FINC6023 Chap.3 VaR Accuracy, Backtesting, Scaling & Expected Shortfall
VaR Accuracy, Backtesting, Scaling & Expected Shortfall
VaR Accuracy, Backtesting, Scaling and Expected Shortfall asks whether a reported VaR is useful and how it changes when the requested confidence level or horizon changes. Confidence conversion uses the ratio of normal quantiles. Horizon conversion uses the square root of time only when returns are independent and identically distributed.
Positive serial dependence or a trend can make the rule understate longer-horizon risk, while mean reversion can make it overstate risk.
Sampling error matters because a risk estimate is built from finite data. Volatility is generally estimated more precisely than expected return, which helps explain why short-horizon VaR often ignores drift. Backtesting compares realised losses with the forecast threshold.
Under a correctly calibrated daily model, exception counts follow a binomial benchmark. The number of exceptions tests unconditional coverage, while their timing matters because clustered exceptions can reveal changing volatility or model failure even when the total count appears acceptable.
VaR also has a conceptual limitation.
It identifies a tail boundary but says nothing about the severity of losses beyond that boundary. VaR can fail sub-additivity, which means the measured risk of a combined position can exceed the sum of separate VaRs.
Expected Shortfall instead averages losses in the tail beyond VaR and satisfies the coherence conditions used in the unit, including sub-additivity.
The available final paper tests the meaning of VaR, the direction of square-root scaling errors, and a joint confidence-and-horizon conversion. To earn calculation marks, separate the confidence multiplier from the horizon multiplier and state the independence assumption.
To earn conceptual marks, distinguish calibration from exception clustering and explain why Expected Shortfall supplies information that the VaR quantile omits.
What this chapter covers
- 01
Scaling confidence: VaR_c2 = VaR_c1 · (z_c2 / z_c1)
- 02
Scaling horizon: the √T rule and when it is valid (i.i.d. returns)
- 03
Trend → √T underestimates; mean reversion → √T overestimates
- 04
Precision of estimates: σ estimated more precisely than μ
- 05
Binomial backtesting: exceptions, E[x] = pT, z = (x − pT)/√(p(1−p)T)
- 06
Type I vs Type II error; Kupiec, Christoffersen, and Basel model controls
- 07
Expected Shortfall (ES/CVaR) = mean tail loss; coherence and sub-additivity
Worked example · free
Backtest a 99% VaR model with a binomial z-test
- 1Set up the binomial benchmark: each day is a Bernoulli trial with exception probability p = 0.01, over T = 500 days.
- 1Expected exceptions E[x] = pT = 0.01 × 500 = 5.
- 2Variance V[x] = p(1 − p)T = 0.01 × 0.99 × 500 = 4.95, so the standard deviation is √4.95 = 2.2249.
- 2Test statistic z = (x − pT)/√(p(1−p)T) = (9 − 5)/2.2249 = 4/2.2249 = 1.798.
- 1Compare to the critical value: 1.798 < 1.96, so we do NOT reject the model at the 5% level.
Key terms
- Square-root-of-time (√T) rule
- Scale a 1-day VaR to T days by multiplying by √T. It is valid only for independent, identically distributed returns; a trend makes it underestimate, mean reversion makes it overestimate.
- Backtesting
- Counting how often the actual loss exceeds the predicted VaR (exceptions). Under the model, exceptions are binomial with E[x] = pT; a large-sample z-test or the Kupiec/Christoffersen tests judge whether there are too many or whether they cluster.
- Type I vs Type II error
- A Type I error rejects a correct VaR model (probability set by the test size); a Type II error fails to reject an incorrect model. Rare exceptions make backtests low-powered, raising Type II risk.
- Expected Shortfall (ES / CVaR)
- The average loss given that the loss exceeds VaR — the mean of the tail beyond the VaR quantile. ES ≥ VaR and is coherent (sub-additive), so it cannot be reduced by splitting a portfolio.
- Coherence / sub-additivity
- A risk measure is coherent if it satisfies monotonicity, translation invariance, positive homogeneity and sub-additivity (the risk of a combined portfolio is no more than the sum of the parts). VaR can violate sub-additivity; ES does not.
- VaR exception
- A VaR exception is a period in which the realised loss exceeds the forecast VaR threshold, supplying the Bernoulli event counted in a coverage backtest.
- Exception clustering
- Exception clustering occurs when VaR breaches arrive close together, indicating time dependence or changing volatility even when the total number of exceptions appears reasonable.
VaR Accuracy, Backtesting, Scaling & Expected Shortfall FAQ
When is the √T rule wrong?
It assumes returns are independent and identically distributed with stable variance. If returns trend, positive autocorrelation can make √T understate long-horizon risk because losses reinforce one another. If returns mean-revert, negative autocorrelation can make it overstate risk because moves partly cancel. State the direction of the bias and use an autocorrelation-aware variance when supplied.
Why is Expected Shortfall preferred to VaR?
VaR ignores how bad the tail is beyond the quantile and is not sub-additive, so a trader can split a position to lower reported VaR. ES averages the whole tail, is always at least as large as VaR, and is coherent — combining portfolios never makes ES look smaller than the sum, so it cannot be gamed the same way.
Why is σ estimated more precisely than μ?
The standard error of the volatility estimate, se(σ̂) = σ̂·√(1/2n), shrinks faster with sample size than the mean's se(μ̂) = σ̂·√(1/n). This is why VaR models lean on volatility (which we can pin down) rather than the drift (which is noisy).
How does the final combine confidence and horizon scaling?
The available paper starts from a daily VaR at one confidence level and requests a multi-day VaR at a higher confidence. Multiply first by the ratio of the new and old normal quantiles, then by the square root of the new horizon. Keep the two multipliers visible so each assumption can be checked.
What does a satisfactory backtest actually show?
A reasonable exception count supports unconditional coverage but does not prove that the model is complete. Inspect whether breaches cluster, whether volatility changed, and whether independence is plausible. A backtest also has limited power when exceptions are rare, so failure to reject a model is not the same as proving that it is correct.
Exam move
Split revision into three routines. For scaling, write the old and new confidence levels, obtain the corresponding normal multipliers, apply their ratio, and then handle the holding period with a separate square-root factor. Add one sentence stating that the horizon rule assumes independent and identically distributed returns.
For backtesting, translate the confidence level into an exception probability, compute the expected exception count and variance, and compare the observed count with the model benchmark. Then ask whether exceptions cluster, because correct unconditional coverage can still hide dependence.
For risk-measure theory, use a small discrete loss distribution to compare VaR and Expected Shortfall, and check sub-additivity by combining positions. Build an error log that records direction: trend tends to make square-root scaling too low, mean reversion tends to make it too high, and a deeper tail raises Expected Shortfall without changing the VaR boundary.
End every calculation with a sentence about model adequacy rather than only a test statistic.
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