Measures of Financial Risk: formula sheet

Key formulas

VaR definition

P(Loss > VaR) = 1 − confidence level

VaR is the quantile of the loss distribution at the chosen confidence level.

Normal VaR (return form)

VaR = (z × σ − μ) × Portfolio value

σ and μ must be for the same horizon as the VaR. Many exam questions set μ = 0.

Common z-values (one-tailed)

90%: 1.28 | 95%: 1.645 | 97.5%: 1.96 | 99%: 2.326

Use one-tailed values for VaR. Using 1.96 for 95% is a classic error.

Square-root-of-time rule

VaR(t days) = VaR(1 day) × √t

Valid for i.i.d. returns with zero mean. Scale σ, not the mean.

Annual to daily volatility

σ(daily) = σ(annual) ÷ √252

Use the trading-day count given in the question; 250 or 252 are common.

Historical simulation quantile

VaR = loss at rank (1 − c) × N in the sorted loss list

With 500 scenarios at 99%, VaR is around the 5th worst loss. Conventions on interpolation vary, so follow the question.

Linear (delta) approximation

ΔP ≈ Δ × ΔS, so VaR(position) ≈ |Δ| × VaR(underlying)

Delta-normal ignores gamma, so it can misstate option VaR.

Definition of ES

ES(α) = E[L | L ≥ VaR(α)]

L is loss, shown as a positive number. α is the confidence level, such as 95% or 99%. The tail has probability 1 − α.

Discrete or historical ES

ES(α) = Σ (tail loss × its probability) ÷ (1 − α)

For N equally likely historical observations, ES is the simple average of the worst N × (1 − α) losses. Dividing by the tail probability is essential.

Normal ES (losses)

ES(α) = μ + σ × φ(z_α) ÷ (1 − α)

z_α = Φ⁻¹(α) and φ is the standard normal density, φ(z) = (1 ÷ √(2π)) × e^(−z²÷2). μ and σ are the mean and standard deviation of the loss.

Normal VaR for comparison

VaR(α) = μ + σ × z_α

Common z values: 1.645 at 95% and 2.326 at 99%.

Normal ES multipliers (μ = 0)

95%: ES ≈ 2.063σ; 99%: ES ≈ 2.67σ

Memorise these. They come from φ(1.645) ≈ 0.1031 and φ(2.326) ≈ 0.0267. Add μ to get the general case.

Ordering of measures

ES(α) ≥ VaR(α)

Holds at the same confidence level for the same loss distribution.

Monotonicity

If X ≤ Y in every state (X has lower losses), then ρ(X) ≤ ρ(Y)

Check the direction in terms of losses. Worse outcomes mean higher risk.

Subadditivity

ρ(X + Y) ≤ ρ(X) + ρ(Y)

The key property. VaR can violate it. ES cannot.

Positive homogeneity

ρ(kX) = k × ρ(X), for k > 0

Ignores liquidity effects, which is a known criticism.

Translation invariance

ρ(X + c) = ρ(X) − c, where c is a sure gain (cash added)

If written with losses, a sure loss of c increases ρ by c.

Expected shortfall

ES = average of losses that are greater than or equal to VaR at the same confidence level

Coherent. ES is always at least as large as VaR at the same confidence level.

Coherent measure

Coherent = monotonicity + subadditivity + positive homogeneity + translation invariance

All four must hold.

Spectral risk measure

M = ∫₀¹ φ(p) q(p) dp

q(p) is the loss quantile at probability p. φ(p) is the risk aversion weighting function.

Conditions for a coherent spectral measure

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

Non-decreasing weights mean worse losses never get less weight. This gives subadditivity.

Expected shortfall as a spectral measure

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

Equal weights on the tail quantiles. Coherent.

VaR as a limiting case

φ(p) = spike at p = α (all weight on q(α))

Not non-decreasing, so VaR is not coherent in general.

Discrete approximation

M ≈ Σ wᵢ qᵢ, with wᵢ ≥ 0, Σ wᵢ = 1, w₁ ≤ w₂ ≤ … ≤ wₙ

Quantiles qᵢ are ordered from best to worst. Weights rise with the loss.

Standard deviation

σ = √[ Σ(Rᵢ − R̄)² ÷ (n − 1) ]

Sample version uses n − 1. Symmetric: gains and losses count equally.

Downside deviation

DD = √[ Σ min(Rᵢ − MAR, 0)² ÷ n ]

Only returns below the minimum acceptable return (MAR) contribute. Divide by the total number of observations, not just the bad ones, unless the question says otherwise.

Normal VaR

VaR = −μ + z × σ (as a loss)

z = 1.645 at 95% and 2.326 at 99% (one-tailed). Valid only under normality.

Horizon scaling

σ(T days) = σ(1 day) × √T

Assumes independent returns with constant volatility.

Expected shortfall

ES = E[ Loss | Loss ≥ VaR ]

Always at least VaR at the same confidence level.

Coherence

Monotonicity, translation invariance, positive homogeneity, subadditivity

ES is coherent. VaR fails subadditivity in general. Standard deviation fails monotonicity.

Standard error of a quantile estimate

se(q̂) = √[p(1−p) ÷ n] ÷ f(q)

p is the cumulative probability of the quantile (e.g. 0.95 for the 95% loss quantile). f(q) is the pdf of the loss at q. This is an approximation that works for large n.

Confidence interval for a quantile

q̂ ± z × se(q̂)

Use z = 1.96 for 95% confidence and z = 1.645 for a 90% two-sided interval. The interval is approximate.

Normal VaR quantile

q = μ + z_p × σ

z_p is 1.645 at 95% and 2.326 at 99%. Losses are measured as positive numbers.

Normal pdf at z

φ(z) = exp(−z² ÷ 2) ÷ √(2π); f(q) = φ(z) ÷ σ

Divide by σ when converting from the standard normal to a loss with standard deviation σ. φ(1.645) ≈ 0.1031 and φ(2.326) ≈ 0.0267.

ES as average of tail VaRs

ES_α ≈ average of VaR at α + (1−α) × i ÷ (k+1), for i = 1 to k

These are k equally spaced interior tail levels, not slice midpoints. A larger k gives a more accurate ES. The midpoint variant uses the levels α + (1−α) × (2i−1) ÷ (2k) instead; follow the question.

ES from a sample

ES ≈ average of the losses greater than the VaR estimate

With n losses and level α, this is roughly the average of the worst n × (1−α) losses.

Quick revision

• VaR is a loss quantile at a given confidence level and horizon; it says nothing about losses beyond that point.
• Normal VaR (loss) = μ + z × σ; with zero mean, VaR = z × σ.
• One-tailed normal z values: 1.645 at 95% and 2.326 at 99%.
• Scaling to T days with iid returns and zero mean: VaR(T) = VaR(1) × √T.
• ES is the expected loss given that the loss exceeds VaR; at the same confidence level ES ≥ VaR.
• Coherent means monotonicity, translation invariance, positive homogeneity and subadditivity.
• Subadditivity: risk of a combined portfolio ≤ sum of the separate risks; it reflects diversification.
• VaR is not always subadditive; ES is coherent.
• Spectral measures are weighted averages of quantiles; they are coherent if the weights are non-negative and non-decreasing in the tail.
• Standard deviation treats gains and losses alike and is not a tail measure; it is not coherent in general.
• Quantile estimates are less precise further into the tail and with fewer observations.
• Check whether a question gives losses or returns before choosing the sign and tail.

Common mistakes

• Using a two-tailed z-value such as 1.96 for 95% VaR. Fix: VaR looks at one tail only. Use 1.645 for 95% and 2.326 for 99%.
• Forgetting to convert volatility to the VaR horizon. Fix: Divide annual σ by √252 for daily, then multiply by √t for longer horizons.
• Forgetting to divide by the tail probability (1 − α). Fix: The weighted sum covers only part of the distribution. Always divide by 1 − α so the tail probabilities act as weights that sum to 1.
• Using z_α instead of φ(z_α) ÷ (1 − α) in the normal formula. Fix: VaR multiplier is z_α. ES multiplier is φ(z_α) ÷ (1 − α). Memorise 2.063 and 2.67 for 95% and 99%.
• Saying VaR violates all four properties or is never coherent for any distribution. Fix: VaR fails only subadditivity, and only in some cases. For normal (elliptical) distributions it is subadditive.
• Confusing homogeneity with subadditivity. Fix: Homogeneity scales one portfolio by k. Subadditivity combines two different portfolios.
• Saying VaR is a spectral risk measure that is coherent. Fix: VaR puts all weight on one quantile, a spike followed by zero weight. That is a degenerate limiting weighting, not a proper non-decreasing spectral φ. So VaR is not a proper spectral risk measure, and it fails subadditivity in general.
• Allowing weights that fall as losses get worse. Fix: Coherence needs non-decreasing weights. Falling weights mean the measure under-weights extreme losses, which breaks subadditivity.
• Saying standard deviation measures only downside risk. Fix: Remember it counts upside deviations too. Use semi-deviation for downside only.
• Treating VaR as the maximum possible loss. Fix: VaR is a threshold at a confidence level. Losses beyond it can be far larger.

Exam tips

• Check whether the question gives annual or daily volatility before touching z. Unit mismatch is the most common trap.
• Know the contrast table cold: delta-normal is fast but assumes normality and linearity; historical simulation is model-free but relies on the past window; Monte Carlo is flexible but model- and compute-heavy.
• Questions often ask which change increases VaR. Higher confidence, longer horizon and higher volatility all do.
• Watch for 'relative VaR' versus 'absolute VaR'. Relative VaR is measured from the mean, while absolute VaR subtracts the mean return from z × σ.
• Use the calculator's square-root key for √t and keep at least four decimals until the final step.
• Expect the question to ask you to compare VaR and ES. The safe statements: ES ≥ VaR, ES captures tail severity, and ES is coherent while VaR can fail subadditivity.
• Memorise the two normal multipliers, 2.063 at 95% and 2.67 at 99%. They save time on every normal-distribution question.
• For historical data, always compute the tail count as N × (1 − α) before touching a calculator.