14.1 Measures of Financial Risk

Measures of Financial Risk

Valuation and Risk Models opens with how institutions quantify “how bad can it get?” GARP’s VRM–1 reading links Markowitz mean-variance thinking to modern regulatory risk metrics—Value at Risk (VaR) and Expected Shortfall (ES)—and forces you to judge those metrics against the formal definition of a coherent risk measure.

Mean-Variance Framework and the Efficient Frontier

Under mean-variance analysis, an investor cares about the expected portfolio return μ_p and the portfolio variance σ_p² (or volatility σ_p). For N assets with expected-return vector μ and covariance matrix Σ, a weight vector w (summing to 1 for a fully invested portfolio) gives:

μ_p = wᵀμ

σ_p² = wᵀΣw

The set of all attainable (μ_p, σ_p) pairs is the feasible region. The upper boundary of minimum-variance portfolios is the efficient frontier: for each target return, no other portfolio has lower risk. Adding a risk-free asset produces the capital market line under CAPM assumptions, with the tangency (market) portfolio maximizing the Sharpe ratio (μ_p − r_f) / σ_p.

Mean-variance is elegant but fragile. It treats variance as the complete risk summary, which is optimal only for elliptical distributions (including the multivariate normal) or quadratic utility. Once returns are skewed or heavy-tailed, two portfolios with identical means and variances can have very different crash losses—so VaR and ES enter as tail-focused complements, not replacements for the frontier story.

Normal Versus Real Return Distributions

Risk models often assume normally distributed returns (or log returns) because the normal is fully described by μ and σ, portfolio sums remain normal, and closed-form VaR formulas are cheap. Empirically, daily and weekly equity and credit-spread returns usually show:

• Leptokurtosis (fat tails): more probability mass in extremes than N(μ, σ²) predicts; sample kurtosis often exceeds 3.
• Negative skewness for equities: large downside moves are more frequent/severe than large upside moves of equal size.
• Volatility clustering: large moves follow large moves, so unconditional samples mix calm and turbulent regimes.

A quick diagnostic: if daily returns were N(0, 1%), a 5% one-day drop is a 5-sigma event with probability about 2.87 × 10⁻⁷ under the normal—essentially “never” in a career. Markets produce such moves repeatedly. Therefore, normal VaR at 95% or 99% understates regulatory-style extreme losses unless you fatten tails (Student-t, mixture models, historical simulation, EVT).

Worked contrast

Portfolio value V = $100 million. Daily σ = 1% under a zero-mean normal. One-day 99% VaR ≈ 2.33 × 0.01 × 100m = $2.33 million. If true returns are Student-t with 4 degrees of freedom scaled to the same σ, the 99% quantile is larger than 2.33σ. Do the scaling carefully: the raw t₄ 99% quantile is 3.75, but a raw t₄ has standard deviation √(ν/(ν−2)) = √2 ≈ 1.41, so the unit-variance quantile is 3.75 / 1.41 ≈ 2.65σ. The same “1% vol” book therefore has VaR near $2.65 million rather than $2.33 million. Same volatility input; different distributional assumption; a ~14% higher capital number. Exam trap: comparing the unscaled t quantile (3.75) against the normal 2.33 double-counts the fatter tail, because the raw t is already more dispersed than a unit-variance variable.

Value at Risk: Definition, Assumptions, and Limits

Define the portfolio loss over horizon H as L = −ΔV (or −return × V, depending on sign convention). VaR at confidence level α (e.g., α = 0.99) is the smallest number L_α such that the probability of exceeding it is at most 1 − α:

P(L > VaR_α) ≤ 1 − α

or, for continuous distributions, VaR_α is the α-quantile of the loss distribution: VaR_α = F_L⁻¹(α).

Common packaging assumptions:

1. Horizon H is fixed (1 day, 10 days, 1 year)—you cannot compare VaRs across horizons without a scaling rule.
2. Confidence level α is fixed (95%, 99%, 99.9%).
3. A model maps positions into a loss distribution (parametric, historical, or Monte Carlo).
4. Often a zero mean approximation over short horizons (drift is second-order versus volatility).

Limits you must recite on the exam:

• VaR is a threshold, not an expected loss in the bad state.
• VaR is silent on tail shape beyond α—two portfolios can share VaR and have wildly different catastrophic losses.
• VaR can discourage diversification when subadditivity fails (see coherence below).
• Estimated VaR is noisy in the far tail; sampling error grows as α → 1.
• Horizon scaling with √T assumes i.i.d. (or at least uncorrelated) returns and breaks under mean reversion, autocorrelation, or liquidity horizons that are not T trading days.

Expected Shortfall: Calculation and Comparison to VaR

Expected Shortfall at level α is the expected loss conditional on exceeding VaR:

ES_α = E[L | L ≥ VaR_α]

For a continuous loss distribution, ES_α equals the average of all VaR_u for u ≥ α:

ES_α = (1 / (1 − α)) ∫_α^1 VaR_u du

Worked ES from a discrete loss table

Suppose five equally likely scenarios for one-day loss ($m): 0.5, 1.0, 1.5, 3.0, 8.0. For α = 0.80 (so 1 − α = 0.20), VaR_0.80 is the 80th percentile of the empirical loss distribution. With five scenarios, the worst 20% is the single worst outcome 8.0, so VaR_0.80 = 8.0 and ES_0.80 = 8.0. For α = 0.60, the worst 40% covers the two worst outcomes {3.0, 8.0}; VaR_0.60 = 3.0 and ES_0.60 = (3.0 + 8.0) / 2 = 5.5. Notice ES ≥ VaR always (for the same α), and ES reacts when you thicken the extreme outcome while holding VaR fixed.

Coherent Risk Measures—and Why VaR Is Not Coherent

Artzner et al. define a risk measure ρ as coherent if it satisfies four axioms (stated for portfolio values or losses with consistent sign conventions):

1. Translation invariance (cash invariance): adding cash c reduces risk by c. ρ(L − c) = ρ(L) − c.
2. Subadditivity: ρ(L₁ + L₂) ≤ ρ(L₁) + ρ(L₂). Merging books never creates extra capital beyond the sum—formalizing diversification.
3. Positive homogeneity: ρ(λL) = λ ρ(L) for λ ≥ 0. Scaling the position scales the risk (no liquidity premium in the axiom itself).
4. Monotonicity: if L₁ ≤ L₂ almost surely (L₁ is a smaller loss), then ρ(L₁) ≤ ρ(L₂).

ES satisfies all four (for continuous distributions / atomless probability spaces under standard technical conditions). VaR satisfies 1, 3, and 4 but can violate subadditivity. Classic counterexample: two independent defaultable bonds, each with default probability 0.6% and loss 100 on default (0 otherwise). At 99% confidence, each bond alone has VaR = 0 (default is outside the 1% tail). The combined portfolio defaults with probability ≈ 1.2% if defaults are independent in a simple Bernoulli setup, so 99% VaR of the sum can be 100 while VaR₁ + VaR₂ = 0—diversification appears to increase measured risk. That pathology is exactly why GARP stresses coherence and why desks pair VaR with ES, stress tests, and capital add-ons.

Exam cue: if a question asks which property VaR can fail, answer subadditivity. If it asks which measure is coherent among common choices, answer ES / CVaR / TailVaR (not VaR, not volatility, not a raw notional).