“market” declines during a bad year), thereby increasing the probability of default; when Z is positive (in a good year), the entire p(Z) distribution shifts upward, thereby decreasing the default probability. The impact of market forces on the conditional default probability depends on the index weight w,such that when wis close to one (zero), values are highly corre- lated (uncorrelated) with the market factor, and the conditional default probability is highly dependent on (independent of) market forces.
The third issue relates to the portfolio of bonds used in calculating the transition matrix. Altman and Kishore (1997) found noticeable impact of bond “aging” on the probabilities calculated in the transition matrix. In- deed, a material difference is noted, depending on whether the bond sample used to calculate transitions is based on new bonds or on all bonds out- standing in a rating class at a particular moment in time. This undermines the assumption of credit risk homogeneity for all obligations in the same ratings classification. Kealhofer, Kwok, and Weng (1998) showed that de- fault rates are skewed within each ratings class so that the mean may be twice as large as median default rates. Simulating Moody’s bond ratings transition matrices 50,000 times using Monte Carlo simulation techniques, they find that approximately 75 percent of borrowers within a rating grade may have default rates below the mean, leading to adverse selection among borrowers; that is, only the riskiest 25 percent of all borrowers within each FIGURE 6.5 Unconditional asset distribution and conditional distributions with positive and negative Z.Source:Finger (1999), p. 16.
Unconditional Conditional,
Z = –2
Conditional, Z = 2
–3 –2 –1
0.5 0.4 0.3 0.2 0.1
1
0 2 3
Market Factor
Relative Frequency
rating classification obtain loans if they are priced at the mean default spread. Moreover, there was such an overlap in the range of default proba- bilities for each class that a bond rated BBB may have a default probability in the AAA rating class range.
The fourth issue relates to the general problem of using bond transition matrices to value loans. As noted earlier, to the extent that collateral, covenants, and other features make loans behave differently from bonds, using bond transition matrices may result in an inherent valuation bias.
Moreover, bond ratings lag market-based measures of default risk, such as KMV’s EDFin forecasting default probabilities (see Chapter 4). This sug- gests that the internal development of loan rating transitions by banks (dis- cussed in Chapter 2) based on EDFs and historic loan databases, might be viewed as crucial in improving the accuracy of VARmeasures of loan risk.25 Valuation
In the VAR calculation shown earlier in this chapter, the amount recover- able on default (assumed to be $51.13 per $100), the forward zero interest rates (1ri), and the credit spreads (si) are all nonstochastic (or at least hedged). Making any or all of them stochastic generally will increase any VARcalculation and capital requirement. In particular, loan recovery rates have quite substantial variability [see Carty and Lieberman (1996)], and the credit spread on, say, an AA loan might be expected to vary over some rat- ing class at any moment in time (e.g., AA+ and AA− bonds or loans are likely to have different credit spreads). More generally, credit spreads and interest rates are likely to vary over time, with the credit-cycle, and shifts in the term structure, rather than being deterministic. One reason for assum- ing that interest rates are nonstochastic or deterministic is to separate mar- ket risk from credit risk,26but this remains highly controversial, especially to those who feel that their measurement should be integrated rather than separated and that credit risk is positively correlated with the interest rate cycle [see Crouhy et al. (2000)]. Kiesel et al. (2001) incorporate spread risk into CreditMetrics, arguing that stochastically varying spreads are strongly correlated across different exposures and thus are not diversified away, and find spread risks of about 7 percent of asset values for a portfolio of five- year maturity bonds. However, Kim (2000) contends, in the limited context of market VAR, that time horizon mismatches (up to 10 days for market risk and up to one year for credit risk) create problems in integrating spread risk and credit migration risk that may lead to overestimation of economic capital requirements.
Regarding recovery rates, if the standard deviation of recovery rates is
$25.45 around a mean value of $51.13 per $100 of loans, it can be shown
that the 99 percent VARfor the BBB loan in our example under the normal distribution will increase to 2.33×$3.18 million=$7.41 million, or a VAR-based capital requirement of 7.41 percent of the face value of the BBB loan (as compared to $6.97 million under the fixed LGD assumption) for unexpected losses only.27 A related question is whether the volatility of LGDs of bonds is the same as for loans given the greater contract flexibility of the latter.28
Mark-to-Market Model versus Default Model
By allowing for the effects of credit rating changes (and hence, spread changes) on loan values, as well as default, CreditMetrics can be viewed as a mark-to-market (MTM) model. Other models—for example, CreditRisk Plus (see Chapter 8)—view spread risk as part of market risk and concen- trate on expected and unexpected loss calculations rather than on expected and unexpected changes in value (or VAR) as in CreditMetrics. This alter- native approach is often called the default model or default mode (DM).
It is useful to compare the effects of the MTM model versus the DM model by calculating the expected and, more importantly, the unexpected losses for the same example (the BBB loan) considered earlier. Table 6.1 shows that, in a two-state, default/no-default world, the probability of de- fault is p=0.18 percent and the probability of no default (1−p) is 99.82 percent. After default, the recovery rate is $51.13 per $100 (see Table 6.3), and the loss given default (LGD) is 1 minus the recovery rate, or $48.87 per
$100. The book value exposure amount of the BBB loan is $100 million.
Given these figures, the expected loss on the loan is:
Expected loss =p×LGD×Exposure
=.0018×.4887×$100,000,000 (6.3)
=$87,966
To calculate the unexpected loss, we have to make some assumptions regarding the distribution of default probabilities and recoveries. The sim- plest assumption is that recoveries are fixed and are independent of the dis- tribution of default probabilities. Moreover, because the borrower either defaults or does not default, the probability of default can (most simply) be assumed to be binomially distributed with a standard deviation of:
)
(6.4) σ = p
(
1−p)
Given a fixed recovery rate and exposure amount, the unexpected loss on the loan is:
)
To make this number comparable with the VARnumber calculated under CreditMetrics for the normal distribution, we can see that the one standard deviation loss of value (VAR) on the loan is $2.99 million versus $2.07 mil- lion under the DM approach.29This difference occurs partly because the MTM approach allows an upside as well as a downside to the loan’s value, and the DM approach fixes the maximum upside value of the loan to its book or face value of $100 million. Thus, economic capital under the DM approach is more closely related to book value accounting concepts than to the market value accounting concepts used in the MTM approach.
SUMMARY
In this chapter, we outlined the VARapproach to calculating the capital re- quirement on a loan or a bond. We used one application of theVAR methodology—CreditMetrics—to illustrate the approach and raise the technical issues involved. Its key characteristics are: (1) it involves a full valuation or MTM approach in which both an upside and a downside to loan values are considered, and (2) the analyst can consider the actual dis- tribution of estimated future loan values in calculating a capital require- ment on a loan. We will revisit VARmethodology and CreditMetrics again in Chapter 11, when we consider calculating the VARand capital require- ments for a loan portfolio.