FINC6023 Chap.2 Value at Risk: Parametric & Empirical
Value at Risk: Parametric & Empirical
Value at Risk: Parametric and Empirical introduces the unit's central market-risk measure. A VaR statement is incomplete unless it gives a loss amount, a confidence level, and a holding period.
A one-day VaR at ninety-nine percent is a quantile, not a ceiling: on about one day in one hundred the loss is expected to meet or exceed that amount, and the loss can be much larger beyond the cut-off.
The parametric or delta-normal method assumes a linear position and normally distributed returns.
It converts volatility into a loss quantile with the position value, the appropriate standard-normal multiplier, and the square root of the holding period. The mean is commonly set to zero over short horizons because expected daily return is small relative to daily volatility. Units must remain consistent, so daily volatility is paired with a horizon measured in days.
The empirical method avoids a normality assumption.
Historical factor changes or profit-and-loss observations are ranked from worst to best, and VaR is read from the required tail percentile. This preserves observed skewness and fat tails but depends on the relevance and size of the historical window.
Parametric VaR is transparent and economical; historical VaR can better represent a non-normal tail when past observations are informative.
The available final paper tests the VaR definition directly and asks students to compare parametric and historical measures. Calculation items also use VaR as the starting value for later confidence and horizon conversions.
For every answer, name the method, show the quantile convention, retain the amount-confidence-horizon triple, and interpret the final number as a tail threshold rather than a maximum possible loss.
What this chapter covers
- 01
The VaR definition and the (amount, confidence, horizon) triple
- 02
Interpreting VaR: 'lose at least X on 1 day in 100', not at most
- 03
Parametric (delta-normal) VaR = W · |z| · σ · √t
- 04
Key z-quantiles: z₉₅ = 1.645, z₉₇.₅ = 1.960, z₉₉ = 2.326
- 05
Threshold-return route R* = μ + z·σ, then VaR = |R*| · W
- 06
Empirical/historical VaR: rank P&L and read the percentile
- 07
Parametric vs historical: fat tails make historical VaR larger at high confidence
Worked example · free
Parametric 1-day 99% VaR and what it means
- 2Select 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.
- 2Substitute: VaR = 1,500,000 × 2.326 × 0.015.
- 1Compute: 1,500,000 × 0.015 = 22,500; 22,500 × 2.326 = $52,335.
- 1Interpret: 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.
- Holding period
- The holding period is the horizon over which the profit-and-loss distribution and VaR are measured, expressed on the same time basis as the volatility input.
- Fat tail
- A fat tail places more probability on extreme outcomes than a normal distribution, which can make empirical tail losses larger than delta-normal estimates at high confidence.
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.
How do I choose between parametric and historical VaR?
Use parametric VaR when a linear exposure and a normal-return approximation are defensible and the required volatility and correlation estimates are available. Use historical simulation when the observed empirical distribution should determine the tail. State the method's assumptions because the same confidence and horizon do not make the two estimates interchangeable.
What should every VaR interpretation contain?
State the currency loss, confidence level, and holding period together. Then describe the number as a tail threshold: losses are expected to meet or exceed it with probability one minus the confidence level. Do not call VaR the maximum possible loss because the distribution beyond the quantile is not bounded by the VaR figure.
Exam move
Practise VaR as a decision sequence rather than a single formula. First write the amount, confidence, and horizon requested. Next decide whether the data and payoff support a delta-normal model or whether an empirical ranking is required. For parametric work, mark the position value, volatility unit, normal quantile, and horizon before multiplying.
For historical work, write the sample size and convert the tail probability into an observation rank before choosing the cut-off. Always read the result aloud as a loss threshold that may be exceeded, not a cap on loss.
Keep a short list of sanity checks: higher volatility, confidence, horizon, or exposure should not reduce VaR; a fat empirical tail can exceed the normal estimate; and a daily volatility cannot be combined with a yearly horizon without conversion. Finish mixed practice by explaining why the rejected method does not fit the stem.
That explanation is valuable preparation for conceptual multiple-choice items as well as calculations.
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