Value at Risk: Measuring How Much You Could Lose

Any smart investor, whether a billion-dollar bank or a retiree with a savings account, should know the answer to one question: how much could I lose? Chapter 19 introduces Value at Risk (VaR), the industry standard for answering exactly that.

Wilmott opens with a sharp observation: there have been plenty of well-publicized cases where institutions had absolutely no clue what their exotic derivatives could cost them. VaR was born from the desire for more transparency.

What VaR Actually Means

The definition is straightforward:

VaR is the maximum loss on a portfolio over a given time horizon, at a given level of confidence.

For example: “Over the next week, there is a 95% probability that we will lose no more than $10 million.” That $10 million is your VaR.

A few important details. The confidence level is typically 95%, 97.5%, or 99%. The time horizon might be one day for traders or months for portfolio managers. It should represent how long it takes to sell your positions in an orderly way, without crashing the market.

VaR assumes normal market conditions. Crashes and panics are treated separately. So VaR tells you what to expect during the regular day-to-day business. The truly catastrophic scenarios need different tools.

VaR for a Single Asset

Let us start simple. You hold a quantity $\Delta$ of a stock with price S and volatility $\sigma$. You want to know with 99% confidence what is the most you can lose over one week.

The stock return is assumed Normal with mean zero (reasonable for short time horizons) and standard deviation $\sigma S \sqrt{\delta t}$, where $\delta t$ is the time period. For a week, $\delta t = 1/52$ of a year.

Now you need the 99th percentile of the Normal distribution. From the standard table:

So at 99% confidence, VaR for a single stock position is:

$$\text{VaR} = 2.33 \times \sigma \times \Delta S \times \sqrt{\delta t}$$

More generally, for confidence level c:

$$\text{VaR} = \alpha(c) \times \sigma \times \Delta S \times \sqrt{\delta t}$$

where $\alpha(c)$ is the inverse cumulative Normal function.

Why zero mean? Because over short time horizons, the standard deviation (which scales with $\sqrt{t}$) dominates the drift (which scales with t). Over longer horizons, you should add back the drift term $\mu \delta t$. And importantly, use the real drift rate, not the risk-neutral one. VaR is about real-world losses, not derivatives pricing.

VaR for a Portfolio

With multiple assets, you need volatilities and correlations. The portfolio VaR is:

$$\text{VaR} = \alpha(c) \sqrt{\delta t} \sqrt{\sum_i \sum_j \Delta_i S_i \Delta_j S_j \sigma_i \sigma_j \rho_{ij}}$$

This looks like the portfolio standard deviation from Chapter 18, just with a confidence multiplier in front. Same math, different application.

The obvious criticisms: returns are not actually Normal, volatilities and correlations are hard to measure, and this formula ignores derivatives entirely. Wilmott addresses the last point next.

VaR for Derivatives: The Delta Problem

Options and other derivatives add a tricky wrinkle. Even if the underlying stock moves follow a Normal distribution, the option value does not change linearly with the stock. A call option is not just “stock times a constant.” The relationship is curved.

Delta Approximation

For small market moves, you can approximate. The sensitivity of an option to the underlying is its delta ($\Delta$). If the stock has standard deviation $\sigma S \sqrt{\delta t}$, then the option position has standard deviation approximately:

$$\Delta \times \sigma S \sqrt{\delta t}$$

Replace stock positions with delta-equivalent positions, and use the same portfolio VaR formula. Simple, fast, works okay for small moves.

Which Volatility to Use?

Here is a subtle point that Wilmott highlights. The delta depends on implied volatility (how the market prices the option). But the stock movement depends on actual (realized) volatility. If implied and actual volatilities differ, you need both: implied for the delta, actual for the stock movement. Using the wrong one gives the wrong VaR.

Delta-Gamma Approximation

For larger moves, the delta-only approach fails. You need the second-order term, gamma ($\Gamma$). The change in option value becomes:

$$\delta V \approx \Delta \cdot \delta S + \frac{1}{2}\Gamma \cdot (\delta S)^2$$

This introduces a quadratic term in the random variable. The distribution of $\delta V$ is no longer Normal, even if $\delta S$ is. Wilmott shows that the resulting distribution gets skewed.

Here is the practical insight: positive gamma is good, negative gamma is bad. With positive gamma, your downside is limited. The quadratic term works in your favor when things go wrong. With negative gamma, the opposite happens. Your upside is capped, but the downside can be much worse than the delta approximation suggests.

The question to ask: does the critical value (where the quadratic turns around) fall within the tail you care about? If yes, the delta-only approximation will badly underestimate your risk.

When All Else Fails: Simulation

For seriously nonlinear portfolios, neither delta nor delta-gamma works well. The brute force solution: simulate thousands of random scenarios for the underlying, reprice everything using full valuation models, and build up a distribution of portfolio changes directly.

This is accurate but slow. If each scenario requires solving a multi-factor PDE, you could be waiting a long time for your VaR number.

Fixed-Income VaR

For bonds and other fixed-income instruments, the trick is to use yield to maturity as the random variable instead of price. Yields are correlated across instruments. The sensitivity of bond price to yield is duration (first order) and convexity (second order). So duration plays the role of delta, and convexity plays the role of gamma. Same framework, different vocabulary.

Simulation Methods

Monte Carlo

Generate random numbers from a Normal distribution. For each scenario, compute what happens to your portfolio. Do this tens of thousands of times to build a distribution of outcomes. Read off the tail to get VaR.

The important detail: use real returns, not risk-neutral. We are modeling what actually happens, not pricing derivatives.

Bootstrapping

Instead of generating artificial random numbers, use actual historical data. Take four years of daily returns for all your assets. Number each day from 1 to 1000. Draw random numbers, pick the corresponding day’s returns, apply them to today’s portfolio. Repeat many times to build up your distribution.

The clever trick: keep all asset changes from the same date together. If on March 15, 2024, stocks fell and bonds rose, you use that whole vector as one scenario. This automatically preserves the real-world correlations between assets.

Advantages: captures real correlations, real fat tails, real non-Normal behavior. No need to estimate correlation matrices.

Disadvantage: your historical data might reflect economic conditions that no longer apply. The past four years might not predict the next four years.

VaR as Performance Measurement

VaR is not just for risk managers. It can measure trader performance too. Traditionally, traders get bonuses based on profit alone. This encourages reckless risk-taking: imagine a coin flip where you keep a percentage of the win but do not pay for the loss (the bank absorbs it). How much would you bet? Everything.

A better approach uses the Sharpe ratio (profit over risk) or profit over VaR. Reward traders who make money efficiently, not just traders who make money by betting the farm.

Extreme Value Theory: When Normal Is Not Enough

This is where the chapter gets interesting. Everything above assumes Normal distributions. But real markets have fat tails. Crashes happen way more often than a Normal distribution predicts. The standard VaR calculation using Normal assumptions systematically underestimates tail risk.

Extreme Value Theory (EVT) focuses specifically on the tails of the distribution. The key results:

Distribution of maxima/minima. If you take the maximum of many independent random variables, its distribution converges to one of three types: Gumbel, Weibull, or Frechet. In finance, the Frechet distribution matters most because it captures fat tails.

Peaks over threshold. Given that losses exceed some high threshold u, how much further can they go? This is modeled by the Generalized Pareto Distribution. For heavy tails, not all moments exist. This means the mean and variance might be infinite, which has serious implications for any VaR model that assumes finite variance.

Wilmott cites a striking example from Alexander McNeil (1998). Fit a Frechet distribution to 28 years of annual maximum daily falls (1960 to October 16, 1987). The model predicted a 50-year return level of 24%. The very next business day, October 19, 1987 (Black Monday), the market fell 20.4%. The model saw it coming, even though the worst fall in the dataset had been “just” 6.7%.

Coherence: Does Your Risk Measure Make Sense?

Artzner, Delbaen, Eber, and Heath proposed four properties that any sensible risk measure should satisfy:

1. Sub-additivity. Combining two portfolios should not make total risk worse. Diversification should help, not hurt.
2. Monotonicity. If portfolio X always performs better than Y in every scenario, then X should have lower risk.
3. Positive homogeneity. Double the portfolio, double the risk.
4. Translation invariance. Adding cash reduces risk by exactly that amount.

A risk measure satisfying all four is called “coherent.” The uncomfortable truth: classical VaR violates sub-additivity. You can construct cases where combining two portfolios gives a VaR worse than the sum of individual VaRs. This means VaR can actually discourage diversification. A risk measure that punishes you for diversifying is not a great risk measure.

The Bottom Line

VaR is like your star sign, as Wilmott puts it in a footnote. You have to tell your investors what your VaR is, just like you have to tell the person you are chatting up your star sign. You do not necessarily believe there is meaning in either.

It is a useful number. It forces you to think about downside risk. It gives regulators and managers a standardized way to compare risk across different portfolios. But it is built on assumptions (Normal distributions, stable correlations) that break down exactly when they matter most, during a crisis.

The more sophisticated approaches (EVT, simulation, coherent risk measures) address some weaknesses but add complexity. In Chapter 42, Wilmott returns to these ideas in more detail.

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