Risk Measures — Cross-Cutting Comparison
This note compares every risk measure used across the Finance library — variance / standard deviation, semi-variance, MAD, VaR (parametric / historical / Monte Carlo), CVaR / Expected Shortfall (ES), spectral risk, distortion risk, entropic risk, Sharpe, Sortino, Treynor, Jensen alpha, Calmar, MAR, Pain, Ulcer, Information ratio, K-ratio, max drawdown, time-under-water, MAE/MFE, beta, idiosyncratic vol — on the axes of coherence (Artzner-Delbaen-Eber-Heath 1999), elicitability (Gneiting 2011), regulatory acceptance (Basel III/IV FRTB, EU Solvency II/III, ICS), tail-sensitivity, and computational cost. Decision tree at the end picks by regulatory regime, reporting requirement, portfolio type, and investor type.
See also
- portfolio-construction-and-risk-deep
- investments-and-portfolio-management
- derivatives-and-quant-finance
- options-pricing-deep
- structured-products-deep
- market-making-and-liquidity-provision-deep
- market-microstructure-and-hft
- insurance-and-actuarial
- probability-distributions
- copulas-and-dependence
1. The taxonomy
DISPERSION COHERENT (Artzner et al 1999)
variance CVaR / Expected Shortfall (ES)
std dev (volatility) spectral risk (Acerbi 2002)
semi-variance distortion (Wang 2000)
mean absolute deviation (MAD) entropic risk
worst-case CVaR
QUANTILE
Value-at-Risk (VaR) CONVEX (Föllmer-Schied 2002 relax PH)
conditional VaR (CVaR / ES) convex risk measure
range (max - min) OCE (Ben-Tal-Teboulle 2007)
shortfall risk
DRAWDOWN RATIO / RISK-ADJUSTED RETURN
max drawdown (MDD) Sharpe (1966)
conditional drawdown (Chekhlov 2005) Sortino (Sortino-Price 1994)
pain index, ulcer index Treynor (1965)
Calmar = ann. return / MDD Jensen alpha (1968)
MAR ratio M-squared (Modigliani-Miller-jr 1997)
K-ratio (Kestner) Information ratio
time-under-water (TUW) Calmar
Omega (Keating-Shadwick 2002)
Kappa (Kaplan-Knowles 2004)
TRADING FACTOR
MAE (Max Adverse Excursion) beta (CAPM)
MFE (Max Favorable Excursion) idiosyncratic vol
R-multiple downside beta
expectancy Treynor-Mazuy / Henriksson-Merton
2. Coherence — the Artzner-Delbaen-Eber-Heath 1999 axioms
A risk measure is coherent iff it satisfies all four axioms:
- Monotonicity — if X ≤ Y pathwise then ρ(X) ≥ ρ(Y) (less return = more risk).
- Translation invariance — ρ(X + c) = ρ(X) - c (adding cash reduces risk by exactly c).
- Positive homogeneity — ρ(λX) = λρ(X) for λ ≥ 0 (scaling positions scales risk).
- Sub-additivity — ρ(X + Y) ≤ ρ(X) + ρ(Y) (diversification can only reduce risk).
| Measure | Coherent? | Why / why not |
|---|---|---|
| Variance / std dev | NO | not monotone, not translation invariant |
| Semi-variance | NO | not monotone |
| MAD | NO | not monotone |
| VaR | NO | fails sub-additivity (Artzner et al 1999 counter-example) |
| CVaR / Expected Shortfall | YES | the canonical coherent measure |
| Spectral risk | YES | weighted average of quantiles w/ non-increasing weight function |
| Distortion risk | YES (if distortion fn is concave) | Wang 2000 |
| Entropic risk | NO (not positive homogeneous; convex) | but convex |
| Worst-case CVaR | YES | over a family of measures |
| Max drawdown | NO | not coherent in the ADEH sense (uses path) |
| Sharpe ratio | not a risk measure (ratio) | uses dispersion |
The single most important consequence: VaR is not coherent. It fails sub-additivity (Artzner et al 1999 counter-example: two long-out-of-the-money options on different underlyings have lower VaR sum than the portfolio VaR). This is why Basel III FRTB (Fundamental Review of the Trading Book) replaced VaR with ES.
3. Convex risk measures — Föllmer-Schied 2002 relaxation
Föllmer-Schied 2002 dropped positive homogeneity, keeping only:
- Monotonicity
- Translation invariance
- Convexity — ρ(λX + (1-λ)Y) ≤ λρ(X) + (1-λ)ρ(Y)
Convex risk measures admit a dual representation:
where is a penalty function. Coherent = convex + positive homogeneous = α takes only values 0 or +∞.
Entropic risk: is convex but not coherent; γ is the risk-aversion parameter. Used in optimal control + reinforcement learning (risk-sensitive RL).
4. Elicitability — Gneiting 2011 framing
A risk measure is elicitable if there exists a scoring function S such that ρ(X) = argmin E[S(X, x)]. Elicitability is what makes backtesting possible — a regulator can score forecasts.
| Measure | Elicitable? |
|---|---|
| Mean (expected value) | YES (squared loss) |
| Median | YES (absolute loss) |
| Quantile / VaR | YES (asymmetric piecewise linear, “pinball loss”) |
| Variance | YES jointly with mean |
| ES / CVaR | NO (Gneiting 2011) — alone |
| (VaR, ES) jointly | YES (Fissler-Ziegel 2016) — jointly with VaR |
| Mean + variance | YES jointly |
The Gneiting 2011 result was a shock: ES is not elicitable alone, which means you cannot backtest ES alone with a scoring rule. Fissler-Ziegel 2016 rescued this: (VaR, ES) is jointly elicitable. Acerbi-Szekely 2014 independently proposed three direct ES backtests (now in Basel III FRTB).
The Basel III FRTB ES backtesting framework uses these — banks must backtest both VaR (at 97.5%) and ES (at 97.5%) jointly.
5. The risk-measure cheat sheet
| Measure | Formula | Coherent | Elicitable | Tail-sensitive | Computational cost |
|---|---|---|---|---|---|
| Variance | E[(X - μ)²] | no | yes (w/ mean) | no | trivial |
| Std dev (volatility) | √variance | no | with mean | no | trivial |
| Semi-variance | E[((X - μ)⁻)²] | no | unclear | partial | trivial |
| MAD | E[|X - μ|] | no | no (with median, yes) | no | trivial |
| VaR_α | -inf{x : P(X ≤ x) ≥ 1-α} | no | yes | yes | medium |
| CVaR_α / ES_α | E[-X | X ≤ -VaR_α] | yes | jointly w/ VaR | yes | medium |
| Spectral risk | -∫_0^1 q_u(X) φ(u) du with φ ≥ 0 ↘ | yes | depends | tunable | medium |
| Distortion risk | -∫_0^1 q_u(X) dg(u) with g concave | yes | depends | tunable | medium |
| Entropic risk | (1/γ) log E[exp(-γX)] | no (convex only) | unclear | yes | trivial |
| Worst-case CVaR | sup_Q CVaR^Q | yes | no | yes | high (robust opt) |
| Maximum drawdown | max_t (cummax X_s - X_t) | no | no (path-dep) | yes | trivial in batch |
| Conditional drawdown | mean drawdown beyond percentile | yes (Chekhlov) | unclear | yes | medium |
| Pain index | mean | drawdown| | no | unclear | yes |
| Ulcer index | √mean(drawdown²) | no | unclear | yes | trivial |
6. Performance-and-risk ratios
| Ratio | Numerator | Denominator | Sensitivity |
|---|---|---|---|
| Sharpe | excess return | std dev | dispersion |
| Sortino | excess return | downside std dev | downside dispersion |
| Treynor | excess return | beta | systematic risk |
| Jensen alpha | actual - CAPM expected | n/a (absolute) | factor-mispricing |
| M-squared | leveraged Sharpe at market vol | std dev | market-equivalent return |
| Information Ratio (IR) | active return | tracking error | active management |
| Calmar | annualized return | max drawdown | tail dispersion |
| MAR | compound annual growth | max drawdown | hedge-fund standard |
| Sterling | return | average top-N drawdowns | tail (smoothed) |
| Burke | return | sqrt sum of drawdown² | tail (smoothed) |
| Pain ratio | return | pain index | path-dep |
| Ulcer Performance Index (UPI) | excess return | ulcer index | path-dep |
| K-ratio | slope of cumulative log-return vs time / std error | linearity of equity curve | volatility of slope |
| Omega | ∫(1-F(x))dx above threshold / ∫F(x)dx below | distributional | full distribution |
| Kappa-n | excess return / lower partial moment | downside | downside (general n) |
| Modified Sharpe | Sharpe w/ Cornish-Fisher VaR | parametric quantile | skew + kurtosis |
| Probabilistic Sharpe Ratio (Bailey-López de Prado 2012) | accounts for skew/kurt + sample size | dispersion w/ stats | uncertainty in Sharpe |
| When | Use |
|---|---|
| Long-only equity, sample period > 5y | Sharpe |
| Asymmetric return (options, hedge funds w/ short vol) | Sortino, Calmar, MAR |
| Active manager vs benchmark | Information Ratio |
| Tracking strategy w/ drawdown discipline | Calmar, MAR |
| Time-varying or vol-targeting fund | UPI, Pain |
| Smoothness of equity curve | K-ratio |
| Survivorship + small samples | Probabilistic Sharpe, Deflated Sharpe |
| Theoretical evaluation of distribution | Omega, Kappa |
7. VaR vs ES — the practical comparison
| Property | VaR_α | ES_α |
|---|---|---|
| Definition | quantile at level α | average loss conditional on > VaR |
| Coherent | NO | YES |
| Elicitable | YES | NO (alone); YES jointly with VaR |
| Reports a number | yes | yes |
| Captures tail beyond threshold | NO | YES |
| Basel III status (post-FRTB) | replaced by ES at 97.5% (in IMA) | the new standard |
| Solvency II (insurance) | VaR 99.5% over 1Y | considered ES; III may adopt |
| Common quantile | 95%, 99%, 99.5%, 99.9% | 97.5% in FRTB |
| 1-day vs 10-day VaR | scale by √10 (Gaussian assumption — sometimes wrong) | similar |
| Backtesting | binary breach test (Kupiec, Christoffersen) | Acerbi-Szekely 2014, Fissler-Ziegel 2016 |
The 2017 Basel III FRTB switch from VaR to ES at 97.5% was the single biggest regulatory risk-measure change in 20 years. The implementation date was repeatedly delayed; phased adoption is happening 2024–2028 across jurisdictions.
8. Three ways to compute VaR / ES
| Method | Compute | Assumptions | Strengths | Weaknesses |
|---|---|---|---|---|
| Parametric (variance-covariance, Riskmetrics 1996) | -μ + σ z_α | Normal/Student-t returns | trivial, analytic Greeks | tail thin (Normal), fragile under regime shift |
| Historical simulation | empirical quantile of N-day returns | data IID + stationary | model-free, captures fat tails | window-dependent, doesn’t extrapolate, ages out |
| Monte Carlo | simulate from model + take quantile | model-dependent | flexible, handles nonlinear (options) | expensive, model risk |
| Filtered historical (Barone-Adesi-Engle 2008) | historical residuals scaled by GARCH vol | GARCH + filter | adapts to volatility regime | GARCH misspec |
| Extreme Value Theory (EVT) | GPD / GEV fit to tail | tail follows EVT | rigorous tail extrapolation | requires careful threshold selection |
| Copula-based (Sklar + Gaussian/Student-t/Archimedean) | marginals + dependence structure | choice of copula | flexible dependence | copula misspecification |
9. Drawdown measures — path-dependent risk
| Measure | Definition | Use |
|---|---|---|
| Max drawdown (MDD) | max_t (peak - X_t) / peak | hedge fund reporting; CTA |
| Time under water | longest duration X_t < cummax | investor-experience proxy |
| Conditional Drawdown at Risk (CDaR, Chekhlov-Uryasev-Zabarankin 2005) | average of worst-α drawdowns | coherent path measure |
| Pain index | mean of | drawdown |
| Ulcer index (Martin 1987) | √mean drawdown² | strategy comparison |
| Sterling | n-period average drawdown | hedge fund (legacy) |
| Burke | sqrt sum drawdown² | hedge fund (legacy) |
| Drawdown ratio | return / MDD | Calmar / MAR variants |
Drawdown is the investor-experience measure — drawdowns trigger redemptions, fire sales, career-ending mistakes. Calmar / MAR ratios are standard in CTA / hedge-fund reporting; institutional allocators read drawdowns before Sharpe.
10. Factor + systematic risk
| Measure | Formula | Use |
|---|---|---|
| Beta (CAPM) | cov(R, R_m) / var(R_m) | systematic risk per unit of market |
| Downside beta | beta conditional on R_m < threshold | bear-market-only beta |
| Idiosyncratic vol | residual std dev of CAPM | active-management diversifiable |
| Tracking error (TE) | std dev of (R - R_benchmark) | active deviation |
| Active share (Cremers-Petajisto 2009) | half of | w - w_benchmark |
| Factor exposures (Fama-French / Carhart / HXZ) | β to factors | factor decomposition |
| Risk contribution | w_i × ∂σ_p / ∂w_i | per-position risk allocation |
| Marginal VaR / component VaR | ∂VaR_p / ∂w_i × w_i | per-position VaR allocation |
| Conditional risk attribution (Tasche 2002, Litterman) | similar to MVaR but for ES | per-position ES allocation |
11. Regulatory frameworks
| Regime | Standard measure | Notes |
|---|---|---|
| Basel III (banks, market risk pre-2024) | VaR 99% 10-day | scaling factor of 3 (or 4 for poor backtesting) |
| Basel III FRTB (banks, market risk post-2024 phased) | ES 97.5% | Standardized Approach (SA) or Internal Models (IMA) |
| Basel III credit | various; PD × LGD × EAD framework + stress | Risk-Weighted Assets (RWA) |
| CCAR / DFAST (US Fed, banks) | Severely-adverse scenario stress | top-down vs bottom-up |
| EU Solvency II (insurance, 2016+) | VaR 99.5% 1-year | Standard Formula or Internal Model; ORSA |
| EU Solvency III (consultation 2024+) | considering ES | not yet adopted |
| ICS (Insurance Capital Standard, IAIS) | VaR 99.5% 1-year | global insurance |
| IFRS 17 (insurance reporting) | risk adjustment for non-financial risk | actuarial CoC + percentile + cost-of-capital |
| SEC Form PF (private funds) | concentration + leverage + liquidity | quarterly / annual report |
| AIFMD (EU alternatives) | leverage + concentration | reporting + manager remuneration |
| MiFID II / MIFIR | best execution + reporting | not a risk-measure regime |
| EMIR (EU OTC) | central clearing + reporting | counterparty risk |
| Dodd-Frank Title VII (US OTC) | central clearing + reporting | counterparty risk |
| CECL / IFRS 9 (credit, accounting) | lifetime expected loss | replaces incurred-loss model |
The Basel III FRTB regime is the most material change to bank market-risk capital in 30 years. ES at 97.5% (one-tailed = approximately 99% one-tailed of VaR), liquidity horizons per risk factor, P&L attribution test (fail = mandatory standardized approach), backtesting of both VaR and ES via Acerbi-Szekely.
12. Insurance + actuarial — separate world
| Measure | Use |
|---|---|
| Probable Maximum Loss (PML) | catastrophe insurance, single-event |
| 1-in-N return period | catastrophe (1-in-200, 1-in-250, 1-in-1000) |
| ULAE / ALAE | Unallocated / Allocated Loss Adjustment Expenses |
| Loss ratio | losses / earned premium |
| Combined ratio | losses + expenses / earned premium |
| Reserve risk | runoff uncertainty of claim reserves |
| Risk margin / risk adjustment | IFRS 17 CoC or quantile-based |
| Tail VaR / TVaR | same as ES; common in actuarial |
| Required capital (SCR) | Solvency II at 99.5% 1Y |
| MCR (Minimum Capital Requirement) | Solvency II floor |
See insurance-and-actuarial for the full insurance stack.
13. Decision tree — pick a risk measure
What's the regime?
├─ Bank market risk (regulated)
│ → Basel III FRTB: ES 97.5% per liquidity horizon
│ → Backtest via Acerbi-Szekely (+ joint VaR-ES Fissler-Ziegel)
│ → SA fallback if IMA fails P&L attribution test
├─ Bank credit risk (regulated)
│ → Basel III: PD × LGD × EAD framework + stress
│ → IFRS 9 / CECL: lifetime expected loss
├─ Insurance / actuarial (regulated)
│ → Solvency II / ICS: VaR 99.5% over 1 year
│ → IFRS 17 risk adjustment: CoC or quantile
├─ Hedge fund / CTA reporting
│ → Max drawdown, Calmar, MAR (track record-friendly)
│ → Sharpe, Sortino, Information Ratio
│ → Pain / Ulcer for smoothness
├─ Long-only mutual fund
│ → Sharpe, IR vs benchmark, tracking error, active share
│ → Volatility for ICR / SDR labeling
├─ Pension fund / endowment
│ → ALM-aware: liability-relative; sometimes 1-in-20 / 1-in-100 stress
│ → Surplus volatility, funding-ratio drawdown
├─ Retail / private wealth
│ → Max drawdown + Calmar (intuitive)
│ → Sharpe (familiar)
│ → Capital protection probability (P(loss > 10%))
├─ Quant strategy backtest
│ → Sharpe + Calmar + MDD + TUW + DDR
│ → Probabilistic Sharpe (small sample bias-aware)
│ → Deflated Sharpe (multiple-testing-aware, Bailey-López de Prado 2014)
├─ Market making / dealer book
│ → VaR + ES + xVA aggregated
│ → Stressed VaR (Basel III stressed scenarios)
│ → Liquidity-adjusted VaR (LVaR)
├─ Options portfolio
│ → ES + Greek Greeks (delta-gamma-vega VaR)
│ → Stress scenarios (1987-like, 2020-like, 2022-like)
└─ Catastrophe / climate risk
→ PML at 1-in-N return period
→ Per-peril ELT (event loss table)
14. Anti-patterns
- Reporting VaR without ES or scenarios — VaR is not coherent and gives no tail info.
- Backtesting ES alone (without joint VaR) — Gneiting non-elicitability; use Acerbi-Szekely or Fissler-Ziegel joint tests.
- Scaling 1-day VaR to 10-day by √10 unconditionally — assumes Normal + IID; can be wrong by 50% in fat-tailed regimes.
- Parametric VaR for options book — non-linear payoffs need full revaluation, not delta-gamma approximation past mild moves.
- Sharpe ratio for short-vol or insurance strategy — left-skewed returns make Sharpe deceptive; use Calmar / Sortino.
- Maximum drawdown from monthly data when daily is available — underestimates MDD by ~30-50%.
- Reporting ES without specifying “above VaR” vs “above zero” — conventions vary.
- Comparing Sharpe across strategies without horizon adjustment — annualization assumption matters.
- CDaR / drawdown-based risk for high-frequency strategies — drawdowns occur within seconds and may be irrelevant to investor experience.
- Using historical VaR with 1-year window after regime change — window is too short to reflect new regime, too long to forget the old.
15. The 2024–2026 frontier
- Basel III FRTB fully phased in by 2026 — banks must run ES daily; P&L attribution tests; standardized fallback. Material capital impact (10-30% RWA increase in many cases).
- Solvency III consultation (EIOPA 2024) — possible ES adoption, climate risk extension.
- EU Sustainable Finance Disclosure Regulation (SFDR) + EU Taxonomy — ESG + climate VaR-style reporting.
- Climate VaR — physical + transition risk integrated into stress; PRA SS3/19 (UK) and ECB climate stress tests.
- Liquidity-adjusted VaR / ES — Basel III FRTB liquidity horizons (10 days for FX, 60+ for credit illiquid).
- Acerbi-Szekely 2014 ES backtests — three direct ES tests now standard in regulatory filings.
- Differentiable VaR / ES (Hong-Hu-Liu 2014, Glasserman gradients) — JAX/PyTorch back-prop through MC; gradient-aware capital optimization.
- Conformal prediction for VaR (Bates-Angelopoulos-Vovk frontier) — distribution-free finite-sample coverage for VaR estimates.
- Risk parity + tail risk parity (Spitznagel-Universa, Roncalli AQR + Edhec) — TR parity allocates to tail-equivalent risk units.
- Climate stress under NGFS scenarios — 2°C, 1.5°C, disorderly transition.
- AI-driven risk attribution — transformer-based scenario generation, agent-based market simulation.
Adjacent
- Portfolio construction — portfolio-construction-and-risk-deep for how risk measures feed into optimization (mean-variance, risk parity, robust opt, hierarchical risk parity).
- Options + Greeks — options-pricing-deep for the input to Greek VaR.
- Derivatives general — derivatives-and-quant-finance.
- Structured products — structured-products-deep for the tail-risk loaded payoffs that motivate ES.
- Market making — market-making-and-liquidity-provision-deep for inventory-risk measures.
- Insurance — insurance-and-actuarial for Solvency II + ICS + IFRS 17.
- Fixed income — fixed-income-deep for credit-risk metrics, DV01, key-rate duration.
- Pricing models — _compare_pricing-models for the model inputs that VaR/ES consume.
- Probability frameworks — _compare_probability-frameworks for the Bayesian vs frequentist views of risk estimation.
- Probability distributions — probability-distributions for the parametric families used.
- Copulas + dependence — copulas-and-dependence for multivariate tail dependence underlying portfolio VaR.
When to pick what
The fastest narrowing: regulated bank → Basel III FRTB ES at 97.5%; regulated insurer → Solvency II VaR 99.5% (or ES if III adopts); hedge fund / CTA → Sharpe + Calmar + MDD + MAR; long-only fund → Sharpe + IR + tracking error; endowment → ALM-aware surplus risk + drawdown; options book → ES + scenario stress; catastrophe → PML at 1-in-200 / 1-in-1000. The single biggest practical lesson from the 1998 LTCM, 2008 GFC, 2020 COVID, and 2022 yen-carry episodes is VaR is insufficient — backstop with stress scenarios + ES + liquidity-adjusted measures. Modern risk reporting layers: headline measure (ES or VaR per regime), stress scenarios (historical + hypothetical), liquidity horizon adjustment, path-dependent drawdown view, and factor decomposition showing which factors drive the risk.