Wednesday, September 29, 2010

Why Default Correlation Matters (Part 2)

In part 1 of this series, I highlighted how important default correlation is to credit risk management. In particular, I showed through some simple examples how default correlation dominates the risk calculus. I provided a simple link between default correlation and asset correlation using a simple one factor model. While this might seem highly stylized, it is worth pointing out that this setup is essentially the mechanism that is used to quote CDO tranches in terms of base correlation.

In this post, I’ll continue the discussion by showing how to pull information from the equity markets in order to estimate asset covariance. The solution for corporate bonds is twofold. First, 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. Second, we employ a factor model structure on the equity returns to dimension reduce the problem.

The original and simplest of the firm value models is the Merton model. In the Merton model, firm asset value is modeled as a geometric Brownian motion, and firm equity is modeled as a European call option on the value of the assets, where the barrier (strike) is the notional K on the outstanding debt. Because assets equal debt plus equity, owning the debt is equivalent to being long the assets and short a call option on the same. Although this model is highly simplified in its assumptions about a firm’s capital structure, the appeal of the model is that it provides a direct tractable link, in the form of the Black-Scholes option pricing, between the debt, the assets, and the equity. Mathematically, the Merton model provides us with two non-linear equations which we can attempt to solve for the asset value and the asset volatility.
Where the A is the value of the assets, D the notional on the debt, E the value of the equity, C is the value of the European call option, and the s’s represent the volatility of the assets and equity respectively. Once we solve the system, we have the pieces needed to determine the probability of default P, and then by adding a recovery assumption, we can approximate the bond spread by
Tunring to factor models, we recall that in a factor model, the goal is to express the N individual returns in terms of a linear combination of common factors as,

The main benefit of this is that the covariance/correlation structure is highly simplified. If the factos are orthogonal (and we can usually perform a little technical magic called "factor rotation" to make this so) then the variance and covariance structure is simply given by,
With this more robust framework I can now show you the effect of a correlation shift on credit VaR in a real world setting. To see this framework in action, we just need a portfolio of credits, and an equity factor model. For the sample portfolio, I chose the Bank of America Merrill Lynch U.S. Corporates Large Cap/Industrials (5-10 Y) (MLCIL6) index, which has a weighted average rating of A3. For the equity factor model, I did a simple principal component analysis (PCA) on the daily returns for the prior 250 days and retained the top 10 principal components. I use PCA for the factor model in my example because the principal components are already orthogonal (uncorrelated), hence it gives me a simple way to alter the correlation structure while preserving the individual variances within a Monte Carlo based simulation by using equations (6) and (7). This will let me show the effect on the 99% VaR of a correlation shift for a real life portfolio, while at the same time not altering any of the marginal probabilities of default.

The scree plot for this is shown below. Effectively the first 10 PCs explain roughly 58% of the daily variation in returns, with the first component accounting for more than 40% of the variation alone. It is worth pausing for a moment to compare this to the simple one factor asset value model presented in part 1. The scree plot indicates that such a simple model may not be terribly inaccurate.

To use the details of how to use the above framework to generate a VaR number is as follows:
  1. Given the observable equity value, equity volatility, and outstanding firm debt, calculate the asset value and asset volatility by solving equations (1) and (2).
  2. Calibrate the debt level to force the spread relation (3) to match the observed starting bond spread.
  3. With the equity factor model in hand, run a bunch of simulated returns of the common and idiosyncratic factors to generate a bunch of equity return scenarios. Assume debt is unchanged, and then calculate asset value changes as A(sim) = A(base) + E(sim) – E(base).
  4. Hold the asset volatility constant and use the asset value changes to compute new probabilities of default, and then through the spread relation (3), compute a spread change.
  5. Use the simulated spread changes to compute the portfolio weighted average returns (due solely to spread moves), and calculate VaR.
To see the effect of an equity return correlation change on my real life portfolio of credits, I did two tests. First, I perturbed the issuer equity factor loadings, while preserving the % of variation due to idiosyncratic risk. In general, this spread the systematic risk out across the factors more evenly, and as a result, lowered the average pair-wise equity return correlations. As expected, the VaR is a function of the average correlation. The results of this are below. Recall that the equity factor model is using daily returns, so the analysis effectively shows the impact on 1-day VaR on the BofA Merrill Lynch U.S. Corporates Large Cap / Industrials (5-10 Y) (MLCIL6) index portfolio, in basis points. Comparing the trend line to the single factor model discussed in part 1, the impact of increasing the correlation from .3 to .4 is roughly 10 bps, which represents an increase of 25% to the VaR. This compares to a 20% increase in VaR when going from a correlation of 0.3 to 0.4, for the simple single factor model.

The second test was to shift variance from the idiosyncratic component to the systematic components in a relatively uniform way. The graph confirms that shifting risk from uncorrelated idiosyncratic components to the systematic factors increases average correlation and the VaR. Slightly different than in the stylized single factor model, the magnitude of the effect declines with rising correlation. This is mostly due to the fact that the idiosyncratic exposures are heterogeneous in our real life portfolio.
Finally, to emphasize the point that correlation dominates, I examined the effect of holding the correlation structure constant, but simply increasing the individual risk neutral probabilities of default (achieved by raising the equity volatilities). The base case weighted average probability of default was roughly 4%. The 95%-VaR level for the base case was 44bps. There are a few really interesting aspects of the graph to note. First, there is a clear pivot point in the risk at about 9%. In a real life portfolio, the linear increase in marginal risk of the stylized example does break down at some point. The second aspect to note is how flat the VaR profile is below 6%. There is very little to be gained risk wise from decreasing the individual probabilities of default. The same probably cannot be said of the yield. Lastly, the probabilities are risk neutral, not physical, so they do not directly translate to empirical default frequencies as measured by the rating agencies without knowledge of investor’s risk aversion. A detailed look into that topic will have to wait for another post. That said, if the pivot point corresponds to a point just beyond the investment grade/high yield breakpoint, a manager might be able to add significant extra yield by selectively crossing the divide. For sake of comparison, the yield on our test portfolio is 3.84 with an average rating of A3, while the yield on the BofA Merrill Lynch U.S. High Yield Master II (MLH0A0) is 8.47, with an average rating of B1. How much yield are you leaving on the table?

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Wednesday, September 1, 2010

Why default correlation matters (Part 1)

In this post, I will focus on the underappreciated topic of default correlation, how it might be related to equity correlation, and what it means for the VaR of your corporate credit portfolio. Balanced fund managers take note!

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 pA and similarly the probability of default of B before T
by pB.

Now for a little pop quiz.

Question: With knowledge of pA and pB you can determine which of the following?
  1. PA|B The conditional probability that A defaults given B has defaulted
  2. PAB The joint probability that both A and B default
  3. rAB The linear correlation coefficient between the default indicator events IA and IB
  4. None of the above
Unfortunately, knowledge of the marginal probabilities of default is insufficient to determine the conditional probabilities of default, the joint probability, or the linear correlation. So the answer is none of the above. Knowledge of the marginal distributions and the linear correlation of default is, however, enough to determine the conditional default probabilities, and the joint probability, of default. The relationships between these quantities are given by: Before discussing how one might go about estimating the linear correlation of default, let's just take a moment to see just how important this number is for credit risk. Let's suppose that the individual default probability for each obligor is 2%, so that pA = pB = 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 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%.
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.

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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