Value at Risk: Why the Model Choice Changes the Loss Number

Value at Risk is a statement about a loss percentile, a holding period and a portfolio. Its credibility depends on the data and modelling choices behind all three. Two systems can report different VaR numbers for the same book without either containing an arithmetic error: they may use different return definitions, volatility dynamics, distributional tails or revaluation methods. A useful comparison isolates those choices instead of ranking the models by whichever produces the preferred number.

REFERENCE · NO CALL

This report consolidates historical simulation exercises, a volatility-comparison workbook and teaching notes dated June 2024. The retained model settings and market observations are historical teaching material. They do not constitute current volatility estimates or forecasts for an identified market participant.

Define the loss before estimating its percentile

Let $L_{t+1}$ denote the portfolio’s loss over the next day, taken as positive when value falls. At confidence level $\alpha$, VaR is a quantile of its conditional distribution:

$$\operatorname{VaR}_{\alpha,t} =\inf\{\ell:\Pr(L_{t+1}\leq\ell\mid\mathcal{F}_t)\geq\alpha\}.$$

In words: Using the information available now, choose the smallest loss level whose probability of not being exceeded meets or exceeds the chosen confidence level.

The information set $\mathcal{F}_t$ contains what was available when the forecast was made. This timing restriction matters. A backtest using today’s realised volatility to improve yesterday’s prediction is no longer testing an implementable forecast.

A 99% one-day VaR is not a maximum potential loss. Losses beyond it remain possible and their severity is not specified by the percentile. Describing it as the maximum without this qualification is particularly misleading for option books, credit portfolios and instruments with jump exposure.

The portfolio definition must also be fixed. A static-book forecast cannot be cleanly judged against P&L that includes subsequent discretionary trades unless the comparison explicitly intends to include that activity. Frozen positions and controlled valuation snapshots make model comparison interpretable.

Historical simulation: preserve the scenario structure

In a simple historical simulation, the analyst calculates historical factor changes, applies each joint change to the current portfolio and sorts the resulting losses. For a linear single-asset position of current value $A$, a simple-return scenario $r_i$ produces approximate loss

$$L_i=-Ar_i.$$

The source exercise uses a position value of 1,000,000 currency units. A retained return of approximately −2.0696756% therefore produces a loss of about 20,696.76 units. This is an arithmetic teaching example; the value is not a confidential position amount.

For a multi-factor portfolio, the scenario is a row of simultaneous market changes. Independently sorting each factor destroys their historical dependence. A model can preserve individual distributions yet invent combinations that did not occur—or remove combinations that mattered—if it mixes dates during the scenario construction.

Nonlinear positions require repricing under the scenario. Applying a stock return to the market value of an option treats the option like stock and is not a valid substitute. Delta approximation may be useful for small shocks, but its scope should be stated and compared with full revaluation.

Sampling conventions are economically visible

A 250-observation window gives a 99% tail only 2.5 observation-equivalents. Quantile interpolation and boundary conventions can therefore change the reported number materially. The calculation should record whether it uses a nearest-rank estimator, linear interpolation or another explicit rule.

The same issue appears in Expected Shortfall. At 97.5%, a 250-day sample contains 6.25 observation-equivalents in the tail. Averaging seven complete observations is a rounded-tail convention, not the unique empirical definition. Different conventions can be legitimate for different uses, but they should not be switched during model comparison.

Missing prices require another explicit decision. Forward-filling can create artificial zero returns and then concentrate several days’ movement into the next available observation. Removing dates can break alignment with other factors. Neither operation is harmless merely because it makes the spreadsheet easier to calculate.

An equally weighted window also has an abrupt memory boundary. When an extreme old return leaves the sample, its influence disappears at once. This can change the estimate without any unusual new market movement. The direction depends on the departing observation and the estimator; it should be diagnosed from the actual window rather than described with a universal direction.

Parametric normal VaR is an exposure model plus a distribution

For a linear portfolio whose conditional P&L is approximated as normal, let $\mu_L$ be expected loss and $\sigma_L$ its standard deviation. Then

$$\operatorname{VaR}_{\alpha}=\mu_L+z_{\alpha}\sigma_L,$$

where $z_{\alpha}$ is the standard-normal quantile. For several factors with signed exposure vector $\mathbf{s}$ and factor covariance matrix $\Sigma$,

$$\sigma_L^2=\mathbf{s}^{\mathsf T}\Sigma\mathbf{s}.$$

The covariance matrix describes joint factor movements; the sensitivities translate them into money. Changing either changes the answer. A sensitivity mapping error cannot be cured simply by selecting a heavier-tailed distribution. A sound factor mapping cannot compensate for a volatility estimate that reacts too slowly to a regime change.

The normal model is attractive because its assumptions are visible and its calculations are easy to replicate. Its limitations become serious when skew, fat tails, jumps or nonlinear payoffs dominate the chosen horizon. It is therefore a useful benchmark even when it is not the final production model.

Heavier tails require correct scaling

The historical workbook compares normal tails with a Student-t specification using three degrees of freedom. That setting produces substantially heavier tails than normality. It also requires a scaling check: a raw Student-t variable with $\nu>2$ degrees of freedom has variance $\nu/(\nu-2)$. To give it unit variance, multiply it by

$$\sqrt{\frac{\nu-2}{\nu}}.$$

Without this adjustment, a comparison may attribute a larger VaR to tail shape when it has also changed the variance. At three degrees of freedom, that distinction is large. The parameter is a retained workbook setting, not evidence that three degrees of freedom is appropriate for another portfolio or period.

Fitting the tail parameter also introduces estimation uncertainty. A small sample can support very different values. A validation report should compare plausible specifications and show the impact on the tail measure, rather than presenting one fitted number as known precisely.

Conditional volatility changes the forecast through time

The workbook uses an exponentially weighted moving average with decay parameter $\lambda=0.94$. A common zero-mean form is

$$\sigma_t^2=\lambda\sigma_{t-1}^2+(1-\lambda)r_{t-1}^2,$$

where $r_{t-1}$ is the previous return and $\sigma_t^2$ is the next conditional variance estimate. Recent squared returns have more weight, while older ones decay smoothly. The chosen value is a historical teaching convention, not an optimised universal parameter.

A GARCH(1,1) formulation adds a long-run component:

$$\sigma_t^2=\omega+a r_{t-1}^2+b\sigma_{t-1}^2.$$

Here $\omega$ is a constant, $a$ measures reaction to the latest squared return and $b$ carries forward past variance. Under the usual positive-parameter stationary setting, $a+b<1$ gives finite unconditional variance. A workbook label containing one persistence or decay number is insufficient to reconstruct all three parameters; the formulas and estimation output must be inspected.

Asymmetric volatility specifications can allow negative and positive returns of equal magnitude to have different effects. That can matter for equity portfolios, but additional flexibility should earn its place through out-of-sample evidence and stable estimation.

Compare models on a controlled experiment

Change one modelling dimension at a time in a controlled comparison

Backtesting should retain both exception frequency and severity. A model that produces fewer breaches only because it is uniformly inflated may waste risk capacity. A model with acceptable frequency can still understate the average loss beyond the threshold. Regime analysis helps reveal whether the apparent full-sample success consists of overprediction in calm periods offsetting underprediction in stress.

Forecast comparisons should also consider operational robustness. A slightly better statistical fit is not useful if calibration fails unpredictably, missing data silently changes the window or a production rerun cannot reproduce the previous number. The decision is about an implementable forecasting process, not a formula in isolation.

The useful deliverable is a decomposition

A strong VaR comparison explains the difference between models in stages: factor coverage, revaluation, volatility dynamics, dependence, tail shape and estimation uncertainty. It preserves the historical settings as evidence while identifying where further calibration would be needed for another book.

The resulting number remains a percentile forecast with a stated scope. Its limitations should lead to complementary controls—tail-severity measures, P&L explanation and stress tests—rather than to a claim that one chosen model has eliminated uncertainty.