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This section provides the theoretical formulas of portfolio loss variance and volatility and their decomposition in absolute risk contributions, and shows how to derive the capital allocation. This is the basic capital allocation model. The next section illustrates these theoretical formulas and provides a fully detailed numerical example using the above two-obligor portfolio example. Readers can move directly to the numerical examples, but the theoretical properties are relatively easy to demonstrate.

Capital Allocation

The ratio of capital to loss volatility depends on the confidence level. With a confidence level of 1%, the loss percentile is 100 and the capital is the loss percentile in excess of the expected loss or, using a rounded value = 100 — 9.5 = 90.5. Since the

portfolio loss volatility is 28.729, the ratio m(1%) of capital to loss volatility is m(1%) = 90.5/28.73 = 3.150. Hence, the absolute risk contributions to the portfolio capital are 23.628 x 3.150 = 74.430 and 5.101 x 3.150 = 16.070, summing to the capital 90.500. These are the capital allocations. We can now complete the previous Table 51.6 with capital allocations (Table 51.7).

The capital allocation uses the risk contributions. Nevertheless, we could also use any other criteria such as standalone risk to perform the same task. What are the drawbacks and benefits of these two alternative schemes? The capital allocations based on risk contributions to portfolio loss volatility differ from allocations on standalone risks. The risk contributions of A and B are respectively 23.628 and 5.101, summing to 28.729. The percentage allocations are 23.628/28.729 and 5.101/28.729, or respectively 74.43% and 16.07%. Similar percentage allocations derive from the standalone risks. The standalone loss volatilities of A and B are respectively 25.515 and 10.897, also summing to 36.412 > 28.729. The corresponding percentage allocations would be 25.515/36.412 and

10. 897/36. 422, or 70.07% and 29.93% respectively. The capital allocation to B with risk contributions is heavier than with standalone risks, and the opposite occurs for A. Both choices are practical.

The major difference is that the risk contributions embed the correlation structure, whereas standalone risks do not. For example, two facilities might have the same standalone risk, with one being highly correlated with the rest of the portfolio and the other having a lower correlation. The former contributes more to the portfolio risk than the latter. Risk contributions differentiate the two facilities accordingly. On the other hand, the drawback of risk contributions is that they are more difficult to interpret than standalone risks. Using standalone risks is simpler, but this reference ignores the differences between two facilities having different correlations with the rest of the portfolio and identical standalone risks.

Calculation of Absolute Risk Contributions from the Variance-Covariance Matrix

The absolute risk contributions derive from the simple matrix formula of the variance of the portfolio loss. This matrix extends to any number of obligors. To proceed, we stick to the default model with two obligors, but the formulas are general and apply to full valuation models as well. The variance of the portfolio loss distribution is:

where X is the row vector of exposures, or 100 and 50, X^T is the column vector of exposures, the transpose of the above, Ј is the unit variance-covariance of the portfolio of exposures, with each term equal to p_abct_act_b. p_AB is the default correlation between A and B (Table 51.8).

The matrix format applies to the calculation of any portfolio loss volatility, as illustrated in Chapter 28. Here, we focus on a specific additional step allowing us to obtain the vector of risk contributions. The sequential steps are:

• Multiplying the row vector of exposures X by the unit variance-covariance matrix Ј results in a row vector of the unit covariances of individual obligors A and B with the portfolio loss, or 6.788 and 2.931. A sample calculation is as follows. The covariance of A with portfolio P = A + B, Cov(A,P), depends only on unit covariances and on exposures: Cov(A,P) = 100 x 6.51% x 50 x 0.5561% = 6.788. In order to find the risk contributions, we simply need to divide these by the portfolio loss volatility obtained in the next step.

• Multiplying this row vector by the transpose of X, we get the variance of the portfolio loss, 825.358, and the square root is the portfolio loss volatility 28.729.

covariance of obligor A with portfolio P. Similarly, X_B x Cov(B,P) = 50 x 2.931 = 147. Obviously, these two covariances add up to the same portfolio variance as before, 825.358.

• Dividing these covariances, 679 and 147, by the portfolio loss volatility, we obtain the absolute risk contributions of A and B in value, as a row vector, 23.628 and 5.101. They are ARCA = 679/28.729 = 23.628 and ARCB = 147/28.729 = 5.101.

• Summing these two absolute risk contributions in value, or 23.628 + 5.101, gives the portfolio loss volatility 28.729.

• Using the absolute risk contributions as a percentage of the portfolio loss volatility, we obtain the capital allocation coefficients based on absolute risk contributions, or 82.24% and 17.76%.

• Finally, the absolute risk contributions to capital are directly derived from the absolute risk contributions to loss volatility using the multiple m(1%) = 3.15, this multiple necessitates a knowledge of the entire loss distribution.

All risk calculations in matrix format are given in Table 51.10. Table 51.11 summarizes the capital allocation process.

Marginal Risk Contributions

This chapter focuses on marginal risk contributions, to portfolio loss volatility or to portfolio capital, and compares them with absolute risk contributions. Marginal risk contributions serve essentially for risk-based pricing with an ex ante view of risk decisions, while absolute risk contributions are the basis for the capital allocation system.

They have a number of specific properties, some of them counterintuitive, which make them somewhat difficult to handle at first sight. The marginal risk contributions to the portfolio loss volatilities are lower than the absolute risk contributions and lower than the standalone loss volatilities. They also depend on the order of entrance of facilities or obligors in the portfolio. They sum to less than the capital, or to the portfolio loss volatility, since the absolute contributions, which are larger, do. Nevertheless, marginal risk contributions to capital are the correct references for risk-based pricing.

Pricing based on marginal risk contributions charges to customers a mark-up equal to the risk contribution times the target return on capital. The mark-up guarantees that the return on capital for the entire portfolio will remain in line with the target return when adding new facilities. However, prices based on marginal risk contributions are lower than prices based on absolute risk contributions. This is a paradox, since the absolute risk contributions are the ones that sum to capital! In fact, the new facility diversifies the risk of those facilities existing prior to entrance of the new one. Therefore, adding a new facility results in a decline in all absolute risk contributions of existing facilities. Because of this decline, the overall return of the portfolio remains on target. However, the ex ante measure of risk-based performance, on the marginal contribution, and the ex post measure, on the absolute contribution, differ for the same facility!

The first section defines marginal and absolute risk contributions, illustrates their calculation using sample calculations based on the simple two-obligor portfolio, and summarizes the properties of risk contributions. The second section demonstrates the relationship, and the ranking, between marginal and absolute risk contributions. The third section deals with marginal risk contributions and risk-based pricing, and addresses the pricing paradox. The last section contrasts the implicit underlying philosophies of absolute versus marginal risk contributions, or the ex post view versus the ex ante view.