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.

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.

To use the details of how to use the above framework to generate a VaR number is as follows:

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?

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,

Where, 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:

- Given the observable equity value, equity volatility, and outstanding firm debt, calculate the asset value and asset volatility by solving equations (1) and (2).
- Calibrate the debt level to force the spread relation (3) to match the observed starting bond spread.
- 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).
- 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.
- Use the simulated spread changes to compute the portfolio weighted average returns (due solely to spread moves), and calculate VaR.

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.

*Receive new blogs by e-mail*

*and follow*

*@FactSet*

*on Twitter.*