Value-at-Risk

The one-line definition

Value-at-Risk is the loss quantile. For a random loss L over horizon h and a confidence level α ∈ (0, 1), VaR is the smallest threshold whose tail probability is at most 1 − α:

VaR_α(L) = inf { x : P(L > x) ≤ 1 − α }

Equivalently — and more memorably — it is the α-quantile of the loss distribution. When someone says "our 1-day 99% VaR is $10 million" they mean: with probability 0.99 the loss tomorrow will be no worse than $10M; with probability 0.01 it will be worse. The metric reduces a whole probability distribution to a single dollar number, which is why bank treasurers, board directors and regulators all use it as a daily-discussion currency. That reduction is also where the trouble lives.

Three numbers together define a VaR figure: the confidence α (typically 95%, 99% or 99.9%), the horizon h (1 day for desk-level trading limits, 10 days for Basel market-risk capital, 1 year for credit and economic capital), and the model used to estimate the loss distribution. A "VaR" quoted without all three is incomplete; a quoted VaR cannot be compared with another quoted VaR unless all three agree.

RiskMetrics, 1994 — how VaR took over the world

The concept was not new in 1994. Brokerage firms had been computing capital-at-risk numbers for decades, and the SEC's "net capital rule" of 1980 was already an explicit haircut-based VaR for broker-dealers. What made 1994 the watershed was that JP Morgan's Chairman Dennis Weatherstone asked his risk staff for a single one-page "4:15 report" delivered each afternoon summarising the firm's market risk for the next day. The internal product became RiskMetrics, and JP Morgan published the methodology and a dataset of variances and correlations free of charge in October 1994. Within a year every major bank had built a VaR system. The 1996 amendment to the Basel I Accord allowed banks to use internal VaR models — calibrated to a 10-day horizon at 99% confidence and multiplied by a regulatory scaling factor (typically 3) — to size trading-book capital. RiskMetrics had moved from JP Morgan management report to global banking standard in two years.

The three estimation methods

There are essentially three production approaches. Banks usually run two or three of them in parallel and reconcile.

Historical simulation

Take the last N daily price moves (Basel suggests at least 250 observations; many shops use 500–1000), apply each to today's portfolio, build the empirical distribution of hypothetical one-day P&Ls, and read off the α-th worst as VaR. The method is non-parametric — no normality assumption, no implicit Gaussian copula — and trivially explained to a board: "this is the worst day we'd have had in the last two years on today's book."

For each historical day t in {1..N}:
   r_t = (P_t - P_{t-1}) / P_{t-1}   for every risk factor
   L_t = -V(portfolio | shocked by r_t) + V(portfolio | today)
Sort L_t ascending.
VaR = L_t at index floor((1-α) N).        # e.g. floor(0.01 × 500) = 5th worst

Weakness: the window is the worst window of the past, not necessarily a worst plausible future. A calm 250 days produce a calm VaR, which is exactly what happened in 2004–2007. The fix is window weighting (give more weight to recent observations), Filtered Historical Simulation (rescale past returns by today's GARCH volatility), or simply forcing inclusion of a stress window (Basel 2.5's Stressed VaR).

Parametric / variance-covariance

Assume returns are jointly normal with mean vector μ and covariance Σ. The portfolio P&L is then approximately normal with mean μ_p = w^T μ and standard deviation σ_p = √(w^T Σ w). VaR is read off the inverse normal CDF:

VaR_α = -μ_p + σ_p · Z_α
       where Z_0.95 = 1.645,  Z_0.99 = 2.326,  Z_0.999 = 3.090

For most short-horizon problems the mean is much smaller than the standard deviation, so practitioners drop μ and write VaR ≈ σ · Z_α — what RiskMetrics's "delta-normal" approach does. The method is analytic, fast, and lets you decompose risk into per-factor contributions cleanly. Its weakness is the assumption it imposes: real returns have fatter tails than normal (kurtosis > 3, often well above 6 for individual stocks), and the assumption that pairwise correlations summarise dependence breaks down in panics.

Monte Carlo

Simulate many synthetic future paths of the risk factors — drawing from any specified distribution (normal, t with low degrees of freedom, GARCH, jump-diffusion, regime-switching) — re-price the portfolio under each scenario, and take the α-quantile of the resulting P&L distribution. Monte Carlo handles non-linear instruments (options, exotics) and arbitrary distributions; the cost is computation (10⁵–10⁶ scenarios for a stable tail estimate) and model risk (your VaR is only as good as your generator).

Method	Distributional assumption	Handles options	Compute	Main risk
Historical simulation	None (empirical)	Yes (full-revaluation)	O(N × instruments)	Window not representative of future
Parametric (delta-normal)	Joint normal	Approximately (linearisation)	Analytic, sub-second	Fat tails, correlation breakdown
Parametric (delta-gamma)	Joint normal, second-order	Better (Greeks)	Analytic	Still tied to normality
Monte Carlo	User-specified	Yes (any payoff)	O(M × instruments)	Generator misspecification

Worked example: a $100M equity book

Suppose a portfolio is long $100M of an equity-only book with an annualised volatility of σ_ann = 25% and negligible drift. To compute a 1-day 99% parametric VaR:

σ_daily = σ_ann / √252  = 0.25 / 15.87 ≈ 0.01575   (1.575% per day)

VaR_0.99 (1-day, %)   = Z_0.99 × σ_daily
                      = 2.326 × 0.01575
                      ≈ 0.0367   (3.67% of book)

VaR_0.99 (1-day, $)   = 0.0367 × $100M
                      ≈ $3.67 M

To scale to a 10-day horizon — the Basel market-risk standard — invoke the "square-root-of-time" rule (valid under i.i.d. normal increments):

VaR_0.99 (10-day) ≈ √10 × VaR_0.99 (1-day)
                  ≈ 3.16 × $3.67M
                  ≈ $11.6 M

Multiplied by the Basel scaling factor (typically 3, sometimes higher under the traffic-light backtest), regulatory market-risk capital on this book is roughly $35M. Note how sensitive the number is to σ_ann: if realised vol drops to 15%, the same calculation gives a $21M capital charge — a 40% reduction with no underlying change in position. That sensitivity is the procyclicality channel discussed below.

Coherent risk measures and where VaR falls short

In a 1999 paper that became the standard reference for risk theory, Philippe Artzner, Freddy Delbaen, Jean-Marc Eber and David Heath asked what axioms a sensible risk measure ρ should satisfy. They wrote down four.

• Monotonicity. If X always loses at least as much as Y, then ρ(X) ≥ ρ(Y). Worse losses, bigger number.
• Translation invariance. ρ(X + c) = ρ(X) − c. Adding c units of cash to a portfolio reduces its required capital by c.
• Positive homogeneity. ρ(λX) = λρ(X) for λ ≥ 0. Doubling the position doubles the risk.
• Subadditivity. ρ(X + Y) ≤ ρ(X) + ρ(Y). Merging two portfolios cannot increase total risk — the diversification axiom.

VaR satisfies the first three. It can violate the fourth. A famous textbook counterexample uses two highly skewed corporate bonds: each individually has a 99% VaR of zero (the rare default is in the 1% tail and so does not exceed VaR), but the combined two-bond portfolio has a positive 99% VaR because the union of the two tails crosses the 1% threshold. By the additivity axiom, diversifying made the measured risk worse — clearly the wrong direction. The defect appears whenever loss distributions are discrete, heavy-tailed, or have material point-masses (defaults, jumps, structured-credit cliffs). In the bulk of trading-book scenarios with continuous near-normal returns, VaR is empirically subadditive; the failure mode is exactly the regime where it matters most.

Expected Shortfall — the coherent successor

Expected Shortfall (ES), also called Conditional VaR (CVaR), Tail VaR, or Expected Tail Loss, fixes both of VaR's defects. It is defined as the expected loss given that loss exceeds VaR:

ES_α(L) = E[ L | L > VaR_α(L) ]
        = (1 / (1 − α)) ∫_α^1 VaR_u(L) du

By integrating across the entire tail, ES is sensitive to losses beyond the quantile and is therefore coherent: it satisfies all four axioms, including subadditivity. The Basel Committee's 2013 consultative paper "Fundamental Review of the Trading Book" (FRTB) proposed replacing 99% VaR with 97.5% ES; the standard was finalised in January 2016 with phased implementation. The 97.5% level was chosen because for a normal distribution it gives roughly the same numerical value as 99% VaR, smoothing the transition. For fat-tailed distributions the two diverge — which is precisely the point of the move.

When VaR has failed — LTCM 1998, 2008

The two clearest case studies are familiar. Long-Term Capital Management's VaR systems in mid-1998 implied an expected worst daily loss of roughly $35M; on 21 August 1998 the fund lost $553M in one day following Russia's debt default, a loss that under their Gaussian assumption was about 15 standard deviations from the mean — a probability so small that it should not occur in the lifetime of the universe. The lesson was not that LTCM ran the math badly; it was that the distributional assumption ruled out the regime that actually arrived. Markets do not produce independent draws from a stable normal distribution; they regime-switch.

In August 2007, Goldman Sachs CFO David Viniar told the Financial Times that the firm's quant funds were "experiencing 25-standard-deviation moves, several days in a row" — a quote that became the canonical critique of risk-model normality. The 2007–2009 crisis exposed three failure modes simultaneously, and they were the same three that had taken down LTCM nine years earlier.

• Fat tails. Real return distributions have far more probability in the extremes than a normal does. A 5σ daily move under a normal has probability about 3 × 10⁻⁷ — once in 13,500 years; in real equity returns 5σ moves occur every few years.
• Correlation breakdown. In calm markets, asset correlations cluster around their long-run averages. In a panic, almost everything correlates near 1: investors sell what they can, not what they want. Parametric VaR built on a 250-day correlation window dramatically understates the risk of a coordinated drawdown.
• Liquidity collapse. A one-day VaR is meaningful only if positions can actually be unwound at quoted prices over one day. In a panic, bid-ask spreads widen by 10× or more, depth disappears, and exit costs dwarf the modelled price move. Basel's response — incorporating "liquidity horizons" into FRTB — partially closes this gap.

The volatility paradox and procyclicality

VaR's most insidious property is that it falls in calm markets and spikes in stressed ones. When realised volatility is low, parametric and historical VaR both shrink. Lower measured risk frees regulatory and economic capital, lowers internal limit usage, and rewards leverage. Hyun Song Shin's work documented how the 2004–2007 build-up was driven precisely by this mechanism: low VaR encouraged bigger positions, which were unwound in 2008 at exactly the worst moment. The phenomenon — sometimes called the volatility paradox after a 2010 paper by Brunnermeier and Sannikov — is a structural property of any risk-sensitive capital regime, not just VaR. Basel III's countercyclical capital buffer is the supervisory antibody: regulators add capital requirements above the baseline when systemic credit growth is high, deliberately leaning against the procyclical wind.

Backtesting

A VaR model that is right would, by construction, see realised loss exceed the prior-day VaR on (1 − α) of days. For a 99% one-day model over a year of 250 trading days, that's about 2 or 3 exceptions. Basel's "traffic-light" framework categorises models:

Zone	Exceptions in 250 days	Capital multiplier	Interpretation
Green	0 – 4	3.0	Acceptable; no supervisory action
Yellow	5 – 9	3.4 – 3.85	Possible problem; investigate
Red	≥ 10	4.0+	Model is unreliable; may be withdrawn

Two formal statistical tests dominate the academic literature. Kupiec's Proportion-of-Failures (POF, 1995) tests whether the unconditional exception rate matches 1 − α. Christoffersen's conditional coverage test (1998) adds an independence component: exceptions should not cluster. A model with the right average exception rate but clustered exceptions is slow to react to volatility shifts and fails Christoffersen even when it passes Kupiec.

Variants and extensions

• Stressed VaR. Basel 2.5 (2009). Same calculation, but the historical window is fixed at a period of significant stress (typically 2007–2009). Market-risk capital became VaR + SVaR, breaking procyclicality with respect to recent calm.
• Incremental Risk Charge (IRC). Basel 2.5 capital add-on for default and migration risk in the trading book, capturing tail events that 10-day VaR misses.
• Expected Shortfall (Basel III / FRTB). 97.5% ES with liquidity-horizon scaling replaces 99% VaR for market-risk capital under FRTB. Phased into force from 2023.
• Conditional Drawdown at Risk (CDaR). Coherent measure on running drawdown; popular in hedge-fund risk reporting.
• Spectral risk measures. A family of coherent measures with user-specified weighting across the tail; ES is a special case (uniform weighting beyond the α-quantile).
• Entropic Value-at-Risk (EVaR). An upper bound on ES derived from the Chernoff inequality; coherent and analytically convenient.

Common pitfalls

• Quoting VaR without h and α. "Our VaR is $10M" is meaningless. The same portfolio can have a 1-day 95% VaR of $4M and a 10-day 99% VaR of $40M.
• Square-root-of-time scaling outside i.i.d. normal. The √h rule assumes independent normal increments. With autocorrelation, fat tails, or mean-reversion, it can over- or under-state multi-period VaR by a factor of two or more.
• Confusing VaR and worst-case loss. VaR is a quantile, not a maximum. A correctly calibrated 99% VaR will be breached on about 2.5 days a year. Treating a breach as a model failure rather than expected behaviour leads to overfitting.
• Adding desk VaRs. Because VaR is not subadditive in general, the firm VaR can exceed or fall below the sum of desk VaRs. Aggregation requires full-revaluation of the joint book, not arithmetic on subtotals.
• Ignoring options gamma. Delta-normal VaR linearises payoffs and badly understates risk for portfolios with material convexity. Delta-gamma or full Monte Carlo is required for option books.
• Trusting the calm. A 250-day historical VaR fit during a tranquil window is structurally blind to regime change. Stress VaR, scenario analysis and reverse stress testing exist precisely to plug this gap.