45%

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Table 13: The P&L of the Statistical Arbitrage Strategy for 10 Firms

This table provides summary statistics of P&L for the statistical arbitrage strategies for ten individual rms, which

deviation, accumulated prot, Sharpe ratio, and Max Drawdown. Panel(a) is the in sample result, Panel(b) is the ou


Sector Rating Max Min Median Mean Std. Skew. Kurt. Accum. Sharpe M

Cytec Inds Inc BM BBB 0.08 -0.04 0.01 0.01 0.02 0.79 4.25 1.20 1.68 0.Procter & Gamble Co CG AA 0.20 -0.08 0.01 0.02 0.05 2.05 8.63 1.75 1.78 0.JetBlue Awys Corp CS CCC 0.85 -0.49 0.04 0.07 0.19 0.91 7.99 6.69 2.35 3.Gen Elec Cap Corp Fin AAA 0.19 -0.20 0.00 0.01 0.04 -0.03 14.39 0.96 0.50 2.Boston Scientic Corp HC BB 0.51 -0.38 0.01 0.02 0.09 1.05 15.06 2.14 1.12 4.Honeywell Intl Inc Ind A 0.10 -0.03 0.01 0.13 0.25 1.41 5.13 1.26 1.93 0.Marathon Oil Corp OG BBB 0.08 -0.04 0.01 0.01 0.02 0.62 3.40 1.24 1.70 0.

Hewlett Packard Co Tec A 0.11 -0.06 0.01 0.01 0.03 0.47 3.39 1.25 1.38 0.Intelsat Ltd Tel B 0.24 -0.07 0.01 0.02 0.05 2.04 10.31 1.75 1.69 1.CMS Engy Corp Uti BB 0.14 -0.06 0.02 0.03 0.04 0.47 3.32 2.64 3.63 0.


Sector Rating Max Min Median Mean Std. Skew. Kurt. Accum. Sharpe M

Cytec Inds Inc BM BBB 0.07 -0.04 0.01 0.01 0.02 0.67 3.98 1.05 1.43 0.Procter & Gamble Co CG AA 0.07 -0.04 0.01 0.01 0.02 0.67 4.32 0.83 0.64 0.JetBlue Awys Corp CS CCC 0.26 -0.11 0.02 0.02 0.05 1.45 8.05 2.37 2.40 1.Gen Elec Cap Corp Fin AAA 0.17 -0.12 0.00 0.01 0.03 1.35 13.88 0.95 0.63 1.Boston Scientic Corp HC BB 0.46 -0.36 0.01 0.02 0.08 0.97 15.14 1.81 0.96 3.Honeywell Intl Inc Ind A 0.09 -0.03 0.01 0.01 0.02 1.37 5.06 1.12 1.69 0.Marathon Oil Corp OG BBB 0.07 -0.03 0.01 0.01 0.02 0.73 3.30 1.06 1.37 0.Hewlett Packard Co Tec A 0.05 -0.03 0.01 0.01 0.02 0.67 3.25 0.88 1.03 0.Intelsat Ltd Tel B 0.16 -0.13 0.01 0.01 0.03 0.38 10.51 1.26 1.22 1.CMS Engy Corp Uti BB 0.16 -0.06 0.03 0.03 0.05 0.48 3.31 3.04 3.87 0.

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Figure 1: Dynamics of Zero Yields

This gure provides dynamics of zero yields bootstrapped from LIBOR and Swap rates with maturities

from 3 months to 30 years with equal increment of 3 months. Sample period spans the begining of 2005 and

the end of 2008.

06

07

08

5

10

15

20

25

30

0.02

0.03

0.04

0.05

0.06

DateMaturities

Zero

Yields

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Figure 2: In Sample Paths of Accumulative Prots for 10 rms

This gure provides the in sample paths of accumulated prots of the statistical arbitrage strategy for

10 rms between January 17, 2007 and December 31, 2008.

Jan07 Jan08 Jan090

0.5

1

Cy tec Inds Inc(BM,BBB)

Jan07 Jan08 Jan090

1

Procter & Gamble Co(CG,AA)

Jan07 Jan08 Jan090

5

JetBlue Awy s Corp(CS,CCC)

Jan07 Jan08 Jan090

0.20.40.60.8

Gen Elec Cap Corp(Fin,AAA)

Jan07 Jan08 Jan090

1

2

Boston Scientif ic Corp(HC,BB)

Jan07 Jan08 Jan090

0.5

1

Honeywell Intl Inc(Ind,A)

Jan07 Jan08 Jan090

0.5

1

Marathon Oil Corp(OG,BBB)

Jan07 Jan08 Jan090

0.5

1

Hewlett Packard Co(Tec,A)

Jan07 Jan08 Jan090

1

Intelsat Ltd(Tel,B)

Jan07 Jan08 Jan090

1

2

CMS Engy Corp(Uti,BB)

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Figure 3: In Sample Time Series of P&L for 10 rms.

This gure provides the in sample time series of the P&L of the statistical arbitrage strategy for 10

rms between January 17, 2007 and December 31, 2008.

Jan07 Jan08 Jan09-0.04-0.02

00.02

0.040.060.08


Jan07 Jan08 Jan09

0

0.1

0.2


Jan07 Jan08 Jan09-0.4-0.2

00.20.40.60.8


Jan07 Jan08 Jan09-0.2-0.1

00.1


Jan07 Jan08 Jan09

-0.20

0.20.4


Jan07 Jan08 Jan09

-0.020

0.020.04

0.060.08

Honey well Intl Inc(Ind,A)

Jan07 Jan08 Jan09-0.04-0.02

00.020.040.06


Jan07 Jan08 Jan09-0.05

0

0.05

0.1


Jan07 Jan08 Jan09

00.1

0.2

Intelsat Ltd(Tel,B)


00.05

0.1

CMS Engy Corp(Ut i,BB)

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Figure 4: In Sample Weekly Return Histograms for 10 rms

This gure plots the histograms of in sample weekly returns of the statistical arbitrage strategy for 10


-0.04 -0.02 0 0.02 0.04 0.06 0.080

10


-0.05 0 0.05 0.1 0.15 0.20

10


-0.4 -0.2 0 0.2 0.4 0.6 0.80

10

20


-0.2 -0.1 0 0.10

10

20


-0.2 0 0.2 0.40

10

20

Boston Scientif ic C orp(HC,BB)

-0.02 0 0.02 0.04 0.06 0.080

5

10


-0.04 -0.02 0 0.02 0.04 0.060

5

10


-0.05 0 0.05 0.10

5

10


-0.05 0 0.05 0.1 0.15 0.20

10

Intelsat Ltd(Tel,B)

-0.05 0 0.05 0.10

5

10


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Figure 5: Out of Sample Paths of Accumulative Prots for 10 rms

This gure provides the out of sample paths of accumulated prots of the statistical arbitrage strategy

for 10 rms between January 17, 2007 and December 31, 2008.

Jan07 Jan08 Jan090

0.5

1


Jan07 Jan08 Jan090

0.5


Jan07 Jan08 Jan090

1

2

JetBlue Awys Corp(CS,CCC)

Jan07 Jan08 Jan090

0.20.40.60.8


Jan07 Jan08 Jan090

1


Jan07 Jan08 Jan090

0.5

1


Jan07 Jan08 Jan090

0.5

1


Jan07 Jan08 Jan090

0.20.40.60.8


Jan07 Jan08 Jan090

0.5

1

Intelsat Ltd(Tel,B)

Jan07 Jan08 Jan090

2


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Figure 6: Out of Sample Time Series of P&L for 10 rms.

This gure provides the out of sample time series of the P&L of the statistical arbitrage strategy for 10


Jan07 Jan08 Jan09

-0.0200.02

0.040.06


Jan07 Jan08 Jan09-0.04-0.02

00.02

0.040.06



0

0.1

0.2


Jan07 Jan08 Jan09

-0.1

0

0.1


Jan07 Jan08 Jan09

-0.20

0.20.4



00.020.04

0.060.08


Jan07 Jan08 Jan09

-0.020

0.020.040.06



00.020.04


Jan07 Jan08 Jan09

-0.1

0

0.1

Intelsat Ltd(Tel,B)


00.05

0.10.15


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Figure 7: Out of Sample Weekly Return Histograms for 10 rms

This gure plots the histograms of out of sample weekly returns of the statistical arbitrage strategy for

10 rms between January 17, 2007 and December 31, 2008.

-0.02 0 0.02 0.04 0.060

510


-0.04 -0.02 0 0.02 0 .04 0 .060

5

10


-0.1 0 0.1 0.20

10


-0.1 -0.05 0 0.05 0.1 0.150

10

20


-0.2 0 0.2 0.40

20


-0.02 0 0.02 0.04 0.06 0 .080

5

10

Honeywell Intl Inc(Ind,A)

-0.02 0 0.02 0.04 0.060

5

10


-0.02 0 0.02 0.040

5

10


-0.1 -0.05 0 0.05 0.1 0.150

10

20Intelsat Ltd(Tel,B)

-0.05 0 0.05 0.1 0.150

5

10


00a exploring statistical arbitrage opps ye-xiaoxia

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