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The marginal contributions of a facility, or a subportfolio, to the portfolio loss volatility or to capital are the differences of the portfolio loss volatility, or the portfolio capital, with and without the facility or subportfolio. We calculate below both marginal risk contributions (MRC) to loss volatility and to capital. All calculations use the example of a two-obligor (A and B) portfolio is in the previous chapter (and Chapter 46). They assume that B is the last transaction entered in the portfolio. Its marginal risk contribution is the difference between the loss volatility, and the capital, of the portfolio A + Band the similar values for A only.

Marginal Contributions to Loss Volatility

The marginal contribution of B to the portfolio loss volatility is the latter minus the loss volatility of A, or 28.73 — 25.51 = 3.21. The marginal risk contribution of A is determined in the same way, 28.73 — 10.90 = 17.83. The sum of these marginal risk contributions is 21.05, significantly less than the portfolio loss volatility. Table 52.1 compares standalone risk, absolute risk contributions and marginal contributions.

The Marginal Risk Contributions to Capital

The portfolio distribution is a prerequisite for defining capital and its relation to loss volatility. Capital derives from the loss distributions and the loss percentiles at various confidence levels. In the preceding chapter, we used the portfolio loss percentile L(1%) = 100. Capital is the loss percentile in excess of expected loss totalling 9.5, or 100 — 9.5 = 90.5. A second calculation at the 0.5% confidence level results in a maximum portfolio loss of 150 and a capital of 150 — 9.5 = 140.5.

Marginal contributions to capital require a two-step procedure, since we need to define the first-in obligor. The first-in obligor in the two-obligor portfolio has a capital identical to his marginal risk contribution to capital by definition, following the with versus without principle. When the second-in obligor comes in, his marginal capital contribution follows the with and without the second-in rule. It is the portfolio capital minus the initial capital used up by the first-in obligor. Therefore, marginal risk contributions to capital depend on the order of entrance of a particular obligor into the portfolio, since each obligor has a different first-in marginal risk contribution to capital.

The first calculation uses a 1% confidence level, leading to a loss percentile of 100 for the portfolio A + B and a capital of 100 — 9.5 = 90.5. A second calculation at a 0.5% confidence level would result in a maximum portfolio loss of 150 and a capital of 150 9.5   140.5.

Marginal Risk Contributions to Capital at 1% of A and B

When A only is in the portfolio, the confidence level of 1% is lower than the standalone default probability of A, which is 7%. Therefore, the loss percentile at 1% is 100 and the expected loss is 7, so that capital at 2% is 100.0 — 7.0 = 93.0. Adding B raises the portfolio capital to 90.5. Hence the marginal contribution to capital of B is 90.5 — 93.0 = —2.5. The diversification effect makes the incremental contribution of B negative. This non-intuitive result depends on the magnitudes of exposures and expected losses.

Economically, negative marginal risk contributions to capital do make sense. This does not happen with contributions to the loss volatility. First, absolute risk contributions to loss volatility and capital are never negative. Second, marginal contributions to loss volatility are always positive because any new incremental exposure increases the loss volatility unless the correlations with the portfolio are negative. This case applies to insurances or credit derivatives.

If we start with B as the first-in exposure, the capital at the 1% confidence level is the entire exposure of B, whose default probability 5% is higher, minus its expected loss, or 50 — 2.5 = 47.5. When adding A to B, we reach the capital level of 90.5, resulting in an incremental risk contribution of A of 43.0. This sequential process implies that marginal risk contributions to capital always sum to the capital of the portfolio grouping all exposures, by definition. Table 52.2 summarizes the calculations.

Marginal Risk Contributions to Capital at 0.5% of A and B

For comparison purposes, we conduct similar calculations following exactly the same two-step procedure, with a tighter confidence level of 0.5% and a capital in excess of expected loss 9.5 equal to 150 — 9.5 = 140.5. Accordingly, all marginal risk contributions to capital increase, and they sum to the capital. However, in this second example, none is negative, whatever the first-in (Table 52.3).

With a very tight confidence level, A and B have positive marginal risk contribution to capital no matter which enters first in the portfolio. The first-in marginal risk contributions of A and B are 93 and 47.5 respectively. In addition, in this case, the marginal capital contributions are identical for a first entrant or a second entrant. For instance, A has 93 when first-in and 93 when second-in as well. The same happens to B. This illustrates why marginal risk contributions to capital are difficult to tackle, because of unexpected or non-intuitive effects.

General Properties of Marginal Risk Contributions

General properties of marginal risk contributions include:

• The marginal risk contributions to the portfolio loss volatility are lower than the absolute risk contributions and lower than the standalone loss volatilities. Accordingly, marginal risk contributions to portfolio loss volatility add up to a value lower than the portfolio loss volatility. The subsequent sections demonstrate that these properties are general.

• On the other hand, marginal risk contributions to portfolio capital can be higher or lower than absolute risk contributions to capital (derived from absolute risk contributions to volatility). In addition, they add up to portfolio capital if we calculate sequentially the marginal risk contributions of each obligor when he enters the portfolio.

The calculation of marginal contributions to capital in excess of expected loss suggests several observations:

• The marginal risk contributions to capital calculated following the two-step procedure are additive. They sum up by construction to the portfolio capital, and this sum is algebraic (negative incremental risk contributions are possible). This contrasts with the finding that the sum of marginal contributions to the portfolio loss volatility is always lower than, or at most equal to, the portfolio loss volatility.

• The marginal contributions to the portfolio capital can be positive or negative. This depends on the shape of the distribution and on the confidence level selected.

• The marginal contributions to the portfolio capital can be bigger or smaller than the absolute risk contributions to capital. This depends also on the shape of the distribution and the confidence level. This finding contrasts with the property that marginal contributions to loss volatility are always lower than the absolute risk contributions.

• The marginal risk contributions of the first entrant and of the second entrant depend on who is first and who is second, and on the confidence level.

Such properties are non-intuitive. They become apparent here because we use a two-obligor portfolio, and because individual exposures are large compared to total exposure. In real cases, the exposures are very small fractions of the portfolio exposure, and we do not have such drastic concentration effects. Considering these observations, the most representative case is the second one, with a tight confidence level of 0.5% making both As and Bs incremental risk contributions positive.

Implications

The discrepancies between absolute and marginal risk contributions and between contributions to loss volatility and capital raise important issues. These include: risk-based pricing (ex ante decision); risk-based performance, once facilities are in the portfolio (ex post measure); consistency of capital allocation rules with pricing rules. The question is which risk contributions should we use and for what purposes?

For example, when considering risk-based pricing, choosing the right measure of risk contribution is critical. Risk-based pricing means that revenues net of expected loss are at least a minimum percentage of capital. This constraint sometimes has unexpected implications. The negative incremental risk contribution of B in the first case above (capital at 1% confidence level) implies that we can afford negative revenues since we save capital thanks to B. All risk-based performance measures and risk-based pricing implications use the same example as above.

In the next sections, we refer only to marginal risk contributions to loss volatility. The discussion of risk contributions to loss volatilities stems from the fact that many transactions are small compared to the reference portfolio and that it is acceptable to use a constant ratio of capital to portfolio loss volatility. Therefore, the marginal risk contributions to capital are always positive, unless we use credit derivatives, which are like negative exposures.