In January 2005, Prof. Robert Jarrow and I published a paper in RISK Magazine entitled "Estimating Default Correlations Using a Reduced Form Model." Since that time, we have seen the management of portfolios of risky credits evolve in one of two directions. The best practice approach is to fully exploit the advantages of reduced form models by using that approach for both default probability estimation and forward-looking simulation of portfolio defaults and credit-adjusted values. A second approach is common among analysts making the transition from Merton default probabilities and Merton-linked copula simulation of credit portfolios. In this note, we show the linkages among three different meanings of "default correlation" for this second approach. We provide a worked example following Jarrow and van Deventer (2005).

**Introduction**

It is well-known that the reduced form approach to default probability estimation is significantly more accurate than a Merton-based approach. Important papers by Bharath and Shumway (2008) and Campbell, Hilscher and Szilagyi (2008, 2011) document the degree of accuracy differential. More detailed documentation is available in Kamakura Risk Information Services Technical Guides beginning in 2002 and updated with each new version of the KRIS reduced form and Merton models.

For this reason, many institutions with credit portfolios have replaced Merton default probabilities with reduced form default probabilities. They are then faced with two intimately related questions:

- Looking forward, how do we model correlated default?
- What is the implied correlation relevant to a Merton credit portfolio simulation?

We answer these questions in this note using a worked example focused on a typical borrower, BP PLC (NYSE:BP), and the "market portfolio" consisting of all observations in the Kamakura Risk Information Services modeling data base for version 6, representing more than 2.2 million observations on public firms including more than 2,600 corporate failures.

The annual 1990-2014 relationship between the average 1 year default rate for BP PLC and our proxy for the market portfolio is shown here:

In providing today's worked example, there are three different but mathematically linked meanings of the phrase "default correlation" that Jarrow and van Deventer explain.

- Correlation in the time series of default probabilities. This is the simple correlation of the default probabilities above.
- Correlation in the "events of default," the forward-looking time series of zeros and ones for BP PLC and other borrowers, where 0 is an observation with no default and 1 is a defaulting observation. Clearly history is no guide to this correlation since, by definition, any firm that is a current borrower is highly unlikely to have defaulted in the past.
- Correlation in the "drivers of default", which can take on three meanings depending on the risk management infrastructure of the firm:
- Correlation in the historical inputs to the reduced form default probability models
- Correlation in the forward looking inputs to the reduced form default probability models
- Correlation in the forward looking value of company assets (in our case for BP PLC) and the sum of the value of company assets for all firms in the Merton/copula framework.

These definitions of correlation are mathematically linked. In this example we focus on definitions 1, 2 and 3c. Correlation in the reduced form framework is a completely transparent multi-factor modeling process that has been fully documented in a large body of work by Prof. Robert Jarrow and in technical guides from Kamakura Risk Information Services. More details are available from info@kamakuraco.com.

Why would one use a Merton framework for credit portfolio management if one already has proven that reduced form default probabilities are more accurate than Merton default probabilities? Simply stated, the process of switching risk systems and risk policies and procedures is a multi-step process and it is common for firms to find themselves in this hybrid state as they make the transition to a full reduced form credit portfolio modeling infrastructure.

**A Worked Example of the Three Meanings of Default Correlation**

The derivation of what follows is explained in Jarrow and van Deventer (2005) and available from info@kamakuraco.com, so we will simply apply the formulas that they provide. We calculate each of our three correlation definitions in turn.

**Correlation in Historical Default Probabilities**

This is a simple Excel-based calculation, as are the other definitions. The simple correlation between annual default probabilities of BP PLC and the market portfolio is 7.75%. Summary statistics are given here:

We will use these summary statistics in what follows.

**Correlation in the 0/1 Events of Default**

The analysis in this note is between BP PLC and the market portfolio, but we could have instead compared BP PLC and another public firm. We assume for expository purposes that we can analyze the market as a public firm (in practice we model the market from the bottom up, firm by firm). What is the correlation between the forward looking annual time series of zeros (no default) and ones (default) for BP PLC and the market portfolio, given their historical default probabilities and the correlation of those two histories? Jarrow and van Deventer give the answer in this formula:

While this looks complicated, it really isn't. The correlation of the events of default is a straightforward function of the correlations in the default probabilities, their means, and standard deviations given above. λA and λB refer to the default probability histories for BP PLC (NYSE:A) and the market portfolio (NYSE:B). "Std" is the standard deviation of their histories and E refers to their averages. T is the length of the period we are analyzing in years, in our case T=1. We first calculate theta as follows from the figures above:

If you are going to try this at home, please be sure to use the statistics expressed in decimal form, not percent. We can now calculate the forward looking correlation in the events of default:

The correlation in the events of default is very low in large part because the historical correlation in the default probabilities is low.

**The Probability of Default in the Same Year**

As a by-product of this calculation, Jarrow and van Deventer provide a formula for the probability of default by both BP PLC and "the market" in the same year, given the correlation in historical default probabilities. If the firms were absolutely uncorrelated in any way, the probability of joint default in the same year would be (0.00694)(0.012186) = 0.0008% using the average default probabilities and (0.000791 )(0.002945) = 0.0002% using the 2014 default probabilities. If we take correlations correctly into account, Jarrow and van Deventer show the probability of joint default must be as follows:

The answer is the sum of the covariance of the default probabilities and the product of their means, adjusted by the length of the time period. The answer for this example is given here:

The existence of correlation means that the probability of joint default is slightly higher than the independent calculation we did above.

**Correlation in the Return on the**

**Value of Company Assets in the Merton Framework**

We now come to the last of our three worked examples. In order to use the reduced form default probabilities and their implied probability of joint default in the Merton framework, we have to answer this question:

*If we are using one factor to drive correlated default, the return on the value of the assets on all companies, what is the beta on that factor that we should use for BP PLC?*

For both BP PLC and the market portfolio, we assume that a normal distribution with mean 0 and standard deviation of 1 is that the heart of the Merton default probability. We let x be the value of company assets for BP PLC and y is the value of total assets for the market. We can imply the value of company assets using the inverse cumulative normal function:

For BP PLC using the most recent 0.0791% default probability

0.000791=norminverse(x,0,1) so x is -3.15914 standard deviations from the mean.

For the market portfolio using its most default probability, 0.2945%,

0.002945 = norminverse(y,0,1) = -2.7538 standard deviations from the mean

We now restate the question above to get a direct answer:

What is the correlation between x and y that gives a joint probability of default of 0.0009% as shown above?

The answer is very straightforward in the reduced form case, but in the Merton approach we must employ the bivariate normal distribution, which takes this form. The author has excerpted from a very clear exposition done by the University of Wisconsin:

One of the nice features of the bivariate normal distribution is that if we know the default probabilities for both firms and their probability of joint default, we can imply the beta which defines the correlation in the value of company assets. In probability terms, we quote from the University of Wisconsin document:

In our example, we know the value of company assets for the market portfolio. It's consistent with a default probability of 0.2945% and implies a y value of -2.7538 standard deviations. We now can calculate the conditional mean and standard deviation of the value of BP PLC company assets, given y, if we assume a starting guess for correlation rho. Let us guess ρ is zero. If rho is zero, the conditional mean of BP PLC assets is still 0 and the conditional standard deviation is still one, implying the conditional BP PLC default probability has the same value, 0.0791%. The joint probability of default is

Pr(joint) = Pr(market)* conditional Pr = 0.000233%.

This figure, however, is below the desired probability of 0.0009% (to be precise, 0.00088887%).

**The Solution**

We now use the "solver" function on common spreadsheet software to imply the beta or correlation value that gives us a joint probability of default equal to 0.00088887%. The result is a beta or rho value of 0.163512. This implies a conditional default probability for BP PLC of 0.3018% and we can confirm that

Pr(joint) = 0.002945(0.003018) = 0.0008887%

**Conclusions**

The last calculation for the correlation in the value of company assets is not necessary when using a true reduced form simulation framework for credit portfolio management. It's needed only in the hybrid situation while one is in a transition from the 1974 Merton framework to a modern reduced form environment.

For information on credit risk management and a supporting spreadsheet underlying this note, please contact us at info@kamakurco.com.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.