FRM Part II · FRM Exam Part II

Portfolio Performance Evaluation: formula sheet

Full chapter guide

Key formulas

Sharpe ratio: SR = (Rp − Rf) ÷ σp; Rp = portfolio return, Rf = risk-free rate, σp = standard deviation of portfolio returns. Uses total risk.
Treynor ratio: TR = (Rp − Rf) ÷ βp; Uses systematic risk. Only meaningful for positive beta; compare among diversified portfolios.
Jensen's alpha: αp = Rp − [Rf + βp × (Rm − Rf)]; Excess return over the CAPM-required return. Positive alpha means outperformance for the beta taken.
M-squared (Modigliani): M² = Rf + SRp × σm, and M² difference = M² − Rm = (SRp − SRm) × σm; σm = market (benchmark) volatility. Result is a return, directly comparable with the benchmark return.
Information ratio: IR = (Rp − Rb) ÷ TE, where TE = standard deviation of (Rp − Rb); Rb = benchmark return. TE = tracking error. Numerator is active return.
Sortino ratio: Sortino = (Rp − MAR) ÷ DD; MAR = minimum acceptable return (often Rf or a target). DD = downside deviation, using only returns below MAR.
Link between Sharpe and Treynor: TR = SR × (σp ÷ βp); Rankings agree only if σp ÷ βp is similar, that is, if portfolios have similar diversification.
Active return: Rₐ = R_P − R_B; Portfolio return minus benchmark return for the same period.
Tracking error (ex-post): TE = √[ Σ(Rₐ,ₜ − R̄ₐ)² ÷ (n − 1) ]; Sample standard deviation of active returns. Use n − 1 unless the question says otherwise.
Information ratio: IR = R̄ₐ ÷ TE; Use mean active return and tracking error over the same horizon, annualised consistently.
Annualising: TE_annual = TE_periodic × √(periods per year); mean active return_annual = mean periodic × periods per year; Monthly: √12. Weekly: √52. The square-root rule assumes independent active returns.
Tracking error from two volatilities: TE² = σ_P² + σ_B² − 2ρσ_Pσ_B; Valid when the active return is exactly R_P − R_B. ρ is the correlation of portfolio and benchmark returns.
Ex-ante tracking error from active weights: TE = √(wₐᵀ Σ wₐ), where wₐ = w_P − w_B; Σ is the covariance matrix of asset returns.
Tracking error VaR: TEVaR = z × TE × portfolio value (over the horizon); Assumes normal active returns and zero mean active return. For 95% one-tailed, z = 1.645.
Active return: Rp − Rb = Σ wp_i × Rp_i − Σ wb_i × Rb_i; Rp and Rb are total portfolio and benchmark returns. Sum over segments i.
Allocation effect (Brinson-Fachler): A_i = (wp_i − wb_i) × (Rb_i − Rb); Rewards overweighting segments that beat the total benchmark.
Allocation effect (BHB): A_i = (wp_i − wb_i) × Rb_i; Segment values differ from Fachler, but the sum across segments is the same when weights each sum to 100%.
Selection effect: S_i = wb_i × (Rp_i − Rb_i); Uses benchmark weight. Measures stock picking within the segment.
Interaction effect: I_i = (wp_i − wb_i) × (Rp_i − Rb_i); Joint effect of weight and return differences.
Total check: Σ (A_i + S_i + I_i) = Rp − Rb; Use this to verify your answer.
Returns-based style regression: R_fund,t = b1·F1,t + b2·F2,t + … + bn·Fn,t + e_t; F are style index returns. Constraints: every b ≥ 0 and Σb = 1 (100%). e_t is the selection residual.
Style R²: R² = 1 − Var(e) ÷ Var(R_fund); Share of fund return variance explained by the style mix. The remainder is selection.
Selection return: Selection return = R_fund − Σ(b_i × F_i); Return not explained by style. It is the active return versus the style benchmark.
Active return: Active return = R_portfolio − R_benchmark; Positive means the manager beat the benchmark in that period.
Properties of a valid benchmark: Unambiguous, investable, measurable, appropriate, specified in advance, reflects current investment opinions, accountable; Know each property and what breaks it.
Time-weighted return (linked): TWR = (1 + R₁) × (1 + R₂) × … × (1 + Rₙ) − 1; Each Rᵢ is the return of a sub-period between cash flows. Annualise a multi-year TWR with (1 + TWR)^(1/years) − 1.
Money-weighted return (IRR): Σ CFₜ ÷ (1 + r)^t = 0; Treat investments as negative flows and the ending value as a positive flow. Solve for r.
Alpha from a regression: Rp − Rf = α + β × (Rm − Rf) + ε; With a multi-factor model, add more factor terms. α is the intercept.
t-statistic of alpha: t = α̂ ÷ SE(α̂); Roughly, |t| above about 2 is significant at 5% (two-sided). The exact critical value depends on degrees of freedom.
Information ratio: IR = α ÷ ω, where ω is the residual (active) risk; Use annual alpha and annual residual risk together. Some texts use tracking error as ω when the benchmark is the reference.
t-statistic from IR: t ≈ IR × √T; T is the number of years when IR is annualised. Years needed for t = 2: T = (2 ÷ IR)².
Fundamental law of active management: IR = IC × √BR; IC is the forecast-outcome correlation. BR is the number of independent bets per year.
Fundamental law with constraints: IR = TC × IC × √BR; TC is the transfer coefficient, between 0 and 1. It measures how fully forecasts are reflected in the actual portfolio.
Expected false positives: Expected false positives = N × significance level; This applies to N managers with no true skill, tested independently.

Quick revision

Sharpe ratio = (Rp − Rf) ÷ σp. It uses total risk.
Treynor ratio = (Rp − Rf) ÷ βp. It uses systematic risk only.
Jensen's alpha = Rp − [Rf + β(Rm − Rf)].
Information ratio = (Rp − Rb) ÷ tracking error.
Tracking error is the standard deviation of active returns, not their average.
Active return = portfolio return − benchmark return.
Attribution splits active return into allocation, selection and interaction effects.
Time-weighted return removes the effect of cash flow timing, so it suits judging a manager.
Money-weighted return (IRR) reflects the investor's timing of flows.
A good benchmark is unambiguous, investable, measurable, appropriate and specified in advance.
A high ratio over a short sample can be luck, so check the number of observations.

Common mistakes

Using standard deviation in the Treynor ratio or beta in the Sharpe ratio. Fix: Remember S for Sharpe, S for Standard deviation (total risk). T for Treynor, T for beta as the sensitivity to the market.
Forgetting to subtract the risk-free rate in the numerator. Fix: Always write Rp − Rf as step one. Only the information ratio uses Rp − Rb.
Treating tracking error as the average active return. Fix: Tracking error is a standard deviation. Average active return is a mean. Only the second tells you whether the manager outperformed.
Using the portfolio's own standard deviation as tracking error. Fix: Compute active returns first, then their standard deviation.
Using portfolio weight in the selection effect Fix: Selection uses benchmark weight. The weight change is picked up in interaction.
Forgetting to subtract total benchmark return in allocation Fix: If the question says Brinson-Fachler or relative to the benchmark, use Rb_i − Rb. Otherwise follow the stated model.
Treating any popular market index as an appropriate benchmark. Fix: Check that the index matches the manager's style, size and region. A large-cap index is wrong for a small-cap value fund.
Saying a benchmark can be chosen after seeing results. Fix: A valid benchmark is specified in advance. Choosing it afterwards invites cherry-picking.
Using MWR to compare managers Fix: MWR reflects the investor's flow timing, which the manager does not control. Use TWR to judge the manager and MWR to describe the investor's outcome.
Averaging sub-period returns instead of linking them Fix: Always multiply (1 + R) across sub-periods and subtract 1 at the end. Annualise by taking the root, not by dividing.

Exam tips

Questions often give both σ and β. Read the name of the measure carefully and pick the matching one.
When two measures give different rankings, expect a conceptual question: the answer usually turns on total versus systematic risk or on diversification.
For the information ratio, the key numbers are active return and tracking error. Ignore the risk-free rate unless the benchmark is cash.
Sortino questions test why it differs from Sharpe: it penalises only downside deviation, so it suits asymmetric or option-like returns.
Keep all inputs in the same units and periodicity. If the data are monthly, annualise consistently before comparing.
Check the units. Questions often give monthly data and ask for annualised answers.
If the stem gives volatilities and correlation, expect the TE² formula. If it gives return series, expect the standard deviation of differences.
Read whether the question wants ex-ante or ex-post tracking error, then pick the weights-based or return-based method.