To begin, let's establish some terminology. Consider obligors A and B, and a time horizon T. Let's denote the probability of default of A before T by p

_{A}and similarly the probability of default of B before T

by p

_{B}.

Now for a little pop quiz.

Question: With knowledge of p

_{A}and p

_{B}you can determine which of the following?

- P
_{A|B}The conditional probability that A defaults given B has defaulted - P
_{AB}The joint probability that both A and B default - r
_{AB}The linear correlation coefficient between the default indicator events I_{A}and I_{B} - None of the above

_{A}= p

_{B}= 2%. Below we plot the joint probability of default, and the (identical) conditional probabilities of default.

It is important to note that the linear default correlation completely dominates both the conditional and joint probabilities of default. The joint probability of default, for example, is roughly 10 times as large under a correlation of 20% as under 0%, and the magnitude of this effect increases as the individual probabilities decrease. This is deferent from our experience with correlation as it applies to equity return variance, where the marginal effect of a lower average correlation is constant on a relative basis. Thus the relative value of proper diversification increases as the as risk decreases for credit sensitive portfolios.

Given that historical analysis by the ratings agencies suggests that the five-year cummulative default rates of the majority of investment grade rated bonds is below 2%, this suggests that for a typical investment grade portfolio of credits, default correlation should dominate the risk calculus. In particular, the use of average credit rating by some bond funds to summarize credit risk is inappropriate, not because it hides a few lower quality bonds, which may skew the loss risk, but because it provides absolutely no information about the dominant risk: the default correlation.

To illustrate the impact of default correlation on the portfolio VaR, consider the case of an equally weighted portfolio of 100 obligors with identical independent individual probabilities of default of p = 5% over a horizon T and zero recovery. Neglecting the interest rate component, the VaR over the horizon of this portfolio is characterized simply by the number of defaults over that horizon. In this example, we can compute the probability of there being

*k*or fewer defaults using the cumulative Binomial distribution functionBelow we display the 99% VaR for this stylized portfolio.

As with equities, fully determining the joint probability of default would require us to estimate all the pair wise correlations within a portfolio. Even for a portfolio of 100 obligors, this would require the specification of 4,950 pair wise correlations, which would in turn require obtaining at a minimum 4,950 default events. This is simply not feasible. Even for the equity world, where there is ample return data, a factor structure is employed to make the correlation problem tractable.

The solution for corporate bonds is twofold. First, we employ a factor model structure on the equity returns to dimension reduce the problem. Second, we will apply a firm value or structural model of default (e.g., Merton Model) to link equity return correlation to asset return correlation and asset return correlation to default correlation.

To keep things simple for now, let's consider a one factor model of the firm assets directly (I will incorporate equity data in a later blog post). Specifically, let's assume that all assets are driven by a single, common factor with a standard normal distribution.

Because the idiosyncratic components are assumed to be uncorrelated, the covariance between the assets will be r. If we normalize the asset values first, then we can view r as an asset correlation. For the structural model part, we will assume a simple barrier model. Namely, an obligor defaults if its asset value at the horizon T falls below some critical level K.

If we assume that all obligors in an equally weighted portfolio have the same barrier K, then uncorrelated idiosyncratic components means that conditional on a realization of F, the probability of having

We can immediately see two things from the figure. First, the VaR converges to the independent binomially distributed case as correlation goes to zero (compare to the prior graph). Second, a 2% individual probability of default with a 20% asset correlation has a 99% VaR of 14. We would get the same VaR by having a 7.5% individual probability of default with a 0% asset correlation. To put this in context, this is like saying that an average BBB rated portfolio would bear the same credit risk (as measured by 99% VaR) as a BB+ porftolio. Three guesses on which portfolio would have a higher yield.If we assume that all obligors in an equally weighted portfolio have the same barrier K, then uncorrelated idiosyncratic components means that conditional on a realization of F, the probability of having

*m*or fewer defaults in a N obligor portfolio is given by the Binomial cumulative distribution function (4) above. Since the common factor has standard normal distribution, the VaR is given by further integrating the Binomial above against the Gaussian density function. Below are the plots of the 99% VaR for a 100 obligor portfolio for different choices of r, where the barrier K is set so that zero correlation corresponds to a marginal probability of default of 2%.Here, I highlighted the importance of default correlation and through a simple factor model, showed how asset correlation dominates the risk equation for a portfolio of investment grade credits. In my next post, I'll go into more detail as to how an equity factor model can be harnassed to estimate the credit risk, and how correlation "stress testing" can be leveraged by looking at some real world examples.

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