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Use Euler's theorem to find how much a given loan adds to the overall risk of a portfolio, and describe how it works.

FRM® Part I · Valuation and Risk Models · Measuring Credit Risk · FRM1.VR.6.i

The idea

Euler's theorem allocates portfolio risk to positions. Suppose a risk measure R is homogeneous of degree one, meaning that doubling every position doubles R. Then Risk = Σ xᵢ × ∂R/∂xᵢ, summed over all loans. Here xᵢ is the exposure to loan i. Each term xᵢ × ∂R/∂xᵢ is the contribution of loan i, and the contributions add up exactly to total risk. For the standard deviation of portfolio loss, the contribution of loan i equals xᵢ × Cov(Lᵢ, LP) ÷ σP. Here LP is the portfolio loss. Standard deviation, VaR and expected shortfall all qualify. A contribution can be small, or even negative for a loan that hedges the rest. Do not confuse it with incremental risk, the portfolio's risk with the loan minus the risk without it, which does not sum to the total.

Formulas

Euler allocation of portfolio risk

R = Σ xᵢ × ∂R ÷ ∂xᵢ

R
a risk measure homogeneous of degree one, such as standard deviation, VaR or expected shortfall
xᵢ
the exposure to loan i
∂R ÷ ∂xᵢ
the change in R per unit increase in that exposure

Each term is a loan's contribution and the terms add exactly to R. A contribution can be negative for a hedge, and it is not the same as incremental risk.

Euler contribution to the standard deviation of portfolio loss

contributionᵢ = xᵢ × Cov(Lᵢ, LP) ÷ σP

xᵢ
the exposure to loan i
Cov(Lᵢ, LP)
the covariance of loan i's loss per unit of exposure with the portfolio loss
σP
the standard deviation of portfolio loss

The contributions sum to σP. A loan correlated with the rest contributes more than its stand-alone share of risk would suggest.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement correctly describes Euler's theorem as used in allocating portfolio risk?

  1. Each loan's contribution is the portfolio's risk with the loan minus the risk without it.Why not: That is incremental risk, which does not sum to total risk.
  2. For a degree-one homogeneous risk measure, total risk equals the sum of each position times the partial derivative of risk with respect to it. - correct
  3. Total risk equals the sum of each loan's stand-alone risk, with no cross terms.Why not: Stand-alone risks add only when losses are perfectly correlated; Euler contributions already include correlation.
  4. It applies only to variance, because other risk measures are not differentiable.Why not: It applies to any degree-one homogeneous measure, including standard deviation, VaR and expected shortfall.

Because doubling every position doubles the risk measure, R = sum x_i dR/dx_i. The terms are the loans' contributions and add to total risk exactly.

A harder one

Two loans have loss standard deviations of USD 20 million and USD 10 million, with a loss correlation of 0.3. Using Euler's theorem on the portfolio standard deviation, the contribution of the first loan is closest to:

  1. USD 18.5 million - correct
  2. USD 16.6 millionWhy not: Splits the portfolio sigma in proportion to stand-alone sigmas, which ignores the covariance structure.
  3. USD 20.0 millionWhy not: Uses the stand-alone standard deviation and ignores the diversification effect.
  4. USD 14.9 millionWhy not: Computes the incremental risk (portfolio sigma minus the second loan's stand-alone sigma), which is not the Euler contribution.

Portfolio sigma = sqrt(400 + 100 + 2 x 0.3 x 200) = 24.90. Contribution of loan 1 = (400 + 0.3 x 200) / 24.90 = 18.47. Loan 2 contributes 6.43; the two sum to 24.90.

The hardest tier

A bank holds three loans whose loss standard deviations are USD 10 million, 20 million and 30 million. Loss correlations are 0.2 between loans 1 and 2, 0.1 between loans 1 and 3, and 0.4 between loans 2 and 3. Using Euler's theorem on the portfolio standard deviation, loan 3 accounts for approximately what percentage of total portfolio risk?

  1. 64%Why not: Uses the share of the sum of variances, which ignores covariances and mixes variances with standard deviations.
  2. 46%Why not: Uses incremental risk (sigma with loan 3 minus sigma without) over total sigma, which is not the Euler allocation.
  3. 50%Why not: Uses the stand-alone sigmas: 30 out of 10 + 20 + 30. This ignores correlations.
  4. 58% - correct

Portfolio sigma = 44.94. Contribution of loan 3 = (Cov(3,1) + Cov(3,2) + Var(3)) / sigma = (30 + 240 + 900) / 44.94 = 26.03, i.e. 58% of 44.94. Contributions are 3.78, 15.13, 26.03, summing to 44.94.

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