Euler Capital Allocation: Decomposing Portfolio Risk

Once a bank has computed the aggregate capital for its portfolio — say

\varrho(L) = \text{ES}_{0.99}(L)

for

L = L_1 + \cdots + L_d

— a natural follow-on question is: how should this capital be disaggregated to individual desks or asset classes? This is not merely an accounting exercise. Capital allocation drives performance measurement (RORAC), investment decisions, and pricing of internal transfer of risk between units.

The classical answer is the Euler principle (also called allocation by the gradient). It rests on a single mathematical fact: Euler's homogeneous-function theorem. If the risk-measure function

r(\lambda) = \varrho(L(\lambda))

is positive-homogeneous and differentiable, the theorem gives

\varrho(L) = \sum_{i=1}^d \frac{\partial r}{\partial \lambda_i}(1) \cdot 1.

Setting the risk contribution of unit

i

\text{AC}_i = \partial r / \partial \lambda_i \big|_{\lambda = 1}

gives a full, exact decomposition. The resulting allocations have clean closed forms for the standard risk measures (standard deviation, VaR, ES) and satisfy two powerful economic properties: RORAC compatibility and no incentive to leave the firm (the diversification benefit is shared with each unit).

Learning objectives. By the end of this note you should be able to:

state Euler's theorem and use it to derive the capital allocation principle;
compute the VaR contribution $\mathbb{E}[L_i | L = \text{VaR}_\alpha(L)]$ and the ES contribution $\mathbb{E}[L_i | L \ge \text{VaR}_\alpha(L)]$ and explain what each measures;
state RORAC compatibility and verify that the Euler principle satisfies it;
explain the diversification benefit inequality $\text{AC}_i \le \varrho(L_i)$ and trace it to convexity of the risk-measure function;
describe how the ES contribution is estimated via Monte Carlo.

In this lesson

1. The allocation problem
1. The Euler principle
1. Euler allocations for specific risk measures
1. Economic properties of the Euler principle
1. Summary table

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