Calculating and Applying VaR: formula sheet

Key formulas

VaR definition

P(Loss > VaR) = 1 − confidence level

VaR is the loss quantile. At 99%, there is a 1% chance of a larger loss over the horizon.

Normal VaR (percentage)

VaR% = z × σ − μ

σ and μ are for the chosen horizon. Use z = 1.645 (95%), 2.326 (99%). With μ = 0, VaR% = z × σ.

Normal VaR (currency)

VaR = Portfolio value × (z × σ − μ)

State it as a positive loss number.

Square-root-of-time rule

VaR(T days) = VaR(1 day) × √T

Valid for i.i.d. returns with zero mean (or normal with mean ignored). Volatility scales the same way: σ(T) = σ(1) × √T.

Scaling volatility to annual

σ(annual) = σ(daily) × √252

252 trading days is a common convention; use the number the question gives.

Change of confidence level (normal)

VaR(new) = VaR(old) × z(new) ÷ z(old)

Only valid under normality and zero mean.

Single-asset parametric VaR (return form)

VaR = (z × σ − μ) × V

Use z = 1.645 for 95% and 2.326 for 99% (one-tailed). μ and σ must be for the same horizon as the VaR.

Zero-mean shortcut

VaR = z × σ × V

Use when the question says to ignore the mean or assumes μ = 0, as is common for daily VaR.

Square-root-of-time scaling

σ(T) = σ(1) × √T; VaR(T) = VaR(1) × √T

Valid for i.i.d. returns with zero mean. If the mean is not zero, scale μ by T and σ by √T separately.

Two-asset portfolio variance

σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ σ1 σ2

Weights are fractions of portfolio value. Take the square root to get σp.

Two-asset portfolio VaR from individual VaRs

VaRp = √(VaR1² + VaR2² + 2ρ × VaR1 × VaR2)

Holds for zero-mean normal returns with the same confidence level and horizon. If ρ = 1, VaRp = VaR1 + VaR2.

Annual to daily volatility

σ(daily) = σ(annual) ÷ √252

Use the trading-day count given in the question; 252 is the common convention.

Delta-normal VaR of a linear position

VaR ≈ |Δ| × S × z × σ(return)

Δ is the delta of the position and S the underlying price. Ignores gamma.

Basic HS VaR

VaR(c) = loss at the (1 − c) quantile of the sorted P&L list

With N observations, about N × (1 − c) observations are worse than VaR. Follow the convention the question gives for choosing the exact observation.

BRW age weight

w(i) = λ^(i − 1) × (1 − λ) ÷ (1 − λ^n)

i = 1 is the most recent observation, n is the sample size, 0 < λ < 1. Weights sum to 1. λ closer to 1 means slower decay; λ = 1 gives equal weights of 1/n.

Volatility-adjusted return

r* (t,i) = r(t,i) × σ(T) ÷ σ(i)

σ(i) is the volatility estimated for day i (for example by EWMA or GARCH). σ(T) is the current volatility. Scale up if volatility is now higher.

Weighted VaR (BRW)

Sort losses from worst; add weights until cumulative weight ≥ 1 − c; that loss is VaR

Replaces counting observations with summing weights. Interpolation between adjacent losses may be used if the question asks for it.

Number in the tail

Tail observations ≈ N × (1 − c)

For example, 500 × 1% = 5 observations in the 1% tail.

VaR from simulated quantile

VaR(c) = − (the (1 − c) quantile of simulated P&L)

With N scenarios sorted worst first, the loss is roughly the k-th worst, where k = N × (1 − c). Quote VaR as a positive loss.

Sampling error scaling

Standard error ∝ 1 ÷ √N

Quadrupling N halves the standard error. This holds for the simulation error only, not for model error.

Correlated normal draws

x = L × z, where Σ = L × Lᵀ and z is a vector of independent standard normals

L is the Cholesky factor of the covariance matrix Σ. This gives the draws the assumed correlations.

Scaling a normal draw to returns

R = μ + σ × z

z is a standard normal draw. For short horizons μ is often set to zero.

Lognormal price step (one step)

S(t+Δt) = S(t) × exp[(μ − σ² ÷ 2) × Δt + σ × √Δt × z]

Used to simulate stock prices under geometric Brownian motion. Prices stay positive.

Expected shortfall definition

ES(α) = E[ L | L ≥ VaR(α) ]

L is the loss. α is the confidence level, e.g. 95% or 99%. It is the average loss in the tail beyond VaR.

Normal VaR (loss as positive number)

VaR(α) = μ + σ × z(α)

z(95%) = 1.645, z(99%) = 2.326. Use μ = 0 if the question assumes zero mean.

Normal ES

ES(α) = μ + σ × φ(z(α)) ÷ (1 − α)

φ is the standard normal density. φ(1.645) ≈ 0.1031 and φ(2.326) ≈ 0.0267. So ES factor = 2.063 at 95% and 2.665 at 99%.

Discrete ES

ES = (sum of the worst losses in the tail) ÷ (number of tail outcomes)

With n equally likely scenarios, the tail has n × (1 − α) outcomes. Average them.

Coherence axioms

Monotonicity; Subadditivity: ρ(X + Y) ≤ ρ(X) + ρ(Y); Positive homogeneity: ρ(λX) = λρ(X) for λ > 0; Translation invariance: ρ(X + c) = ρ(X) − c

Here X and Y are returns or P&L, with cash c added. Learn the intent of each, not only the symbols.

Delta approximation

ΔV ≈ δ × ΔS

Linear in the underlying move. Use for small moves. Portfolio delta is the sum of position deltas on the same underlying.

Delta-gamma approximation

ΔV ≈ δ × ΔS + ½ × Γ × (ΔS)²

Second-order Taylor expansion. Gamma is positive for long options and negative for short options.

Delta-normal VaR for an option

VaR = |δ| × S × z × σ (σ is the volatility of the return over the horizon)

Worst-case move in S at the confidence level, times |δ|. z is 1.645 at 95% and 2.326 at 99% (one-tailed).

Bond duration-convexity approximation

ΔP/P ≈ −D_mod × Δy + ½ × C × (Δy)²

Δy in decimals. Same logic as delta-gamma.

Full revaluation loss

Loss = V(S₀) − V(S*)

S* is the shocked risk factor. Reprice with the full model.

Exception probability

p = 1 − confidence level

For 99% VaR, p = 0.01. For 95% VaR, p = 0.05.

Expected number of exceptions

E(X) = n × p

250 days at 99% gives 2.5 exceptions.

Binomial probability of x exceptions

P(X = x) = C(n, x) × p^x × (1 − p)^(n − x)

Assumes exceptions are independent with constant probability.

Standard deviation of exceptions

σ = √(n × p × (1 − p))

Used in the normal approximation to the binomial.

Z-score for exception count

z = (x − n × p) ÷ √(n × p × (1 − p))

Compare with the critical value, such as 1.96 for a two-sided 5% test.

Basel traffic light zones

Green: 0-4; Yellow: 5-9; Red: 10 or more exceptions (99% VaR, 250 days)

Capital multiplier increases from green to yellow to red.

Kupiec unconditional coverage idea

Reject the model if the exception count is too high or too low

Tests whether the exception frequency equals p. It does not test whether exceptions cluster.

Quick revision

• VaR = loss threshold at a stated confidence level and horizon; it says nothing about the size of losses beyond it.
• One-tailed z-values: 1.645 at 95%, 2.326 at 99%.
• Square-root-of-time scaling: σ over t days = σ daily × √t, assuming independent returns and constant volatility.
• Parametric VaR assumes normal returns and linear exposure to risk factors.
• Historical simulation uses actual past changes; no distribution is assumed, but the past must represent the future.
• Monte Carlo is the most flexible and the most computationally costly method; results depend on the model.
• Expected shortfall is the average loss given that the loss exceeds VaR.
• Coherent measures satisfy monotonicity, translation invariance, positive homogeneity and subadditivity.
• VaR is not always subadditive; ES is subadditive.
• Delta-only VaR can misstate risk for options; gamma adds curvature and full revaluation reprices the position.
• Backtesting compares exceptions with the expected count: at 99% over 250 days, about 2.5 exceptions are expected.
• Too many exceptions suggest the model understates risk.

Common mistakes

• Scaling VaR by the number of days instead of its square root. Fix: Independent daily shocks partly offset, so volatility grows with √T. Multiply by √10, about 3.162, for 10 days.
• Reading 99% VaR as the maximum possible loss. Fix: VaR is a threshold. Losses exceed it with probability 1 − confidence. It says nothing about the size of those larger losses.
• Using the two-tailed z-score (1.96) for 95% VaR. Fix: VaR is a one-tailed loss measure. Use 1.645 for 95% and 2.326 for 99%.
• Adding individual VaRs to get portfolio VaR. Fix: Add only if ρ = 1. Otherwise use the correlation formula, which gives a smaller number.
• Picking the wrong observation in the tail, for example the 1st worst loss for 99% VaR with 500 data points. Fix: Compute N × (1 − c) first. For 500 observations at 99%, that is 5 in the tail. Then apply the convention in the question.
• Labelling the oldest observation as i = 1 in the BRW formula. Fix: In BRW, i = 1 is the most recent observation and has the largest weight. Check that weights fall as you go back in time.
• Saying Monte Carlo makes no distributional assumption. Fix: Remember that Monte Carlo scenarios come from an assumed model. Historical simulation draws from actual past data.
• Believing more simulations remove model risk. Fix: More runs reduce only sampling error. A wrong model or wrong correlation stays wrong however large N is.
• Saying ES is smaller than or equal to VaR. Fix: ES averages losses beyond VaR, so it is always at least as large as VaR at the same confidence level.
• Claiming VaR is never subadditive. Fix: VaR is not always subadditive. It fails in some cases, such as non-elliptical distributions, but it is subadditive for normal distributions.

Exam tips

• Always check the stated assumptions (normal, i.i.d., zero mean) before using √T scaling.
• Memorise 1.645 and 2.326, and be ready to get the z-ratio between 95% and 99% (2.326 ÷ 1.645 ≈ 1.414).
• Read the final sentence of the question for the target horizon and confidence level before calculating.
• Conceptual questions often test what VaR does not tell you: loss beyond the threshold.
• Work in the same unit throughout: percent or currency, daily or annual.
• Memorise 1.645 and 2.326. Questions rarely give a z-table.
• Read whether the question says to ignore the mean. If not and μ is given, subtract it.
• Check the time units before calculating. Annual to daily conversions are a favourite trap.