VaR and risk measures

~9 min read

🎯 Interview frequent

Why it matters

Value-at-Risk is the industry’s default one-number market-risk quote: “we don’t expect to lose more than \(X\) in \(N\) days more than \(\alpha\) of the time.” Interviewers want a precise definition, the three standard ways to compute it, and the punchline that VaR is not coherent — it can ignore the tail beyond the quantile and can discourage diversification. Expected shortfall (ES / CVaR) is the usual repair.

Core concepts

Definition. Let \(L\) be the loss over horizon \(h\). The \(\alpha\)-VaR (e.g. 99% 1-day) is the \(\alpha\)-quantile of \(L\): \[ \operatorname{VaR}_\alpha = \inf\{ x : P(L \le x) \ge \alpha \}. \] In words: a number \(x\) such that \(P(L > x) \le 1-\alpha\). It does not say how bad the exceedances are.
Three computational families.
1. Parametric / variance-covariance / delta-normal: assume \(L\) is (locally) linear in risk factors, factors jointly normal, \(\operatorname{VaR} = \sigma_L \Phi^{-1}(\alpha)\) (plus mean if you use relative VaR). Fast; blind to options’ gamma and to fat tails.
2. Historical simulation: revalue the book under each of the last \(N\) historical factor moves; read the empirical quantile. No distributional assumption; very dependent on the window and on “the future looking like the past.”
3. Monte Carlo: simulate factors from a model (often with fat tails, stochastic vol, jumps), full reval, quantile. Flexible and expensive; model risk replaces window risk.
Horizon scaling. Naive: \(\sigma_{h} = \sigma_1 \sqrt{h}\) if i.i.d. returns. This fails with autocorrelation, mean reversion, or nonlinear options (gamma scales closer to \(h\), not \(\sqrt{h}\)). Don’t square-root a 1-day options VaR to 10-day without thinking.
Coherent risk measures (Artzner et al.): translation invariance, positive homogeneity, monotonicity, subadditivity \(\rho(X+Y) \le \rho(X)+\rho(Y)\). Subadditivity = “diversification should not increase measured risk.” VaR can violate it. Expected shortfall \(\operatorname{ES}\alpha = E[L \mid L \ge \operatorname{VaR}\alpha]\) (average of the tail) is coherent (for continuous distributions) and is what Basel uses in the Fundamental Review of the Trading Book.
What VaR is used for anyway. Limits, regulatory capital (still, in places), backtesting (Kupiec, Christoffelsen: count exceptions vs. \(1-\alpha\)). A 99% 1-day VaR should “break” about 2–3 times per year. Clustering of breaks means your independence assumption is wrong.
P&L explain vs. VaR. A good risk system maps the book to factors (see duration, delta, vega in earlier topics) so you can explain P&L with the same Greeks that fed VaR. If unexplained P&L is large, the VaR map is a toy.

Mental model

  loss density
    |
    |     ****
    |    *    *
    |   *      *
    |  *        *   *
    | *           *    *||||  tail
    +------------------|-------------------> L
                      VaR_99%
                      <---- ES = average of this tail ---->

  VaR = a point on the axis
  ES  = how bad, given you're past that point

VaR is a quantile. Treating it as “maximum loss” is the most common verbal fail in interviews.

Interview questions

State 99% 1-day VaR in one sentence that a desk head would accept, and one mistake that sentence must not make. Answer: “On 99% of days, the 1-day loss should not exceed \(X\); on about 1% of days it will be worse.” Mistake: calling \(X\) the maximum loss, or saying the probability of losing \(X\) is 99%.
Why can 99% VaR of a portfolio be higher than the sum of the standalone 99% VaRs? Sketch a counterexample. Answer: Subadditivity failure. Two bonds that each default with 0.6% probability, independently: standalone 99% VaR can be ~0 (no default in the 99% mass), but the portfolio can have \(P(\text{exactly one default}) > 1\%\), so portfolio VaR includes a default loss. Digital / jump risks at frequencies near \(1-\alpha\) are the usual examples.
Delta-normal VaR for a deep short-dated option book: what’s wrong? Answer: The P&L is not linear in \(S\) (gamma, vega) and not normal (truncated, jumps). A 1-day 99% move can be a gamma blow-up the linear map never sees. Full reval historical or MC, and/or delta-gamma-vega analytic with a better distribution, is required.
Historical simulation: last 250 days, no crisis in the window. What is the risk of your 99% VaR? Answer: The 99% sample quantile is the 2nd or 3rd worst day — extremely noisy — and if the window is calm, the quantile is too small. You are assuming the next year looks like this year. Stress overlays and longer/weighted windows exist because of this.
Why did regulation move from VaR to expected shortfall for internal models? Answer: ES is subadditive (coherent), sensitive to the shape of the tail beyond the quantile, and harder to game by stuffing risk just past the VaR threshold. Backtesting ES is harder than counting VaR exceptions — that’s the tradeoff.

Watch

What is value at risk (VaR)? FRM T1-02 — Bionic Turtle. Definition and the “how bad in the tail” caveat.
FRM: Three approaches to value at risk (VaR) — Bionic Turtle. Parametric, historical, Monte Carlo.
Coherent risk measures and why VaR is not coherent (FRM T4-5) — Bionic Turtle. Subadditivity and the case for ES.