A Probability Distribution Functions For Portfolio Losses

Credit loss distributions computed for large homogeneous portfolios of exposures often look something like the function shown below:

This particular distribution function has a number of obvious features:

• It is continuous and unimodal.
• It is 'skewed' - asymmetric and fat-tailed.

Computed distributions for smaller and/or inhomogeneous portfolios are sometimes multimodal and/or discontinuous, which is a result of the nature of the portfolio, together with the modelling assumptions that are made.

You can generate some portolio-loss distributions, and experiment with the effects of changing the input parameters:

• Using our simple: On-Line Portfolio Credit Risk Model.

Some Simple Distribution Statistics and Measures.

Averages - The Mean, Mode and Median.

The Mean :
The Mode :	A quantile for which the pdf is a local/global maximum. (The 'peak' of the distribution.)
The Median :	The quantile that has half of the (cumulative) probability below it, an half above it.

Variance, Standard Deviation and Root Semi-Variance.

The Variance :
The Standard Deviation :	σ: the (positive) square root of the variance.
The Upper Semi-Variance :
The Lower Semi-Variance :
The Root Semi-Variances :	The upper and lower root semi-variances are simply given by the (positive) square roots of the corresponding semi-variances.

Moments of the Distribution Function.

Moments about zero :
Moments about the mean :

Percentiles and Confidence Levels.

Conditional Shortfall and Tail-Moments.

Conditional Shortfall :
	The conditional shortfall for threshold t is given by this expression with r = 1. We also refer to values obtained with r > 0 as the normalised tail moments.

Entropy and Information Content.

Entropy :

Visualising the 'Fat Tail' of a Skewed Portfolio Loss Distribution.

The shape of the tail of a skewed portfolio loss distribution is often of particular interest to risk analysts. Some insight into its shape can be gained, for example, by comparing the spread of a set of chosen confidence levels, but a graphical representation can be invaluable. Unfortunately, if we look at the raw distribution function (or worse, the cumulative function), the tail structure can be very difficult to discern.

An alternative is to plot the logarithm of the (right) tail integral vs loss. To the practiced eye, his curve provides detailed information about the structure of the tail (although 'error' bars must be taken into account when using this technique with model results).

In addition, if we use base-10 logarithms, a value such as '-2' corresponds directly to the 99% confidence level for the distribution.

Such a plot is shown below for the distribution function at the top of the page.

Dr Andrew Gray