Value at Risk explained for FRM candidates: three methods and their limits
Value at Risk (VaR) answers one question: over a stated horizon, what loss will we not exceed with a stated probability? A one-day 99% VaR of 200,000 means that on 99 days out of 100 the portfolio should lose less than 200,000. It says nothing about how bad the other day is. VaR appears in FRM® Part I (Valuation and Risk Models) and Part II (Market Risk). For the shape of the exam, see how the FRM exam is structured, and for the full list of guides see the FRM page.
What VaR is, and is not
VaR needs a horizon, a confidence level and a distribution of profit and loss. It is a quantile of that distribution, quoted as a positive loss, not the worst case and not a statement about the tail beyond the cut-off.
Method 1: parametric (variance-covariance)
Assume returns are normal, so the whole distribution is described by a mean and a standard deviation. With the mean set to zero over a short horizon:
VaR = z x sigma x portfolio value
The usual z-values are 1.645 at 95% and 2.326 at 99% (one-tailed). For several positions, sigma comes from the covariance matrix: for two positions with dollar volatilities a and b and correlation r, portfolio sigma = square root of (a^2 + b^2 + 2rab).
It is fast and transparent, but real returns have fat tails and options are non-linear, so it can understate risk. See fat tails, skewness and kurtosis.
Method 2: historical simulation
Take the last N days of changes in risk factors, apply each one to today's portfolio, revalue, and sort the resulting profits and losses. The VaR is the loss at the chosen percentile. With 500 days at 99%, that is roughly the fifth-worst outcome, though practitioners differ on interpolation and on whether to take the fifth or sixth.
No distribution is assumed, and fat tails come through if they occurred in the window. But a calm window gives a low VaR, and each observation counts equally however old it is.
Method 3: Monte Carlo simulation
Specify a model for the risk factors, draw many random scenarios, revalue the portfolio in each, and read off the percentile as in historical simulation. It handles options and non-normal models well, at the cost of computing time and model risk.
Expected shortfall
Expected shortfall (ES, also called conditional VaR) is the average loss in the tail beyond the VaR level. Where VaR is a threshold, ES is the mean of everything past it. ES is always at least as large as VaR at the same confidence level, and unlike VaR it is subadditive, so diversification never makes it look worse.
Scaling by the square root of time
If daily returns are independent with the same volatility, the standard deviation over T days is sigma x square root of T. So:
10-day VaR = 1-day VaR x square root of 10, about 3.16 times.
It fails when returns are autocorrelated, when volatility is not constant, when positions change during the period, or when options make the loss non-linear in the risk factors. It also should not be applied blindly to historical VaR, which has no distributional model behind it.
Worked example
An invented portfolio holds 6 million in equity A (daily volatility 1.2%) and 4 million in equity B (daily volatility 0.8%), with correlation 0.3.
- Dollar volatilities: A = 0.072 million, B = 0.032 million.
- Portfolio sigma = square root of (0.072^2 + 0.032^2 + 2 x 0.3 x 0.072 x 0.032) = 0.0871 million.
- One-day 99% VaR = 2.326 x 0.0871 = 0.203 million.
- Ten-day 99% VaR = 0.203 x 3.162 = 0.641 million.
- Adding the two stand-alone VaRs (2.326 x 0.104) gives 0.242 million, so diversification saves about 0.039 million.
- Under normality, one-day 99% ES = sigma x pdf(2.326) / 0.01 = 0.232 million, a little above the VaR.
Common mistakes
- Using the one-tailed z-value of 1.645 for 99%, or 1.96 (two-tailed) for 95%.
- Adding VaRs of positions as if correlation were 1, or ignoring correlation.
- Scaling 1-day to 10-day by multiplying by 10 instead of its square root.
- Reading 99% VaR as the maximum loss, or as a loss expected once a year (it is about 2.5 days in 250).
How the exam typically tests it
Typically you compute a parametric VaR, scale it across horizons, or compare the methods on assumptions and weaknesses. Checking a VaR model against outcomes is covered in backtesting VaR and the two errors. Charta's practice questions for this topic let you rehearse these calculations.
Common questions
What does a 99% one-day VaR mean?
It is the loss that should be exceeded on about 1 day in 100. It does not say how large the loss is on those days.
Which VaR method is best?
None is best everywhere. Parametric is quick but assumes normality, historical simulation uses real data but depends on the window, and Monte Carlo is flexible but depends on its model.
How does expected shortfall differ from VaR?
VaR is a cut-off point in the loss distribution. Expected shortfall is the average loss beyond that point, so it captures tail severity and is at least as large as VaR.
When does the square root of time rule fail?
When returns are not independent and identically distributed, for example with autocorrelation, changing volatility, trading during the period or option positions.
Sources
- GARP, the FRM programme: what the exam is and its two parts.
- Basel Committee on Banking Supervision, the committee's website: its market risk standards discuss VaR and expected shortfall.
The exam and the syllabus can change. Check GARP's current details before you book.