Jun 25

Capital allocation to business units (paper)

by David Harper, CFA, FRM, CIPM

Here is a tight overview of Euler allocation as a basis for economic capital (EC) allocation. In the FRM, Ong's "Portfolio Effects" (Chapter 6) introduces the concept of risk contribution which he says is "the single most important risk measure for assessing credit risk."

Portfolio unexpected loss (portfolio UL) is a VaR concept: the worst expected loss given some confidence in excess of the expected loss (EL). We can think of at least three levels of unexpected loss:

Ong's unexpected loss (UL) = one standard deviation (i.e., the "volatility" of the loss distribution). Note this reflects a relatively low level of confidence and could be called capital at risk. For example, if the distribution were normal (clearly incorrect for a credit portfolio but just to illustrate), one standard deviation corresponds to a confidence of only 84%: =NORMSDIST(1) = 84%.
Internal unexpected loss: some multiple of (multiplier on) Ong's UL since the confidence will be higher. That's why we don't put Ong's UL directly into the denominator of RAROC: it is not enough risk capital. Instead, RAROC = risk-adjusted return / [(UL)(multiplier to scale the confidence)]
Regulatory unexpected loss: also much higher than Ong's as Basel II confidence levels are 99%/99.9% under the advanced approaches

Then we are tempted to breakdown the portfolio UL into pieces and we can think of at least two ways:

Incremental UL. This is analogous to Jorion's Incremental VaR: Portfolio VaR â€“ Portfolio VaR (without the asset). That is, how much does Portfolio UL/VaR drop if we remove the asset. The weakness of this measure is that it is not additive. Sum of Incremental UL does not equal Portfolio UL.
Risk contribution (a.k.a., marginal UL). Analogous to Jorion's Component VaR. This is desirable because it is additive: by design, risk contributions sum to portfolio unexpected loss.