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This section uses a simple example to calculate various loss statistics including risk contributions. The example uses a pure default model, building up on the example of the two-obligor portfolio with 10% default correlation identical to that of Chapter 46. We do not replicate all detailed calculations. Rather, we detail the comparison of standalone and portfolio risk measures.

The Portfolio Used as Example

To illustrate the above risk measures, we use the example of a portfolio of two obligors with correlated defaults. Tables 51.2 and 51.3 provide the details of exposures and the loss distribution.

The cumulated loss probabilities provide the loss percentiles. For instance, the loss at the 7% confidence level is 50, and the loss at the 0.906% confidence level is 100. When we use the second percentile, we consider as a rough proxy of the loss at the 1% confidence level. For confidence levels lower than or equal to 0.906%, the loss is maximum, or 150. Between 7% and less than 0.906%, the loss is 100. Between 11.094% and less than 7.00%, the loss is 50.

Standalone Expected Losses and Portfolio Expected Losses

The expected loss of obligor i is EL;- = d_; x Lgd_;- in value. The expected loss for the portfolio of obligors is the sum of individual obligor expected losses: EL_P = J2_; d x Lgd_;-. The expected losses of A and B are the default probabilities times the exposure, or 100 x 7% = 7 and 50 x 5% = 2.5 respectively for A and B. The portfolio expected loss also results directly from the portfolio loss distribution considering all four possible events, with single default probabilities lower than standalone default probabilities. The probability weighted average of the four loss values is also 9.5.

Portfolio Capital

Capital derives from the loss distributions and the loss percentiles at various confidence levels. The portfolio loss percentile at 1% is approximately L(1%) = 100. The expected losses of A and B are respectively 7.0 and 2.5, totalling 9.5 for the portfolio. Capital is the loss percentile in excess of expected loss, or 100 — 9.5 = 90.5. If the confidence level changes, the loss percentiles do as well.