Portfolio Risk: Analytical Methods: formula sheet

Key formulas

Portfolio variance (two assets)

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

Use weights with return volatilities, or dollar positions with dollar volatilities. Do not mix them.

Portfolio variance (matrix form)

σp² = wᵀ Σ w

Σ is the covariance matrix. Use this form for three or more assets.

Relative VaR (to the mean)

VaR(relative) = z × σp × V

z is the one-tailed standard normal value. V is portfolio value.

Absolute VaR (to zero)

VaR(absolute) = (z × σp − μp) × V

μp is the expected portfolio return over the same horizon. A positive mean lowers absolute VaR.

Common z-values (one-tailed)

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

Use one-tailed values. 1.96 is the 95% two-tailed value and is the 97.5% one-tailed value.

Horizon scaling

σ(T) = σ(1) × √T and μ(T) = μ(1) × T

The √T rule assumes independent, identically distributed returns.

Undiversified VaR bound

VaR(p) ≤ VaR1 + VaR2 when ρ ≤ 1; equality when ρ = 1

For normal returns, portfolio VaR equals the sum of stand-alone VaRs only if all correlations are 1 (with a zero mean).

Portfolio VaR (normal)

VaR = z × σp × W, with σp² = Σi Σj wi wj σi σj ρij

z is the normal quantile (1.645 at 95%, 2.326 at 99%). W is portfolio value. Use weights that sum to 1.

Beta of position i to the portfolio

βi = Cov(Ri, Rp) ÷ σp², where Cov(Ri, Rp) = Σj wj σi σj ρij

Σi wi βi = 1. The weighted average beta of the portfolio is always 1.

Marginal VaR (per unit of currency)

Marginal VaR_i = ∂VaR/∂(position i) = z × Cov(Ri, Rp) ÷ σp = (VaR ÷ W) × βi

Gives the VaR change for one extra unit of position i. Some texts quote it per unit of weight; check the unit.

Component VaR

Component VaR_i = position_i × Marginal VaR_i = VaR × wi × βi

Same currency as VaR. Can be negative for hedges.

Additivity (Euler) property

Σi Component VaR_i = Portfolio VaR

Use this to check your work. Percent contribution = wi × βi.

Incremental VaR

Incremental VaR = VaR(portfolio with change) − VaR(portfolio before)

Exact for any size of change. Needs two full VaR calculations.

Delta VaR approximation

Incremental VaR ≈ Marginal VaR × change in position

Good for small changes only. It equals the component VaR scaled by the fractional change in the position.

Marginal VaR (normal)

Marginal VaR_i = z × Cov(R_i, R_p) ÷ σ_p = z × β_i × σ_p

β_i is the beta of position i against the portfolio. z is the confidence multiplier, for example 1.645 at 95%.

Component VaR

Component VaR_i = w_i × Marginal VaR_i

Here w_i is the position's weight, or its dollar amount if marginal VaR is per dollar. Sum of components = portfolio VaR.

Percent contribution to risk

% contribution_i = Component VaR_i ÷ Portfolio VaR = w_i × β_i

Contributions sum to 100% with betas measured against the portfolio.

Optimality condition

(E[R_i] − R_f) ÷ Marginal VaR_1 = (E[R_i] − R_f) ÷ Marginal VaR_2 = … for all i

Excess return per unit of marginal VaR is equal across all positions at the optimum. Equivalent to equal (E[R_i] − R_f) ÷ β_i under normality.

Risk parity rule

Component VaR_i = Portfolio VaR ÷ N for all N positions

A special risk budget with equal contributions. It is not the same as the optimality condition.

Active return

Rₐ = R_P − R_B

Portfolio return minus benchmark return for the same period.

Active weights

wₐ,i = w_P,i − w_B,i

Sum to zero if both portfolio and benchmark are fully invested.

Tracking error (two-step)

TE = σ(Rₐ) = √(σ_P² + σ_B² − 2ρσ_Pσ_B)

ρ is the correlation between portfolio and benchmark returns. Equals √(wₐ′Σwₐ).

Tracking error (active weights)

TE = √(wₐ′ Σ wₐ) = √(ΣᵢΣⱼ wₐ,i wₐ,j σᵢⱼ)

Use the covariance matrix of asset returns. Result is in the same time unit as the covariances.

Relative VaR (normal)

Relative VaR = z × TE × V

Assumes zero mean active return. z = 1.645 at 95%, 2.326 at 99% (one-tailed). V is portfolio value.

Information ratio

IR = E(Rₐ) ÷ TE

Both measured over the same period, usually annualised.

Time scaling

TE(annual) = TE(monthly) × √12

Valid when active returns are independent over time.

Single-factor model

R_i = α_i + β_i × F + ε_i

ε_i has mean zero and is uncorrelated with F and with other assets' residuals.

Single-factor variance of an asset

σ_i² = β_i² × σ_F² + σ_εi²

First term is systematic, second is idiosyncratic.

Portfolio beta

β_p = Σ w_i × β_i

Applies to each factor separately. Weights include shorts as negative values.

Portfolio variance, single factor

σ_p² = β_p² × σ_F² + Σ w_i² × σ_εi²

Assumes uncorrelated residuals.

Multi-factor portfolio variance

σ_p² = β_p' Σ_F β_p + Σ w_i² × σ_εi²

Σ_F is the factor covariance matrix. With two factors the systematic part is β1²σ1² + β2²σ2² + 2β1β2ρσ1σ2.

Systematic share of risk (R²)

R² = β² σ_F² ÷ σ²

Idiosyncratic share is 1 − R².

Equal-weight idiosyncratic variance

σ_ε²(p) = σ_ε² ÷ n

Holds for n equally weighted assets with equal, uncorrelated residual variance.

Two-asset portfolio VaR

VaR_p = √(VaR_1² + VaR_2² + 2ρ × VaR_1 × VaR_2)

Normal returns, same confidence level, zero mean. Equals VaR_1 + VaR_2 only when ρ = 1.

Delta-normal parametric VaR (normal, zero mean)

VaR = z(α) × σ(P) × V

z = 1.645 at 95% and 2.326 at 99% (one-tailed). σ(P) is portfolio return volatility, V is portfolio value.

Delta approximation for an option

ΔP ≈ δ × ΔS

Linear in the underlying move. Ignores curvature, so it errs for large moves.

Delta-gamma approximation

ΔP ≈ δ × ΔS + ½ × Γ × (ΔS)²

Positive gamma adds to P&L in both directions. Negative gamma subtracts from it.

Expected shortfall under normality

ES = σ × φ(z) ÷ (1 − α) (zero mean)

φ is the standard normal density. At 99%, ES ≈ 2.665σ versus VaR ≈ 2.326σ.

Expected shortfall definition

ES(α) = E[Loss | Loss ≥ VaR(α)]

Always at least as large as VaR at the same confidence level.

Quick revision

• Portfolio VaR = z × σp × portfolio value, scaled by √t for the horizon (assuming independent returns).
• z-values for one-tailed normal VaR: 1.645 at 95% and 2.326 at 99%.
• Two-asset σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2.
• Lower correlation means lower portfolio VaR; diversification benefit = sum of standalone VaRs − portfolio VaR.
• Marginal VaR is the change in portfolio VaR for a small change in a position.
• Component VaR = position × marginal VaR, and the components sum to total VaR.
• Incremental VaR is the full change in VaR from adding or removing a position, and it can differ from component VaR for large trades.
• Optimal allocation: excess return ÷ marginal VaR is equal across positions.
• Tracking error is the standard deviation of active return (portfolio minus benchmark).
• Information ratio = active return ÷ tracking error.
• Factor model: total variance = systematic variance + idiosyncratic variance.
• Analytical VaR assumes normality and linear exposures, so it understates tail risk and option risk. Stress tests cover what it misses.

Common mistakes

• Adding the stand-alone VaRs of the assets to get portfolio VaR. Fix: Always combine standard deviations with the correlation term, then apply z once.
• Using weights and volatilities in the formula and then forgetting to multiply by portfolio value. Fix: Finish every answer with × V and write the currency unit.
• Treating marginal VaR and component VaR as the same thing. Fix: Marginal VaR is per unit and does not add up. Component VaR is marginal VaR times position size, and it adds up to total VaR.
• Assuming incremental VaR equals component VaR. Fix: Component VaR is a decomposition of current VaR. Incremental VaR is the actual change after adding or removing the position. They match only approximately, for small changes.
• Dividing total expected return, not excess return, by marginal VaR. Fix: Always subtract the risk-free rate first.
• Using standalone VaR instead of marginal VaR in the ratio. Fix: Standalone VaR ignores diversification. Use marginal or component VaR, which reflect correlation with the portfolio.
• Using portfolio weights instead of active weights in the covariance formula. Fix: Subtract benchmark weights first. Then use wₐ′Σwₐ.
• Treating tracking error as the same thing as information ratio. Fix: TE is a standard deviation (risk). IR is active return divided by TE (reward per unit risk).
• Adding volatilities instead of variances when combining systematic and idiosyncratic risk. Fix: Convert to variance, add, then take the square root.
• Forgetting to square the weights in the idiosyncratic term. Fix: Betas are linear in w. Residual variance uses w² because residuals are uncorrelated.

Exam tips

• Look at the last line of the question first. It tells you whether you need relative or absolute VaR and which horizon.
• Memorise 1.645 and 2.326 and the √T rule. Most numerical items need only these.
• Expect interpretation items: a 99% one-day VaR is exceeded about once in 100 days, and VaR says nothing about the size of losses beyond it.
• Know the assumptions: normal returns, stable correlations, linear exposures. Questions often ask which one fails for options or in a crisis.
• When a question lowers correlation or adds an asset, reason first: lower correlation lowers portfolio VaR, and that can save you a calculation.
• Questions often give betas or covariances with the portfolio. Go straight to VaR × weight × beta and skip the full variance calculation.
• Use the sum rule as a check. Component VaRs must add to portfolio VaR, and weighted betas must add to 1.
• Read the wording. Add or remove a position points to incremental VaR. Share of risk or decomposition points to component VaR. Per extra dollar points to marginal VaR.