Pricing CDX Credit Default Swaps using the Hull-White Model

Pricing CDX Credit Default Swaps using the Hull-White Model Bastian Hofberger and Niklas Wagner September 2007 Abstract We apply the Hull and White (2000) model with its standard intensity and its approximate no-arbitrage valuation approach to the pricing of credit default swaps (CDSs). Based on a representative sample of individual obligors from the DJ CDX.NA.IG index universe, we evaluate the pricing performance using an overall of 63,460 quotes during the period January 1, 2002 to July 7, 2006. Given that liquid bond data are available, both valuation approaches on average provide satisfying results as measured by spread change correlations and pricing error statistics. Testing for cointegration of spreads generally confirms a stable pricing relationship. However, in about 25 percent of our sample obligors the model is obviously weak. Comparing the results to those of Stewart and Wagner (2008) tentatively suggests that, on average, the Hull-White model is not dominated by CreditGrades or trinomial trees. Forthcoming in: Wagner N. (ed.) (2008): Credit Risk Models, Derivatives and Management, Financial Mathematics Series Vol. 12, Chapman & Hall / CRC, Boca Raton, London, New York. Acknowledgements: The authors would like to thank Philip Gisdakis for his very valuable support with the data set. Corresponding author: Niklas Wagner, Passau University, Chair in Finance and Financial Control, 94032 Passau, Germany. E-mail: niklas.wagner@uni-passau.de. Electronic copy available at: http://ssrn.com/abstract=1096366

1. Introduction The market for credit default swaps shows an enormous growth as documented by the international swaps and derivatives association (ISDA) in mid 2006. This is in line with the exponential growth of the market since the early years of this century. Taking a closer look on the structure of the various products, we see that two third of the notional amount outstanding are products on single-name entities, while the remainder are multi-name instruments. Most singlename credit default swaps (CDSs) are traded on investment grade companies (84 percent), with a relatively large proportion (23 percent) referenced to sovereigns. The average rating for the underlying firms is BBB. The dominant maturity is five years, covering 80 percent of the market. The growth in the credit derivatives market since the year 2000 makes obvious that researchers like Hull and White (2000) not even had the chance to perform a reliable testing of their timely theory. Probably due to a lack in sufficient data, a broad empirical testing of CDS pricing models has not yet been conducted in the literature. Houweling and Vorst (2005) represent one of the largest studies of a reduced from model so far, with about 23,000 CDS quotes used, but the study includes CDS data only up to January 2001, when the CDS market was not yet fully developed. The last few years with increased bond market liquidity and a well-developed CDS market now provide sufficient time-series data for a comprehensive empirical study of CDS pricing models. To our knowledge, this contribution is the first to study the Hull and White (2000) model s empirical performance based on a broad sample of liquid CDS spread quotes. The model s attractiveness as a reduced form approach is that it can be implemented based on observable market data. We use both the standard and the approximate Hull-White pricing approach. The standard intensity based no-arbitrage approach requires data from at least three corresponding 1 Electronic copy available at: http://ssrn.com/abstract=1096366

liquid corporate bonds. The approximate no-arbitrage approach requires data from two corresponding liquid corporate bonds. Using an overall of 63,460 quotes during the period January 1, 2002 to July 7, 2006, we study 47 names in the standard pricing approach and 64 names in the approximate pricing approach, all from the Dow Jones CDX North America Investment Grade (DJ CDX.NA.IG) index universe. We evaluate the pricing performance using spread change correlations and squared relative pricing errors. Given that liquid bond data with appropriate maturities are available for the respective companies, both approaches provide satisfying result as measured by spread change correlations and squared relative pricing errors. Confirming previous results by Houweling and Vorst, we find that swap rather than Treasury rates should be used as a proxy for the risk free rate. Tests on cointegration of Hull-White model and market spreads show that a stable pricing relationship is present for about 75 percent of our sample companies, while the model faces seriously weak performance for the remainder cases. The literature on credit default swap valuation can be divided into three major groups: Theory and models that started with the implementation of structural models, the later developed intensity based models and finally empirical evidence including commercial implementations of credit pricing models. The family of intensity based models, also known as reduced form models, assumes that default is an unpredictable event, which is driven by some latent default intensity process. Well-known representatives are for example the Jarrow and Turnbull (1995), the Duffie and Singleton (1999) and the Hull and White (2000) model, denoted Hull-White hereafter. Intensity based models show several attractive features. A great benefit lies in their flexibility and mathematical tractability. Also, Jarrow and Protter (2004) point out that while structural models assume complete information, they are in fact suffering a lack of information concerning default points and expected recovery. Furthermore, a modeler who investigates CDS spreads is unlikely 2

to have superior information with respect to the market, so again reduced form models seem more appropriate. However, flexibility, which may be an advantage in that a model can be well fitted to some given data sample, may also prove to be a weakness. Many papers demonstrate that their approach yields decent pricing results, using one or a few sample companies. However, insample flexibility for some examples of course does not predict how a model will perform out-ofsample and for a broader cross-section of cases. The rest of the contribution is as follows. Section 2 describes our data set. In Section 3, we sketch important ingredients and results of the Hull-White model. Section 4 contains our empirical pricing results for both Hull-White valuation approaches. Section 5 concludes. 2. The CDS Data Set We pick suitable credit default swap names from the 125 companies universe of the 6th revision of the Dow Jones CDX North America Investment Grade Index. This provides us with a broad coverage of the U.S. market, with mostly liquid credit default swaps. For our sample, we downloaded CDS quotes from Open Bloomberg. The remainder data, including bond, treasury and swap data, are from Thomson Financial Datastream. Model implementation requires market data for at least two actively traded bonds for each obligor name. This reduces our sample to 47 names for the standard Hull-White valuation approach and to 64 names for the approximate valuation approach. After this reduction, there are still 63,460 quotes of credit default swaps remaining, namely 31,860 quotes for the standard and 41,576 quotes for the approximate valuation method. The data cover the period from January 1, 3

2002 to July 7, 2006. Given this, the study is one of the broadest empirical studies on the applicability of intensity based models for the pricing of CDS spreads. Our data represent U.S. investment grade issuers. The sectors Consumer Products & Retail together with Basic Industrials represent somewhat more than 25 percent of the market capitalization of our sample. These sectors are followed by Financials (21 percent), Technology, Media & Telecommunication (15 percent) and finally Energy (8 percent). This is quite in line with the overall composition of the CDX-Index and demonstrates that the choice of our sample does not yield a substantial bias with respect to industry composition. Concerning the rating classes, most of our sample obligors fall in the A and BBB rating category, which is in line with the rating of the majority of the indexed companies. In detail, 30 percent of our obligors have an A-rating and nearly 65 percent have a BBB -rating. The 5 percent remainder is rated AAA or AA, while lower rating classes are negligible (as they are removed halfannually due to the fact that the index is based on investment grade companies only). Table 1 shows the CDS quotes rating categories in the overall sample as well as by sample year. We see that the AA -rating class somewhat loses in share while the other rating classes remain pretty stable. Quote availability raises remarkably from 2002 to 2003. Note that a lower number of quotes in the year 2006 results due to the fact that we cover less than half of that year within our sample. 2002 2003 2004 2005 2006 avg. AAA 208 261 262 260 135 1,126 AA 192 522 524 520 270 2,028 A 2,304 4,452 4,978 4,940 2,565 19,239 BBB 3,972 9,196 10,914 11,180 5,805 41,067 6,676 14,431 16,678 16,900 8,775 63,460 Table 1: CDS quotes rating categories during the period January 1, 2002 to July 7, 2006 4

Since our sample commences with relatively recent data, starting as late as January 1, 2002, the problem of missing CDS-quotes was rather negligible. Less than 0.2 percent of the data were missing, normally not more than one or two days in a row, so we could use linear interpolation to close these minor gaps. Weekends and holidays are not contained in the sample as only trading days are reported. Note that the CDS data represent averages between quoted bid and ask prices since actual transaction are not publicly available. 1 For the default-free interest rates, we choose treasury rates as well as swap rates as provided by Thomson Financial Datastream. The data was complete and hence no interpolation was needed. The same holds for the corporate bond data. We choose bonds with complete time series within our sample period, with no data missing nor any obvious data errors appearing. 3. Hull-White CDS Valuation For the valuation of credit default swaps, we will use the Hull-White model. This includes the Hull-White standard intensity no-arbitrage pricing approach as well as their approximate noarbitrage pricing approach. Skipping most of the derivation, we give a brief review of both valuation approaches. 3.1 Hull-White Intensity Model Valuation In the Hull-White intensity model, the set of assumptions contains that default events, treasury rates and recovery rates are all mutually independent and that the claim made in the event of a 1 For some additional details on the data see Stewart and Wagner (2008) in Chapter 11 of this volume who start from the same initial CDS universe. 5

default is the face value plus the accrued interest. The notional principal is one currency unit. The maturity of the credit default swap is T and q (t) describes the risk-neutral probability of default at any given point of time t as determined in time zero. Subsequently, π describes the riskneutral probability of no credit event occurring between time zero and time T. The expected recovery rate in a risk-neutral world Rˆ is independent of the default time and the same for all bonds of an issuer. The present value of payments at the rate of one currency unit per year on payment dates between time zero and time t is u (t). The term e (t) represents the present value of an accrual payment with a payment date immediately preceding time t. The present value of one currency unit received at time t will be v (t). The total payments per year made by a CDS buyer (i.e. the insurance buyer) are denoted as w. The payments last until maturity T or a credit event, whichever occurs sooner. Finally, A (t) is the time t accrued interest on the reference obligation. Collecting all payment streams that a CDS will expectedly provide, the present value of the expected payoffs minus the present value of the expected payments is T 0 T [ 1 Rˆ A( t) Rˆ] q( t) v( t) dt w q( t)[ u( t) + e( t)] dt wπ u( T ). The fair spread of the credit default swap, s, is the value of w that makes this expression equal to zero, i.e. T [1 Rˆ A( t) Rˆ ] q( t) v( t) dt 0 s = 0 T. (3.1) q( t)[ u( t) + e( t)] dt + πu( T ) 0 6

The spread s represents the total sum of payments per year for a newly issued credit default swap as a relative of the swap s notional principal. 3.2 Hull-White Approximate Valuation Put very simple, a portfolio consisting of a CDS and an underlying bond of the same obligor both having maturity T, should be risk-free. It should therefore have the same payoff as a treasury bond with maturity T. Assuming for simplicity that the treasury curve is flat and that interest rates are constant, a simple no-arbitrage argument then results in the spread * s = maturity T corporate bond yield maturity T treasury bond yield which will typically overestimate the true spread s, see Hull and White (2000). However, we are able to close most of the gap between * s and s. Let * ( t) A represent the time- t accrued interest as a percentage of the face value on a T-year par yield bond that is issued at time zero by the reference entity with the same payment dates as the swap. We may refer to this bond as the underlying par yield corporate bond. As an approximation, use a (t) as the average value for A (t) under the integral in (3.1) and analogously define a * ( t) as the average value for A * ( t ), 0 < t < T. This yields an approximate formula for s, where * s (1 Rˆ arˆ) s =. (3.2) * (1 Rˆ)(1 + a ) 7

4. Empirical Results This section summarizes our empirical results from an analysis of the data set previously described in Section 2. We calculate the theoretical CDS spread for both Hull-White pricing approaches as given in Section 3 and then evaluate the results from various perspectives. In their paper, Hull and White come to the conclusion that the approximate valuation (3.2) ends up with results only differing marginally from the intensity model spread (3.1). This would constitute a quick and simple method for the applied valuation of credit default swaps. Hence, we will address both, respective model performance as compared to the market as well as relative pricing performance of the two Hull-White approaches. Additionally, we consider the swap rate as a proxy for the risk-free interest rate as an alternative to the treasury rate, which may be a bad proxy due to tax and other reasons, see Elton, Gruber, Agrawal and Mann (2001). We start our exposition with the approximate approach then turn to the standard intensity based approach and finally discuss some model sensitivity examples. 4.1 Approximate Valuation Results We start with the calculation of CDS spreads with the Hull-White approximate valuation first using the treasury rate as a proxy for the risk free interest rate. For our calculations, we need the five-year CDS spread and, in theory, a five-year corporate bond. As this is rather unlikely to find, we decided to interpolate and hence require at least one bond maturing in between one and five years and one bond maturing in five to ten years. We decline shorter maturities as we find prices obviously not to reflect usual market spread. Also, we decline bonds that would fit the maturity 8

range, but obviously show not to be liquid as their prices are subject to infrequent and large daily price jumps. This results in a data sample of 64 companies and 41,567 quotes. We find that using the treasury rate mis-estimates the CDS-spread while using the swap rate comes to better results. We calculate mean squared relative pricing errors (MSRPE) via deriving model versus market spreads for each quote and setting them relative to the observed market spread before taking squares. We sum up the MSRPE results for the whole sample in Table 2. It becomes obvious from the relatively large MSRPE statistic of 0.874 versus 0.273 that using the treasury rate yields inferior, i.e. biased, CDS spreads. 2 Treasury Swap Correlation Median 0.549 0.632 Mean 0.526 0.586 Standard Deviation 0.325 0.294 Minimum -0.236-0.062 Maximum 0.973 0.970 MSRPE Median 0.797 0.232 Mean 0.874 0.273 Standard Deviation 0.367 0.132 Minimum 0.135 0.108 Maximum 1.878 0.706 Table 2: Correlation and MSRPE statistics for Hull-White approximate valuation based on treasury and swap rates In a further investigation on the performance of the models, we consider whether model and market spreads tend to move together and if they do, to what extent. To this aim, the average 9

correlation between model and market spread changes is reported in Table 2 as well. We see a slight but obvious advantage of the performance again under the swap rate. Table 2 allows us to have a closer look at the cross-sectional variation of the spread change correlations and MSRPE statistics in our sample. It turns out that the correlation statistics end up in a large corridor between zero or even negative correlation and a nearly perfect correlation of more than 97 percent. The same results for the spread of the squared differences. The overall outcome with a mean correlation of nearly 60 percent and median in MSRPE of 23.2 percent (i.e. a root-msrpe of 48.2 percent) is not an excellent, but a satisfying result for the simple model (under the swap rate). Note however that the numbers have to be seen in the context of declining spreads and a sample that is completely in the investment grade sector. On July 7 2006, 60 percent of our sample obligors end up with a market CDS spread of less than 40 bps, where small absolute spread changes, that may be related to market issues other than credit risk, imply large relative changes. 4.2 Intensity Model Valuation Results We now implement the Hull-White intensity based model and calculate theoretical CDS spreads. Since we found above that the swap rate is the appropriate proxy for the risk free rate, we perform all calculations using the swap rate only. As it is not possible to observe the default intensity q (t) of Section 3.1, we use discrete data of all available bonds and interpolate inbetween. Therefore, we need liquid bonds to get an appropriate estimate for the yield and default 2 Unreported results show that this holds throughout all industry sectors where the treasury rate performance is between a factor of 2.3 to 5.1 times worse. 10

intensity curves. The minimum number is three bonds. Again, we require at least one bond with maturity less than five years and one with maturity beyond five years. We always include all outstanding bonds of the respective obligor. The beneficial effect is that we avoid bias by not choosing only some bonds, while, as an adverse effect, we may also include bonds, which more or less obviously lack liquidity. As a result, 47 companies providing 31,860 quotes are included in our sample. We point out that 46 of the 47 companies were contained in our study of the approximate Hull-White approach. Hence, our samples largely overlap and we now basically study a subset of the previous names. We start by taking a look at the correlation coefficient between our model spreads and the market spread changes. The results are presented in Table 3. Once the results are put in contrast to our results from the approximate valuation above, it turns out that the intensity model valuation in fact yields higher average spread change correlation results (65.6 percent) than the simple approximate valuation approach (58.7 percent). A median correlation of 71.2 percent is a highly satisfactory result with a standard deviation, which is remarkably smaller (21.9 versus 29.4 percent). 3 However, the situation changes once we consider the squared relative pricing errors in Table 3. In this case, a mean error of 27.3 percent for the approximate valuation approach clearly dominates a mis-pricing measure of 43.8 percent for the intensity approach. Also the results on crosssectional pricing stability here do not seem to support the intensity approach. As it occurs, the intensity based approach seems to be more prone to pricing bias than the approximate approach. 11

Approx. Intensity Correlation Median 0.632 0.712 Mean 0.587 0.656 Standard Deviation 0.294 0.219 Minimum -0.062 0.114 Maximum 0.970 0.948 MSRPE Median 0.232 0.372 Mean 0.273 0.438 Standard Deviation 0.132 0.212 Minimum 0.108 0.167 Maximum 0.706 1.014 Table 3: Correlation and MSRPE statistics for Hull-White approximate and intensity based valuation using swap rates. Given the above results, we may conclude that there is no clear answer to the question whether intensity valuation dominates approximate valuation or vice versa. Still we have to point out that, to some extend, our results may relate to sample selection bias. With the approximate pricing we choose the most liquid and perfect fitting bonds only, while with intensity pricing we always use the whole set of available bonds in order to derive the default intensity curve. It could be that pricing bias in the intensity approach is related to the use of mis-priced low liquidity bonds. In order to investigate this issue, we next consider our pricing statistics for different sectors and rating classes in Table 4 and in Table 5. 3 Looking at single cross-sectional spread change correlations, we observe that in some cases one model approach obviously fails while the other provides excellent results. Apparently, this was a matter of one bond being seriously mis-priced due to liquidity problems or other factors we could not monitor. 12

In Table 4 the spread correlation results clearly favor the Basic Industrials and the Technology, Media & Telecommunication sectors which, due to intensity modeling, show absolute improvements in correlation of 15.4 and 9.2 percent, respectively. We point out that that these sector companies often have a wide range of bonds outstanding, giving us plenty of data for calculating default density curves. On the rating side, lower ratings improve their performance under intensity modeling especially with the BBB-rating now reaching 67.6 percent spread change correlation. 4 Approx. Intensity Correlation Consumer Products 0.575 0.580 Energy 0.698 0.636 Financials 0.544 0.606 Basic Industrials 0.563 0.717 Technology 0.653 0.744 AAA 0.847 0.533 AA 0.550 0.576 A 0.477 0.613 BBB 0.627 0.676 mean 0.526 0.656 Table 4: Correlation statistics for Hull-White approximate and intensity based valuation according to sectors and ratings The MSRPE results of Table 5 confirm the bias issue with the intensity model approach. For the sectors, best performance now is in Consumer Products and Retail as well as in Energy. Basic Industrials and Technology, Media & Telecommunication now perform mediocre when compared to the approximate pricing approach. Hence, this supports the hypothesis that a 4 The decline in the AAA is to be considered as random due to the fact that only one company is contained in that subsample. 13

large number of available bonds may to improve spread correlation while it does not reduce pricing bias. In Table 5, lower-rated bonds appear to be the best performers for intensity valuation. Approx. Intensity MSRPE Consumer Products 0.258 0.369 Energy 0.180 0.368 Financials 0.274 0.489 Basic Industrials 0.266 0.462 Technology 0.310 0.484 AAA 0.528 1.014 AA 0.195 0.763 A 0.337 0.571 BBB 0.230 0.368 mean 0.273 0.438 Table 5: MSRPE statistics for Hull-White approximate and intensity based valuation according to sectors and ratings 4.3 Spread Cointegration: Are there Stable Pricing Relationships? The results of Sections 4.1 and 4.2 showed relatively satisfying pricing results based on average spread change correlations and squared pricing errors. We examine cointegration in market versus Hull-White approximate model spreads. In case we find cointegration being present, this confirms a stable pricing relationship and supports a potential use of the model for trading and arbitrage strategies. Differences between model spread and market spread then have a stationary distribution during time, including constant mean and variance. 14

We test the null hypothesis that there is cointegration including a constant term on the 10, 5 and 1 percent signification level. The estimation is done using the classical Dickey-Fuller test for a unit-root on the differences between market and model spread. Once we run the test over the 64 obligors sample, we find that for 75 percent of our obligors we can confirm the existence of cointegration at the 95 or even 99 percent confidence level. 5 This result supports the general validity of the pricing model. However, it also points out that the Hull-White approximate model (due to various disturbing factors as given during our sample period) fundamentally fails for the remainder 25 percent of our 64 obligors. We observe that spread cointegration is supported especially for companies providing sufficient bond data. In such individual cases, trading strategies could be derived given that the market spread deviates from the model spread. The spreads of AutoZone Inc. in Figure 1 may serve as an example, where price deviations appear stable during time and occasionally vanish over the longer run. Figure 1: AutoZone Inc., Market and Hull-White approximate CDS spreads 5 These results go in hand with the previous results. Once we find individual spread change correlations reaching 15

4.5 Sensitivities for CDS Valuation Examples In this section we briefly present three companies that turned out to provide a decent model fit with a spread change correlation of more than 90 percent. We consider high, medium and low CDS market spread levels and chose Walt Disney Co., Clear Channel Inc. and Supervalu Inc. as example companies. On September 28, 2006, these had CDS spreads of 27.7, 116.3 and 185.3 bps, respectively. Figure 2: Sensitivity of the model spread with respect to the recovery rate: three examples. levels of 80 percent or more, we can in most cases confirm cointegration at least at the 5 percent level. 16

Figure 3: Sensitivity of the model spread with respect to the risk-free rate: three examples. We perform an analysis of the sensitivity of the Hull-White approximate model with respect to the main input-parameters, namely the recovery rate (see Figure 2) and the risk-free interest rate (see Figure 4). In Figure 2, the recovery rate is shown to be of subordinated importance, at least as long as it does not exceed values of 65 to 70 percent, a scenario, which normally seems rather unlikely. Inside this range, assuming recovery rates from zero to 60 percent, the impact on the estimated CDS spreads is in the range of one to five basis points, i.e. less than 5 percent of the respective absolute CDS spreads. In contrast to that, the level of the risk-free interest rate has a remarkable influence on the calculated CDS spreads in Figure 3. An interest rate reduction of 50 bps results in a spread increase of around 48 bps for Clear Channel Inc. and Supervalu Inc. This result is intuitive, since ceteris paribus CDS model spreads are proportional to bond minus risk-free rates. As can also be seen from Figure 3, a lower barrier is present in that the interest rate sensitivity shrinks to zero as the spread approaches zero. 17

5. Conclusion While the calibration of an intensity model may provide brilliant results for several specific obligors, the present study shows that CDS pricing faces problems when applied to a broader cross-section of names. These problems start with liquidity issues and the availability of input variables and continue with a large cross-sectional dispersion of the pricing performance of a given model. We suggest that an important factor, which determines Hull-White pricing performance is first of all the input data, while other factors such as sector specification and rating class may also play a role. This could be addressed in future research. Despite these shortcomings, we stress that even the approximate Hull-White valuation approach can indeed produce satisfactory results. This is a remarkable result as pricing is achieved without any in-sample fitting. Especially in the BBB rating class, the approximate approach competes very well to the intensity approach, which requires a larger number of input data. The latter approach may achieve superior pricing results especially for the high-quality AAA to A rating classes. All in all, both Hull-White approaches represent powerful pricing tools, ready to calculate fair CDS spreads. Given partly overlapping CDX samples, we may compare the results of this study to those of Stewart and Wagner (2008) in Chapter 11. Using percentage median spread change correlations as the aggregate pricing performance measure, indicates that a tentative performance ranking would imply the following order: Hull-White intensity (71 percent) Hull-White approximate (63 percent) trinomial trees (60 percent) CreditGrades (42 percent). This suggests that Hull- White pricing may on average well compete with structural pricing approaches such as 18

CreditGrades or with hybrid valuation via trinomial trees. Future research may address CDS pricing models and their performance in more detail. References Duffie D., Singleton K. (1999): Modeling the Term Structure of Defaultable Bonds, Review of Financial Studies 12: 687-720 Elton E. J., Gruber M. J., Agrawal D., Mann C. (2001): Explaining the Spread on Corporate Bonds, Journal of Finance 56: 247-277 Houweling P., Vorst T. (2005): Pricing Default Swaps: Empirical Evidence, Journal of International Money and Finance 24: 1200-1225 Hull J. C., White A. (2000): Valuing Credit Default Swaps: No Counterparty Default Risk, Journal of Derivatives 8: 29-40 Jarrow R. A., Protter P. (2004): Structural versus Reduced Form Models: A New Information Based Perspective, Journal of Investment Management 2: 1-10 Jarrow R. A., Turnbull S. (1995): Pricing Derivatives on Financial Securities Subject to Default Risk, Journal of Finance 50: 53-86 Stewart C., Wagner N. (2008): Pricing CDX Credit Default Swaps with CreditGrades and Trinomial Trees, in: Wagner N. (ed.) (2008): Credit Risk Models, Derivatives and Management, Chapman & Hall CRC, Boca Raton, pp. 181-196 19