FINC6023 · Financial Risk Management
Value at Risk: Parametric & Empirical
Value at Risk: Parametric & Empirical introduces the unit's central number. VaR is the loss not exceeded with a stated confidence over a stated horizon, quoted as a (currency amount, confidence, horizon) triple. You compute it two ways: the parametric (delta-normal) route assumes normal returns and reads the loss off a z-quantile, VaR = W · |z| · σ · √t; the empirical (historical) route ranks actual profit-and-loss and reads the loss at the (1 − c) percentile with no distributional assumption. The contrast matters because with fat tails the historical VaR usually exceeds the delta-normal one at high confidence.
What this chapter covers
- 01The VaR definition and the (amount, confidence, horizon) triple
- 02Interpreting VaR: 'lose at least X on 1 day in 100', not at most
- 03Parametric (delta-normal) VaR = W · |z| · σ · √t
- 04Key z-quantiles: z₉₅ = 1.645, z₉₇.₅ = 1.960, z₉₉ = 2.326
- 05Threshold-return route R* = μ + z·σ, then VaR = |R*| · W
- 06Empirical/historical VaR: rank P&L and read the percentile
- 07Parametric vs historical: fat tails make historical VaR larger at high confidence
Parametric 1-day 99% VaR and what it means
- 2 marksSelect the delta-normal formula VaR = W · |z| · σ · √t and identify W = 1,500,000, z = 2.326 (99%), σ = 0.015, t = 1 so √t = 1.
- 2 marksSubstitute: VaR = 1,500,000 × 2.326 × 0.015.
- 1 markCompute: 1,500,000 × 0.015 = 22,500; 22,500 × 2.326 = $52,335.
- 1 markInterpret: on the worst 1 day in 100 we expect to lose AT LEAST about $52,335.
Key terms
- Value at Risk (VaR)
- The loss not exceeded with confidence c over horizon t, quoted as (amount, confidence, horizon). It is a quantile of the loss distribution, read as a minimum loss in the tail.
- Parametric (delta-normal) VaR
- VaR = W · |z| · σ · √t, assuming normally distributed returns. W is the position value, σ the per-period volatility, and z the standard-normal quantile for the confidence level.
- Empirical (historical) VaR
- Rank the historical profit-and-loss worst to best and read the loss at the (1 − c) percentile. It makes no normality assumption, so it captures observed fat tails.
- z-quantile
- The standard-normal multiplier for a confidence level: z₉₅ = 1.645, z₉₇.₅ = 1.960, z₉₉ = 2.326. Provided in the exam's normal table.
Value at Risk: Parametric & Empirical FAQ
Why is historical VaR often bigger than parametric VaR at 99%?
Because real return distributions usually have fatter tails than a normal curve. The delta-normal method assumes normality and so understates how often extreme losses occur, whereas the historical method reads the actual observed tail — at high confidence the gap widens.
Does a 99% 1-day VaR mean I cannot lose more than that amount?
No — that is the most common error. VaR is the loss you expect to be exceeded only about 1 day in 100; on those bad days the actual loss can be far larger. VaR says nothing about how deep the tail goes (that is Expected Shortfall's job).
When do I set the mean to zero?
Over short horizons (a day or a few days) the expected return is negligible next to the volatility, so the convention is μ = 0. Over longer horizons the drift can matter and you should include μ via the threshold-return form R* = μ + z·σ.
Exam move
Memorise the three z-values and the delta-normal formula, and practise reading a VaR aloud as a minimum tail loss. Be ready to compute it both ways and to say in one line why fat tails make the historical number larger.
