VaR, CVaR, and drawdown

Variance is a symmetric, one-shot measure of risk. The real risks portfolio managers worry about are asymmetric — losses, not gains — and path-dependent — drawdowns, not single-day moves. VaR, CVaR, and drawdown measures address these limitations of pure variance-based risk.

VaR (recap from Stats Module 12)

VaR_α is the α-quantile of the loss distribution. 'We are α-confident we won't lose more than this in the next period.' Standard convention: VaR_99 = 99% confidence one-day loss bound.

Three ways to compute portfolio VaR

• Variance-covariance: assume MVN returns; VaR = z_α · √(wᵀΣw). Closed-form and fast.
• Historical simulation: compute portfolio returns over T historical days; take the α-quantile.
• Monte Carlo: simulate returns from a calibrated model; take the simulated α-quantile.

CVaR / Expected Shortfall

CVaR_α = E[L | L > VaR_α]

Average loss in the worst (1 - α) fraction of days. CVaR is coherent (sub-additive), unlike VaR — combining portfolios cannot increase CVaR beyond the sum. Basel III mandates CVaR at 97.5%.

Maximum drawdown

The largest peak-to-trough decline in cumulative returns over a period. Path-dependent — depends on the order of returns, not just their distribution. Important for investors with redemption risk (hedge funds) or psychological capital constraints (retail).

P_t = cumulative value at time t
DD_t = max_{s ≤ t}(P_s - P_t) / max_{s ≤ t} P_s
MDD = max_t DD_t

Time to recover

Drawdowns matter not just in magnitude but in duration. A 50% drawdown that recovers in 3 months is qualitatively different from one that lingers for 5 years. The 'Calmar ratio' (annual return / MDD) and the 'ulcer index' (RMS drawdown over time) attempt to quantify drawdown pain.

Coherent risk measures (Artzner-Delbaen-Eber-Heath 1999)

Four axioms a 'good' risk measure ρ should satisfy:

1. Monotonicity: more loss → more risk. X ≤ Y ⟹ ρ(X) ≥ ρ(Y).
2. Sub-additivity: diversification doesn't increase risk. ρ(X + Y) ≤ ρ(X) + ρ(Y).
3. Positive homogeneity: scaling positions scales risk. ρ(αX) = α ρ(X) for α ≥ 0.
4. Translation invariance: adding cash reduces risk. ρ(X + c) = ρ(X) - c.

VaR satisfies axioms 1, 3, 4 — but fails sub-additivity in general. CVaR satisfies all four. Hence the regulatory shift toward CVaR-based capital requirements.

Marginal CVaR for risk attribution

Analogous to marginal contribution to variance, marginal CVaR captures how each position contributes to tail loss. Useful for tail-risk budgeting at fund-of-funds and large institutional portfolios.

Stress testing

Statistical risk measures (VaR, CVaR) are estimated from sample data and cannot foresee events outside the sample. Stress testing addresses this by simulating historical or hypothetical extreme scenarios — Lehman week, October 1987, COVID-March-2020, a 30% USD/KES move. Required for regulated banks, dollar-prudent for everyone else.