MR 10 – Financial Correlation Modeling – Bottom-Up Approaches

A copula is a function that takes several separate one-variable distributions and stitches them into one joint distribution, carrying a chosen dependence structure along the way. Risk managers rarely struggle to describe a single asset on its own; the hard part is describing how many assets behave together. A copula splits that job in two: you model each marginal distribution however you like, and the copula supplies the correlation that binds them.

The mathematics is older than its use in finance. Abe Sklar laid the foundation in 1959, with extensions by Schweizer and Wolff (1981) and by Schweizer and Sklar (1983). Oldrich Vasicek brought one-factor copulas into finance in 1987, and David Li applied more flexible multivariate copulas to credit in 2000. The appeal was obvious: a single, multidimensional function seemed able to correlate a large pool such as the 125 names in a collateralized debt obligation (CDO). The approach was adopted with great enthusiasm, yet its reputation collapsed once the 2007 global financial crisis struck, largely because the convenience of one function had been mistaken for an accurate picture of how defaults cluster.

The formal definition

Stated compactly, a copula C squeezes an n-dimensional input, with every coordinate living on the unit interval, into a single number that is also between zero and one.

Now let G_i(u_i) be univariate uniform distributions, one for each of the variables u₁ to u_n, with i a natural number. Sklar showed that a copula function C exists satisfying the relation below.

The G_i(u_i) are the marginal distributions, F_n is the joint cumulative distribution function, F_i^-1 is the inverse of F_i, and ρ_F is the correlation structure of F_n. Read from left to right, each marginal is pushed through an inverse function to obtain an abscissa (x-axis) value, and those abscissa values are then fed jointly into F_n, which imposes ρ_F. When the mapped values F_i^-1(G_i(u_i)) are continuous, the copula is unique. Full properties and proofs appear in Sklar (1959) and Nelsen (2006).

Copulas used in finance are usually grouped by how many parameters drive them. One-parameter copulas include the Gaussian copula, whose single parameter is the correlation ρ, and the Archimedean family, which is built from a generator function φ_a(t). Gumbel, Clayton and Frank are the best known Archimedean members. Among two-parameter copulas, the ones cited most often are the Student’s t, the Frechet and the Marshall-Olkin.

One refinement matters for the Gaussian case. Only the bivariate Gaussian copula is strictly a one-parameter copula, because with two assets there is just one correlation coefficient. Once more assets are added, the multivariate Gaussian copula carries a full correlation matrix holding many coefficients.

The Gaussian copula C_G uses the standard normal distribution as the bridge between the marginals and the joint distribution, and its convenient properties have made it one of the most widely applied copulas in finance. For n variables it is written as follows.

Here N^-1 is the inverse of the univariate standard normal distribution, M_n is the cumulative standard normal distribution of all n variables taken jointly, and ρ_M is its n × n correlation matrix, which must be symmetric and positive-definite. If every G_x(u_x) is uniform, the transformed values N^-1(G_x(u_x)) are standard normal and M_n is a standard multivariate normal (a proof is in Cherubini et al. 2005).

From a generic copula to correlated default times

Li’s step in 2000 was to choose the cumulative default probability of each entity as the marginal. Writing Q_i(t) for the probability that entity i has defaulted by a fixed time t, the Gaussian default time copula C_GD follows directly.

The inner step, N^-1(Q_i(t)), is a percentile to percentile mapping. A cumulative default probability sitting at the 5th percentile is sent to the 5th percentile of the standard normal, one at the 10th percentile goes to the 10th percentile, and so on, so every output is an abscissa value on the standard normal x-axis. The outer step then joins those abscissa values inside M_n using the correlation matrix ρ_M, and the result is the probability that the n correlated entities have all defaulted by time t.

Notice what the model assumes. The correlation structure is borrowed from the multivariate normal and applied to marginals that came from default curves, not from normal variables. The reason is purely that the mathematics and computation become easy, and this choice is exactly the assumption that later drew criticism.

Take two distressed companies, B and Caa, each with an estimated annual default probability for years 1 to 10. For an investment-grade firm the default probability would usually rise over time because uncertainty grows with the horizon. These two firms are in trouble now, so their nearest years carry the highest risk, and the annual default probabilities shrink if they survive. Adding the annual figures gives the cumulative default probability Q(t), and applying N^-1 to each cumulative figure gives its standard normal percentile. In a spreadsheet the mapping is =NORMSINV(Q(t)) in Excel or norminv(Q(t)) in MATLAB; for instance, NORMSINV(0.0651) returns -1.5133.

Source: Meissner, Correlation Risk Modeling and Management, 2nd edition, Chapter 5; two source tables combined into one view.

With only two companies, the correlation matrix collapses to a single coefficient ρ, and equation 10.4 reduces to the bivariate cumulative normal M₂.

M₂ has no closed form, but code for the bivariate cumulative normal is widely available, and any statistics package will evaluate it. The density behind it is the familiar bell-shaped surface over two axes; the copula only needs the area under that surface to the lower left of the two mapped points.

These results explain the name default time copula: the quantities being correlated are the default times t_i of the assets, not their returns. The sensitivity is steep. Holding the year 1 inputs fixed and varying only ρ gives the joint default probabilities below.

Two assets can be handled analytically, but a portfolio needs simulation. The goal is the default time τ_i of asset i in a way that reflects its correlation with the default times of every other asset, i = 1 to n.

The steps

First, draw a sample M_n(·) from the n-variate Gaussian copula, the right-hand side of equation 10.5 extended to n assets, so the draw lies in [0, 1]. Correlated draws are produced through Cholesky decomposition of the default correlation matrix ρ_M, which is how the dependence enters. Second, set the draw equal to the asset’s cumulative default probability curve and solve for the time.

Neither equation has a closed-form solution, so τ_i is found numerically, for example with a Newton-Raphson search or simply with a lookup function in Excel. Third, repeat the whole process many times, say 100,000, and average the τ_i across simulations to estimate the default time. Because the correlation matrix feeds M_n, the estimate already carries the default correlation with the rest of the portfolio.