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A simple continuous measure of credit risk
Available online at www.sciencedirect.com
International Review of Financial Analysis 16 (2007) 508 – 523
A simple continuous measure of credit risk Hans Byström a,⁎, Oh Kang Kwon b,1 b
a Department of Economics, Lund University, Box 7082, 220 07 Lund, Sweden Discipline of Finance H69, The University of Sydney, Sydney NSW 2006, Australia
Received 9 February 2007; accepted 4 March 2007 Available online 14 March 2007
Abstract This paper introduces a simple parameterization for the risk-neutral default probability distributions for risky firms that are easily computed from quoted bond prices. The corresponding expected times to default have a particularly simple form and are proposed as a measure for credit risk. Being continuous in nature, times to default provide a much finer measure of risk than those provided by ratings agencies. Comparison with the ratings provided by Moody's and the distance to default measures calculated using the Merton [Merton, R. (1974). On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 2(2), 449–470] model shows that the highest rank correlation is found between the proposed time to default measure and Moody's ratings. © 2007 Elsevier Inc. All rights reserved. JEL classification: G33; C20 Keywords: Default probability; Continuous credit rating; Expected time to default; Corporate bonds
1. Introduction Credit risk, or the risk of counterparty default, is an important factor in the valuation and risk management of financial assets. The losses arising from the string of recent corporate collapses, including Enron, WorldCom and Kmart, provide ample evidence of its importance. However, the risk of default, by its very nature, is difficult to quantify, and the traditional credit ratings supplied by agencies such as Moody's and Standard & Poor's are far from ideal. For example, the discrete nature of these ratings result in groups of firms being assigned the same rating despite the
⁎ Corresponding author. Tel.: +46 46 2229478; fax: +46 46 2224118. E-mail addresses: [email protected] (H. Byström), [email protected] (O.K. Kwon). 1 Tel.: +61 2 9351 6448; fax: +61 2 9351 6461. 1057-5219/$ - see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.irfa.2007.03.002
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differences in their credit worthiness, and the changes in the credit worthiness of firms are not immediately reflected in their ratings. Moreover, the measures of credit risk implied from models, be it traditional scoring models or structural models like the Merton (1974) model, typically rely on balance sheet data and historical stock volatilities. Consequently these measures tend to rely as much on historical events as on the current market situation, and are subject to the availability of accurate and up to date balance sheet data. This paper introduces a new measure of credit risk that complements the traditional credit ratings in several ways. Firstly, being continuous in nature, the measure allows a finer separation of firms according to their credit worthiness. Secondly, being computed from current bond data, it adjusts immediately to changes in the market's assessment of the credit worthiness of firms. Finally, unlike the symbolic ratings that are without economic meaning, it represents the expected time to default under the risk-neutral measure. As pointed out in Delianedis and Geske (1998), for example, the risk-neutral default probabilities are upper bounds for the actual default probabilities, and so the times to default we compute are useful indicators of changes in creditworthiness and likelihood of default for risky firms. Since expected time to default is a time weighted average of the default probabilities, it gives more weight to the risk of default in the longer term. This long term view is similar to the approach taken by Moody's whose ratings typically focus on at least one entire business cycle. Merton's model, on the other hand, is typically used to calculate one to two year default probabilities. Nevertheless, a comparison of the measures reveals a fairly high degree of correlation, and, in any case, for practical purposes they should probably be seen as complements rather than substitutes for one another. The data requirements to compute the measure introduced in this paper are minimal. The only requirements are the bonds issued by the firms under consideration and proxies for the risk-free bonds, which may be the treasury bond or the bonds issued by a firm of highest credit rating. However, to ensure sensible default intensities, we found it necessary to construct the risk-free yield curve with the condition that the risk-free yields lie below the yields of all corporate bond yields. The model is also quite simple. In particular, it is shown that a firm's entire inter-temporal distribution of default intensities can be inferred from instantaneous forward rates. This distribution is then used to extract the expected time to default. The structure of the remainder of the paper is as follows. Section 2 gives a brief review of the existing measures of credit risk, Section 3 defines the new credit risk measure and outlines a method for its computation, Section 4 presents the data and the methodology used in the empirical study of this paper, and Section 5 provides the results. The paper finally concludes with Section 6. 2. Traditional measures of credit risk All investors, regardless of whether they invest in equities or fixed income instruments, need to take into consideration the risk of counterparty default when making their investment decisions, and the ability to quantify this so called credit risk is important for both pricing and risk management purposes. There are several ways in which the risk of default can be gauged. One way is to rely on rating agencies, such as Moody's and Standard & Poor's, that rate individual firms' capability to service and repay their debt. An alternative way is to rely on models that attempt to quantify the level of credit risk using accounting information and stock price data.
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Unfortunately, the precise processes by which the ratings agencies arrive at their credit ratings remain the proprietary information of the respective agencies. However, it is most likely that they incorporate the analysts' views formed from public and private information. A particularly simple and popular approach for inferring credit risk information using accounting and stock market information is the contingent claims based model of Merton (1974). This model views a firm's liabilities (equity and debt) as contingent claims issued against the firm's underlying assets. By backing out asset prices and volatilities from quoted equity prices and balance sheet information, it provides estimates of the firm's default probabilities. This model is now described in detail. 2.1. Review of the Merton model Recall that equity holders have the residual claim on a firm's assets while being subject to limited liability. Merton (1974) recognised that equity in a firm is equivalent to a long position in a call option on the firm's assets, and used this correspondence to derive the market value and volatility of the firm's underlying assets. More precisely, Merton used the Black and Scholes (1973) framework to solve for the asset value and volatility implied by the option price and the option price volatility.2 The asset value and the asset volatility can then be combined into a risk measure called distance to default that is directly related to the credit worthiness of the equity issuing firm. At the heart of the Merton (1974) model lies a modified version of the Black–Scholes formula linking the market value of equity and the market value of assets VE ¼ VA N ðd1 Þ−e−rðT −tÞ DN ðd2 Þ
ð1Þ
where N ðdÞ is the cumulative normal distribution, and VE VA D T−t r d1 d2
Market value of the firm's equity Market value of the firm's assets Total amount of the firm's debt Time of maturity of the firm's debt Risk-free interest rate lnðVA =DÞ þ r þ 12 r2A ðT −tÞ pffiffiffiffiffiffiffiffi rA T −t pffiffiffiffiffiffiffiffi d1 −rA T −t
Moreover, it is easily shown that the equity and asset volatility are related by the expression rE ¼
VA N ðd1 ÞrA VE
ð2Þ
where σE and σA are volatilities of the firm's equity and asset returns respectively. Solving the nonlinear system of Eqs. (1) and (2) gives VA and σA, and the distance to default is defined by the expression lnðVA =DÞ þ r− 12 r2A ðT −tÞ pffiffiffiffiffiffiffiffi g¼ ð3Þ rA T −t 2
Note that in this case the option is equity and the underlying asset is the firm's assets.
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This is simply the number of standard deviations that the firm value is from the threshold point, and the smaller the value of γ the larger the probability that the firm will default on its debt. In this paper we compare the rankings implied by this risk measure to the rankings obtained from our continuous risk measure and those provided by Moody's. Note that while the risk measure implied by Merton's model is based on observations from the stock market, the measure introduced in this paper will be based on observations from the bond market. Consequently, the comparison of the two results could shed some light on whether the stock and the bond markets carry similar information about the credit worthiness of firms. 2.2. Motivations for a new risk measure Although useful, traditional credit ratings and scoring models relying on accounting information have several drawbacks as outlined above. The most obvious limitations with the rankings provided by the ratings agencies include the low frequency with which they are updated, often only once or twice a year, and the discrete nature of the rankings. Accounting based scoring models, on the other hand, rely on infrequently updated data that are not only released with a time lag, but also with possible accounting manipulations. Moreover, accounting information is inherently backward looking, based on historic information rather than the market's assessment of the future. Both approaches also suffer from producing measures without meaningful economic interpretations. Neither a credit rating nor a credit score is easily transformed into an actual probability of default and neither gives much information about the actual loss given such a default. An alternative to these credit ratings and scores involves the extraction of information about credit risk from up to date market data. If credit risk is incorporated into market prices then there must be ways of filtering the information contained in these prices to extract the component that can be attributed to credit risk. Various ways of extracting this information from either the stock or the bond data have been suggested. Public information is likely to be instantaneously and simultaneously embedded into individual security prices in the stock and the bond markets. This makes stock and bond prices contemporaneously correlated. In contrast, private information conveyed by informed investors that systematically trade in either the stock or the bond market is transmitted more through one of the markets than through the other.3 Stock and bond markets consequently carry somewhat different information regarding credit risk. The most well known stock market based approach of providing estimates of the firm's default probability, the Merton (1974) model, was described above. Now, data from the bond market can also be used as the basis for extracting credit risk information, and the method proposed in this paper assumes a parametric form for the risk-neutral default probability distributions under which the expected times to default have a simple closed form expression. The expected times to default are then proposed as a summary statistic, or measure, of credit risk of firms. As with most credit risk measures based on debt market data, the underlying idea is to compare bonds with credit risk (corporate bonds) to those essentially without credit risk (treasury bonds or high quality corporate bonds), and transform yield spreads into default probabilities.
3
The reasons for informed traders to systematically prefer one market to the other include difference in risk aversion, transaction costs, insider-trading laws or different institutional constraints.
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3. Expected time to default This section provides a detailed description of the method used to compute the expected time to default under the risk-neutral measure. As mentioned previously, being continuous, this measure avoids the problems of traditional ratings regarding widely different (ex post) default rates within the same rating class and the large overlaps in default rates between different rating classes. It also assigns a single number to each firm rather than the multiple ratings of ordinary rating agencies, due mainly to their focus on debt issues instead of the actual firm. In computing the expected time to default, we make the simplifying assumption that the risk-free interest rate process and the default processes of the firms are independent. This assumption is common in the literature, and is made for example in Das and Tufano (1996), Jarrow, Lando and Turnbull (1997) and Leland and Toft (1996), but whether it is reasonable or not is an outstanding empirical issue. It should be noted that several studies indicate violations of this assumption, most notably Duffee (1998) and Longstaff and Schwartz (1995) who find that spreads on corporate bonds fall when risk-free interest rates rise. An important practical issue is the choice of risk-free bonds from which the corresponding risk-free interest rates must be inferred. Although treasury bonds and the corporate bonds of the highest quality are both commonly used proxies for the risk-free bonds in the literature, the latter is often the better choice, as suggested for example in Kwan (1996), since a large part of the spread between the corporate and treasury bond yields arises from differences in factors such as taxation and liquidity (Huang & Huang, 2002), and using high quality corporate bonds allows the impact of some of these non-credit related causes of the credit spread to be minimized. In this paper, we construct the risk-free forward rate curves with the primary goal of ensuring that they lie below all the risky forward rate curves. The risk-free forward rates are then used to derive the risk-free zero-coupon prices and the default probabilities. This procedure is necessary if the risk-free bonds are to be priced above their risky counter parts, and this in turn is necessary to ensure the positivity of default probabilities. Finally, it is assumed that the recovery rate, or the fraction of the par value that is recovered by the bondholder in case of default is constant and firm independent. For the purposes of this paper, we use an averaged historical value quoted in Jarrow et al. (1997). Although it is possible to relax this assumption by estimating the recovery rate from the bond price data, we nevertheless treat it as an exogenous constant for simplicity since the recovery rate is of less importance for the purposes of determining the relative credit rankings of firms in this paper. 3.1. Expected time to default under the risk-neutral measure The first step in determining the expected time to default involves inferring the risk-neutral survival probabilities from the bond market data. For convenience, assume that the firms under consideration are indexed by the set {1, 2, …, n} for some integer n and let 0 be the index for the risk-free proxy. Let p0 (t, T) denote the time t price of a T-maturity risk-free zero coupon bond and let pi (t, T) be the corresponding price for bonds issued by firm i. Then, as in Jarrow et al., 1997 the assumption of independence of the risk-free interest rate process and the default processes implies pi ðt; T Þ ¼ p0 ðt; T Þ½dFi ðT jtÞ þ ð1−Fi ðT jtÞÞ
ð4Þ
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where δ is the constant recovery rate for all firms, and Fi(T|t) is the risk-neutral conditional probability of default prior to time T for firm i given that it has survived to time t ≤ T. Rearranging this equation gives the expression, Fi ðT jtÞ ¼
1−pi ðt; T Þ=pi ðt; T Þ 1−d
ð5Þ
for the risk-neutral conditional default probabilities. Since it is assumed that the recovery rate is given exogenously, the only task that remains in order to calculate the default probabilities, Fi(T|t), is to extract zero coupon bond prices from traded coupon bond prices, and a detailed description of the method used for this task in this paper is given in the next subsection. It follows from Eq. (5) that the conditional default density, fi(T|t), corresponding to Fi(T|t) is given by the equation fi ðT jtÞ ¼
AFi ðT jtÞ 1 A pi ðt; T Þ ¼− AT 1−d AT p0 ðt; T Þ
ð6Þ
Now, if we denote for each firm i the corresponding T-maturity forward rate at time t by ri(t,T), then Z T pi ðt; T Þ ¼ exp − ri ðt; uÞdu ð7Þ t
and so we have Z T 1 A exp − ðri ðt; uÞ−r0 ðt; uÞÞdu 1−d AT t Z T ri ðt; T Þ−r0 ðt; T Þ exp − ¼ ðri ðt; uÞ−r0 ðt; uÞÞdu 1−d t ri ðt; T Þ−r0 ðt; T Þ pi ðt; T Þ ¼ 1−d p0 ðt; T Þ
fi ðT jtÞ ¼ −
The time to default for firm i under the risk-neutral measure is hence given by Z l si ¼ ufi ðujtÞdu
ð8Þ
ð9Þ
0
In the following subsections, we propose a parameterization for the conditional default densities, fi(T|t), that facilitates the computation of the above integral. 3.2. Forward rate curves In constructing the forward rate curves, we face a few problems. Firstly, bonds traded in the market are all of finite maturity, and moreover the longest dated bond for each firm differ in general. Next, since only a few bonds are traded for each firm, some form of interpolation is needed to construct the entire curve of forward rates. Finally, the forward rate curves constructed from most bootstrapping type techniques are not smooth enough for the purposes of this paper.
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In order to overcome these problems, we assume a parametric form for the forward rate curves proposed in Nelson and Siegel (1987). Under this parameterization, the instantaneous forward rate curve, riNS(t,T), is assumed to be given by the expression riNS ðt; T Þ ¼ ai;0 þ ai;1 e−ji ðT−tÞ þ ai;2 ðT −tÞe−ji ðT −tÞ
ð10Þ
where ai,0, ai,1, ai,2 and κi are parameters satisfying the constraints ai;0 N0
ai;0 þ ai;1 N0
and
ji N0
ð11Þ
Note that ai,0 = lim(T−t)↑∞ riNS(t, T) corresponds to the long rate for firm i, and ai,0 + ai,1 = riNS (t, t) corresponds to the short rate. By fitting a Nelson–Siegel curve to each firm and date, we ensure that forward rates are available for all maturities for each firm and date, and the forward rate curves, riNS(t, T), are smooth. The standard approach for fitting a Nelson–Siegel curve to bond data, as proposed for example in Bolder and Stréliski (1999), is to determine the parameters by minimizing the total weighted4 squared error in the bond prices implied by Eq. (10) using some optimization routine. However, as pointed out by the authors, the parameters thus obtained are highly sensitive to the initial values. To overcome this problem, they approximate ai,0 and ai,1 using the yields on the longest and the shortest dated bonds, and compute remaining parameters by minimizing the total squared error as described above. We construct the Nelson–Siegel curves using an alternative approach which avoids approximating ai,0 and ai,1 using bond yields. First, we use a technique from Kwon (2002) to compute the forward rate curves that maximize a given smoothness measure. These curves turn out to be splines with certain functional forms. The technique also adjusts the market bond prices, within their bid–ask spreads, to improve the smoothness of the resulting forward rate curves. The smoothness measure used in this paper is Z T¯ ri Vðt; uÞ2 du ð12Þ 0
where T¯ is the maturity of the longest dated bond, and results in quadratic splines. This is the smoothness measure used in Frishling and Yamamura (1996) in constructing their discrete forward rate curves from bond data. Since the resulting forward rate curves, si(t, T), from this approach have finite horizon, T¯, and are often oscillatory in nature, they are unsuitable for the purposes of this paper. Nevertheless, they play a key role in determining the Nelson–Siegel parameters. More specifically, in contrast to the approach of fitting the parameters directly to bond data as outlined in Bolder and Stréliski (1999), we fit the parameters to si(t, T). Under our approach, the Nelson–Siegel parameters ai,0, ai,1, ai,2, κi are obtained by minimizing the squared error N X
ðriNS ðt; Tk Þ−si ðt; Tk ÞÞ2
ð13Þ
k¼1
for some N, where riNS(t, Tk) are the Nelson–Siegel forward rates, si(t, Tk) are the spline forward rates, and Tk are suitably chosen maturities. For simplicity, we set N equal to the number of knot points in the spline si(t, T), and Tk to the k-th knot point in our calculations. In contrast to the standard approach, it was found that the parameters computed by our approach were not very sensitive to the initial values. 4
Squared errors are weighted by the square of the duration of the bonds.
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Fig. 1. Nelson–Siegel forward rate curves for September 28, 2001. Pale curves in the background are the splines to which Nelson–Siegel curves were fitted.
As discussed earlier, common proxies for the risk-free bonds in the literature are the treasury bonds or the bonds from a firm with the highest credit rating. However, with the data used in this paper, neither of these were satisfactory since in many cases the corresponding risk-free forward rates were higher than their risky counterparts, resulting in negative default probabilities as is evident from Eq. (8). To overcome this serious problem, we constructed the risk-free forward rate curves with the main aim of ensuring that they remained below all the risky forward rate curves. For each date, we divided the maturity interval into 0.25-year subintervals, identified the lowest forward rate in each subinterval among all the forward rate curves for that date including the Treasury curve, and then fit a Nelson–Siegel curve through all these minimal forward rates. Fig. 1 gives examples of typical Nelson–Siegel curves that result from the above construction procedure. It can be seen that the Nelson–Siegel parameterization does a good job of fitting the splined forward rate curves for the firms.5 3.3. Inter-temporal default densities It is assumed that risky firms eventually default so that limT↑∞ Fi(T|t) = 1. However, this is incompatible with using the Nelson–Siegel parameterization for the forward rate curves. To see this, note that if the forward rate curve, riNS(t, T), is given by Eq. (10), then the corresponding bond prices are given by ai;1 −ji t −ji T ai;2 −ji t −ji T ai;2 −ji t −ji T pNS ðt; T Þ ¼ exp −a ðT −tÞ− ðe −e Þ− ðe −e Þ− ðte −Te Þ i;0 i ji ji j2i ð14Þ 5
These firms are chosen among the 23 firms in the empirical study in Section 4. They are chosen from the top and bottom end of firms ranked according to creditworthiness.
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Hence, in view of condition (11) and the fact that ai,0 N a0,0, we have pNS i ðt; T Þ ¼ lim e−ðai;0 −a0;0 ÞðT −tÞ ¼ 0 NS Tzl p0 ðt; T Þ T zl lim
and so NS 1−pNS 1 i ðt; T Þ=p0 ðt; T Þ ¼ p1 1−d 1−d T zl
lim Fi ðT jtÞ ¼ lim
Tzl
ð15Þ
unless δ = 0. It follows that Fi (T |t) implied by the Nelson–Siegel forward rate curves is not a proper probability distribution. In order to overcome this problem, we use the Nelson–Siegel parameterization once again, but this time for the conditional default probability distribution Fi(T |t). That is, it is assumed that Fi(T |t) has the form FiNS ðT jtÞ ¼ bi;0 þ bi;1 e−ki ðT−tÞ −bi;2 ðT −tÞe−ki ðT −tÞ
ð16Þ
and to ensure that FiNS(T |t)is a proper probability distribution, we impose that the parameters satisfy the constraints ki N0;
bi;0 ¼ 1;
bi;1 ¼ −1;
ð17Þ
0Vbi;2 bki FiNS(T |t) ≥ 0
These conditions follow from imposing the requirements for all T ≥ t, and limT↑∞ FiNS(T |t) = 1. Hence, the valid Nelson–Siegel conditional default probability distributions are FiNS ðT jtÞ ¼ 1−ð1 þ bi;2 ðT −tÞÞe−ki ðT −tÞ
FiNS(t|t) = 0,
ð18Þ
with 0 ≤ b2,i b λi. The parameters bi,2 and λi are finally computed by minimizing the squared error 2 N NS X 1−pNS i ðt; Tk Þ=p0 ðt; Tk Þ FiNS ðTk =tÞ− ð19Þ 1−d k¼1 for some N and suitably chosen maturities Tk. Here, piNS(t, Tk) and p0NS(t, Tk) are the risky and the risk-free bond prices implied by the Nelson–Siegel forward rate curves. In this paper, we have chosen N = 80 and Tk = 0.125k for 1 ≤ k ≤ 80. The conditional default density function, fi NS(T |t), corresponding to FiNS(T |t) is then given by AFiNS ðT jtÞ ¼ ðki −bi;2 þ ki bi;2 ðT −tÞÞe−ki ðT−tÞ AT and the expected time to default for firm i is given by Z l bi;2 þ ki si ¼ ufiNS ðujtÞdu ¼ k2i 0 fiNS ðT jtÞ ¼
ð20Þ
ð21Þ
Hence, the expected time to default, τi, are easily computed once the Nelson–Siegel parameters for FiNS(T |t) have been determined. 4. The data The data used in this paper consists of daily closing mid prices of 23 firms in various industries that are traded at the NYSE from January 3, 2001 to December 31, 2001. A list of these firms is
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Table 1 Average daily ranking of the firms over the year Moody's
Merton
Time to default
GE (Aaa) Walmart (Aa2) Coca Cola (Aa3) Bell South (Aa3) Pacific Bell (Aa3) Bank One (Aa3) Allstate (A1) Target (A2) Phillip Morris (A2) Sears Roebuck (A3) Carolina P.L. (A3) Time Warner (Baa1) Seagram (Baa2) Illinois power (Baa2) Coastal (Baa2) Pohang Steel (Baa2) Lockheed Martin (Baa3) Occidental (Baa3) Union Pacific (Baa3) MGM (Baa3) Marriott (Ba2) SCI (B1) Owens Illinois (B3)
Coca Cola Coastal Walmart Bell South Allstate Occidental Carolina P.L. Pacific Bell Marriott Phillip Morris Seagram Lockheed Martin Time Warner Union Pacific Target Bank One GE Illinois power Pohang Steel Sears Roebuck MGM SCI Owens Illinois
GE Seagram Target Walmart Coca Cola Phillip Morris Bell South Allstate Union Pacific Lockheed Martin Pacific Bell Sears Roebuck Time Warner Marriott Occidental Illinois power Coastal Carolina P.L. Pohang Steel MGM Bank One SCI Owens Illinois
given in Table 1. Most of the firms are large multinationals and for the time period considered they all had at least three outstanding bonds with maturities less than ten years. All the corporate bonds in our sample are bonds without any contractual provisions such as convertible, callable, or putable features. The bond price data was obtained from DataStream, and represent actual trade prices rather than matrix prices. In the statistical study of the three risk measures in Section 4, we were forced to shorten our sample significantly from 261 days to 50 days. The reason being that many bonds were not traded on a daily basis and there are only 50 days when all 23 firms have at least three traded bonds.6 Finally, following Jarrow et al. (1997) we have set δ = 0.3265 as the exogenous recovery rate for all firms. This assumption can easily be relaxed by setting a different recovery rate for the firms. For comparative reasons we also obtained daily credit ratings from Moody's (senior unsecured debt ratings), and the daily stock prices and debt levels for each firm from DataStream. Moody's ratings were then converted to numerical values between 1 and 23 with the larger numerical value corresponding to better credit quality.7 The debt levels required for Merton's model were approximated by interpolating the total debt figures contained in the year-end balance sheet reports and the maturities of these debts were assumed to be one year. Market capitalizations were 6 An alternative would have been to reduce the number of firms, but in addition to reducing an already fairly small sample of firms it would also have removed nonproportional amounts of firms with low credit ratings. A second alternative would have been to interpolate between two daily measures to compute the risk measure on the date in between. This would, however, create its own problems and we therefore settled on dates when all 23 firms had at least three traded bonds. 7 This cardinalization of course imposes an implicit assumption that Moody's ratings are linearly related to the actual risk of the firm.
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Table 2 Correlation between the risk measures Moody's
Moody's 1
Merton 0.382
Time to default 0.513
Spearman rank correlation Merton Time to default
0.382 0.513
1 0.435
0.435 1
Correlation (Pearson) Merton Time to default
0.440 0.553
1 0.341
0.341 1
calculated from quoted stock prices and the number of outstanding shares, equity volatilities were estimated using the 3-month historical sample volatilities, and the risk-free rates were proxied by the 3-month US Treasury bill rates. These choices are all in line with other studies in the literature. 5. Empirical results This section gives the results obtained from implementing the technique described in Section 2 on the bond data described in Section 3. Since the ultimate task of our measure is to rank different firms according to their credit worthiness, an evaluation of the measure's performance is warranted. However, due to the well known difficulties associated with evaluating credit risk models (default is a very rare event), we have decided on a somewhat indirect approach of comparing our measure with other well known, and thoroughly tested, credit measures. Considering the fact that we treat our risk measure primarily as a rating tool giving information on the relative health of a group of firms, we chose to concentrate on the measure's performance in producing relative rankings. The first issue we consider is the degree of correlation among the three risk measures.8 Despite the differences among the measures in their methodologies and target horizons, it would not be unreasonable to expect a certain level of correlation among the measures. Since the data representing the risk measures consists of ranks and the assumption of normality without outliers does not hold for this data set, we present both the ordinary Pearson correlation coefficients and Spearman's rank correlation coefficients in Table 2. Among the measures, Merton's distance to default measure is least correlated with the other two. The level of correlation between the distance to default risk measure and the other two measures is approximately 0.4 for both the ordinary Pearson correlation coefficient and for the Spearman rank correlation coefficient. Meanwhile, the correlation between the expected time to default and Moody's ratings is slightly higher; the Pearson's correlation coefficient is 0.55 and the Spearman's rank correlation coefficient is 0.51. This fairly high correlation between the expected time to default and Moody's ratings is not very surprising considering both measures' long term view, and it also supports the popular view that the rating agencies' interests are more closely aligned with the interests of bond holders than with those of equity holders. The rankings of the individual firms using the three different measures in our study are shown in Table 1. These rankings are the averages of the 50 daily rankings in the sample. The strongest agreement between the three measures occur at the extremes. For example, all three measures
8
In order to calculate correlations Moody's ratings are first cardinalized as described above.
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Fig. 2. Expected time to default as a function of Moody's ratings.
suggest that Walmart and Coca Cola are very stable firms while they all suggest the opposite for Owens Illinois, SCI and MGM. We next consider in greater detail the relationship between our expected time to default measure and Moody's ratings. In Fig. 2 and Table 3, the relationship between the two measures of credit worthiness are presented on an aggregated level. The two curves in Fig. 2 represent the mean and the median times to default estimates among all the firms over the entire year within the different Moody's rating classes. Overall, it is evident that the expected time to default decreases with deteriorating Moody's credit ratings. For some rating classes the monotonic relationship is broken, however. A2 rated firms, for instance, are on average considered safer by our credit measure than both the Aa3 rated and the A1 rated firms. Overall, however, our continuous credit measure very much confirms Moody's views on the credit worthiness of firms, at least at the aggregate level.
Table 3 Expected time to default as a function of Moody's ratings
Aaa Aa2 Aa3 A1 A2 A3 Baa1 Baa2 Baa3 Ba2 B1 B3 All
Mean
Median
σ
Max
Min
Observations
36.43 31.68 23.57 25.21 32.90 22.25 21.63 23.95 21.98 21.51 8.50 8.72 23.59
36.41 31.62 23.79 23.92 31.19 18.38 21.30 21.00 22.00 19.77 7.087 8.277 22.63
5.32 1.79 4.17 4.91 10.03 8.33 2.31 8.90 3.33 4.41 3.14 1.83 8.74
52.80 35.38 39.42 54.75 66.25 39.71 25.97 54.75 30.68 35.47 15.85 12.52 66.25
28.87 27.21 16.04 19.73 21.29 13.03 18.68 13.17 15.89 14.23 5.245 6.736 5.245
50 50 200 50 100 100 50 214 186 50 68 32 1150
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Fig. 3. Default probability densities for September 28, 2001. Pale curves in the background are the default densities according to the Nelson–Siegel forward rate curves.
Recall that we assumed a parametric form for the default probability distributions of firms. The shapes of these distributions contain additional information not reflected in expected time to default, such as the relative likelihood of default for various time horizons. Fig. 3 shows the
Fig. 4. Short term cumulative default probabilities for September 28, 2001. Pale curves in the background are the default densities according to the Nelson–Siegel forward rate curves.
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Fig. 5. Long term cumulative default probabilities for September 28, 2001. Pale curves in the background are the default densities according to the Nelson–Siegel forward rate curves, and shows that they are not true probability distributions.
default density function for Walmart, Sears Roebuck, MGM and Owens Illinois, and Figs. 4 and 5 show the corresponding cumulative default distribution functions. An interesting observation is the clear dependence of the shape of the inter-temporal default distributions on the riskiness of the
Fig. 6. Time series plot of the cumulative default probability distribution for MGM between January 3, 2001 and December 31, 2001.
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firm. The riskier firms have a skewed distribution with most of the mass at short maturities. This is consistent with the view that although the riskier firms are more likely to default in the short term, if they somehow manage to survive the current problems, then the likelihood of default over the subsequent periods are likely to decrease (the cumulative probabilities are of course steadily increasing). Safer firms, on the other hand, exhibit a more symmetrical, slightly bellshaped, distribution. The safer firms are unlikely to default in the short term, with the likely time of default some distance away. Fig. 3 shows that the probability of immediate default for Owens Illinois, the riskiest firm on September 28, is approximately 15%, while the corresponding value for Walmart, the safest firm, is close to 0%. It can also be seen from Fig. 4 that the riskiest firm only has a 20% chance of survival beyond ten years while the safest firm has close to 90% chance of survival. Finally, the time series of inter-temporal default intensity curves for MGM is shown in Fig. 6. Such time series plots provide an opportunity to trace the changes in the inter-temporal risk profile of a firm over time, and should be useful in predictions of default probabilities over various time horizons. 6. Conclusions This paper introduced a simple parameterization for the risk-neutral default probability distribution of risky firms, and proposed expected time to default as a measure of their credit risk. Expected time to default has advantages over traditional ratings in that they allow finer separation of firms, incorporates up to date market information, and is easily computed from readily available data. The parameterized default probability distributions also provide useful information about the evolution of the credit worthiness of firms over time. A comparison of the relative rankings according to the expected time to default measure with those produced by Moody's and the distance to default type measure based on the Merton (1974) model for a set of US firms showed that the three measures produce fairly similar results, particularly at the extremes. Rank correlation estimates revealed a closer relationship between our measure and Moody's ratings than that between our measure and the Merton distant to default measure. Due to the lack of actual default events in our sample, it was not possible to investigate the relative merits of the various measures as predictors of future default. Acknowledgements Financial assistance from STINT and Bankforskningsinstitutet is gratefully acknowledged. References Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637−659. Bolder, D., & Stréliski, D. (1999). Yield curve modelling at the Bank of Canada. Technical report. Bank of Canada. Das, S., & Tufano, P. (1996). Pricing credit-sensitive debt when interest rates, credit ratings and credit spreads are stochastic. Journal of Financial Engineering, 5(2), 161−198. Delianedis, G., & Geske, R. (1998). Credit risk and risk neutral default probabilities: Information about rating migrations and defaults. Working paper. The Anderson School at UCLA. Duffee, G. (1998). The relation between treasury yields and corporate bond yield spreads. Journal of Finance, 53(6), 2225−2241. Frishling, V., & Yamamura, J. (1996, September). Fitting a smooth forward rate curve to coupon instruments. Journal of Fixed Income, 97−103.
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Huang, J.-Z., & Huang, M. (2002). How much of the corporate-treasury yield spread is due to credit risk? Working paper. Stanford University. Jarrow, R., Lando, D., & Turnbull, S. (1997). A Markov model for the term structure of credit risk spreads. Review of Financial Studies, 10(2), 481−523. Kwan, S. (1996). Firm-specific information and the correlation between individual stocks ad bonds. Journal of Financial Economics, 40, 63−80. Kwon, O. (2002). A general framework for the construction and the smoothing of forward rate curves. QFRG working paper 73. School of Finance and Economics, University of Technology Sydney. Leland, H., & Toft, K. (1996). Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads. Journal of Finance, 51(3), 897−1019. Longstaff, F., & Schwartz, E. (1995). A simple approach to valuing risky fixed and floating rate debt. Journal of Finance, 50(3), 789−818. Merton, R. (1974). On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 2(2), 449−470. Nelson, C., & Siegel, A. (1987). Parsimonious modeling of yield curves. Journal of Business, 60, 473−489.
International Review of Financial Analysis 16 (2007) 508 – 523
A simple continuous measure of credit risk Hans Byström a,⁎, Oh Kang Kwon b,1 b
a Department of Economics, Lund University, Box 7082, 220 07 Lund, Sweden Discipline of Finance H69, The University of Sydney, Sydney NSW 2006, Australia
Received 9 February 2007; accepted 4 March 2007 Available online 14 March 2007
Abstract This paper introduces a simple parameterization for the risk-neutral default probability distributions for risky firms that are easily computed from quoted bond prices. The corresponding expected times to default have a particularly simple form and are proposed as a measure for credit risk. Being continuous in nature, times to default provide a much finer measure of risk than those provided by ratings agencies. Comparison with the ratings provided by Moody's and the distance to default measures calculated using the Merton [Merton, R. (1974). On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 2(2), 449–470] model shows that the highest rank correlation is found between the proposed time to default measure and Moody's ratings. © 2007 Elsevier Inc. All rights reserved. JEL classification: G33; C20 Keywords: Default probability; Continuous credit rating; Expected time to default; Corporate bonds
1. Introduction Credit risk, or the risk of counterparty default, is an important factor in the valuation and risk management of financial assets. The losses arising from the string of recent corporate collapses, including Enron, WorldCom and Kmart, provide ample evidence of its importance. However, the risk of default, by its very nature, is difficult to quantify, and the traditional credit ratings supplied by agencies such as Moody's and Standard & Poor's are far from ideal. For example, the discrete nature of these ratings result in groups of firms being assigned the same rating despite the
⁎ Corresponding author. Tel.: +46 46 2229478; fax: +46 46 2224118. E-mail addresses: [email protected] (H. Byström), [email protected] (O.K. Kwon). 1 Tel.: +61 2 9351 6448; fax: +61 2 9351 6461. 1057-5219/$ - see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.irfa.2007.03.002
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differences in their credit worthiness, and the changes in the credit worthiness of firms are not immediately reflected in their ratings. Moreover, the measures of credit risk implied from models, be it traditional scoring models or structural models like the Merton (1974) model, typically rely on balance sheet data and historical stock volatilities. Consequently these measures tend to rely as much on historical events as on the current market situation, and are subject to the availability of accurate and up to date balance sheet data. This paper introduces a new measure of credit risk that complements the traditional credit ratings in several ways. Firstly, being continuous in nature, the measure allows a finer separation of firms according to their credit worthiness. Secondly, being computed from current bond data, it adjusts immediately to changes in the market's assessment of the credit worthiness of firms. Finally, unlike the symbolic ratings that are without economic meaning, it represents the expected time to default under the risk-neutral measure. As pointed out in Delianedis and Geske (1998), for example, the risk-neutral default probabilities are upper bounds for the actual default probabilities, and so the times to default we compute are useful indicators of changes in creditworthiness and likelihood of default for risky firms. Since expected time to default is a time weighted average of the default probabilities, it gives more weight to the risk of default in the longer term. This long term view is similar to the approach taken by Moody's whose ratings typically focus on at least one entire business cycle. Merton's model, on the other hand, is typically used to calculate one to two year default probabilities. Nevertheless, a comparison of the measures reveals a fairly high degree of correlation, and, in any case, for practical purposes they should probably be seen as complements rather than substitutes for one another. The data requirements to compute the measure introduced in this paper are minimal. The only requirements are the bonds issued by the firms under consideration and proxies for the risk-free bonds, which may be the treasury bond or the bonds issued by a firm of highest credit rating. However, to ensure sensible default intensities, we found it necessary to construct the risk-free yield curve with the condition that the risk-free yields lie below the yields of all corporate bond yields. The model is also quite simple. In particular, it is shown that a firm's entire inter-temporal distribution of default intensities can be inferred from instantaneous forward rates. This distribution is then used to extract the expected time to default. The structure of the remainder of the paper is as follows. Section 2 gives a brief review of the existing measures of credit risk, Section 3 defines the new credit risk measure and outlines a method for its computation, Section 4 presents the data and the methodology used in the empirical study of this paper, and Section 5 provides the results. The paper finally concludes with Section 6. 2. Traditional measures of credit risk All investors, regardless of whether they invest in equities or fixed income instruments, need to take into consideration the risk of counterparty default when making their investment decisions, and the ability to quantify this so called credit risk is important for both pricing and risk management purposes. There are several ways in which the risk of default can be gauged. One way is to rely on rating agencies, such as Moody's and Standard & Poor's, that rate individual firms' capability to service and repay their debt. An alternative way is to rely on models that attempt to quantify the level of credit risk using accounting information and stock price data.
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Unfortunately, the precise processes by which the ratings agencies arrive at their credit ratings remain the proprietary information of the respective agencies. However, it is most likely that they incorporate the analysts' views formed from public and private information. A particularly simple and popular approach for inferring credit risk information using accounting and stock market information is the contingent claims based model of Merton (1974). This model views a firm's liabilities (equity and debt) as contingent claims issued against the firm's underlying assets. By backing out asset prices and volatilities from quoted equity prices and balance sheet information, it provides estimates of the firm's default probabilities. This model is now described in detail. 2.1. Review of the Merton model Recall that equity holders have the residual claim on a firm's assets while being subject to limited liability. Merton (1974) recognised that equity in a firm is equivalent to a long position in a call option on the firm's assets, and used this correspondence to derive the market value and volatility of the firm's underlying assets. More precisely, Merton used the Black and Scholes (1973) framework to solve for the asset value and volatility implied by the option price and the option price volatility.2 The asset value and the asset volatility can then be combined into a risk measure called distance to default that is directly related to the credit worthiness of the equity issuing firm. At the heart of the Merton (1974) model lies a modified version of the Black–Scholes formula linking the market value of equity and the market value of assets VE ¼ VA N ðd1 Þ−e−rðT −tÞ DN ðd2 Þ
ð1Þ
where N ðdÞ is the cumulative normal distribution, and VE VA D T−t r d1 d2
Market value of the firm's equity Market value of the firm's assets Total amount of the firm's debt Time of maturity of the firm's debt Risk-free interest rate lnðVA =DÞ þ r þ 12 r2A ðT −tÞ pffiffiffiffiffiffiffiffi rA T −t pffiffiffiffiffiffiffiffi d1 −rA T −t
Moreover, it is easily shown that the equity and asset volatility are related by the expression rE ¼
VA N ðd1 ÞrA VE
ð2Þ
where σE and σA are volatilities of the firm's equity and asset returns respectively. Solving the nonlinear system of Eqs. (1) and (2) gives VA and σA, and the distance to default is defined by the expression lnðVA =DÞ þ r− 12 r2A ðT −tÞ pffiffiffiffiffiffiffiffi g¼ ð3Þ rA T −t 2
Note that in this case the option is equity and the underlying asset is the firm's assets.
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This is simply the number of standard deviations that the firm value is from the threshold point, and the smaller the value of γ the larger the probability that the firm will default on its debt. In this paper we compare the rankings implied by this risk measure to the rankings obtained from our continuous risk measure and those provided by Moody's. Note that while the risk measure implied by Merton's model is based on observations from the stock market, the measure introduced in this paper will be based on observations from the bond market. Consequently, the comparison of the two results could shed some light on whether the stock and the bond markets carry similar information about the credit worthiness of firms. 2.2. Motivations for a new risk measure Although useful, traditional credit ratings and scoring models relying on accounting information have several drawbacks as outlined above. The most obvious limitations with the rankings provided by the ratings agencies include the low frequency with which they are updated, often only once or twice a year, and the discrete nature of the rankings. Accounting based scoring models, on the other hand, rely on infrequently updated data that are not only released with a time lag, but also with possible accounting manipulations. Moreover, accounting information is inherently backward looking, based on historic information rather than the market's assessment of the future. Both approaches also suffer from producing measures without meaningful economic interpretations. Neither a credit rating nor a credit score is easily transformed into an actual probability of default and neither gives much information about the actual loss given such a default. An alternative to these credit ratings and scores involves the extraction of information about credit risk from up to date market data. If credit risk is incorporated into market prices then there must be ways of filtering the information contained in these prices to extract the component that can be attributed to credit risk. Various ways of extracting this information from either the stock or the bond data have been suggested. Public information is likely to be instantaneously and simultaneously embedded into individual security prices in the stock and the bond markets. This makes stock and bond prices contemporaneously correlated. In contrast, private information conveyed by informed investors that systematically trade in either the stock or the bond market is transmitted more through one of the markets than through the other.3 Stock and bond markets consequently carry somewhat different information regarding credit risk. The most well known stock market based approach of providing estimates of the firm's default probability, the Merton (1974) model, was described above. Now, data from the bond market can also be used as the basis for extracting credit risk information, and the method proposed in this paper assumes a parametric form for the risk-neutral default probability distributions under which the expected times to default have a simple closed form expression. The expected times to default are then proposed as a summary statistic, or measure, of credit risk of firms. As with most credit risk measures based on debt market data, the underlying idea is to compare bonds with credit risk (corporate bonds) to those essentially without credit risk (treasury bonds or high quality corporate bonds), and transform yield spreads into default probabilities.
3
The reasons for informed traders to systematically prefer one market to the other include difference in risk aversion, transaction costs, insider-trading laws or different institutional constraints.
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3. Expected time to default This section provides a detailed description of the method used to compute the expected time to default under the risk-neutral measure. As mentioned previously, being continuous, this measure avoids the problems of traditional ratings regarding widely different (ex post) default rates within the same rating class and the large overlaps in default rates between different rating classes. It also assigns a single number to each firm rather than the multiple ratings of ordinary rating agencies, due mainly to their focus on debt issues instead of the actual firm. In computing the expected time to default, we make the simplifying assumption that the risk-free interest rate process and the default processes of the firms are independent. This assumption is common in the literature, and is made for example in Das and Tufano (1996), Jarrow, Lando and Turnbull (1997) and Leland and Toft (1996), but whether it is reasonable or not is an outstanding empirical issue. It should be noted that several studies indicate violations of this assumption, most notably Duffee (1998) and Longstaff and Schwartz (1995) who find that spreads on corporate bonds fall when risk-free interest rates rise. An important practical issue is the choice of risk-free bonds from which the corresponding risk-free interest rates must be inferred. Although treasury bonds and the corporate bonds of the highest quality are both commonly used proxies for the risk-free bonds in the literature, the latter is often the better choice, as suggested for example in Kwan (1996), since a large part of the spread between the corporate and treasury bond yields arises from differences in factors such as taxation and liquidity (Huang & Huang, 2002), and using high quality corporate bonds allows the impact of some of these non-credit related causes of the credit spread to be minimized. In this paper, we construct the risk-free forward rate curves with the primary goal of ensuring that they lie below all the risky forward rate curves. The risk-free forward rates are then used to derive the risk-free zero-coupon prices and the default probabilities. This procedure is necessary if the risk-free bonds are to be priced above their risky counter parts, and this in turn is necessary to ensure the positivity of default probabilities. Finally, it is assumed that the recovery rate, or the fraction of the par value that is recovered by the bondholder in case of default is constant and firm independent. For the purposes of this paper, we use an averaged historical value quoted in Jarrow et al. (1997). Although it is possible to relax this assumption by estimating the recovery rate from the bond price data, we nevertheless treat it as an exogenous constant for simplicity since the recovery rate is of less importance for the purposes of determining the relative credit rankings of firms in this paper. 3.1. Expected time to default under the risk-neutral measure The first step in determining the expected time to default involves inferring the risk-neutral survival probabilities from the bond market data. For convenience, assume that the firms under consideration are indexed by the set {1, 2, …, n} for some integer n and let 0 be the index for the risk-free proxy. Let p0 (t, T) denote the time t price of a T-maturity risk-free zero coupon bond and let pi (t, T) be the corresponding price for bonds issued by firm i. Then, as in Jarrow et al., 1997 the assumption of independence of the risk-free interest rate process and the default processes implies pi ðt; T Þ ¼ p0 ðt; T Þ½dFi ðT jtÞ þ ð1−Fi ðT jtÞÞ
ð4Þ
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where δ is the constant recovery rate for all firms, and Fi(T|t) is the risk-neutral conditional probability of default prior to time T for firm i given that it has survived to time t ≤ T. Rearranging this equation gives the expression, Fi ðT jtÞ ¼
1−pi ðt; T Þ=pi ðt; T Þ 1−d
ð5Þ
for the risk-neutral conditional default probabilities. Since it is assumed that the recovery rate is given exogenously, the only task that remains in order to calculate the default probabilities, Fi(T|t), is to extract zero coupon bond prices from traded coupon bond prices, and a detailed description of the method used for this task in this paper is given in the next subsection. It follows from Eq. (5) that the conditional default density, fi(T|t), corresponding to Fi(T|t) is given by the equation fi ðT jtÞ ¼
AFi ðT jtÞ 1 A pi ðt; T Þ ¼− AT 1−d AT p0 ðt; T Þ
ð6Þ
Now, if we denote for each firm i the corresponding T-maturity forward rate at time t by ri(t,T), then Z T pi ðt; T Þ ¼ exp − ri ðt; uÞdu ð7Þ t
and so we have Z T 1 A exp − ðri ðt; uÞ−r0 ðt; uÞÞdu 1−d AT t Z T ri ðt; T Þ−r0 ðt; T Þ exp − ¼ ðri ðt; uÞ−r0 ðt; uÞÞdu 1−d t ri ðt; T Þ−r0 ðt; T Þ pi ðt; T Þ ¼ 1−d p0 ðt; T Þ
fi ðT jtÞ ¼ −
The time to default for firm i under the risk-neutral measure is hence given by Z l si ¼ ufi ðujtÞdu
ð8Þ
ð9Þ
0
In the following subsections, we propose a parameterization for the conditional default densities, fi(T|t), that facilitates the computation of the above integral. 3.2. Forward rate curves In constructing the forward rate curves, we face a few problems. Firstly, bonds traded in the market are all of finite maturity, and moreover the longest dated bond for each firm differ in general. Next, since only a few bonds are traded for each firm, some form of interpolation is needed to construct the entire curve of forward rates. Finally, the forward rate curves constructed from most bootstrapping type techniques are not smooth enough for the purposes of this paper.
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In order to overcome these problems, we assume a parametric form for the forward rate curves proposed in Nelson and Siegel (1987). Under this parameterization, the instantaneous forward rate curve, riNS(t,T), is assumed to be given by the expression riNS ðt; T Þ ¼ ai;0 þ ai;1 e−ji ðT−tÞ þ ai;2 ðT −tÞe−ji ðT −tÞ
ð10Þ
where ai,0, ai,1, ai,2 and κi are parameters satisfying the constraints ai;0 N0
ai;0 þ ai;1 N0
and
ji N0
ð11Þ
Note that ai,0 = lim(T−t)↑∞ riNS(t, T) corresponds to the long rate for firm i, and ai,0 + ai,1 = riNS (t, t) corresponds to the short rate. By fitting a Nelson–Siegel curve to each firm and date, we ensure that forward rates are available for all maturities for each firm and date, and the forward rate curves, riNS(t, T), are smooth. The standard approach for fitting a Nelson–Siegel curve to bond data, as proposed for example in Bolder and Stréliski (1999), is to determine the parameters by minimizing the total weighted4 squared error in the bond prices implied by Eq. (10) using some optimization routine. However, as pointed out by the authors, the parameters thus obtained are highly sensitive to the initial values. To overcome this problem, they approximate ai,0 and ai,1 using the yields on the longest and the shortest dated bonds, and compute remaining parameters by minimizing the total squared error as described above. We construct the Nelson–Siegel curves using an alternative approach which avoids approximating ai,0 and ai,1 using bond yields. First, we use a technique from Kwon (2002) to compute the forward rate curves that maximize a given smoothness measure. These curves turn out to be splines with certain functional forms. The technique also adjusts the market bond prices, within their bid–ask spreads, to improve the smoothness of the resulting forward rate curves. The smoothness measure used in this paper is Z T¯ ri Vðt; uÞ2 du ð12Þ 0
where T¯ is the maturity of the longest dated bond, and results in quadratic splines. This is the smoothness measure used in Frishling and Yamamura (1996) in constructing their discrete forward rate curves from bond data. Since the resulting forward rate curves, si(t, T), from this approach have finite horizon, T¯, and are often oscillatory in nature, they are unsuitable for the purposes of this paper. Nevertheless, they play a key role in determining the Nelson–Siegel parameters. More specifically, in contrast to the approach of fitting the parameters directly to bond data as outlined in Bolder and Stréliski (1999), we fit the parameters to si(t, T). Under our approach, the Nelson–Siegel parameters ai,0, ai,1, ai,2, κi are obtained by minimizing the squared error N X
ðriNS ðt; Tk Þ−si ðt; Tk ÞÞ2
ð13Þ
k¼1
for some N, where riNS(t, Tk) are the Nelson–Siegel forward rates, si(t, Tk) are the spline forward rates, and Tk are suitably chosen maturities. For simplicity, we set N equal to the number of knot points in the spline si(t, T), and Tk to the k-th knot point in our calculations. In contrast to the standard approach, it was found that the parameters computed by our approach were not very sensitive to the initial values. 4
Squared errors are weighted by the square of the duration of the bonds.
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Fig. 1. Nelson–Siegel forward rate curves for September 28, 2001. Pale curves in the background are the splines to which Nelson–Siegel curves were fitted.
As discussed earlier, common proxies for the risk-free bonds in the literature are the treasury bonds or the bonds from a firm with the highest credit rating. However, with the data used in this paper, neither of these were satisfactory since in many cases the corresponding risk-free forward rates were higher than their risky counterparts, resulting in negative default probabilities as is evident from Eq. (8). To overcome this serious problem, we constructed the risk-free forward rate curves with the main aim of ensuring that they remained below all the risky forward rate curves. For each date, we divided the maturity interval into 0.25-year subintervals, identified the lowest forward rate in each subinterval among all the forward rate curves for that date including the Treasury curve, and then fit a Nelson–Siegel curve through all these minimal forward rates. Fig. 1 gives examples of typical Nelson–Siegel curves that result from the above construction procedure. It can be seen that the Nelson–Siegel parameterization does a good job of fitting the splined forward rate curves for the firms.5 3.3. Inter-temporal default densities It is assumed that risky firms eventually default so that limT↑∞ Fi(T|t) = 1. However, this is incompatible with using the Nelson–Siegel parameterization for the forward rate curves. To see this, note that if the forward rate curve, riNS(t, T), is given by Eq. (10), then the corresponding bond prices are given by ai;1 −ji t −ji T ai;2 −ji t −ji T ai;2 −ji t −ji T pNS ðt; T Þ ¼ exp −a ðT −tÞ− ðe −e Þ− ðe −e Þ− ðte −Te Þ i;0 i ji ji j2i ð14Þ 5
These firms are chosen among the 23 firms in the empirical study in Section 4. They are chosen from the top and bottom end of firms ranked according to creditworthiness.
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Hence, in view of condition (11) and the fact that ai,0 N a0,0, we have pNS i ðt; T Þ ¼ lim e−ðai;0 −a0;0 ÞðT −tÞ ¼ 0 NS Tzl p0 ðt; T Þ T zl lim
and so NS 1−pNS 1 i ðt; T Þ=p0 ðt; T Þ ¼ p1 1−d 1−d T zl
lim Fi ðT jtÞ ¼ lim
Tzl
ð15Þ
unless δ = 0. It follows that Fi (T |t) implied by the Nelson–Siegel forward rate curves is not a proper probability distribution. In order to overcome this problem, we use the Nelson–Siegel parameterization once again, but this time for the conditional default probability distribution Fi(T |t). That is, it is assumed that Fi(T |t) has the form FiNS ðT jtÞ ¼ bi;0 þ bi;1 e−ki ðT−tÞ −bi;2 ðT −tÞe−ki ðT −tÞ
ð16Þ
and to ensure that FiNS(T |t)is a proper probability distribution, we impose that the parameters satisfy the constraints ki N0;
bi;0 ¼ 1;
bi;1 ¼ −1;
ð17Þ
0Vbi;2 bki FiNS(T |t) ≥ 0
These conditions follow from imposing the requirements for all T ≥ t, and limT↑∞ FiNS(T |t) = 1. Hence, the valid Nelson–Siegel conditional default probability distributions are FiNS ðT jtÞ ¼ 1−ð1 þ bi;2 ðT −tÞÞe−ki ðT −tÞ
FiNS(t|t) = 0,
ð18Þ
with 0 ≤ b2,i b λi. The parameters bi,2 and λi are finally computed by minimizing the squared error 2 N NS X 1−pNS i ðt; Tk Þ=p0 ðt; Tk Þ FiNS ðTk =tÞ− ð19Þ 1−d k¼1 for some N and suitably chosen maturities Tk. Here, piNS(t, Tk) and p0NS(t, Tk) are the risky and the risk-free bond prices implied by the Nelson–Siegel forward rate curves. In this paper, we have chosen N = 80 and Tk = 0.125k for 1 ≤ k ≤ 80. The conditional default density function, fi NS(T |t), corresponding to FiNS(T |t) is then given by AFiNS ðT jtÞ ¼ ðki −bi;2 þ ki bi;2 ðT −tÞÞe−ki ðT−tÞ AT and the expected time to default for firm i is given by Z l bi;2 þ ki si ¼ ufiNS ðujtÞdu ¼ k2i 0 fiNS ðT jtÞ ¼
ð20Þ
ð21Þ
Hence, the expected time to default, τi, are easily computed once the Nelson–Siegel parameters for FiNS(T |t) have been determined. 4. The data The data used in this paper consists of daily closing mid prices of 23 firms in various industries that are traded at the NYSE from January 3, 2001 to December 31, 2001. A list of these firms is
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Table 1 Average daily ranking of the firms over the year Moody's
Merton
Time to default
GE (Aaa) Walmart (Aa2) Coca Cola (Aa3) Bell South (Aa3) Pacific Bell (Aa3) Bank One (Aa3) Allstate (A1) Target (A2) Phillip Morris (A2) Sears Roebuck (A3) Carolina P.L. (A3) Time Warner (Baa1) Seagram (Baa2) Illinois power (Baa2) Coastal (Baa2) Pohang Steel (Baa2) Lockheed Martin (Baa3) Occidental (Baa3) Union Pacific (Baa3) MGM (Baa3) Marriott (Ba2) SCI (B1) Owens Illinois (B3)
Coca Cola Coastal Walmart Bell South Allstate Occidental Carolina P.L. Pacific Bell Marriott Phillip Morris Seagram Lockheed Martin Time Warner Union Pacific Target Bank One GE Illinois power Pohang Steel Sears Roebuck MGM SCI Owens Illinois
GE Seagram Target Walmart Coca Cola Phillip Morris Bell South Allstate Union Pacific Lockheed Martin Pacific Bell Sears Roebuck Time Warner Marriott Occidental Illinois power Coastal Carolina P.L. Pohang Steel MGM Bank One SCI Owens Illinois
given in Table 1. Most of the firms are large multinationals and for the time period considered they all had at least three outstanding bonds with maturities less than ten years. All the corporate bonds in our sample are bonds without any contractual provisions such as convertible, callable, or putable features. The bond price data was obtained from DataStream, and represent actual trade prices rather than matrix prices. In the statistical study of the three risk measures in Section 4, we were forced to shorten our sample significantly from 261 days to 50 days. The reason being that many bonds were not traded on a daily basis and there are only 50 days when all 23 firms have at least three traded bonds.6 Finally, following Jarrow et al. (1997) we have set δ = 0.3265 as the exogenous recovery rate for all firms. This assumption can easily be relaxed by setting a different recovery rate for the firms. For comparative reasons we also obtained daily credit ratings from Moody's (senior unsecured debt ratings), and the daily stock prices and debt levels for each firm from DataStream. Moody's ratings were then converted to numerical values between 1 and 23 with the larger numerical value corresponding to better credit quality.7 The debt levels required for Merton's model were approximated by interpolating the total debt figures contained in the year-end balance sheet reports and the maturities of these debts were assumed to be one year. Market capitalizations were 6 An alternative would have been to reduce the number of firms, but in addition to reducing an already fairly small sample of firms it would also have removed nonproportional amounts of firms with low credit ratings. A second alternative would have been to interpolate between two daily measures to compute the risk measure on the date in between. This would, however, create its own problems and we therefore settled on dates when all 23 firms had at least three traded bonds. 7 This cardinalization of course imposes an implicit assumption that Moody's ratings are linearly related to the actual risk of the firm.
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Table 2 Correlation between the risk measures Moody's
Moody's 1
Merton 0.382
Time to default 0.513
Spearman rank correlation Merton Time to default
0.382 0.513
1 0.435
0.435 1
Correlation (Pearson) Merton Time to default
0.440 0.553
1 0.341
0.341 1
calculated from quoted stock prices and the number of outstanding shares, equity volatilities were estimated using the 3-month historical sample volatilities, and the risk-free rates were proxied by the 3-month US Treasury bill rates. These choices are all in line with other studies in the literature. 5. Empirical results This section gives the results obtained from implementing the technique described in Section 2 on the bond data described in Section 3. Since the ultimate task of our measure is to rank different firms according to their credit worthiness, an evaluation of the measure's performance is warranted. However, due to the well known difficulties associated with evaluating credit risk models (default is a very rare event), we have decided on a somewhat indirect approach of comparing our measure with other well known, and thoroughly tested, credit measures. Considering the fact that we treat our risk measure primarily as a rating tool giving information on the relative health of a group of firms, we chose to concentrate on the measure's performance in producing relative rankings. The first issue we consider is the degree of correlation among the three risk measures.8 Despite the differences among the measures in their methodologies and target horizons, it would not be unreasonable to expect a certain level of correlation among the measures. Since the data representing the risk measures consists of ranks and the assumption of normality without outliers does not hold for this data set, we present both the ordinary Pearson correlation coefficients and Spearman's rank correlation coefficients in Table 2. Among the measures, Merton's distance to default measure is least correlated with the other two. The level of correlation between the distance to default risk measure and the other two measures is approximately 0.4 for both the ordinary Pearson correlation coefficient and for the Spearman rank correlation coefficient. Meanwhile, the correlation between the expected time to default and Moody's ratings is slightly higher; the Pearson's correlation coefficient is 0.55 and the Spearman's rank correlation coefficient is 0.51. This fairly high correlation between the expected time to default and Moody's ratings is not very surprising considering both measures' long term view, and it also supports the popular view that the rating agencies' interests are more closely aligned with the interests of bond holders than with those of equity holders. The rankings of the individual firms using the three different measures in our study are shown in Table 1. These rankings are the averages of the 50 daily rankings in the sample. The strongest agreement between the three measures occur at the extremes. For example, all three measures
8
In order to calculate correlations Moody's ratings are first cardinalized as described above.
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Fig. 2. Expected time to default as a function of Moody's ratings.
suggest that Walmart and Coca Cola are very stable firms while they all suggest the opposite for Owens Illinois, SCI and MGM. We next consider in greater detail the relationship between our expected time to default measure and Moody's ratings. In Fig. 2 and Table 3, the relationship between the two measures of credit worthiness are presented on an aggregated level. The two curves in Fig. 2 represent the mean and the median times to default estimates among all the firms over the entire year within the different Moody's rating classes. Overall, it is evident that the expected time to default decreases with deteriorating Moody's credit ratings. For some rating classes the monotonic relationship is broken, however. A2 rated firms, for instance, are on average considered safer by our credit measure than both the Aa3 rated and the A1 rated firms. Overall, however, our continuous credit measure very much confirms Moody's views on the credit worthiness of firms, at least at the aggregate level.
Table 3 Expected time to default as a function of Moody's ratings
Aaa Aa2 Aa3 A1 A2 A3 Baa1 Baa2 Baa3 Ba2 B1 B3 All
Mean
Median
σ
Max
Min
Observations
36.43 31.68 23.57 25.21 32.90 22.25 21.63 23.95 21.98 21.51 8.50 8.72 23.59
36.41 31.62 23.79 23.92 31.19 18.38 21.30 21.00 22.00 19.77 7.087 8.277 22.63
5.32 1.79 4.17 4.91 10.03 8.33 2.31 8.90 3.33 4.41 3.14 1.83 8.74
52.80 35.38 39.42 54.75 66.25 39.71 25.97 54.75 30.68 35.47 15.85 12.52 66.25
28.87 27.21 16.04 19.73 21.29 13.03 18.68 13.17 15.89 14.23 5.245 6.736 5.245
50 50 200 50 100 100 50 214 186 50 68 32 1150
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Fig. 3. Default probability densities for September 28, 2001. Pale curves in the background are the default densities according to the Nelson–Siegel forward rate curves.
Recall that we assumed a parametric form for the default probability distributions of firms. The shapes of these distributions contain additional information not reflected in expected time to default, such as the relative likelihood of default for various time horizons. Fig. 3 shows the
Fig. 4. Short term cumulative default probabilities for September 28, 2001. Pale curves in the background are the default densities according to the Nelson–Siegel forward rate curves.
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Fig. 5. Long term cumulative default probabilities for September 28, 2001. Pale curves in the background are the default densities according to the Nelson–Siegel forward rate curves, and shows that they are not true probability distributions.
default density function for Walmart, Sears Roebuck, MGM and Owens Illinois, and Figs. 4 and 5 show the corresponding cumulative default distribution functions. An interesting observation is the clear dependence of the shape of the inter-temporal default distributions on the riskiness of the
Fig. 6. Time series plot of the cumulative default probability distribution for MGM between January 3, 2001 and December 31, 2001.
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firm. The riskier firms have a skewed distribution with most of the mass at short maturities. This is consistent with the view that although the riskier firms are more likely to default in the short term, if they somehow manage to survive the current problems, then the likelihood of default over the subsequent periods are likely to decrease (the cumulative probabilities are of course steadily increasing). Safer firms, on the other hand, exhibit a more symmetrical, slightly bellshaped, distribution. The safer firms are unlikely to default in the short term, with the likely time of default some distance away. Fig. 3 shows that the probability of immediate default for Owens Illinois, the riskiest firm on September 28, is approximately 15%, while the corresponding value for Walmart, the safest firm, is close to 0%. It can also be seen from Fig. 4 that the riskiest firm only has a 20% chance of survival beyond ten years while the safest firm has close to 90% chance of survival. Finally, the time series of inter-temporal default intensity curves for MGM is shown in Fig. 6. Such time series plots provide an opportunity to trace the changes in the inter-temporal risk profile of a firm over time, and should be useful in predictions of default probabilities over various time horizons. 6. Conclusions This paper introduced a simple parameterization for the risk-neutral default probability distribution of risky firms, and proposed expected time to default as a measure of their credit risk. Expected time to default has advantages over traditional ratings in that they allow finer separation of firms, incorporates up to date market information, and is easily computed from readily available data. The parameterized default probability distributions also provide useful information about the evolution of the credit worthiness of firms over time. A comparison of the relative rankings according to the expected time to default measure with those produced by Moody's and the distance to default type measure based on the Merton (1974) model for a set of US firms showed that the three measures produce fairly similar results, particularly at the extremes. Rank correlation estimates revealed a closer relationship between our measure and Moody's ratings than that between our measure and the Merton distant to default measure. Due to the lack of actual default events in our sample, it was not possible to investigate the relative merits of the various measures as predictors of future default. Acknowledgements Financial assistance from STINT and Bankforskningsinstitutet is gratefully acknowledged. References Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637−659. Bolder, D., & Stréliski, D. (1999). Yield curve modelling at the Bank of Canada. Technical report. Bank of Canada. Das, S., & Tufano, P. (1996). Pricing credit-sensitive debt when interest rates, credit ratings and credit spreads are stochastic. Journal of Financial Engineering, 5(2), 161−198. Delianedis, G., & Geske, R. (1998). Credit risk and risk neutral default probabilities: Information about rating migrations and defaults. Working paper. The Anderson School at UCLA. Duffee, G. (1998). The relation between treasury yields and corporate bond yield spreads. Journal of Finance, 53(6), 2225−2241. Frishling, V., & Yamamura, J. (1996, September). Fitting a smooth forward rate curve to coupon instruments. Journal of Fixed Income, 97−103.
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