VRM 2: Calculating and Applying VaR
A portfolio is linear when its value responds in strict proportion to the market variables underneath it. Hold 100 shares priced at USD 50 apiece: the holding is worth USD 5,000, and whatever the share price does passes straight through, scaled by the share count.
Mixing long and short positions changes nothing. With n the share count in stock i, negative when short, and S its price, the money committed is their product and the portfolio moves by the sum of money committed times return.
Options break the proportionality. At maturity a call option pays nothing when the stock finishes below the strike price X, and pays S minus X above it, so the payoff kinks at the strike, and before maturity the kink rounds into a smooth curve. A tangent at the current stock price describes tiny movements well, but widen the move and it pulls away. That gap is the error a linear model commits.
Derivatives are not automatically non-linear, which is the point students most often miss. An obligation to purchase an asset at time T for a fixed price K amounts, in substance, to holding it now and settling later, so a forward contract is worth the asset less the present value of the sum owed. The slope of that expression in S is one at every level.
A forward contract commits its holder to buy an asset in three years for USD 12,000. The asset is worth USD 10,000 today and pays no income. Interest rates are 3% with annual compounding.
Historical simulation is the leading non-parametric route to value at risk and expected shortfall, and nothing is assumed about the shape of any distribution. The recent past becomes a catalogue of what could happen next, and each past day is replayed against the portfolio held today.
Work begins by naming the market variables the portfolio depends on: equity prices, exchange rates, interest rates, commodity prices, credit spreads and volatilities. These are the risk factors. With today as Day 500, a run of 501 daily observations labelled Day 0 through Day 500 yields 500 scenarios, one fewer than the days because a scenario comes from the move between consecutive days.
Percentage changes and actual changes
Risk factors split into two groups, and the split governs how a past move is transplanted onto tomorrow. In the first, a percentage change in the past becomes a percentage change in the future, and stock prices and exchange rates sit here. In the second, an actual change becomes an actual change, and interest rates and credit spreads sit here: what matters is the difference in percentage points.
Scenario 1 asks what happens if every factor repeats its behaviour between Day 0 and Day 1. Suppose the stock price gained 4% across those days. It stands at 63 today, so the scenario carries it to USD 65.52, that is 63 multiplied by 1.04. Scenario 2 uses Day 1 to Day 2, and so on until all 500 exist.
Scenario 2 is worth spelling out across several factors at once. The stock price lands at USD 55.73. The exchange rate moves from 1.2500 to 1.2503, a proportional move carried over from the past. The interest rate goes from 2.36 to 2.37, an absolute rise of one basis point, and the credit spread climbs from 47 to 48 basis points.
Each scenario is then priced. The portfolio is revalued at the scenario factor levels and the result subtracted from the current value to give a loss, with gains entered as negative losses so everything sits on one scale. Full revaluation under all 500 scenarios is accurate and slow, which is why the next section exists.
Once every scenario has produced a loss, the losses are ranked from worst to best and both risk measures are read off that ranking. With 500 scenarios and 99% confidence the figure sits at the first percentile, and since 5 divided by 500 equals 0.01, that is the fifth worst loss. Expected shortfall averages the losses worse than the value at risk, here the four entries above the fifth.
Neither convention is the only defensible one: a case can be made for the sixth worst loss, or for averaging the fifth and sixth. The percentile also need not land on a whole observation, and with 250 observations at 99% it falls between the second and third worst loss, so the midpoint is taken.
| Rank | Scenario number | Loss (USD millions) | Average loss down to this rank |
|---|---|---|---|
| 1 | 210 | 7.8 | 7.80 |
| 2 | 195 | 6.5 | 7.15 |
| 3 | 2 | 4.6 | 6.30 |
| 4 | 23 | 4.3 | 5.80 |
| 5 | 48 | 3.9 | 5.42 |
| 6 | 367 | 3.7 | 5.13 |
| 7 | 235 | 3.5 | 4.90 |
Source: losses and scenario numbers as tabulated in the chapter. The fourth column is derived here and gives expected shortfall directly at rank 4.
A historical simulation on 500 scenarios has produced the ranked losses above, in millions of United States dollars.
Revaluing a large book 500 times is expensive, and for a bank with tens of thousands of positions it can be prohibitive. The standard shortcut borrows the Greek letters traders already compute for hedging. Delta is the first: the change in portfolio value divided by the small factor change that caused it.
Read it as a conversion rate. A portfolio with a delta of USD 50,000 to a stock price gains that much for every USD 1 the stock adds, so a move of USD 3.2 implies an estimated USD 160,000 swing. Real portfolios answer to many factors, and the estimated move totals across them.
Adding curvature through gamma
Gamma measures how fast delta itself changes as the factor moves, so it measures the curvature the linear approximation ignores. A term in the square of the factor change upgrades the estimate from a straight line to a parabola.
That multi-factor form assumes each portfolio component answers to one factor only; where a position depends on several, cross gamma terms enter.
Take scenario 2 from the historical simulation, with four risk factors only. Their deltas, in millions of United States dollars per unit of the factor, are 0.5 for the stock price, 2,000 for the exchange rate, minus 10 for the interest rate and 0.1 for the credit spread.
A stock price or a commodity price is a single number and needs no special treatment. Interest rates do not work that way. A government borrowing curve is a family of rates, one per maturity, so a modeller carries a chosen set of maturities and treats each point as its own risk factor.
Rates between the chosen maturities come from linear interpolation. Put the six-month rate at 2.5 in percentage terms, with 2.8 at one year. The nine-month rate then sits halfway between them at 2.65, and the seven-month rate at 2.55. Moves interpolate in the same way: a scenario that lifts the six-month point 10 basis points and the one-year point 8 lifts the nine-month rate by 9. Beyond the ends of the curve the convention is a flat extension.
Credit spreads, the excess a company pays over a benchmark such as the Treasury rate, are handled the same way. A borrower might face 100 basis points at one year, 150 at five years and 180 at ten years, meaning it pays 1% above the risk-free rate for one-year money, 1.5% for five-year money and 1.8% for ten-year money. A seven-year spread interpolates to 162 basis points. Implied volatility is a third such factor.
| Maturity | Spread (basis points) | Excess over the risk-free rate | Steepening per extra year |
|---|---|---|---|
| One year | 100 | 1% | |
| Five years | 150 | 1.5% | 12.5 |
| Ten years | 180 | 1.8% | 6.0 |
Source: spread levels as given in the chapter. The final column is derived here and shows the curve flattening as maturity lengthens.
Stressed VaR and stressed expected shortfall
Everything above assumed the historical window sat immediately behind today. Regulators require a second calculation on a different window. Stressed value at risk and stressed expected shortfall draw scenarios from a one-year period chosen because it would be especially punishing for the portfolio held now, and a bank exposed to a credit crisis might select March 2008 to February 2009. Typically 250 scenarios come from that year, against the 500 two years of ordinary data give, and every other step is unchanged.
The delta-normal model abandons scenarios and reaches the answer through a formula. It rests on the linear approximation plus an assumption that the risk factor changes are multivariate normal. A linear combination of jointly normal variables is itself normal, so the portfolio change is described entirely by its mean and standard deviation.
To cover both categories of factor at once, write the portfolio change as a weighted sum of standardised factor moves. For a stock price the move is the percentage change and the weight is delta multiplied by the current level. For an interest rate the move is the actual change and the weight is delta itself.
Nothing distributional has been used yet: those estimates pin down the mean and standard deviation whatever shape the factors follow. Normality is what turns the two numbers into risk measures. Writing U for the point on the standard normal distribution with an X probability of being exceeded, U is minus 1.645 at 95% confidence and minus 2.326 at 99%.
Dropping the mean is standard for daily work, since over one day the drift is tiny beside the standard deviation. A linear derivative fits without difficulty: a forward contract has a delta of one to the asset, so its value at risk equals that of holding the asset outright. By the Central Limit Theorem, non-normal factors can still give an approximately normal portfolio.
A position holds USD 10,000 of asset X and USD 20,000 of asset Y. Daily volatilities are 1% for X and 2% for Y, and the correlation between their returns is 0.3.
Delta-normal performs well on linear portfolios whose factors are roughly normal. Portfolios containing options can be run through it too, but the linear expression is then an approximation whose error is governed by gamma. Small gamma means gentle curvature and a tolerable answer; large gamma means the tangent departs quickly.
Two effects compound. The first is that pricing error. The second is distributional and more serious: feed a normal distribution for the asset price through a curved value relationship and what emerges is skewed. For a call option the upper tail stretches and the lower tail compresses, since losses flatten towards zero while gains keep running.
The model replaces the curve with its tangent, which forces a normal distribution out the other side. Set beside the true skewed one, it puts too little weight on high option values and too much on low ones. For a long position in a call option both risk measures come out too high, and for a short position both come out too low, which is the dangerous direction.
Practice makes this worse. Derivatives desks routinely run close to delta neutral, since traders take offsetting positions at the end of a day to bring their deltas near zero. Curvature is then the principal risk left in the book, and it is exactly the risk the model cannot see. A butterfly spread is the extreme case: a modest profit if the factor barely moves and a loss if it moves sharply either way. The quadratic approximation prices this far better, but closed-form results exist for the linear case and not the quadratic one, so accuracy costs tractability.
Two call options on a stock trading at USD 80 have six months to run, volatility of 24% and no interest rate. Option A is at the money, strike USD 80, worth USD 5.4097 with a delta of 0.5338 and a gamma of 0.0293. Option B is in the money, strike USD 64, worth USD 16.5342 with a delta of 0.9192 and a gamma of 0.0110. The stock rises to USD 86.
Monte Carlo simulation is the third route and a close relative of historical simulation. The two differ in one place: historical simulation lifts its scenarios from days that actually occurred, while Monte Carlo draws them at random from distributions the analyst has specified. Everything downstream is identical, and the approach handles linear and non-linear portfolios alike.
The procedure step by step
Assume the factor changes are multivariate normal. Value the portfolio today at current factor levels. Take one draw from that distribution for the factor moves, consistent with the standard deviations and correlations estimated from historical data. As in historical simulation, the sampled move is a percentage change for equity prices and exchange rates and an absolute change for interest rates and credit spreads. Apply the moves to get the factor levels one day out, revalue the portfolio, subtract from the current value to obtain a loss, and repeat many times. The collection of losses is the distribution the risk measures come from.
Reading them is the same counting exercise as before. Run 1000 trials and the 99% value at risk is the tenth worst loss, while expected shortfall averages the nine losses worse than it.
What Monte Carlo buys and what it costs
The freedom to choose distributions is the main prize. Delta-normal requires normality, while Monte Carlo accepts any distribution provided the correlations between factors can be specified. The usual construction samples from a multivariate normal reproducing the historical correlations, then converts each value on a percentile-to-percentile basis: a draw at the p-th percentile of the normal maps to the p-th percentile of the target. This is the Gaussian copula, and it lets fat tailed factor distributions coexist with an ordinary correlation estimate.
Cost is the weakness. Full revaluation of a large portfolio on thousands of trials is slow, and slowness bites when risk numbers are needed overnight. The remedy is the one historical simulation uses: replace full repricing with the delta-gamma approximation, known here as partial simulation. A second weakness is that the output is only as good as the assumed distributions.
Both the delta-normal model and Monte Carlo simulation need standard deviations and correlations for the factor moves, and historical data supplies them. Weighting all observations equally is not compulsory: the exponentially weighted moving average and the GARCH family both favour recent observations, which lets an estimate respond when volatility shifts. For a stressed measure the parameters come instead from a period chosen for how badly it would treat the current portfolio.
Correlation breakdown
Volatilities rise in stressed markets, which surprises nobody. Correlations rise as well, and that is the finding with teeth. The claim that all correlations go to one under stress is an exaggeration, but the direction is genuine. During the crisis of 2007-2008 mortgage default rates climbed together across every region of the United States, and when money runs for safety credit spreads widen almost everywhere at once.
This is correlation breakdown. Value at risk and expected shortfall are statements about extreme conditions, yet correlations estimated from calm periods describe calm periods. Using them assumes diversification that will not be there when it is needed, so a risk manager should estimate what correlations will do under stress.
Scaling to longer horizons
Most of these calculations use a one-day horizon, because one day is where the data is richest. Longer horizons are reached by scaling rather than by rebuilding the model.
Two assumptions sit behind the rule. Daily changes in portfolio value are taken to be normally distributed with a mean of zero, and successive daily changes are taken to be independent. Independence is what makes variances additive across days, so standard deviations grow with the square root of the number of days. Both are reasonable often enough for the rule to be the market convention, and both can fail. Autocorrelation in returns breaks additivity and pushes the scaled figure wrong. Volatility clustering means a quiet day tends to follow a quiet day and a violent one another violent one, which the rule cannot represent, and a zero mean is harmless over a day but less so as T grows.
Worst-case scenario analysis asks a different question from value at risk. Instead of fixing a horizon and a confidence level and asking how bad one period could be, it fixes a run of periods and asks how bad the worst of them will be: a manager reporting weekly might want to know what the worst week out of 52 looks like.
Given a distribution for the return in one week, Monte Carlo simulation answers this directly. Simulate 52 weekly returns, record the worst, repeat many times, and a distribution for the worst-case result appears, giving the expected worst-case result over 52 weeks and its 95th percentile. Taking the minimum of many draws pulls the answer deep into the left tail, so worst-case statistics tend to be overly pessimistic and are not substitutes for value at risk or expected shortfall.
Choosing among the three approaches
| Historical simulation | Delta-normal | Monte Carlo simulation | |
|---|---|---|---|
| Where scenarios come from | Actual moves on past days | No scenarios; an analytic formula | Random draws from assumed distributions |
| Distribution assumed | None | Multivariate normal factors | Any, given a correlation structure |
| Main advantage | Captures fat tails and observed correlations without estimating them | Fastest by a wide margin, and transparent | Most flexible, and handles non-linearity properly |
| Main disadvantage | Assumes the past window represents the future, and one window gives limited tail data | Poor for portfolios containing options, worst when delta neutral | Computationally intensive and dependent on assumed distributions |
| Non-linear portfolios | Full revaluation | Approximate at best | Full revaluation |
Source: comparison assembled from the descriptions of the three methods in the chapter.
Historical simulation remains the most widely used, because the means, standard deviations and correlations observed in the market are carried into the calculation automatically, with no estimation step to get wrong. The delta-normal model is the fastest, since an analytic formula replaces the scenario loop, and it descends from the portfolio theory of Markowitz. Managers holding mostly long and short stock positions use it heavily, but it is the wrong tool for a book of options and at its worst for the delta-neutral portfolio where curvature is all that remains. Monte Carlo buys that flexibility back at the cost of computing time. Since the crisis of 2008-2009, regulators have layered stressed versions of both measures on top.