DEFAULT BONDS AND RISKY DEBT

8.3 DEFAULT BONDS AND RISKY DEBT

Bonds are rated to qualify their standard risks. Standard and Poors, Moody’s and other rating agencies use for example, AAA, AA, A, BB, etc. to rate bonds as more or less risky. We shall see in section 8.4 that these rating agencies also provide Markov chains, expressing the probabilities that rated firms switch from one rating to another, periodically adapted to reflect market environment and the conditions particularly affecting the rated firm (for example, the rise and fall of the technology sector, war and peace, and their likes).

Consider a portfolio of B-rated bonds yielding 14 %; typically, these are bonds which currently are paying their coupons, but have a high likelihood of defaulting or have done so in the recent past. A Treasury bond of similar duration yields

5.5 %. Thus, in this example, the Junk–Treasury Spread (JTS) is 8.5 %. Now, let us take a look at the spread’s history over the past 13 years (Jay Diamond, Grant’s Interest Rate Observer data).

The spread depicted in Figure 8.4 corresponds roughly to a B-rated debt. Note the very wide range of spreads, from just below 3 % to almost 10 %. What does

a JTS of 3 % mean? Very bad news for the junk buyer, because he or she will have been better off in Treasuries if the loss rate exceeds 3 %. And even if the loss rate is only half of that, a 1.5 % return premium does not seem adequate to compensate for this risk. There is a wealth of data on the bankruptcy/default rate, allowing us to evaluate whether the prevailing risk premium amounts to adequate compensation.

DEFAULT BONDS AND RISKY DEBT

Figure 8.4 Junk–Treasury spread 1988–2000 (Jay Diamond, Grant’s Interest Rate Observer data).

Rating agencies often use terms such as default rate and loss rate which are important to understand. The former defines the proportion of companies default- ing per year. But not all companies that default go bankrupt. The recovery rate is the proportion of defaulting companies that do not eventually go bankrupt. So

a portfolio’s reduction in return is calculated as the default rate times one minus the recovery rate: if the default rate is 4 % and the recovery rate is 40 %, then the portfolio’s total return has been reduced by 2.4 %. The loss rate, how much of the portfolio actually disappears, is simply the default rate minus the abso- lute percentage of companies which recover. According to Moody’s, the annual long-term default rate of bonds rated BBB/Baa (the lowest ‘investment grade’) is about 0.3 %; for BB/Ba, about 1.5 %; and for B, about 7 %. But in any given year, the default rate varies widely. Further, because of the changes in the high-yield market that occurred 15 years ago, the pre-1985 experience may not be of great relevance to high-yield investing today.

Prior to the use of junk bonds the overwhelming majority of speculative issues were ‘fallen angels’, former investment-grade debt which had fallen on hard times. But, after 1985, most high-yield securities were speculative right from their initial offering. Once relegated to bank loans, poorly rated companies were for the first time able to issue debt themselves. This was not a change for the better. Similar to speculative stock IPOs, these new high-yield bond issues tended to have less secure ‘coverage’ (based on an accounting term defined as the ratio of earnings- before-taxes-and-interest to total interest charges) than the fallen angels of yore, and their default rates were correspondingly higher.

FIXED INCOME , BONDS AND INTEREST RATES

Many financial institutions hold large amounts of default-prone risky bonds and securities of various degrees of complexity in their portfolios that require a reliable estimate of the credit exposure associated with these holdings. Models of default-prone bonds fall into one of two categories: structural models and reduced- form models . Structural models specify that default occurs when the firm value falls below some explicit threshold (for example, when the debt to equity ratio crosses a given threshold). In this sense, default is a ‘stopping time’ defined by the evolution of a representative stochastic process. Merton (1974) first considered such a problem; it was studied further by many researchers including Black and Cox (1976), Leland (1994), and Longstaff and Schwartz (1995). These models determine both equity and debt prices in a self-consistent manner via arbitrage, or contingent-claims pricing. Equity is assumed to possess characteristics similar to a call option, while debt claims have features analogous to claims on the firm’s value. This interpretation is useful for predicting the determinants of credit-spread changes, for example.

Some models assume as well that debt-holders get back a fraction of the debt, called the recovery ratio. This ratio is mostly specified a priori, however. While this is quite unrealistic, such an assumption removes problems associated to the debt seniority structure, which is a drawback of Merton’s (1974) model. Some authors, for example, Longstaff and Schwartz (1995), argue that, by looking at the history of defaults and recovery ratios for various classes of debt of comparable firms, one can find a reliable estimate of the recovery ratio. Structural models are, however, difficult to use in valuing default-prone debt, due to difficulties associated with determining the parameters of the firm’s value process needed to value bonds. But one may argue that parameters could always be retrieved from market prices of the firm’s traded bonds. Further, they cannot incorporate credit-rating changes that occur frequently for default-prone (risky) corporate debts.

Many corporate bonds undergo credit downgrades by credit-rating agencies before they actually default, and bond prices react to these changes (often brutally) either in anticipation or when they occur. Thus, any valuation model should take into account the uncertainty associated with credit-rating changes as well as the uncertainty surrounding default and the market’s reactions to such changes. These shortcomings make it necessary to look at other models for the valuation of defaultable bonds and securities that are not predicated on the value of the firm and that take into account credit-rating changes. For example, a meltdown of financial markets, wars, political events of economic importance are such cases, where the risk is exogenous (rather than endogenous). This leads to reduced-form models.

The problem of rating the credit of bonds and credit markets is in fact more difficult than presumed by analytical models. Information asymmetries compound these difficulties. Akerlof in his 2001 Nobel allocution pointed to these effects further.

A bank granting a credit has less information than the borrower, on his actual default risk. . . . On the same token, banks expanding into new, unknown markets are at a particular risk. On the one hand, due to their imperfect market knowledge, they must rely on the equilibrium between supply and demand to a large extent. On the other hand, under asymmetric information, it is very

DEFAULT BONDS AND RISKY DEBT

Value

Default level

Time

Figure 8.5 Structural models of default.

easy for clients to hide risks and to give too optimistic profit estimates, possibly approaching fraud in extreme cases. Adverse selection then implies a markedly increased default risk for such banks. Banks can use interest rates and additional security as instruments for screening the creditworthiness of clients when they estimate that their information is insufficient. Credit risk and pricing models, of course, are complementary tools. Based on information provided by the client, they produce risk-adjusted credit spreads and thus may set limits to the principle of supply and demand. On the other hand, borrowers with a credit rating may use this rating to signal the otherwise private information on their solvency, to the bank. In exchange, they expect to receive better credit conditions than they would if the bank could only use information on sample averages.

Technically, the value process is defined in terms of a stochastic process {x, t ≥ 0} while default is defined by the first time τ (the stopping time) the process reaches

a predefined threshold-default level. In other words, let the threshold space be ℜ, then:

τ = Inf {t > 0, x(t) / ∈ ℜ}

where ℜ is used to specify the set of feasible states for an operating firm. As soon as the firm’s value is out of these states, default occurs.

Reduced-form models specify the default process explicitly, interpreting it as an exogenously motivated jump process, usually expressed as a function of the firm value. This class of models has been investigated, for example by Jarrow and Turnbull (1995), Jarrow et al. (1997) and others. Although these models are useful for fitting default to observed credit spreads, they mostly neglect the underlying value process of the firm and thus they can be less useful when it is necessary to determine credit spread variations. Jarrow et al. (1997) in particular have adopted the rating matrix used by financial institutions such as Moody’s, Standard and Poors and others as a model of credit rating (as we too shall do in the next section).

Technically, default is defined exogenously by a random variable ˜ T where t<˜ T<T , with T , the bond expiry date. The conditional probability of default is assumed given by:

P (˜ T ∈ (t + dt) t<˜ T<T ) = q(x) dt + 0(dt)

FIXED INCOME , BONDS AND INTEREST RATES

Value

Jump time

Default level

to default

Time

Figure 8.6 Reduced-form models default.

This means that the conditional probability of default q in a small time interval (t + dt), given that no default has occurred previously, is a function of an underlying

stochastic process {x, t ≥ 0}. If the probability q is independent of the process {x, t ≥ 0}, this implies that the probability of default is of the exponential type. That is to say, it implies that at each instant of time, the probability of default is time-independent and independent of the underlying economic fundamentals. These are very strong assumptions and therefore, in practice, one should be very careful in using these models.

A comparison between structural and reduced-form models (see Figure 8.6) is outlined in Table 8.2. Selecting one model or the other is limited by the underlying risk considered and the mathematical and statistical tractability in applying such

a model. These problems are extensively studied, as the references at the end of the chapter indicate.

A general technical formulation, combining both structural and reduced-form models leads to a time to default we can write by Min(τ, ˜ T,T ) where T is the maturity reached if no default occurs, while exogenous and endogenous default are given by the random variables (τ, ˜ T ). If the yield of such bonds at time t for a payout at s is given by, Y (t, s) ≡ y(τ, ˜T , s), the value of a pure default-prone bond paying $1 at redemption is then E exp(−Y (t, T ) Min(τ, ˜T , T )). Of course if there was no default, the yield would be y(t, s) and therefore Y (t, s) > y(t, s) in order to compensate for the default risk. The essential difficulty of these problems is to determine the appropriate yield which accounts for such risks, however. For example, consider the current value of a bond retired at Min(τ, ˜ T,T ) and paying an indexed coupon payout indexed to some economic variable or economic index (inflation, interest rate etc.). Uncertainty regarding the coupon payment, its nominal value and the bond default must then be appropriately valued through the bond yield.

When a bond is freely traded, the coupon payment can also be interpreted as

a ‘bribe’ paid to maintain bond holding. For example, when a firm has coupon payments that are too large, it might redeem the bond (provided it incurs the costs associated with such redemption). By the same token, given an investor with other opportunities, deemed better than holding bonds, it might lead the investor to forgo future payouts and principal redemption, and sell the bond at its current

DEFAULT BONDS AND RISKY DEBT

A comparison of selected models. Model

Table 8.2

Advantages Drawbacks Merton

Simple to implement. (a) Requires inputs about the firm (1974)

value. (b) Default occurs only at debt maturity. (c) Information about default and credit-rating changes cannot be used.

Longstaff (a) Simple to implement. (a) Requires inputs related to the and Schwartz (b) Allows for stochastic term

firm value. (1995)

structure and correlation between (b) Information in the history of defaults and interest rates.

defaults and credit-rating changes cannot be used.

Jarrow, Lando, (a) Simple to implement. (a) Correlation not allowed between and Turnbull

(b) Can match exactly existing prices default probabilities and the (1997)

of default-risky bonds and thus level of interest rates. infer risk-neutral probabilities for

(b) Credit spreads change only default and credit-rating changes.

when credit ratings change. (c) Uses the history of default and credit-rating change.

Lando (a) Allows correlation between default Historical probabilities of defaults (1998)

probabilities and interest rates. and credit-rating changes are used (b) Allows many existing

assuming that the risk premiums due term-structure models to be easily

to defaults and rating changes, is embedded in the valuation

null.

framework. Duffie and

(a) Allows correlation between default Information regarding credit-rating Singleton

probabilities and the level of history and defaults cannot be used. (1997)

interest rates. (b) Recovery ratio can be random and depend on the pre-default value of the security.

(c) Any default-free term-structure model can be accommodated, and existing valuation results for default-free term-structure models can be readily used.

Duffie and (a) Has all the advantages of Duffie (a) Information regarding Huang

and Singleton. credit-rating history and defaults (1996) (swaps) (b) Asymmetry in credit qualities is

cannot be used. easily accommodated.

(b) Computationally difficult to (c) ISDA guidelines for settlement

implement for some swaps, such upon swap default can be

as cross-currency swaps, if incorporated.

domestic and foreign interest rates are assumed to be random.

FIXED INCOME , BONDS AND INTEREST RATES

market value. The number of cases we might consider is very large indeed, but only a few such cases will be considered explicitly here.

Structural and reduced-form models for valuing default-prone debt do not in- corporate financial restructuring (and potential recovery) that often follows de- fault. Actions such as renegotiating the terms of a debt by extending the maturity or lowering/postponing promised payments, exchanging debt for other forms of security, or some combination of the above (often being the case after default), are not considered. Similarly, institutional and reorganization features (such as bankruptcy) cannot be incorporated in any of these models simply. Further, an- ticipated debt restructurings by the market is priced in the value of a defaultable bond in ways that none of these models captures. In fact, many default-prone se- curities are also thinly traded. Thus, a liquidity premium is usually incorporated into these bond prices, hiding their risk of default. Finally, empirical evidence for these models is rather thin. Duffie and Singleton (1997, 1999) find that reduced- form models have problems explaining the observed term structure of credit spreads across firms of different credit qualities. Such problems could arise from incorrect statistical specifications of default probabilities and interest rates or from models’ inability to incorporate some of the features of default/bankruptcy mentioned above. Bond research, just like finance in general, remains therefore

a domain of study with many avenues to explore and questions that are still far from resolved.