FRM Part II · FRM Exam Part II

Portfolio Construction: formula sheet

Full chapter guide

Key formulas

Portfolio expected return: E(Rp) = Σ wᵢ × E(Rᵢ) = w′μ; Weights wᵢ usually sum to 1 under the full-investment constraint.
Portfolio variance: σp² = w′Σw = Σᵢ Σⱼ wᵢ wⱼ σᵢⱼ; Σ is the covariance matrix; σᵢⱼ = ρᵢⱼ × σᵢ × σⱼ.
Two-asset variance: σp² = w₁²σ₁² + w₂²σ₂² + 2 w₁ w₂ ρ σ₁ σ₂; Lower correlation gives lower portfolio variance for the same weights.
Global minimum variance weight (two assets): w₁ = (σ₂² − ρσ₁σ₂) ÷ (σ₁² + σ₂² − 2ρσ₁σ₂); w₂ = 1 − w₁. Valid with full investment and shorting allowed.
Mean-variance utility: U = E(Rp) − (λ ÷ 2) × σp²; Higher λ means more risk aversion and a lower-risk optimal portfolio.
Sharpe ratio: SR = (E(Rp) − Rf) ÷ σp; The tangency portfolio maximizes this ratio when a risk-free asset exists.
Risk parity condition: wᵢ × (Σw)ᵢ ÷ σp = same for all assets i; Each asset contributes equal risk. With zero correlations, weights are proportional to 1 ÷ σᵢ.
Constrained optimisation: max w'μ − (λ ÷ 2) w'Σw subject to constraints such as w ≥ 0, Σw = 1, Σ|w − w0| ≤ T; w0 is the current portfolio and T the turnover cap. Constraints can only lower the objective or leave it unchanged.
Turnover: Turnover = ½ × Σ|w_new − w_old|; One-way turnover uses the half. Some questions define it as the full sum, so check the definition given.
Transaction cost drag: Net return = Gross return − (Turnover × cost per unit traded); Cost must be applied to the amount actually traded, on both buy and sell sides if stated.
Leverage ratio: Leverage = Gross exposure ÷ Equity capital; Portfolio return = L × asset return − (L − 1) × funding cost, for a simple borrowed position.
Information ratio impact of constraints: IR_constrained ≤ IR_unconstrained; The transfer coefficient measures how much of the unconstrained efficiency is kept after constraints.
Risk-aversion coefficient: δ = (E(R_m) − R_f) ÷ σ_m²; Market excess return divided by market variance. Use excess returns and the market portfolio's variance.
Implied equilibrium excess returns (reverse optimization): Π = δ × Σ × w_mkt; Σ is the covariance matrix, w_mkt the market-cap weights. Π is a vector of excess returns.
Unconstrained optimal weights: w* = (δΣ)⁻¹ × μ; With μ = Π you recover w_mkt. With no views, Black-Litterman returns the market portfolio.
Views: P × μ = Q + ε, with ε ~ N(0, Ω); P picks the assets in each view, Q is the view return, Ω is the view uncertainty. For a relative view, the entries in a row of P sum to zero.
Posterior expected returns (precision-weighted form): μ_BL = [(τΣ)⁻¹ + P'Ω⁻¹P]⁻¹ × [(τΣ)⁻¹Π + P'Ω⁻¹Q]; Prior and views weighted by their precision (inverse variance).
Posterior expected returns (update form): μ_BL = Π + τΣP' × (P τΣ P' + Ω)⁻¹ × (Q − PΠ); Equilibrium plus an adjustment based on how far the view is from the prior. For one asset with P = 1: μ_BL = Π + [τσ² ÷ (τσ² + Ω)] × (Q − Π).
Portfolio volatility: σp = √(wᵀ Σ w); w is the weight vector and Σ is the covariance matrix. For two assets: σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂.
Marginal contribution to risk (MCR): MCRᵢ = ∂σp/∂wᵢ = (Σw)ᵢ ÷ σp = Cov(Rᵢ, Rp) ÷ σp; Equivalent form: βᵢ × σp, where βᵢ is asset i's beta to the portfolio.
Component contribution to risk (CCR): CCRᵢ = wᵢ × MCRᵢ, and Σ CCRᵢ = σp; Percentage contribution = CCRᵢ ÷ σp = wᵢ × βᵢ. Percentages sum to 100%.
Marginal VaR (normal): Marginal VaRᵢ = z × MCRᵢ; z is the confidence-level quantile, e.g. 1.645 at 95% and 2.326 at 99%. Use the same time horizon as σ.
Component VaR: Component VaRᵢ = wᵢ × Marginal VaRᵢ (if weights, in currency: positionᵢ × marginal VaR per unit); Component VaRs sum to total portfolio VaR. Valid under the normal, linear model.
Risk parity condition: wᵢ × (Σw)ᵢ = wⱼ × (Σw)ⱼ for all i, j; Equal risk contribution. With zero correlations, wᵢ ∝ 1/σᵢ.
Risk budget check: Optimal when (excess returnᵢ ÷ MCRᵢ) is equal across assets; Return per unit of marginal risk should be equal. Otherwise shift risk toward the higher ratio.
Active return (alpha): α = Rp − Rb; Portfolio return minus benchmark return. A beta-adjusted alpha subtracts β × (Rm − Rf) instead if the question says so.
Tracking error: TE (ω) = standard deviation of (Rp − Rb); Annualise monthly TE by multiplying by √12, and quarterly by √4.
Information ratio: IR = α ÷ ω = mean active return ÷ tracking error; Use the same time basis for numerator and denominator.
Sharpe ratio: SR = (Rp − Rf) ÷ σp; Uses the risk-free rate and total volatility, not a benchmark.
Fundamental law: IR = IC × √BR; Assumes unconstrained, independent bets and consistent skill.
Law with transfer coefficient: IR = TC × IC × √BR; TC is between 0 and 1. Lower TC means constraints reduce realised IR.
Expected alpha from IR: Expected α = IR × TE; Rearranged IR formula. Use it to size alpha for a given risk budget.
Single-factor/multifactor return: Rp = α + β1·F1 + β2·F2 + … + βk·Fk + ε; Fk are factor returns, βk are exposures, ε is idiosyncratic return.
Portfolio factor exposure: βp,k = Σ wi · βi,k; Weights times each holding's exposure to factor k. Use active weights (portfolio minus benchmark) for active exposure.
Long-short factor return: RLS = R(long leg) − R(short leg); Approximately market-neutral. Excludes financing and borrowing costs unless stated.
Systematic variance: σp² = β′ Σf β + σε²; Σf is the factor covariance matrix. Two factors: β1²σ1² + β2²σ2² + 2β1β2σ1σ2ρ.
Share of risk from factors: Factor share = systematic variance ÷ total variance; Equals R² of the factor regression when estimated that way.

Quick revision

Portfolio variance for two assets: w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2.
Mean-variance optimization is highly sensitive to expected return inputs; small changes can produce large weight shifts.
Estimation error often leads optimizers to produce concentrated, extreme positions.
Constraints such as long-only, position limits and turnover limits stabilize weights but can reduce theoretical efficiency.
Transaction costs and market impact reduce net returns and argue for less frequent or smaller rebalancing.
Black-Litterman starts from market-implied equilibrium returns and tilts them toward investor views.
Higher confidence in a view moves the posterior returns closer to that view.
Risk contribution of an asset = weight × marginal contribution to risk; contributions sum to portfolio risk.
Risk parity targets equal risk contributions, not equal capital weights, and often needs leverage to reach return targets.
Information ratio = active return ÷ tracking error.
Fundamental law: IR ≈ IC × √breadth, assuming independent bets, and a transfer coefficient below one reduces realized IR.
Factor portfolios target systematic return drivers such as value, momentum or low volatility, and factor premia can underperform for long periods.

Common mistakes

Forgetting the correlation term or using 2ρσ₁σ₂ without the weights Fix: Write all three terms every time: w₁²σ₁², w₂²σ₂² and 2w₁w₂ρσ₁σ₂.
Reporting variance as if it were volatility Fix: Check whether the question asks for σp or σp². Take √ before answering volatility.
Saying a constraint always changes the optimal portfolio. Fix: Check whether the unconstrained solution already satisfies the limit before recomputing.
Applying transaction costs to the whole portfolio value. Fix: Apply the cost only to the amount traded, which is the weight change times portfolio value.
Treating a larger Ω as higher confidence in a view. Fix: Ω is uncertainty. Larger Ω means a weaker view and a posterior closer to equilibrium. Smaller Ω means a stronger view.
Using total returns instead of excess returns when reverse optimizing. Fix: Subtract the risk-free rate first. Π = δΣw_mkt gives excess returns. Add Rf back only if the question asks for total returns.
Treating equal weight as equal risk, or equating risk parity with equal weight. Fix: Risk parity equalises wᵢ × MCRᵢ. Equal weight only equals risk parity if all assets have the same volatility and the same pairwise correlations.
Using wᵢ ÷ σᵢ-style weights (1/σ) when correlations are not zero. Fix: State that inverse volatility is exact only with zero correlations (or equal correlations across all pairs). Otherwise solve using the covariance matrix.
Using total volatility instead of tracking error in the IR denominator. Fix: IR always uses the standard deviation of active return. Sharpe uses the standard deviation of portfolio return.
Scaling breadth linearly, so doubling breadth doubles IR. Fix: Doubling breadth multiplies IR by √2 ≈ 1.41. Doubling IC doubles IR.

Exam tips

Expect questions asking which input change most affects optimal weights. The usual answer is expected returns.
Know that constraints lower the frontier but can reduce estimation-error damage.
Be ready to contrast MVO, minimum variance and risk parity by inputs used: returns, volatilities and correlations.
In numeric questions, check the answer options for variance versus volatility traps.
Remember that MVO uses variance, so it does not capture skewness or tail risk.
Start every answer with direction: constraints never improve the in-sample optimum.
Read the turnover definition. Questions often test one-way versus two-way.
Look for the phrase binding. If it is not binding, the answer is unchanged.