FRM Exam Part II · Portfolio Performance Evaluation

Return Measurement and Manager Skill Evaluation for FRM Part 2

Updated 11 October 2026 · Fact-checked

Time-weighted return (TWR) removes the effect of cash flows and judges the manager. Money-weighted return (MWR, the IRR) includes flow timing and judges the investor's outcome. To test skill, divide alpha by its standard error (the t-statistic). A high t-statistic, adjusted for how many managers were screened, points to skill rather than luck.

Understand Return Measurement and Manager Skill Evaluation

Start with a simple question: how did the fund do? If no cash moves in or out, the answer is one number. Once investors add or withdraw money, there are two honest answers. One tells you how good the manager's decisions were. The other tells you what the investor actually earned.

Time-weighted return (TWR) splits the period at every cash flow, finds the return of each sub-period, and links them by compounding. Flow size does not matter, so the manager cannot be blamed or praised for when clients invested. This is why TWR is the standard for comparing managers. Money-weighted return (MWR) is the internal rate of return of the cash flows, with the ending value treated as a final inflow. It is larger when money was added before good periods. It is the right measure of the investor's own experience.

Next, skill. A manager's alpha is the return left after you remove the return explained by risk exposures, for example from a CAPM or multi-factor regression. A positive alpha in a sample is not proof of skill, because returns are noisy. The t-statistic of alpha = alpha ÷ standard error of alpha shows how large alpha is relative to that noise. Roughly, a t-statistic above about 2 is significant at the 5% level, but the exact cutoff depends on the degrees of freedom. The t-statistic grows with the square root of the number of observations, so short track records rarely prove anything.

Luck is the main trap. If you test many managers, some will look skilled by chance. At a 5% significance level, you expect about 5 false positives for every 100 managers with zero true skill. This is the multiple-testing problem. Good evaluation looks at the whole group, not just the winners. It also checks that performance persists out of sample and that the alpha is not just a payoff for taking hidden risk, such as illiquidity or tail risk.

Finally, the fundamental law of active management links skill to results. The information ratio (alpha per unit of active risk) is roughly skill times the square root of breadth. Skill is the information coefficient (IC), the correlation between forecasts and outcomes. Breadth is the number of independent bets per year. Constraints reduce the result through the transfer coefficient (TC).

Key formulas to remember

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.

How to solve Return Measurement and Manager Skill Evaluation questions

Use this order for any question on return measurement or skill evaluation.

1Read what is being judged. If the question is about the manager's ability, think TWR. If it is about the investor's realised outcome, think MWR.
2For TWR, split the period at each cash flow. Compute each sub-period return from values just before the flow and the starting value of the sub-period. Link them by multiplying (1 + R).
3For MWR, list all dated flows with signs, treat the ending value as an inflow, and solve for the IRR. Look for a quadratic you can solve directly.
4For skill, find alpha and its standard error, or the information ratio and the number of years. Compute t = α ÷ SE, or t = IR × √T.
5Compare t with the critical value (about 2 for 5% in large samples). State whether alpha is statistically significant.
6Adjust for luck. If many managers or strategies were screened, expect false positives and treat the best performer's t-statistic with caution.
7For fundamental-law questions, multiply IC, √BR and TC if given. Check whether bets are independent, because correlated bets reduce effective breadth.
8State the interpretation in one sentence, naming the measure used.

Quickest way: Fast triage for TWR, MWR and skill questions

When to use it: Use this when you have about 2 minutes per question and need to eliminate options quickly.

If a large cash flow came just before a strong period, MWR should be above TWR. If it came before a weak period, MWR should be below TWR. Use this to eliminate options.
For TWR, just multiply the (1 + R) terms. Ignore flow sizes.
For significance, compute t = α ÷ (σ ÷ √T). Compare with 2.
For the fundamental law, multiply: IC × √BR, then TC if given. If breadth is 100, √BR is 10.
If the question mentions picking the best out of many funds, look for the answer about data mining or multiple testing.

Common mistakes in Return Measurement and Manager Skill Evaluation

Using MWR to compare managers
MWR looks like the 'real' return, so it feels more accurate.
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
Adding or averaging is quicker than compounding.
Fix: Always multiply (1 + R) across sub-periods and subtract 1 at the end. Annualise by taking the root, not by dividing.
Treating any positive alpha as skill
Alpha is reported as a single number, so it looks definitive.
Fix: Check the t-statistic. A 2.4% alpha with a large residual risk and a short history can be pure noise.
Mixing frequencies in t = IR × √T
Monthly alpha is paired with annual risk, or the other way round.
Fix: Keep IR annual and T in years, or use monthly IR with T in months. Never mix them.
Ignoring multiple testing when the best fund is selected
Students judge each fund alone and forget how many were screened.
Fix: Remember that with many managers, about the significance level times N will pass by chance. Look for persistence out of sample and a higher threshold.
Forgetting the transfer coefficient in the fundamental law
The basic formula IR = IC × √BR is memorised without constraints.
Fix: When a question mentions constraints such as long-only or position limits, use IR = TC × IC × √BR.

Worked examples

Example 1

A client invests ₹100 lakh in a fund. After year 1 the fund is worth ₹120 lakh. The client then adds ₹80 lakh, making ₹200 lakh. At the end of year 2 the account is worth ₹180 lakh. Compute the two-year TWR (annualised) and the MWR, and explain the difference.

Show the solution

Year 1 return: 120 ÷ 100 − 1 = 20%.
Year 2 return: 180 ÷ 200 − 1 = −10%. The ₹80 lakh addition is excluded from the return calculation because it was added at the start of year 2.
Linked TWR: 1.20 × 0.90 − 1 = 0.08, or 8% over two years.
Annualised TWR: √1.08 − 1 = 1.0392 − 1 ≈ 3.92% a year.
MWR flows: −100 at time 0, −80 at time 1, +180 at time 2. Set −100 − 80 ÷ (1 + r) + 180 ÷ (1 + r)² = 0.
Let x = 1 ÷ (1 + r). Then 180x² − 80x − 100 = 0, which simplifies to 9x² − 4x − 5 = 0.
x = [4 + √(16 + 180)] ÷ 18 = (4 + 14) ÷ 18 = 1. So 1 + r = 1 and r = 0%.
The investor added the larger amount just before the loss year, so MWR (0%) is below TWR (about 3.92%). The manager's decisions earned about 3.92% a year, but the investor's money did not grow.

Answer: Annualised TWR ≈ 3.92%; MWR = 0%. MWR is lower because the large addition came before the weak year.

Example 2

A manager has an annual alpha of 2.4% against a benchmark-adjusted model. The annualised residual risk is 4%. The record is 9 years. (a) Compute the t-statistic. (b) State whether alpha is significant at the 5% level using a cutoff of about 2. (c) How many years would be needed for t = 2 if alpha and residual risk stay the same?

Show the solution

Information ratio: IR = 2.4% ÷ 4% = 0.6.
Standard error of alpha: 4% ÷ √9 = 4% ÷ 3 = 1.333%.
t-statistic: 2.4 ÷ 1.333 = 1.8. Check: IR × √T = 0.6 × 3 = 1.8.
1.8 is below about 2, so alpha is not significant at the 5% level. The result is consistent with luck.
Years for t = 2: T = (2 ÷ IR)² = (2 ÷ 0.6)² = 3.333² ≈ 11.1 years.

Answer: (a) t = 1.8. (b) Not significant at 5%. (c) About 11.1 years, so roughly 12 years of data are needed.

Exam tips

Look for the word that tells you who is being judged. 'Manager' points to TWR. 'Investor' or 'client experience' points to MWR.
Questions on skill often hide a multiple-testing point: 'the best of 500 funds' or 'top decile'. Expect the right answer to mention luck or selection bias.
Always check units: annual vs monthly alpha and risk. Mixing them is the usual trap in t-statistic calculations.
In fundamental-law questions, compute √BR first. Then check if a transfer coefficient or correlated bets change the answer.
If MWR is asked and the flows are simple, set up a quadratic in 1 ÷ (1 + r). It is often designed to have a clean root.

Practice questions from Portfolio Performance Evaluation

Return Measurement and Manager Skill Evaluation in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CFA Level IMoney-Weighted vs Time-Weighted Return

Return Measurement and Manager Skill Evaluation: frequently asked questions

What is the difference between time-weighted and money-weighted return?

Time-weighted return links sub-period returns and removes the effect of cash flow timing, so it measures the manager. Money-weighted return is the IRR of all cash flows, so it also reflects when the investor added or withdrew money. Use TWR to compare managers and MWR to describe what the investor earned.

How do I test whether a manager's alpha is skill?

Estimate alpha from a regression on the relevant risk factors, then divide it by its standard error to get the t-statistic. A value above about 2 is significant at the 5% level. Then ask whether many managers were screened and whether the result persists out of sample.

How does luck get mistaken for skill in fund manager results?

When many managers are tested, some will show significant alpha by chance. At a 5% level, about 5 in 100 zero-skill managers will pass. This is why the best performers in a large group need stricter evidence, such as a higher threshold or persistence.

What is the fundamental law of active management?

It says the information ratio is approximately the information coefficient times the square root of breadth: IR = IC × √BR. With constraints, multiply by the transfer coefficient: IR = TC × IC × √BR. More independent bets or better forecasts raise the expected IR.