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Copulas as a risk management tool Back  
One of the most interesting new ideas to enter finance in recent years is the copula. Copulas are widely used in financial engineering to value credit-linked products and manage credit risk, and the growth in the collateralised debt obligations (CDOs) market has led to an increase in their use. In this article, Finbarr Murphy and Dr. Bernard Murphy give an introduction to how they are used.
The market for collateralised debt obligations (CDOs) has continued to expand in the past year, as has the credit derivatives market in general. The International Swaps and Derivatives Association (ISDA) estimate that the credit derivatives market grew by 40 per cent to 50 per cent in 2004, to an outstanding notional of over $5 trillion. And it is expected to double in size again over the next two years.

Copula functions are widely used to value credit-linked products and manage credit risk, particularly those credit products whose payoff depends on the performance of several constituent entities. The valuation of the products is very dependent on the default correlation of the constituents, hence the generic title of correlation products. This article explains the use of (Gaussian) Copulas in the valuation of a simple correlation product, specifically a Basket Default Swap (BDS).

Overview of Basket Default Swaps (BDS)
Before discussing Copulas, we must discuss a simple correlation product, a First To Default (FTD) Basket Default Swap (BDS).

A BDS is an agreement where one party (A), pays a second party (B), a fixed periodic fee for a specified period. In return, party B pays a defined amount subject to a credit event on an underlying basket. Therefore B buys protection from A, or looking at it another way, A invests in the credit exposure of the basket receiving a yield from B. Typically payment is made by a physical delivery in return for payment of the par amount of the defaulted asset.

Consider a simple portfolio with 5 equally weighted names, each trading at 50bps above LIBOR. In this case, the yield will be a simple weighted average. With a FTD basket, the probability of a credit event is greater and a higher yield can be expected on the portfolio. Leverage is achieved through this simple mechanism; the resultant risk can be tailored by changing the maturity, number and name of constituents and by changing the trigger from a FTD to an 2nd, …, nth to default.

The basket performance will also depend on the default correlation of the constituent entities. A higher default correlation suggests that if one reference name defaults, this will lead to an increased likelihood that the other components will default. The reverse is also true; a higher correlation suggests that all names are likely to survive together.

Default risk
Before considering default correlation, we need to define and then measure default risk. Default risk is the probability of a definable credit event occurring during a certain time. This can be characterised by a continuous distribution of a random variable, TA, where TA represents the time-to-default of an entity A. This distribution cannot be assumed to be normal. Implicit in this approach is the assumption that a credit event will occur at some point in the future, even for the most secure entities. In practice, we only need to measure the risk for the next five to ten years so we are therefore only interested in the left hand tail of the continuous distribution.

Dropping the subscript for company A, let F(t) denote the distribution function of T. Giving: | t > 0
F(t) tells us the probability of a security defaulting up to, and including, time t.
The default probability measurement for an individual name is extracted from market information in three main ways. Firstly. It can be extracted from stock market information with a variant of Mertons 1974 model. Secondly, corporate bond market yields can be decomposed into risk-free (LIBOR) plus default risk, thereby generating a default risk probability. Finally, the most common method uses the implied default probability from single name Credit Default Swaps (CDSs). Naturally a combination of the above methods can be used.

About correlation
Correlation is a much-misused word in finance. In practitioner terms, by correlation, one usually implies Pearson’s Linear Correlation. I.e. the linear relationship between two variables. Looking at figure 1 below, the first graph shows two independent standard normal variables. As one would expect, the linear correlation is zero. The second graph also has a near zero correlation but clearly there is a strong (polynomial) relationship.
Default occurrence can be described by numerous inter-relationships, linear correlation being just one of them. We will use the more accurate word, dependency, for the remainder of this article. Some other measures of dependency include concordance that, loosely speaking, describe the probability that having large X’s and large Y’s is greater than the probability of having large X’s and small Y’s and tail dependency which describes the amount of asymptotic dependence in the lower-left-quadrant tail or upper-right-quadrant tail of a bivariate distribution.

Bivariate distributions
We will now describe a very simple FTD BDS with just two constituent names. The fixed fee paid by the basket holder effectively protects against the either of the names defaulting. Without getting into the mathematics, it is evident that the fixed fee will be subject to the joint (bivariate) default probability. As discussed, we know the probability of default of each name; we need a method to combine these single univariate marginal distributions into a bivariate distribution.

Firstly, let F1(t1) and F2(t2) denote the univariate marginal distribution functions of the default times for the two entities. I.e.
F1(t1) = Pr(t1≤ t1) and F2(t2) = Pr(t2≤ t2)
Where Pr(tn≤ tn) is the probability of the nth entity defaulting before time tn.
The bivariate distribution will have the following form
F(t1,t2) = Pr(t1≤ t1,t2≤ t2) ≤
We need to somehow link the bivariate distribution with the marginal distributions.

Introducing Gaussian Copulas
A Copula function links univariate marginal distributions to their bivariate (and multivariate) distributions. In general mathematical terms we have
C[F1(x1), F2(x2)] = F(x1, x2)

Sklar’s Theorem establishes the reverse, i.e. he showed that a bivariate distribution can be expressed as a Copula function
F(x1, x2) = C[F1(x1), F2(x2)]

There are many different types of Copula functions; you might be familiar with some of the names, Student-t, Frank and Archimedean. However, the most commonly used is the Gaussian Copula, which has the form:
C(x1, x2) = Fn(F-1(x1), F-1(x2), rn
where Fn is the bivariate normal distribution with a dependency matrix rn.F-1(.) is the inverse of the univariate normal distribution function.

Now, let’s relate this back to our two marginal distributions, F1(t1) and F2(t2). We transform t1 and t2 into x1 and x2 respectively where
x1 = F-1[F1(t1)], x2 = F-1[F2(t2)]

Because F-1(.) is the inverse of the univariate normal distribution function, both x1 and x2 have standard normal distributions by construction. Looking at this graphically, see figure 2 below) we are transforming the univariate distributions into normal distributions

The dependency matrix, rn represents the correlation of the asset returns of the two entities. As these are not directly observable, it is common to use equity correlations as a proxy for this matrix.

Monte Carlo simulation
There is no closed mathematical solution to the Gaussian Copula function described above. An intuitive, tractable solution involves using Monte Carlo simulation. This is a sequential process involving the following steps:
1. Take a sample draw from the bivariate normal distribution with correlation matrix rn. This yields x1 and x2
2. Convert x1 and x2 into t1 and t2 using the transformation described above
3. Find the mint1 and t2.This is the time of the first default, if this is less than the maturity of the BDS, then a credit event has occurred during the life of the BDS
4. If mint1 and t2 calculate the present value (PV) of this payoff otherwise the basket matures without a credit event.
5. Calculate the PV of the fixed premiums (protection payments) to this default or maturity, whichever is first

Repeat the above steps for several thousand simulations; the average value of the default payments over the premiums represents the fair value spread.

Advantages and disadvantages of Copulas
The model described above has two key advantages. Firstly, there were no assumptions made about the marginal distributions F1 and F2. There was no requirement that they should be normal distributions or that they should have the same distributions. The second key advantage is the ability to separate the dependence structure from the marginal distributions.

These advantages allow us to describe the same marginal distributions through difference Copulas functions and dependency structures. In the example above, we have described a Guassian Copula for a bivariate distribution. The bivariate distribution can be easily extended to the multivariate (i.e., the number of basket entities) and the Gaussian Copula can be interchanged for different Copula functions. Whilst Gaussian Copulas remain the market standard in the same way that stock returns are assumed to be lognormal in the Black-Scholes model, different Copula functions can capture different dependency characteristics.

One-factor models
Whilst the Monte Carlo simulations are mathematically tractable, they are computationally very slow. To understand this, consider a five year BDS with five entities, each with an AAA ratings agency grade. The Monte Carlo simulation described above requires several thousand simulations to mimic default events in the next five years. This would require extensive computational resources. Risk management and Greek letter calculations will magnify this effort considerably.

In 2004 Hull & White described a semi-analytical method that eliminates the need for Monte Carlo simulations, the so-called one-factor model. We won’t delve to far into the mathematics of this model but just outline the methods used.

Let pN(k,t | M) represent the probability that k defaults will occur before time t in a portfolio of N names. This probability is conditional on a common (one-factor) M, which can be assumed to be a zero-mean, unit variance random variable. Starting with a portfolio of just one name and recursively adding additional names, we can build a probability distribution conditional on the common factor, M.

This unconditional probability can then be “integrated out” by solving where p(k,t) = is the Gaussian density.

This one-factor method is especially efficient in calculating Greek letters. The one-factor model can be extended to multiple factors but computational efficiency will be sacrificed and model stability becomes an issue.

The Monte Carlo and One-Factor models have been implemented with Gaussian Copulas this means that, assuming the same marginal distributions, the results for both methods should be exactly the same. So, assuming the same Copula, different products can be priced consistently using different default models.

Basket Default Swaps (BDSs) and Synthetic Collateralised Debt Obligations (CDOs) are leveraged credit products, typically tailored to meet specific risk requirements. The premium paid reflects the probability of a credit event of one or more of the constituent entities (along with recovery rates, notional amounts, interest rate term structures, etc). Copula functions allow us to marry non-normal marginal default probability distribution with a dependence structure to calculate the multivariate probability distribution.
In a follow-up article, we shall continue this discussion to calculate the fair value of multiple tranche synthetic CDOs.

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