Heavy tails, EVT, VaR and CVaR

Risk lives in the tails. The normal distribution that fits the centre of a return distribution well typically dramatically underestimates the probability of extreme losses. Extreme-value theory (EVT) provides the asymptotic framework for modelling the tail directly, and Value-at-Risk and Conditional VaR are the regulatory and practitioner standards for quantifying tail risk.

Why returns aren't normal

• Empirical kurtosis of daily equity returns is typically 5-15; normal is 3.
• 1987 SPX one-day move was -20%, a >22σ event under historical normal vol — once-per-10^100-years if normal.
• Conditional dependence: volatility clusters; correlations rise during crises.
• Drawdowns are path-dependent and have heavier tails than single-day losses.

Value-at-Risk (VaR)

VaR_α(L) = inf{x : P(L ≤ x) ≥ α}

The α-quantile of the loss distribution. VaR at 99% says: 'we expect to lose at most VaR_99 on 99 of 100 days'. Conventions vary — sometimes VaR is reported as a positive number (loss magnitude), sometimes as a negative number (P&L).

Three ways to compute VaR

1. Parametric (variance-covariance): assume returns ~ N(μ, σ²); VaR_α = -μ + z_α σ. Trivially fast; underestimates tail risk.
2. Historical simulation: empirical quantile of past returns. Non-parametric; captures realised tail behaviour; assumes stationarity.
3. Monte Carlo: simulate returns from a calibrated model (often with stochastic vol or jump components); take the simulated quantile.

Coherent risk measures and the case against VaR

Artzner-Delbaen-Eber-Heath (1999) define a coherent risk measure as one satisfying four axioms: monotonicity, sub-additivity (diversification doesn't increase risk), positive homogeneity, translation invariance. VaR is NOT sub-additive in general — combining two portfolios can produce higher VaR than the sum, which violates the diversification axiom. Conditional VaR (= Expected Shortfall) IS coherent.

Conditional VaR / Expected Shortfall

CVaR_α(L) = E[L | L > VaR_α(L)]

The expected loss conditional on a tail event. Captures the size of bad days, not just their frequency. Basel III mandates Expected Shortfall at 97.5% confidence for market-risk capital, replacing 99% VaR.

Extreme-value theory — the two pillars

Block maxima — Fisher-Tippett-Gnedenko

The maximum of n i.i.d. random variables, suitably normalised, converges in distribution (for any underlying distribution with regular tails) to a Generalised Extreme Value (GEV) distribution — Gumbel (light tails), Fréchet (heavy tails), or Weibull (bounded). The shape parameter ξ identifies which.

Peaks over threshold — Pickands-Balkema-de Haan

More data-efficient than block maxima. Look at excesses over a high threshold u: Y = L - u given L > u. As u → ∞, Y converges to a Generalised Pareto Distribution with shape ξ and scale β. Fit ξ and β to threshold exceedances; VaR and CVaR at any tail probability follow from the GPD CDF.

Back-testing VaR

Kupiec proportion-of-failures test: under H₀ (correctly-specified VaR), the number of breaches over T days is Binomial(T, 1-α). Reject H₀ if observed breaches deviate too far. Christoffersen test adds an independence test — breaches should not cluster. Basel three-zone framework grades VaR models by breach counts; persistent failure increases capital multipliers.