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Key Takeaways
- VaR is a loss quantile. A complete VaR figure specifies the portfolio, loss amount, confidence level and horizon.
- At confidence c, at least c of the loss distribution lies at or below VaR. For a continuous loss distribution, the probability above it is 1 − c.
- The three standard estimation methods are parametric (variance-covariance), historical simulation, and Monte Carlo simulation, and they can produce different numbers for the same portfolio.
- Delta-normal VaR uses normal risk-factor changes and a linear exposure approximation. Other parametric VaR models use different distributions.
- Use VaR for the loss threshold, expected shortfall for the average of the worst selected fraction, and scenario analysis for the payoff at specific market moves.
What is value at risk (VaR)?
Value at Risk, or VaR, is a quantile of a portfolio loss distribution: the loss amount that a chosen share of outcomes stays at or below over a set horizon. A one-day 95% VaR of $12,337.50 means at least 95% of modeled one-day outcomes lose no more than that amount. A complete VaR figure states the loss amount, confidence level and time horizon.
At confidence c, VaR is the smallest loss threshold whose cumulative probability is at least c. For a continuous distribution, the probability above that threshold is 1 − c. A finite set of scenarios can place several outcomes exactly at the same loss.
Formula
Normal one-period VaR = V × (z × σ − μ), with z = Φ⁻¹(c). For n independent equal-distribution normal additive periods: VaR(n) = V × (z × σ × √n − n × μ).- V is the portfolio value in currency units.
- z is the confidence z-score: about 1.645 for 95 percent and about 2.326 for 99 percent.
- σ is the standard deviation of returns over the chosen horizon, and μ is the expected return over that horizon, often set near zero for short horizons.
- The square-root shortcut VaR(n) = VaR(1) × √n follows when these additive normal increments have zero mean. Compounded simple returns require their compounded distribution.
How to calculate VaR: a normal-return example
A trader holds a $500,000 equity portfolio with an estimated daily return standard deviation of 1.5 percent and an expected daily return near zero. The trader wants the one-day 95 percent VaR, which uses a z-score of 1.645.
- Daily VaR percent: 1.645 × 1.5% − 0% = 2.4675%
- Dollar VaR: 2.4675% × $500,000 = $12,337.50
- Ten-day zero-mean additive normal VaR: $12,337.50 × √10 = $39,014.60, or $39,015 to the nearest dollar.
The one-day normal model places approximately 95% of outcomes at or below a $12,337.50 loss. The remaining approximately 5% lie above that threshold. At ten days, the stated zero-mean additive normal model gives $39,014.60.
For normally distributed portfolio return, VaR = V × (z × sigma − mu). V is starting portfolio value, sigma is return standard deviation, mu is mean return and z is the standard normal quantile at confidence c.
With V = $500,000, sigma = 1.5%, mu = 0 and the rounded 95% quantile z = 1.645, the return threshold is 2.4675% and dollar VaR is $12,337.50.
The chart marks the corresponding −2.4675% return. The tail to its left holds 5% of this normal return distribution.
VaR calculation methods: parametric, historical and Monte Carlo
Delta-normal VaR combines normal factor changes with linear position exposures. Other parametric distributions have their own quantile formulas.
Historical simulation applies observed factor moves to current positions. Their assigned scenario weights define an empirical loss distribution.
Monte Carlo generates factor moves from the selected dynamics and dependence structure, then revalues the positions. Full repricing connects option losses with changes in price, volatility and time.
How VaR scales with the time horizon
For n independent normal additive periods with the same mean and variance, aggregate mean is n × mu and standard deviation is sigma × √n. Normal VaR becomes V × (z × sigma × √n − n × mu).
At zero mean, the ten-day figure in the example is $12,337.50 × √10 = $39,014.60. With a nonzero mean, the mean term scales separately. Compounded simple returns use the resulting compounded distribution.
VaR vs expected shortfall
VaR identifies the selected loss percentile. Expected shortfall averages the worst selected fraction of the distribution. A scenario payoff shows the position result at a specific market move.
Use the same portfolio, horizon and confidence level when comparing VaR and expected shortfall. Recompute portfolio VaR after a proposed trade to measure its effect on the book, and compare the trade’s payoff across relevant price levels in the Options Profit Calculator.
Limitations of value at risk
- Pair VaR with expected shortfall. VaR marks a threshold, not a worst case, and says nothing about how large losses beyond it can be; expected shortfall measures that tail.
- Check normal VaR against historical or Monte Carlo VaR. Normal VaR assumes a bell-shaped return distribution, and market returns can have fatter tails, so a normal model can understate the size and frequency of large losses.
- Reprice option positions in full. Delta-normal VaR uses linear exposures, which miss the curvature of options; full repricing connects option losses with changes in price, volatility and time.
- State the method and its inputs with every VaR figure. Parametric, historical and Monte Carlo methods can produce different numbers for the same portfolio, and each depends on its look-back window or model inputs.
- Recalculate VaR over the full horizon when the mean is nonzero or returns compound. The square-root-of-time shortcut assumes independent, zero-mean normal additive increments.
Bottom line
Value at risk puts every position on one loss scale: the loss a chosen share of outcomes stays within over a set horizon. Use it for limits and comparisons on a consistent confidence level and horizon, and pair it with expected shortfall and stress scenarios to see how bad the losses beyond it can get.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- CFA Institute: Measuring and Managing Market Risk
- Federal Reserve: How Accurate Are Value-at-Risk Models at Commercial Banks?
- Aswath Damodaran (NYU Stern): Value at Risk (Chapter 7)
How entries are sourced, calculated and updated: editorial methodology.