Estimating Market Risk Measures: An Introduction and Overview: formula sheet

Key formulas

Definition of VaR

P(Loss > VaR) = 1 − confidence level = α

VaR is the loss quantile at the (1 − α) level. At 99% confidence, α = 1%.

VaR from P/L data

VaR = −(P/L quantile at α)

The P/L quantile at 5% or 1% is usually negative. Flip the sign to quote VaR as a positive loss.

VaR from return data

VaR = −(return quantile at α) × portfolio value

Use this when you are given returns in percent and a portfolio value.

Order-statistic rule for historical data

With n observations, the α quantile is roughly the (n × α)th worst outcome

For 500 observations at 99%, VaR is about the 5th worst loss. Conventions on interpolation vary, so follow the question.

Square-root-of-time scaling

VaR(h days) = VaR(1 day) × √h

Valid only for i.i.d. returns with zero mean (or a negligible mean). Not exact otherwise.

Normal VaR (reference)

VaR = −μ + z × σ, with z = 1.645 at 95% and 2.326 at 99%

Used when P/L or returns are assumed normal. μ and σ must match the holding period.

Scenario P&L

P&L(i) = Portfolio value today × return(i)

For a linear position. For options or nonlinear positions, reprice the portfolio under each historical scenario instead.

Number of tail observations

k = n × (1 − confidence level)

With n = 500 at 99%, k = 5. VaR is read near the 5th worst loss. Follow the convention given in the question.

HS VaR

VaR = −(P&L at the (1 − c) quantile of the sorted scenario P&Ls)

Reported as a positive loss number.

Scaling to longer horizons (rule of thumb)

VaR(h days) ≈ VaR(1 day) × √h

Valid only if returns are independent and identically distributed with zero mean. It is an approximation, not a property of HS.

Normal VaR (return terms)

VaR% = zα × σ − μ

Use one-tailed z: 1.645 at 95%, 2.326 at 99%. Result is a positive loss rate.

Normal VaR (value terms)

VaR = P0 × (zα × σ − μ)

P0 is the current portfolio value. If the mean is assumed zero, VaR = P0 × zα × σ.

Lognormal VaR

VaR = P0 × (1 − exp(μ − zα × σ))

μ and σ are the mean and standard deviation of the log return over the holding period.

Lowest price at the cut-off

P* = P0 × exp(μ − zα × σ)

Lognormal VaR = P0 − P*.

Time scaling

μT = μ × T; σT = σ × √T

Valid under i.i.d. returns. Convert to the holding period before applying z.

Common one-tailed z-scores

90%: 1.282; 95%: 1.645; 97.5%: 1.960; 99%: 2.326

Do not use two-tailed values such as 1.96 for 95% VaR.

Expected shortfall (continuous)

ES(α) = E[L | L ≥ VaR(α)]

Average loss given that loss is at or beyond the VaR at confidence α. ES ≥ VaR at the same α.

ES from n equally likely historical losses

ES(α) = average of the worst n × (1 − α) losses

If n × (1 − α) is not a whole number, a common treatment is to weight the boundary observation fractionally, or follow the convention the question states.

ES for normal losses

ES(α) = μ + σ × φ(z_α) ÷ (1 − α)

φ is the standard normal density and z_α the α quantile. For α = 99%, φ(2.326) ÷ 0.01 ≈ 2.665; for 95%, ≈ 2.063.

Subadditivity

ρ(X + Y) ≤ ρ(X) + ρ(Y)

Diversification should not increase risk. VaR can violate this; ES cannot.

Positive homogeneity

ρ(λX) = λ × ρ(X), for λ > 0

Scaling a position scales risk by the same factor.

Translation invariance

ρ(X + c) = ρ(X) − c

Where X is a gain/P&L and c is a risk-free cash amount. In loss terms, adding cash c reduces the loss by c.

Monotonicity

If X ≤ Y in every state, then ρ(X) ≥ ρ(Y)

X is P&L. A worse outcome in every state means higher risk.

Spectral risk measure

M_φ = ∫₀¹ φ(p) · q_p dp

q_p is the loss quantile at probability p. φ is the risk-aversion weighting function.

Admissibility conditions on φ

φ(p) ≥ 0; φ(p) non-decreasing in p; ∫₀¹ φ(p) dp = 1

Non-decreasing weights reflect risk aversion. If all three hold, the measure is coherent.

Expected shortfall as a spectral measure

φ(p) = 1 ÷ (1 − α) for p ≥ α; φ(p) = 0 for p < α

Equal weights on all tail quantiles beyond the confidence level α.

Discrete approximation

M_φ ≈ Σ wᵢ · qᵢ, with wᵢ ≥ 0, Σ wᵢ = 1

Weights must be non-decreasing as the quantile gets worse.

Exponential weighting idea

weight rises exponentially toward the worst tail; larger risk aversion → heavier tail weight

Know the behaviour, not a memorised constant.

Standard error of a quantile estimate

se(q̂) = √(p(1 − p)) ÷ (f(q) × √n)

p is the tail probability (1 − confidence level). f(q) is the pdf at the quantile. This is an asymptotic result for large n.

Confidence interval for VaR (normal approximation)

q̂ ± z × se(q̂)

z = 1.645 for 90%, 1.96 for 95%, 2.576 for 99% two-sided intervals.

Order-statistic count distribution

Number of observations below the true p-quantile ~ Binomial(n, p)

This is the basis of distribution-free confidence intervals for a quantile.

Approximate rank limits

Ranks ≈ np ± z × √(np(1 − p))

Count from the worst loss. Round the lower rank up and the upper rank up or down as the question states. Needs np large enough for the normal approximation.

Effect of sample size

se ∝ 1 ÷ √n

Quadrupling n halves the standard error. Doubling n cuts it by about 29%.

Plotting position

p(i) = (i − 0.5) ÷ n

One common choice for the probability assigned to the i-th smallest of n sorted observations. Other conventions exist; the exam focuses on interpretation.

Reference quantile

x(i) = Φ⁻¹(p(i))

Φ⁻¹ is the inverse standard normal CDF. This is the horizontal-axis value for the i-th point.

Standardisation

z(i) = (r(i) − mean) ÷ standard deviation

Standardising lets you compare against the standard normal. A good fit lies near the 45-degree line.

Linear relation under same shape

Data quantile = μ + σ × reference quantile

If data are normal with mean μ and standard deviation σ, the QQ plot against the standard normal is a straight line with intercept μ and slope σ.

Quick revision

• VaR is the loss level exceeded with probability 1 − confidence level over the chosen holding period.
• Normal VaR = −μ + σ × z; z is 1.645 at 95% and 2.326 at 99% (one-tailed).
• Historical simulation VaR is a quantile of sorted past returns; it assumes the past sample represents the future.
• Lognormal VaR keeps asset values positive and works from log returns.
• Expected shortfall is the average loss in the tail beyond VaR, so it is at least as large as VaR.
• A coherent measure satisfies monotonicity, subadditivity, positive homogeneity and translation invariance.
• VaR is not always subadditive; ES is.
• A spectral risk measure is a weighted average of quantiles; it is coherent when weights do not fall as losses worsen.
• Heavier weight on worse losses means greater risk aversion.
• Standard errors shrink as sample size grows; estimates deeper in the tail are less precise.
• A QQ plot compares sample quantiles with a reference distribution; a straight line suggests a good fit.
• S-shaped or curving tails on a QQ plot against the normal point to fat tails or skewness.

Common mistakes

• Reporting VaR as a negative number. Fix: VaR is quoted as a positive loss. Change the sign of the P/L or return quantile.
• Using the confidence level as the tail probability. Fix: Compute α = 1 − confidence first. A 99% VaR looks at the worst 1%.
• Counting from the best outcome instead of the worst Fix: Always locate the worst losses first, then count k observations in from that end.
• Using the confidence level instead of the tail probability to find the position Fix: Use k = n × (1 − c). Then read the kth worst loss.
• Using 1.96 for 95% VaR Fix: VaR is a one-tailed measure. Use 1.645 at 95% and 2.326 at 99%.
• Scaling volatility by T instead of √T Fix: Mean scales with T, standard deviation with √T. Variance scales with T.
• Saying ES is the loss at the 97.5% or 99% quantile. Fix: VaR is a single quantile. ES is the average of all losses beyond that quantile.
• Claiming VaR is never subadditive. Fix: VaR can violate subadditivity. For normally distributed losses it is subadditive. State the condition.
• Treating VaR as a spectral measure that is coherent. Fix: VaR is not a spectral measure and is not coherent because it is not sub-additive in general. It puts all weight on one quantile, which is not an admissible weighting function.
• Forgetting that weights must sum to one. Fix: Always add the weights. If they do not total 1, the number is not a proper spectral measure.

Exam tips

• Write confidence level and holding period at the top of your rough work before you compute anything.
• Check the sign and units of the answer: positive loss, in currency or in percent as asked.
• If options show a number and its negative, VaR is the positive one.
• For 95% versus 99% questions, expect the larger confidence level to give the larger VaR. Eliminate options that go the other way.
• Watch for traps where the data are weekly or monthly and the question asks for a different horizon.
• Check which quantile convention the question uses before you count.
• Expect conceptual MCQs on strengths and limits. Link each limit to equal weighting, window length or sample tail data.
• If the position has options, the right answer usually involves full revaluation under each scenario.