Value at Risk

Every VaR statement is meaningless without three specifications: the loss amount, the time horizon, and the probability. Change any one and the number changes. A firm reporting a one-day 1% VaR will show a larger figure than the same portfolio's one-day 5% VaR, because a more extreme tail is being cut off. Likewise, lengthening the horizon from one day to ten days raises the figure, since losses accumulate over time.

The parametric route makes this explicit. Estimate the portfolio's expected return and standard deviation over the chosen horizon, assume returns are normally distributed, then subtract from the expected return a multiple of the standard deviation corresponding to the chosen tail: roughly 1.65 standard deviations for a 5% probability and roughly 2.33 for 1%. Multiply the resulting return threshold by portfolio value to convert it into a currency loss. Annual inputs are scaled down by dividing expected return by the number of trading days and the standard deviation by the square root of that number.

The parametric or variance-covariance approach is fast and transparent but imposes a distributional assumption. Because financial returns exhibit fat tails and negative skew, and because options and other convex positions have non-linear payoffs, the normal assumption tends to understate genuine tail loss. Analysts sometimes substitute a distribution with heavier tails to compensate.

Historical simulation imposes no distribution. It reprices the current portfolio using actual returns observed over a lookback window and reads the loss threshold directly from the ordered outcomes. Its weakness is that it treats the past as an exhaustive catalogue of the possible, and it is sensitive to window length: a calm sample produces a comfortable number.

Monte Carlo simulation generates large numbers of return paths from assumptions the analyst specifies, which allows complex instruments, path dependency and non-normal distributions to be handled. The cost is computational intensity and complete dependence on the input assumptions chosen.

VaR became the standard market-risk metric because it collapses heterogeneous exposures into one currency figure that senior management and boards can compare across desks. Trading operations allocate risk budgets in VaR terms, set position limits against them, and monitor utilisation daily. Investment managers use it to communicate downside exposure to clients in language less abstract than standard deviation.

Bank capital regulation embedded VaR for market risk, typically at a high confidence level over a multi-day horizon, with backtesting requirements: if actual losses breach the reported threshold more often than the probability level implies, the model is penalised or must be revised. Subsequent regulatory reform shifted the emphasis toward expected shortfall, precisely because VaR says nothing about the size of losses beyond the cut-off. Published annual reports of large dealers commonly disclose average, high and low VaR by risk category.

Conditional VaR, also called expected shortfall or expected tail loss, answers the question VaR leaves open by averaging the losses that occur beyond the threshold. Incremental VaR measures how total portfolio VaR changes when a position is added or resized; marginal VaR captures the sensitivity of portfolio VaR to a small change in a holding. Ex ante tracking error applies the same logic relative to a benchmark. None of these dispenses with stress testing and scenario analysis, which deliberately examine specific severe outcomes rather than probability-weighted ones.

The core weaknesses are estimation sensitivity, false comfort in a single number, and the tendency of correlations to converge toward one in crises, which breaks the diversification assumed in the covariance inputs. At Level I, expect to compute a parametric VaR from expected return and standard deviation, convert between horizons, state a VaR result in correct language, distinguish the three estimation methods, and identify VaR's advantages and limitations relative to its extensions.