Risk Measures: VaR, CVaR, and Coherent Measures

To say "the portfolio is risky" is not enough for a regulator, shareholder, or risk manager. Numbers are needed: "with 99% probability, the daily loss will not exceed X", "capital must be at least Y." Risk measures are formal numerical characteristics of the “danger” of a loss distribution. Their selection determines how much capital a bank holds, which investments are permitted, and how an insurance policy is priced. Basel III (banks), Solvency II (insurance), and ICAAP (internal models) prescribe specific measures—thus, understanding VaR and CVaR is mandatory for any financial professional.

Value at Risk (VaR)

Definition: VaR_α(X) = inf{x: P(X ≤ x) ≥ α} — the α-quantile of the loss distribution X.

In loss terms: VaR_α is the smallest loss that is exceeded with probability no greater than 1 − α. For example, VaR_{0.99} = 1 million rubles means: "with probability 99% the loss ≤ 1 million." On average, once every 100 days it will be exceeded.

Symbol explanation. α — confidence level (typically 0.95, 0.99, 0.995). X — random loss (positive values = losses).

Computation methods.

Parametric (variance-covariance). For X ~ N(μ, σ²): VaR_α = μ + σ·Φ^{−1}(α), where Φ^{−1} is the inverse standard normal function. Φ^{−1}(0.95) ≈ 1.645, Φ^{−1}(0.99) ≈ 2.326.
Historical (historical simulation). We have 1000 historical daily returns. Sort losses in descending order. VaR_{0.99} — the 10th largest loss (1% of 1000).
Monte Carlo. Simulate M = 10,000 scenarios of joint risk factor movements (rates, FX, stocks), revalue the portfolio in each, take the (1 − α)-quantile of the loss distribution.

Numerical Example

Stock portfolio worth 100 million rubles, daily volatility σ = 1.5%.

VaR_{0.95} = 1.645·1.5% × 100 million = 2.47 million rubles. VaR_{0.99} = 2.326·1.5% × 100 million = 3.49 million rubles.

If using historical simulation with 250 days: 250·0.01 = 2.5, so take the average between the 2nd and 3rd worst losses. This may turn out higher than the parametric estimate—real data have fat tails.

VaR Shortcomings

Does not consider “disaster scale” beyond the threshold. Two portfolios with the same VaR can have radically different losses in the worst 1%: one loses 5 million, the other — 50 million.
Not subadditive. There exist examples X, Y: VaR(X + Y) > VaR(X) + VaR(Y). Violates the "diversification reduces risk" principle—undesirable for risk management.
Ignores tail correlation structure. In crises, correlations between assets grow—VaR does not capture this.

Conditional Value at Risk (CVaR / Expected Shortfall)

Definition: CVaR_α(X) = E[X | X > VaR_α(X)] — expected loss given it exceeds VaR_α.

Also called Expected Shortfall (ES), Tail VaR (TVaR), Average VaR.

Properties. CVaR_α(X) ≥ VaR_α(X) (always). CVaR considers the "average severity" of tail events. CVaR is subadditive: CVaR(X + Y) ≤ CVaR(X) + CVaR(Y). Coherent (see below).

Calculation for normal: CVaR_α = μ + σ·φ(Φ^{−1}(α))/(1 − α), where φ is the standard normal density. For α = 0.99: multiplier ≈ 2.665 (vs. 2.326 for VaR).

In our example (100 million, σ = 1.5%): CVaR_{0.99} = 1.5%·2.665·100 = 3.997 million (about 14% higher than VaR_{0.99}).

Coherent Risk Measures (Artzner et al., 1999)

Coherency axioms. A risk measure ρ(X) is coherent if:

Monotonicity: X ≤ Y a.s. → ρ(X) ≤ ρ(Y). Smaller loss means lower risk.
Subadditivity: ρ(X + Y) ≤ ρ(X) + ρ(Y). Diversification reduces risk.
Positive homogeneity: ρ(λ·X) = λ·ρ(X) for λ > 0. Double position — double risk.
Translational invariance: ρ(X + c) = ρ(X) + c. Adding cash reduces risk by exactly the cash amount.

VaR is not coherent — violates subadditivity. CVaR is coherent.

Regulatory standards.

Solvency II (Europe, insurance): SCR = VaR_{0.995} over 1 year for each risk.
Basel III (banks, market risk): transition from VaR to ES (CVaR_{0.975}) in FRTB (since 2023).
IORP II (pensions): combination of VaR, ES, and stress-tests.

Spectral Risk Measures

Generalization of CVaR: ρ(X) = ∫_0^1 VaR_u(X)·φ(u) du, where φ is a weighting function (risk premium). For φ(u) = 1/(1−α) on u ∈ [α, 1] — this is CVaR. Spectral measures are always coherent if φ is decreasing.

Distortion measures (Wang transform). ρ(X) = ∫ x·dT(F(x)), where T is a distortion function. Standard: T(p) = Φ(Φ^{−1}(p) − λ). Used for pricing exotic insurance risks.

Numerical Example: Subadditivity of VaR vs. CVaR

X = Y — independent, P(X = 0) = 0.96, P(X = 100) = 0.04 (same for Y). VaR_{0.95}(X) = 0 (96th percentile = 0). VaR_{0.95}(Y) = 0. VaR_{0.95}(X) + VaR_{0.95}(Y) = 0.

But for X + Y: P(X+Y = 0) = 0.96² = 0.9216. P(X+Y = 100) = 2·0.96·0.04 = 0.0768. P(X+Y = 200) = 0.04² = 0.0016. P(X+Y > 0) = 0.0784 > 0.05 → VaR_{0.95}(X+Y) = 100.

Result: VaR(X+Y) = 100 > 0 + 0 = VaR(X) + VaR(Y). Subadditivity violated!

CVaR_{0.95}(X) = E[X | X > 0] = 100. CVaR(X+Y) = (0.0768·100 + 0.0016·200)/0.0784 ≈ 102. Here 102 ≤ 100 + 100 = 200, subadditivity holds.

Real Applications

Trading floor (banks). Daily VaR limit per desk: 5–10 million USD is typical for a major bank. Exceeding this → escalation, investigation of causes.
Insurance companies (Solvency II). SCR = VaR_{0.995} over a 1-year horizon. Crédit Agricole Assurances: SCR ≈ €15 billion, capital ratio 240%.
Pension funds. UK USS, Dutch ABP estimate liability-VaR under pessimistic interest rate and longevity scenarios.
Cryptocurrency exchanges. Risk engine for margin trading uses VaR in real time to calculate margin requirements.

Exercise. For two losses X ~ N(0, 1), Y ~ N(0, 1), Cov(X, Y) = ρ·σ_X·σ_Y. (a) For ρ = −0.5 compute VaR_{0.95} and CVaR_{0.95} for X, Y, and X+Y (use that X+Y ~ N(0, 1+1+2·(−0.5)) = N(0, 1)). (b) Check subadditivity of CVaR. (c) Construct a discrete example (as above) where VaR clearly violates subadditivity. (d) Using simulation (10,000 points), verify CVaR_{0.95}(X) and CVaR_{0.95}(X+Y) for ρ = 0.7.