Risk models: covariance estimation, VaR and expected shortfall

SIG · Chapter 512 min readAsked at Two Sigma, Citadel, AQR, Point72

Assumes Factor models: CAPM, Fama–French and statistical factors.

After this lesson you should be able to

  • Say why a sample covariance matrix is unusable and what replaces it.
  • Compute VaR and expected shortfall, and state their properties.
  • Decompose portfolio risk into its contributors.

A risk model answers two questions: how much could we lose, and what is driving it. The first needs a distributional assumption you know to be wrong, and the second needs a covariance matrix you know to be badly estimated — so the craft is in managing both failures rather than pretending they are absent.

Proposition 5.1

Why the sample covariance fails

With nn assets you need n(n+1)/2n(n+1)/2 parameters and you have TT observations per asset. When nn approaches TT the estimate is singular and its smallest eigenvalues are pure noise — and an optimiser inverts the matrix, loading heavily into exactly those directions.

Holds when

  • 500 assets need 125,250 parameters; a year of daily data supplies 250 observations each.
  • Shrinkage (Ledoit–Wolf) pulls the estimate toward a structured target and fixes conditioning and noise together.
  • A factor model is the other standard answer: Σ=BΣFB⊤+D\Sigma = B\Sigma_F B^\top + D needs far fewer parameters and is always invertible.

Definition 5.2

Value at risk

VaR, Pr⁡(loss>VaRα)=1−α\Pr(\text{loss} > \mathrm{VaR}_\alpha) = 1 - \alpha — The loss that is exceeded with probability 1−α1-\alpha over a stated horizon. It is intuitive, it is embedded in regulation, and it has a serious defect: it says nothing about how bad the exceedances are, and it is not subadditive, so the VaR of a combined portfolio can exceed the sum of its parts.

Equation 5.3

Expected shortfall

The average loss in the tail beyond VaR. It is coherent — in particular subadditive — which is why regulation has moved toward it.

ESα=E[loss∣loss>VaRα]\mathrm{ES}_\alpha = \mathbb{E}\big[\text{loss} \mid \text{loss} > \mathrm{VaR}_\alpha\big]
subadditive\text{subadditive}
Combining portfolios never increases the measured risk, which is what diversification ought to mean.
-3-2-1012399% VaRStandardised return
Figure 5.4 · Value at risk stops exactly where the problem starts. Value at risk is the edge of the shaded region: the loss you beat 9999 days in 100100. It says nothing whatever about the shape inside it. Expected shortfall is the average of that region — 2.672.67 sigma for a normal — and it is the number that changes when the tail gets fatter.

Why non-subadditivity is disqualifying. A risk measure that can *rise* when you merge two portfolios tells you that diversification made things worse, which is nonsense and also actionable nonsense: it creates an incentive to split a book across desks to report less risk. VaR can do this because it looks at a single quantile and ignores the shape beyond it, so two positions whose tails do not overlap can each show a small VaR while their combination shows a large one. Expected shortfall averages over the whole tail and cannot behave that way.

Example 5.5

A $100\$100m portfolio has 15%15\% annual volatility. What are the daily 99%99\% VaR and expected shortfall under a normal assumption?

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Worked solution

  1. Formula
    VaR=z0.99σdV,ES=φ(z)1−ασdV\mathrm{VaR} = z_{0.99}\sigma_d V, \qquad \mathrm{ES} = \frac{\varphi(z)}{1-\alpha}\sigma_d V
  2. Substitute
    σd=15%/252=0.945%\sigma_d = 15\%/\sqrt{252} = 0.945\%
  3. Solve
    VaR=2.326×0.00945×100m=$2.20m\mathrm{VaR} = 2.326 \times 0.00945 \times 100\text{m} = \$2.20\text{m}
  4. φ(2.326)/0.01=0.02665/0.01=2.665\varphi(2.326)/0.01 = 0.02665/0.01 = 2.665
  5. ES=2.665×0.00945×100m=$2.52m\mathrm{ES} = 2.665 \times 0.00945 \times 100\text{m} = \$2.52\text{m}
  6. Answer
    VaR≈$2.2m, ES≈$2.5m\mathrm{VaR} \approx \$2.2\text{m}, \ \mathrm{ES} \approx \$2.5\text{m}

Sanity check. Expected shortfall is about 15%15\% above VaR under normality. With real fat tails the gap is much larger, which is the practical reason the normal assumption is dangerous at the 99%99\% level and worse beyond it.

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