Recently an interesting, problem came to light while we were working with on of our clients in the area of first to default basket (F2D) pricing. The problem was with the correlation matrix. The correlation matrix that was going into a Monte Carlo simulator was not a valid one and the simulator was repeatedly breaking donw. As we all know that default time correlation matrix is the crucial input in any F2D (or S2D and Nth to default basket). The default time correlations between obligors (names) are an important input in the Monte Carlo simulation to calculate the F2D premium.

Suppose we have three names in an F2D. The first name is a bank in India and the second name is a large financial institution in Asia with a sizeable Indian retail and corporate customer base and both the first and the second name are highly correlated with a default correlation of 0.80; the third name is an Indian software company which has a very low correlation with the first name (Indian bank) and the default correlation between the second name and the third can be taken as zero. However, it is not clear (for whatever reasons) what the default correlation between the second name (the financial institution in Asia ) and the third name (the Indian software company) is. But to price the F2D on this basket one has to have the correct default time correlation matrix.

We shall not go into the detailed issues of F2D pricing here, but just concentrate on the bounds of the default time correlation matrix which will also form the theoretical bounds of F2D premium. We need to find out what are the bounds, or the restrictions on the correlation between the second and the third name in the correlation matrix shown below:

What values can take to make the above a valid correlation matrix. Actually, cannot take any arbitrary values. The first two correlations will put a definite bound on the third correlation-the value of - and if the third correlation deviates (due to wrong calculation from historical data or faulty historical data, error by the model builder, etc.) from that bound then the above correlation matrix will become nonsensical and the entire pricing of F2D will have to be thrown out the window. In reality, will be bound between minus 0.60 and plus 0.60. Outside that range, the correlation matrix will become nonsensical. Why?

This is because if the value of is outside the range of minus 0.60 and plus 0.60 the above default time correlation matrix between three names will not have a Cholesky factor, i.e. a Cholesky matrix of the above correlation matrix will not exist. This means that we are in the realm of the negative "generalized variances", which are undefined (of course a Monte Carlo simulation cannot be performed on a multi-asset basket if the Cholesky matrix does not exist). If you program a cholesky code in *Excel/VBA * (which is really very simple) and then run it on the above correlation matrix with any values outside the range of minus 0.60 and plus 0.60 you will get an output of:

Now input any value between minus 0.60 and plus 0.60 in the above correlation matrix and then do the Cholesky decomposition. Say, you input either plus 0.60 or minus 0.60 (the extreme values of the correlations) then you will get a cholesky like this:

Actually, strictly speaking the above correlation matrix, though not nonsensical, is not a valid one. Why? Because the determinant of the above correlation matrix is zero, which makes the matrix singular and hence non-invertible (i.e. the inverse of the correlation matrix does not exist). This condition violates the existence of correlation. Therefore, the practical bounds of the third correlation will be from minus 0.59 to plus 0.59. Any number between minus 0.59 and plus 0.59 will make the default time correlation matrix valid, robust and workable.

You can experiment with other correlation values yourself using Excel spreadsheet. For a correlation matrix to be valid the following conditions must hold:

- the matrix must be invertible (i.e. the matrix should not be singular and determinant of the correlation matrix should not be exactly equal to zero);

- the Cholesky matrix of the correlation matrix must exist (i.e. the square root of the matrix must exist). The Chloseky decomposition conditions is expressed as:

The above conditions must be borne in mind while generating default time correlations, or performing stress tests, etc. otherwise we will not be able to price F2D or do proper sensitivity analysis.

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