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SAMPLE CALCULATIONS OF RISK CONTRIBUTIONS

This section uses the example of the two-obligor portfolio and a default correlation of 10%. All calculations are in Chapter 48, where we detail the loss distribution of a portfolio of two obligors, with default correlation. We only need to replicate the results here. Tables 53.1 and 53.2 replicate the details of exposures and the loss distribution for this same portfolio.

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. The second percentile is roughly 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.

The sequence of calculations is as follows:

• calculate standalone loss volatilities and expected loss for each obligor

• calculate the portfolio (A + B) loss volatility, given diversification effect: this is the starting point for risk allocation

• define portfolio capital according to loss percentiles at various confidence levels

• finally, derive the risk contributions to portfolio loss volatility and to portfolio capital.

TABLE 53.1 Standalone default probabilities and default correlations

Standalone default probabilities and default correlations

TABLE 53.2 Loss distribution (default correlation = 10%)

Loss distribution (default correlation = 10%)

Standalone Expected Loss and Portfolio Expected Loss

The expected loss of obligor i is E(L) = d.xL. in value. The expected loss for the portfolio of obligors is the summation of individual obligors expected losses: ELp — E.(<i x L). The expected losses are the exposures times the standalone default probabilities. The expected loss of the portfolio is independent of the correlation and is the sum of standalone expected losses. The expected losses of A and B are the default probability multiplied by the exposure or 100 multiplied by 7% — 7 and 50 multiplied by 5% = 2.5 respectively for A and B.

 
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