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# Specific Cases

A specific case is that of independence, where all covariances terms are zero. Then:

The total portfolio variance is the sum of all individual variances and the portfolio loss volatility becomes the square root of this summation. The risk contribution becomes:

or the ratio of the standalone loss variance of the facility to the loss volatility of the portfolio. The undiversifiable risk ratio becomes RR o/o„, the ratio of the standalone risk of the facility to the portfolio loss volatility.

Another reference case is the uniform portfolio, with uniform equal exposures, loss given default, and the same correlation across all pairs of facilities. Such portfolio has a loss volatility and loss percentiles that depends on its correlation. According to the above, all risk contributions in the limit distribution obtained with standard normal variables become equal since RC p = pipa becomes simply the uniform correlation. Capital allocation, or risk contributions, under uniform correlation, is equivalent to allocation of overall economic capital at the pro-rata of standalone loss volatilities of facilities.

# From Risk Contributions to Capital Allocation

The risk contributions sum up to the portfolio loss volatility, but do not add up to capital. In order to proceed to capital allocation, we use the multiple of loss volatility providing the capital at a given confidence level. The loss distribution provides both loss volatility and capital at a given confidence level, from which the multiple derives. If capital K(a) is the capital at the confidence level a, there is always a multiple m(a) such that K(a) — m(a)USp. We also know that LVp = E RC p. For obtaining the capital allocations, we multiply the risk contributions by the multiple m(a), the ratio of portfolio capital to loss volatility, both resulting from the loss distribution:

The capital allocation system simply allocates the fraction m(a)RC p of capital to the facility or the sub-portfolios i. For a sub-portfolio, such as those of business units, the capital allocation

FIGURE 53.1 Risk contributions to portfolio loss volatility and capital allocation

simply sums up all m(aJRC p over all facilities of the sub-portfolio, using the additive property of risk contributions (Figure 53.1).

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