The empirical relationship between the spreads of Credit Default Swaps and Bonds

Index

1. The overview of credit default swaps
1.1 The development of credit default swaps
1.2 Structure of Credit Default Swaps
1.3 Regulation of CDS
1.4 Participants and usage of CDS

2. Theoretical framework
2.1 valuation of credit default swap
2.2. No-arbitrage approach
2.3 Equivalence relationship between CDS spread and bond spread
2.4 Methods of econometrics for the analysis

3. Explaining the data
3.2 Bond Data selection criteria
3.3 Risk free interest rates selection criteria

4. Empirical analysis
4.1 Basis spread
4.1.1 Average basis spread
4.1.2 Factors of the basis spread
4.1.3 Test on the existence of basis
4.2 Long-term relationship between the spread on both markets

5. Concluding comments

Appendix

Literature

The empirical relationship between the spreads of
Credit Default Swap and Bond

Warren Buffet, the world’s richest man, once said that derivatives are financial “weapons of mass destruction.” a term popularized by George W. Bush to describe nuclear arms. Indeed financial derivatives have a far greater impact on the market than their underlying due to their leverage effect. And the most popular and important credit derivatives nowadays are credit default swaps with a current notional value of over 60 trillion US dollars according to ISDA^[1] (International Swaps and Derivatives Association) and 58 trillion US dollars according to BIS^[2] (Bank for international settlement) respectively. That is more than the whole world’s gross domestic product in the same year!^[3]

This paper examines the empirical relationship of CDS premium and credit spread by testing on their theoretical equivalence derived by Duffie (1999). It begins with an overview of CDS followed by the theoretical framework. The analysis starts with explanation of testing methods and description of data. After confirming the existence of the basis spread, this paper goes on to analyse the interactions of CDS spread and Bond spread using econometrics methods like Cointegration and Granger Causality tests. Also examined is the leadership of price discovery process between CDS market and traditional bond market.

1. The overview of credit default swaps

1.1 The development of credit default swaps

Credit default swaps first came out on the market in the mid 90s of the last century in the United States and have been growing like mushrooms after the rain. The market of this derivative has been making enormous progress in the past years. Being the fastest-

growing financial instrument of all time, the annual growth rate of CDS market volume was 81% at the end of 2007. The figures in the following graphic according to the numbers from ISDA Market Survey show clearly the explosive growth of CDS from 2001 to 2007: (Notional amounts outstanding in billions of US dollars, adjusted for double­counting)

Abbildung in dieser Leseprobe nicht enthalten

Figure1. Notional volume of outstanding CDS from 2001 to 2007 in billions of US dollars

The major markets for CDS are in the United States and in Europe, with London being the leading market holding alone almost half of the market share in 2004^[4] One of the major reasons of high market concentration lies in the fact, as pointed out by the German Central Bank in one of its monthly reports, “that the use of CDS requires a disciplined analytical management of risks as well as a sophisticated technical infrastructure.”^[5]

Abbildung in dieser Leseprobe nicht enthalten

Figure 2. Geographic market share of credit derivatives according to data from British Banker’s Association - Credit Derivatives Report 2003/2004)

1.2 Structure of Credit Default Swaps

A credit default swap is in simple terms a contract between two counterparties trading on credit risk of a third party. It is to some extent similar to an insurance contract, with one party selling the protection against credit risk for a certain premium, and the other party paying the price for protection and thus receiving a payoff if a third party defaults. The third party here is referred to as the reference entity. The typical maturity of credit default swaps is 5 years, although 3, 7, and 10 year contracts are also traded. Theoretically it could be any other length of time.

The buyer of a CDS pays a periodical premium on an underlying e.g. Bonds, for a certain period of time in exchange for a payment in case of credit event. If the company which issued the bonds defaults within the runtime of the CDS contract, the seller pays the buyer the settlement as initially arranged. There are two types of settlement: physical settlement and cash settlement. When the physical settlement is chosen, which is still more often the case though its usage is declining6; the buyer delivers the defaulted bonds to the seller and receives from him its par value. The CDS buyer does not need to be in possession of the bonds upon entering the contract and only has to acquire them, when it comes to physical settlement. If the cash settlement is required, the seller pays the buyer the difference^[6] between the par value and the recovery amount of the defaulted bonds, and no delivery of any bonds is needed. In such case a calculation agent is involved to decide on the amount of payout. There is no payment transfer upon entering a CDS contract, thus a CDS contract has zero value at initiation.

Abbildung in dieser Leseprobe nicht enthalten

CDS makes it thus possible for the market participants to trade on the credit risks separately without entering into the credit relationship. This is also one of its major benefits. With its rapid growth and importance, CDS and therefore its premium (CDS spread) offers an alternative in valuating the risks of the underlying bonds to the bond spread itself.

1.3 Regulation of CDS

Though credit default swaps are over-the-counter financial instrument, they are regulated and standardised by ISDA, a global trade organization of financial market participants for OTC derivatives, which offers definitions of CDS terms and conditions, including definitions of credit events. The ISDA was chartered in 1985, and has more than 590 member institutions from 46 countries today.

A CDS confirmation set after the ISDA standards contains typically six parts, which are:

1. General terms (including calculation agent, reference entity and reference obligation)
2. Fixed payment
3. Floating payment (including conditions to settlement, credit events and specifications of obligations)
4. Settlement terms (including settlement method and specifications)
5. Notice and account details
6. Offices

According to ISDA following cases are defined as credit events:^[7]

a. Bankruptcy,
b. Failure to pay,
c. Obligation default,
d. Obligation Acceleration,
e. Repudiation / Moratorium,
f. Restructuring.

1.4 Participants and usage of CDS

The major market participants are banks, insurance companies, securities houses and hedge funds. As predicted in the British Bankers’ Association Survey, the hedge funds have become a major force in the credit derivatives market^[8]. According to a survey^[9] conducted by the ISDA, over 90% of the world’s 500 largest companies use derivative instruments to manage and hedge their risks more effectively. Those risks include interest rate risk, currency risk, commodity price risk and equity price risk.

Many firms like insurance companies buy CDS to limit risk exposure to certain bonds. Instead of selling the bonds that are threatened by credit erosion, the bonds holders can simply buy protection in the CDS market. This is especially useful when buyers of downgrading bonds are hard to find. Banks can buy CDS to eliminate its credit exposure instead of securitizing loans. CDS are also typically used for capital relief, yield enhancement and arbitrage, and of course for speculation.

2. Theoretical framework

2.1 valuation of credit default swap

Most valuation models of credit default swaps fall under two categories. One is the structural models, which are based on the ideas from Black and Scholes (1973) and Merton (1974). Those models treat debt of a company as an option on its asset. According to Merton a company has two types of assets: equity and bonds. The firm defaults if and only if its asset value falls below the face value of its debt. Since they treat debt as option, they use accordingly the option valuation model to determine the value of bonds. However a major problem of those models lies in the fact, that it is very difficult to accurately determine a company’s firm value. The poor empirical performance of those models is another drawback. The other group are the reduced-form models, which are represented by Fon (1994), Jarrow and Turnbull (1997), Duffie and Singleton (1999) and Hull and White (2000). Based on the no-arbitrage and risk neutral default possibility assumption, these models build a direct connection between CDS risk premiums and bond spreads.

One of the most popular reduced-form models on valuation of credit default swaps is provided by Hull and White (2000)^[10]. They made several assumptions for evaluating a plain vanilla credit default swap. First of all, the default events, risk-free interest rates and recovery rates are mutually independent. Second, the claim in the event of default is the face value plus accrued interest.

Hull and White used following notions:

T: Life of credit default swap

q(t): Risk-neutral default probability density at time t

R : Expected recovery rate on the reference obligation in a risk-neutral world. This is assumed to be independent of the time of the default and the same as the recovery rate on the bonds used to calculate q(t).

u(t): Present value of payments at the rate of $1 per year on payment dates between time zero and time t

e(t): Present value of an accrual payment at time t equal to t t£ where t£ is the payment

date immediately preceding time t.

v(t): Present value of $1 received at time t

w: Total payments per year made by credit default swap buyer

^: Value of w that causes the credit default swap to have a value of zero

n: The risk-neutral probability of no credit event during the life of the swap

A(t): Accrued interest on the reference obligation at time t as a percent of face value

Abbildung in dieser Leseprobe nicht enthalten

The payments last either until a credit event or until time T, whichever is sooner. If a default occurs at time t (t < T), the present value of the payments is w[u(t) + e(t)]. If there is no default prior to time T, the present value of the payments is wu(T). The expected present value of the payments is, therefore:Given the assumption about the claim amount, the risk-neutral expected payoff from the

Abbildung in dieser Leseprobe nicht enthalten

The value of the credit default swap to the buyer is the present value of the expected payoff minus the present value of the payments made by the buyer. Since there is no payment exchange at the beginning, the CDS has a value of zero at time 0.

Abbildung in dieser Leseprobe nicht enthalten

2.2. No-arbitrage approach

Another major reduced-form model based on risk neutral default probability and no­arbitrage is represented by Duffie (1999).11 He made for his model the following assumptions:

1. There is no embedded interest-rate swap.
2. There is no payment of accrued credit swaps premium at default.
3. The underlying is a par floating rate note with the same maturity as the CDS.
4. Shorting the underlying FRN costs nothing
5. The underlying FRN has coupon payment of R plus a fixed spread of s, with R being the rate of a risk-free FRN at time t.
6. There are no transaction costs.
7. The payoff in case of credit event is made on the immediately following coupon date of the underlying FRN
8. Tax effects can be ignored
9. The payoff in case of credit events is 100 - D, which is the difference between the face value of the bond (100) and the market value of the bond at default (D).

An arbitrage strategy is structured as following:

Table 1. Arbitrage strategy based on Duffie (1999)

Abbildung in dieser Leseprobe nicht enthalten

One can invest e.g. 100 EUR, which is received by shorting a par FRN, in buying a par risk-free FRN at time 0. During the run time, one gets coupon payment from the risk-free FRN R1 , R2 , R3 ... at times of t1 , t2 , t3 ... and pays out for the shorted FRN

correspondingly R1 + s, R2 + s, R3 + s ... If no credit event takes place before maturity, no

payment takes place. If however the FRN defaults before maturity, one buys immediately the FRN at its market value of D to end the short position and at the same time sells the risk-free FRN, which is still 100, to end the long position. Altogether, one pays a constant amount of s and gets in the end either a net payment of 100 - D in case of credit event or nothing. Since 100 - D is the exact amount specified in the CDS contract as the payoff amount at default, s has to be exactly the CDS premium.

If the real CDS premium s' is higher than the specified spread s, one can sell the CDS and gets a profit equal to the amount of s' - s from the arbitrage strategy. Vice versa, one gets a profit of s - s', if the real CDS premium s' is lower that the specified spread s. Under the no-arbitrage assumption that there is no free lunch, the CDS premium has to be s.

2.3 Equivalence relationship between CDS spread and bond spread

A simple mathematical equivalence relationship can be established with the help of two simplifying assumptions.^[11] The par risk-free note has a fixed coupon rate of r and the par defaultable bond has a fixed coupon rate of c. Both bonds have face value of 100.

Abbildung in dieser Leseprobe nicht enthalten

The arbitrage strategy above can thus be mathematically illustrated. An investor buys a par emitted risk-free bond with fix coupon payment at 100 Euro which he gets from shorting a par emitted defaultable bond also with fix coupon payment. The par risk-free bond can be sold any time at face value, in case the risky bond defaults. Since there is no net payment exchange initially, following equation must hold in order to exclude a profitable arbitrage.

Abbildung in dieser Leseprobe nicht enthalten

This indicates that the CDS premium should be approximately the same as the credit spread which comes from the difference between return and risk-free interest rate. This result serves as the theoretical basis of the empirical analysis.

As Duffie pointed out in his work “Credit Default Valuation” that this no-arbitrage situation only works under ideal conditions which barely occur in practice. For example, transaction costs are not always negligible, as well as the accrued swap premium at default. There are seldom bonds which serve as reference obligation, that have exactly the same maturity as the CDS. In addition to that, there are also problems of termination rights with bonds and aspects of taxation differences and short-sale constraints in bond markets.

In case of physical settlement the Cheapest-To-Deliver (CTD) option is possible. This leads to a higher CDS premium than the credit spread. Furthermore, liquidity in both markets is different. Bonds buyers tend to hold them until maturity, in order to for example avoid price fluctuation, which is not the case with CDS buyers.

2.4 Methods of econometrics for the analysis

Since this paper examines the empirical relationship of spreads in CDS market and bonds market, a proper method would be that of time series models.

An often observed case for economical time series, like stock prices or bond prices, is stochastic trend. A stochastic trend follows a random movement and changes over time. The simplest version of a time series model with a stochastic trend is the random walk. When the path of the series is a random walk, it means that the value of this variable tomorrow is unpredictable; it is its value today plus an unknown portion. Formally a time series Yt is said to follow a random walk if the change in Yt is independent and identically distributed.

Yt = Yt-1 + ut, where ut is independent and identically distributed and has a conditional mean of zero, i.e. £'[Mt|Yt-1,Yt-2,...] = 0. The conditional mean of Yt on the basis of data until t -1 is therefore Yt-1 , i.e. £'[Yt|Yt-1, Yt-2,...]= Yt-1 . This means, when Yt follows a random walk, the best prediction of its value tomorrow is its value today. A random walk

time series is non-stationary. Its variance increases over time, so that the distribution of Yt also varies over time. The time series in this paper are assumed to follow the random walk.

An economic method for testing the hypothesis, that a statistically significant relationship between two time series exists, for example between 3-month interest rates and 12-month interest rates, is the cointegration test. Cointegration is the situation when two or more time series variables share a common stochastic trend. In other words, cointegration is the correlation between non-stationary time series, which characterizes a state of equilibrium in the long run. A major method to test cointegration is contributed by Granger and Engle (1987)^[12], who shared the Nobel Prize in economic science in 2003. When a linear combination of 2 or n vectors of time series, which are themselves non-stationary, is stationary, these times series are then said to be cointegrated.

To test the interaction of two time series in shorter period of time, we use the Granger causality test. A time series Xt is said to Granger-cause another time series Yt, if it can be

proved, that the lagged value of Xt und Yt can better predict the value of variable Yt than it would be without the taking into account of the value of Xt.

3. Explaining the data

All data for this paper are provided by Bloomberg. Only those CDS and Bonds were selected that are quoted in the same currency, either both in USD or both in EUR, in order to eventually avoid calculation mistakes while converting exchange rate. Bonds issued by sovereign entities are excluded. Also excluded are CDS on those bonds. Furthermore, only CDS quotes or bond quotes are accepted when there are quotations on at least 200 days per year.

Due to limited quality of data and their availability^[13], we selected CDS prices and bonds prices from 20 reference entities from US and European investment-grade companies with two intended exceptional cases, i.e. Ford and General Motors. These two shall serve as example for the so-called “fallen angels”. “Fallen angels” are those bonds that were formerly investment grade but have since declined in quality to below investment grade (BB or lower).

The sample period is set from the beginning of 2005 until the end of 2006. Though the data cover only a period of two years, however, as Blanco et al (2005) pointed out in their paper, the analysis is about an arbitrage relation and “a relatively rapid reversion to equilibrium is expected”^[14]. The still missing data are approximated by means of last- observation-carried-forward method. This means, a missing quote will be replaced by the most recent available price. This method is also consistent with the assumption of random walk without drift, without trend. That is, the last observed price is the best approximation for the following missing data, so far as there is no new information available.^[15]

[...]

---

^[1] See ISDA Market Survey historical data (annual only), 1987-present

^[2] See BIS OTC derivatives market activity in the second half of 2007

^[3] Compared to the GDP data of 2007 from the World Bank. According to the World Development Indicators database, World Bank, 1 July, 2008, the world’s GDP of 185 countries of the world sums up to around 54 trillion US dollars in 2007.

^[4]

According to British Bankers’ Association - Credit Derivatives Report 2006, Executive Summery, the estimates of London market share “have dipped slightly to just under 40% from the 2004 survey”.

^[5] See Deutsche Bundesbank Monatsbericht 2004: Credit Default Swaps - Funktionen, Bedeutung und Informationsgehalt. S. 45

^[6] According to British Bankers’ Association - Credit Derivatives Report 2006, Executive Summery, 73% of transactions used physical settlement whereas 86% used physical settlement in the 2004 Survey. Cash settlement has shifted from 11% to 23%. Fixed amount settlement remained 3%.

^[7] See 2003 ISDA Credit Derivatives Definitions

^[8] According to the British Bankers’ Association - Credit Derivatives Report 2006, the share of market volume for hedge funds in both buying and selling credit protection have almost doubled since the 2004 survey.

^[9] ISDA 2003 Derivative Usage Survey

^[10] See Hull, John; White, Alan (2000): "Valuing Credit Default Swaps I: No Counterparty Default Risk"

^[11] See Zhu, Haibin (2004): "An empirical comparison of credit spreads between the bond market and the credit default swap market", BIS Working Papers No.160, Aug. 2004

^[12] See Engle, Robert F.; Granger, C. W. J.(1987): Co-Integration and Error Correction: Representation, Estimation, and Testing, Econometrica, Vol. 55, No. 2. (Mar., 1987), pp. 251-276.

^[13] It was noticed at first glance that prices of bonds are constant over long periods of time. Those bonds were discarded during the first selection process. Due to this knowledge, one can assume that the accepted data may not have all been properly processed. If the distortion of data was not apparent (e.g. identical quotations over long periods of time), we’d have no other chance to detect the discrepancy of the data, if those data were inputted arbitrarily.

^[14] See Blanco, Roberto; Brennan, Simon; Marsh, Ian W. (2005): "An Empirical Analysis of the Dynamic Relation between Investment-Grade Bonds and Credit Default Swaps", The Journal of finance, Vol.LX, No.5, Octorber 2005

^[15] It is assumed, that the missing quotes reflect the missing market information for them.