Value-at-Risk for FX: Parametric, Historical Simulation and Monte Carlo

Value-at-Risk (VaR) estimates a loss threshold for a portfolio over a stated horizon and confidence level. The number is only meaningful when its methodology, horizon, confidence level, data window and model assumptions are clear.

For FX portfolios, three approaches dominate the practical toolkit: parametric VaR, historical simulation and Monte Carlo simulation. They answer the same broad question in different ways. Parametric VaR is fast and transparent but assumption-heavy; historical simulation uses observed market moves but is constrained by the history selected; Monte Carlo can represent complex portfolios and changing dependence structures but is computationally heavier and highly sensitive to model design.

No one method removes tail risk. A robust framework uses methodological diversity, backtesting and stress analysis to identify where a convenient daily risk number stops being reliable.

A 99% one-day VaR is a threshold statement. Under the model and data used, losses are estimated to exceed that threshold on roughly 1% of trading days.

VaR does not tell you how large losses may be once the threshold is breached. It also does not guarantee that observed market behaviour will continue to match the assumptions used in the model.

For that reason, a VaR number should never be presented without:

• confidence level;
• holding period;
• valuation date;
• methodology;
• lookback or calibration window;
• treatment of volatility and correlation;
• treatment of non-linear positions;
• model version.

Parametric, or variance-covariance, VaR reduces the portfolio loss estimate to a statistical distribution and its parameters.

For a normally distributed return with mean $\mu$, volatility $\sigma$ and confidence level $\alpha$:

where $\Phi^{-1}$ is the inverse cumulative normal distribution function.

For short holding periods, implementations often treat the expected return as approximately zero, leaving volatility as the dominant input.

Why firms use it

Parametric VaR is attractive because it is:

• fast enough for frequent monitoring;
• easy to explain;
• easy to decompose;
• straightforward to compare with an operating limit.

Those strengths are also why it is easy to over-trust.

Principal model risks

The method is vulnerable when:

• return distributions have fat tails;
• volatility clusters rather than remaining constant;
• correlations change during stress;
• large option positions introduce non-linear P&L;
• square-root-of-time scaling is applied where independence assumptions do not hold.

A statistical refinement such as a Cornish-Fisher adjustment can incorporate skewness and kurtosis, but it does not remove the need to challenge the underlying distributional assumptions.

Illustrative calculation

Illustrative Example / Hypothetical Scenario

Illustrative Example / Hypothetical Scenario.

Assume a USD 500 million FX position with daily volatility of 1.2% and zero expected drift. At 99% confidence, using a normal quantile of approximately 2.33:

If daily returns are assumed independent and identically distributed, a ten-day estimate may be scaled using $\sqrt{10}$:

The calculation is simple. The difficult question is whether the independence, distribution and volatility assumptions remain credible for the portfolio and market regime being measured.

Historical simulation re-prices the current portfolio using observed historical changes in risk factors. It does not require a normal distribution.

For each observation in the lookback window:

1. apply the historical market move to the current portfolio;
2. revalue the portfolio;
3. record the hypothetical profit or loss;
4. sort the resulting loss distribution;
5. select the percentile corresponding to the chosen confidence level.

This approach naturally captures non-linear portfolio behaviour if the revaluation process itself captures that behaviour.

Strengths

Historical simulation:

• uses observed joint market moves;
• avoids imposing a normal distribution;
• can capture option convexity through full revaluation;
• is easy to explain to governance committees.

Weaknesses

Its central limitation is the history selected.

A severe event that is absent from the lookback window is absent from the risk estimate. A long window may contain useful crisis observations but react slowly to current conditions. A short window reacts faster but may omit rare events.

Weighted historical simulation changes the importance assigned to observations so that recent market moves receive more weight.

One exponential form uses:

where $\lambda$ controls the rate of decay.

The objective is to make the estimate respond more quickly when volatility clusters or the market enters a new regime.

The trade-off is important: faster response to current conditions can reduce the influence of older tail events. Weighting is therefore a calibration choice, not a substitute for stress testing.

Monte Carlo VaR generates a large number of simulated joint movements in market risk factors and revalues the portfolio under each path.

The process is:

1. define the stochastic behaviour of the relevant risk factors;
2. specify volatility and dependence assumptions;
3. simulate many joint scenarios;
4. revalue the portfolio under each scenario;
5. construct the loss distribution;
6. select the required percentile.

For correlated normally distributed factors, a covariance matrix may be decomposed using Cholesky decomposition:

and the resulting factor can be used to transform independent random variables into correlated simulations.

FX correlations can change materially between normal and stressed regimes. A Monte Carlo model using a stable historical correlation matrix can therefore produce a precise-looking result that is poorly suited to a crisis environment.

A more responsive framework can use:

• rolling correlation estimates;
• regime-dependent correlation assumptions;
• volatility models such as GARCH;
• dedicated stress scenarios outside the VaR engine.

The important governance point is that a simulated distribution is only as credible as the process that generates it.

A sensible choice depends on the portfolio.

A largely linear FX book may support frequent parametric monitoring, provided its limitations are visible. A material options book may require historical or Monte Carlo full revaluation. A portfolio exposed to regime-sensitive correlations needs explicit dependence controls regardless of the VaR engine.

The operating metric should also be separated from any regulatory capital metric. A one-day desk VaR limit and a regulatory market-risk capital calculation can coexist without being the same measure or sharing the same horizon.

A defensible VaR framework should retain:

• methodology and model version;
• risk-factor data and lookback window;
• volatility and correlation parameters;
• treatment of outliers;
• holding period and confidence level;
• valuation approach for non-linear instruments;
• model-change records;
• backtesting results;
• comparison with alternative methodologies;
• identified limitations and compensating controls.