VaR and Risk Budgeting in Investment Management: formula sheet

Key formulas

Active return

Active return = Portfolio return − Benchmark return

Use the policy benchmark for sponsor-level questions and the manager benchmark for manager-level questions.

Tracking error (active risk)

TE = standard deviation of (Rp − Rb)

Also called active risk or relative risk. It is a standard deviation, not a VaR, unless you convert it.

Active risk from two components

σA² = σ1² + σ2² + 2ρσ1σ2

σ1 and σ2 are the active risks (or weighted active risks) of the two parts and ρ is their correlation. If ρ = 0, σA = √(σ1² + σ2²).

Relative VaR from tracking error

Relative VaR = z × TE × portfolio value

Assumes normal active returns with zero mean. z = 1.645 at 95% and 2.326 at 99% (one-tailed).

Information ratio

IR = Expected active return ÷ Tracking error

A higher IR means more active return per unit of active risk.

Component VaR

Component VaR(i) = wi × (marginal VaR of i); Σ component VaR = total VaR

Components add to total VaR, so they work as budget shares.

Active return

R_A = R_P − R_B

Portfolio return minus benchmark return.

Active weights

a_i = w_i(P) − w_i(B)

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

Tracking error (two assets)

TE = √(a₁²σ₁² + a₂²σ₂² + 2·a₁·a₂·ρ·σ₁·σ₂)

Include the covariance term. Active weights can be negative.

Tracking error (general)

TE = √(a′Σa)

Σ is the covariance matrix of asset returns.

TE from portfolio and benchmark

TE² = σ_P² + σ_B² − 2·ρ_PB·σ_P·σ_B

Useful when you are given volatilities and their correlation.

Absolute VaR (normal)

VaR = z × σ_P × V (less any expected-return term)

z = 1.645 at 95%, 2.326 at 99% one-tailed.

Relative VaR (tracking error VaR)

Relative VaR = z × TE × V

Same z, but TE replaces σ_P.

Time scaling

σ_T = σ_annual × √(T in years)

For example, monthly = annual ÷ √12. Assumes independent returns.

Information ratio link

IR = expected active return ÷ TE

TE is the denominator of the information ratio.

Portfolio VaR (normal)

VaR_p = z × σ_p × V

z is the confidence z-score (1.645 at 95%, 2.326 at 99%). V is portfolio value. Mean assumed zero unless told otherwise.

Beta of position to portfolio

β_i = Cov(R_i, R_p) ÷ σ_p²

Measures how much position i moves with the portfolio.

Marginal VaR

MVaR_i = z × Cov(R_i, R_p) ÷ σ_p = β_i × VaR_p ÷ V

VaR change per unit of money added to position i. Often quoted as β_i × (VaR_p ÷ V).

Component VaR

CVaR_i = MVaR_i × (w_i × V) = w_i × β_i × VaR_p

w_i is the portfolio weight. Can be negative for a hedging position.

Additivity

Σ CVaR_i = VaR_p

Holds because Σ w_i × β_i = 1. Use it as a check on your answer.

Percentage contribution

CVaR_i ÷ VaR_p = w_i × β_i

Compare with the weight w_i to see whether a position takes more or less than its share of risk.

Incremental VaR

IVaR_i = VaR_with position − VaR_without position

An exact change from a large trade. Approximately CVaR_i for small positions only.

Sharpe ratio

SR = (Rp − Rf) ÷ σp

Rf is the risk-free rate; σp is the standard deviation of portfolio returns.

Information ratio

IR = (Rp − Rb) ÷ TE

TE is tracking error, the standard deviation of (Rp − Rb). Numerator is the mean active return (alpha against the benchmark).

Tracking error

TE = σ(Rp − Rb)

Use the volatility of the active return, not the volatility of the portfolio.

Optimal risk budget condition

(Ri − Rf) ÷ MVaRi = same value for every position i

MVaRi is the marginal VaR of position i. If ratios differ, reallocate towards the higher ratio.

Marginal VaR link to beta

MVaRi = (VaRp ÷ Wp) × βi

βi is the beta of position i against the portfolio, and Wp is portfolio value. So the condition can be stated as excess return ÷ beta being equal.

Risk-adjusted value of a ratio

Annualised IR ≈ monthly IR × √12

Only valid if active returns are independent and identically distributed.

Fundamental law (basic)

IR = IC × √BR

BR is independent bets per year. IR is annualised if BR is per year.

Fundamental law with transfer coefficient

IR = TC × IC × √BR

TC is between 0 and 1. It measures how well forecasts convert into portfolio weights after constraints.

Information ratio

IR = α ÷ ω = active return ÷ tracking error

α is expected active return. ω is tracking error, the standard deviation of active return.

Information coefficient

IC = correlation(forecast return, realised return)

Ranges from -1 to +1. Positive and above zero shows skill.

Implied active return

α = IR × ω

Use this to get expected active return from a target tracking error.

Breadth scaling

BR needed = (IR ÷ IC)²

Rearranged law to solve for the number of independent bets.

Two-manager total VaR

VaR_p = √(VaR₁² + VaR₂² + 2ρ·VaR₁·VaR₂)

Assumes normal, elliptical-type behaviour and the same confidence level and horizon. If ρ = 1, VaR_p = VaR₁ + VaR₂.

Diversification benefit

Benefit = ΣVaRᵢ − VaR_p

Undiversified sum minus portfolio VaR. It is zero only when all pairwise correlations equal 1.

Component VaR

CVaRᵢ = wᵢ × βᵢ × VaR_p, where wᵢ = manager i's fraction of fund value and βᵢ = Cov(Rᵢ, R_p) ÷ σ_p²

Because Σwᵢβᵢ = 1, the component VaRs sum exactly to VaR_p. They can be negative for a hedging manager.

Marginal VaR (parametric)

Marginal VaRᵢ = z × Cov(Rᵢ, R_p) ÷ σ_p

Change in VaR for a small change in exposure to manager i.

Information ratio

IR = α ÷ tracking error

Used to compare risk budget efficiency. Optimal allocation sets alpha per unit of marginal risk equal across managers.

VaR scaling

VaR(T days) = VaR(1 day) × √T

Valid only when returns are independent with constant volatility.

Quick revision

• Absolute VaR measures loss in value; relative VaR measures loss against a benchmark.
• Tracking error is the standard deviation of active return (portfolio minus benchmark).
• Tracking error VaR = z × tracking error × portfolio value, for a chosen confidence level.
• Marginal VaR is the change in portfolio VaR for a small change in one position.
• Component VaR = position × marginal VaR, and component VaRs sum to total portfolio VaR.
• Incremental VaR is the change in VaR from adding or removing a whole position, so it is not the same as component VaR.
• A negative marginal VaR means the position acts as a hedge to the portfolio.
• At the optimal allocation, excess return ÷ marginal VaR is the same for every asset.
• Sharpe ratio = (Rp − Rf) ÷ σp; information ratio = active return ÷ tracking error.
• Fundamental law: IR ≈ IC × √breadth, where breadth means independent bets per year.
• More skill or more independent bets raises the information ratio; correlated bets do not add breadth.
• Diversifying across managers lowers total tracking error unless active bets are positively correlated.

Common mistakes

• Adding tracking errors of managers directly. Fix: Combine variances with correlation. Only a correlation of 1 makes risks add.
• Confusing policy risk with active risk. Fix: Policy risk is the risk of the strategic benchmark mix. Active risk is the deviation from a benchmark.
• Using the portfolio's own volatility for relative VaR. Fix: If the question mentions a benchmark or active risk, use TE, not σ_P.
• Using portfolio weights instead of active weights in the TE formula. Fix: Subtract benchmark weights first, then apply the same formula to the active weights.
• Treating marginal VaR and incremental VaR as the same thing. Fix: Marginal is a rate for a tiny change. Incremental is the actual change from a full add or removal. Marginal is an approximation tool.
• Expecting component VaRs to sum to the sum of stand-alone VaRs. Fix: Component VaRs sum to the diversified portfolio VaR. Stand-alone VaRs sum to a larger number unless correlations are all 1.
• Using portfolio volatility instead of tracking error in the information ratio. Fix: For IR, the denominator is the standard deviation of the difference between portfolio and benchmark returns.
• Using the benchmark return as Rf in the Sharpe ratio, or Rf as the benchmark in the IR. Fix: Sharpe subtracts the risk-free rate. IR subtracts the benchmark return.
• Using BR instead of √BR Fix: Always take √BR first. Write it as a separate line before multiplying.
• Counting correlated bets as breadth Fix: Breadth counts only independent bets. Correlated positions give less breadth than their count.

Exam tips

• Always read which benchmark the question uses. Policy and manager benchmarks give different active risks.
• Expect questions that ask you to pick the right interpretation, such as why total active risk is below the sum of manager limits.
• Memorise z values of 1.645 and 2.326 and check whether the question uses one-tailed values.
• When a question asks how to reallocate budget, think information ratio and component VaR, not just total return.
• Watch for wording that separates risk budgeting from asset allocation: risk budgeting allocates risk, not capital.
• Read for the word benchmark, active or relative. It tells you to use TE, not portfolio volatility.
• Check the horizon and confidence level in the final line of the question before computing.
• Compute TE² first and take the square root once. Rounding early often moves you to a wrong option.