- A client invested $100,000 at the beginning of year one. At the end of year one the account was worth $200,000, and the client contributed an additional $200,000. At the end of year two the account was worth $200,000. The client asks the IAR which return best measures the portfolio manager's performance over the two years. Which return should the IAR report?
Answer: The time-weighted return, approximately 0%.
The time-weighted return is the correct performance measure because it removes the effect of external client cash flows and isolates the portfolio manager's performance. Year one return = ($200,000 − $100,000) / $100,000 = 100%. After the client's $200,000 contribution, the year-two starting value is $400,000, so year two return = ($200,000 − $400,000) / $400,000 = −50%. The cumulative wealth factor is 2.0 × 0.5 = 1.0, so the annualized time-weighted return is (1.0)1/2 − 1 = 0%. The dollar-weighted (IRR) return of approximately −26.8% reflects the client's actual experience, not manager skill, because the contribution was made just before the losing year. The 25% figure is the simple arithmetic average of +100% and −50%, ignoring compounding. A dollar-weighted result of approximately 0% is wrong because the contribution before the losing year makes the IRR negative, not zero. An IAR is evaluating two large-cap equity funds for a client. The client's existing portfolio holds more than 100 stocks across all major sectors and is fully diversified, and whichever fund is chosen will be held as part of that portfolio rather than as a standalone holding. Over the past three years, Fund A returned 10% annually with a standard deviation of 10% and a beta of 1.2. Fund B returned 11% annually with a standard deviation of 15% and a beta of 1.0. The risk-free rate averaged 2% and the market returned 9% over the same period.
On a risk-adjusted basis, which fund should the IAR recommend? Answer: Fund B, because its Treynor ratio of 9.0 exceeds Fund A's Treynor ratio of 6.7.
Governing concept: risk-adjusted performance measures (Sharpe ratio vs. Treynor ratio). Because the client's portfolio is already fully diversified, the unsystematic component of each fund's total risk will be diversified away, so the relevant risk is each fund's systematic risk, measured by beta. The Treynor ratio (excess return divided by beta) is therefore the appropriate measure; the Sharpe ratio (excess return divided by standard deviation) is used when a fund is evaluated as a standalone holding. Fund B's Treynor ratio is (11% - 2%) / 1.0 = 9.0, while Fund A's is (10% - 2%) / 1.2 = 6.7, so Fund B is superior on a risk-adjusted basis. The option citing Fund A's higher Sharpe ratio (0.80 vs. 0.60) uses total risk even though the client's diversification makes systematic risk the relevant risk. The option citing Fund A's alpha of 0.4% reverses the signs of the correct alphas: Fund A's alpha is -0.4% and Fund B's is +2.0% (alpha = R - [Rf + beta × (Rm - Rf)]). The option citing Fund B's lower beta as less total risk confuses beta with total risk; Fund B actually has the higher standard deviation (15% vs. 10%).
- Which of the following statements regarding time-weighted and dollar-weighted returns is correct?
Answer: For a given measurement period with no external cash flows, the time-weighted return and the dollar-weighted return are equal.
Under standard portfolio performance measurement conventions, the time-weighted return links subperiod returns geometrically, while the dollar-weighted return is the internal rate of return on actual external cash flows. With no external cash flows, the dollar-weighted return reduces to the same compound return as the time-weighted return, so that statement is correct. A contribution does not always make the dollar-weighted return lower than the time-weighted return; the direction depends on the timing and subsequent returns, so the 'always' statement is wrong. The IRR measure is not the time-weighted return, and the dollar-weighted return is not an arithmetic average; it is affected by both the amount and timing of client cash flows.
- An IAR is comparing two candidate equity portfolios for a client whose investment portfolio will consist solely of the selected portfolio. The risk-free rate is 4%. Portfolio X is expected to return 20% with a standard deviation of 25% and a beta of 0.8. Portfolio Y is expected to return 16% with a standard deviation of 18% and a beta of 1.3. Which portfolio offers the superior risk-adjusted return, and why?
Answer: Portfolio Y, because its Sharpe ratio of 0.67 exceeds Portfolio X's Sharpe ratio of 0.64.
Sharpe ratio = (return − risk-free rate) / standard deviation. Because the selected portfolio represents the client's entire investment, total risk (standard deviation) is the relevant risk measure, making the Sharpe ratio the proper comparison. Portfolio X: (20 − 4) / 25 = 0.64. Portfolio Y: (16 − 4) / 18 = 0.67. The Treynor ratio uses beta in the denominator and is appropriate only when the portfolio is one component of a larger, diversified portfolio, not the client's sole holding. The nominal-return comparison ignores risk entirely, and the lower-standard-deviation comparison ignores return; neither is a risk-adjusted measure.
An IAR is preparing a performance report for a client who holds a technology-sector mutual fund. Measured against the S&P 500, the fund has an R² of 0.55 and an alpha of +1.8%. Measured against a technology-sector index, the fund has an R² of 0.92 and an alpha of -0.3%.
Which of the following statements is the most accurate assessment of the fund's performance? Answer: The technology index is the better benchmark, so the fund's -0.3% alpha indicates risk-adjusted underperformance.
Governing concept: benchmark selection and the reliability of alpha. A benchmark should match the fund's investment style, and R² shows how much of the fund's return variability the benchmark explains. For a technology-sector fund, the technology-sector index (R² = 0.92) is the more appropriate benchmark than the S&P 500 (R² = 0.55). The fund's -0.3% alpha against the technology index means it earned less than the return expected for its level of systematic risk, so the manager underperformed on a risk-adjusted basis. The +1.8% alpha relative to the S&P 500 is not a reliable measure of skill because that broad index does not capture the fund's sector exposure; a positive alpha against an inappropriate benchmark does not show outperformance or provide a better skill measure. A lower R² makes a benchmark less appropriate, not more.
- An adviser calculates that a client portfolio produced an average monthly excess return over T-bills of 0.60% and a monthly standard deviation of 2.4%. Assume the square root of 12 is approximately 3.46. What is the portfolio's annualized Sharpe ratio?
Answer: 0.87
Sharpe ratio = excess return / standard deviation. To annualize monthly data, multiply the monthly excess return by 12 and multiply the monthly standard deviation by the square root of 12. Annualized excess return = 0.60% × 12 = 7.2%. Annualized standard deviation = 2.4% × 3.46 ≈ 8.30%. Sharpe ratio = 7.2 / 8.30 ≈ 0.87. The 3.00 figure annualizes the return but leaves the standard deviation at its monthly value. The 0.25 figure is the monthly Sharpe ratio (equivalently, it uses 12 rather than the square root of 12 to annualize the standard deviation). The 0.07 figure divides the monthly excess return by the annualized standard deviation.
- An IAR is preparing a performance review for a client whose account holds a portfolio of small-cap growth equities. Which statement regarding performance measurement and benchmarks is correct?
Answer: Comparing the client's small-cap growth portfolio against the S&P 500 can misstate the portfolio's performance because the benchmark does not match the portfolio's size and style characteristics.
A benchmark must match the portfolio's investment style and objectives; the S&P 500 is a large-cap blend index, so using it to evaluate a small-cap growth portfolio mismatches size and style and can distort the performance conclusion. Survivorship bias overstates, not understates, the average return of funds in a database because failed and poorly performing funds disappear. The Sharpe and Treynor ratios can rank portfolios differently because the Sharpe ratio uses total risk (standard deviation) while the Treynor ratio uses systematic risk (beta). Beta measures a portfolio's sensitivity to market movements, not performance relative to the risk-free rate; alpha is a risk-adjusted measure of excess return relative to the return expected for the portfolio's level of systematic risk.
- Which of the following statements regarding Jensen's alpha is correct?
Answer: It is the intercept from regressing excess portfolio returns on excess market returns, and a positive value means the portfolio beat its required return for its beta.
Under portfolio performance measurement conventions, Jensen's alpha is the intercept of the regression of excess portfolio returns on excess market returns. Equivalently, it is actual return minus the CAPM required return: alpha = portfolio return - [risk-free rate + beta × (market return - risk-free rate)]. A positive alpha means the portfolio earned more than the return required for its systematic risk. The statement that divides the difference from the benchmark by beta omits the risk-free rate and CAPM structure. The slope of the regression is beta, not alpha, and a negative slope indicates the portfolio moves opposite the market, not that alpha is negative. Excess return divided by standard deviation is the Sharpe ratio, not alpha.
- An IAR is evaluating two mutual funds against the same benchmark, the S&P 500, which returned 9.4% over the year. Fund 1 returned 11.6% with a 4.0% tracking error. Fund 2 returned 10.2% with a 2.2% tracking error. Based on the information ratio, which fund demonstrates superior performance?
Answer: Fund 1, because its information ratio of 0.55 exceeds Fund 2's information ratio of 0.36.
Information ratio = active return / tracking error. Fund 1: (11.6 - 9.4) / 4.0 = 2.2 / 4.0 = 0.55. Fund 2: (10.2 - 9.4) / 2.2 = 0.8 / 2.2 = 0.36. Because 0.55 exceeds 0.36, Fund 1 demonstrates superior performance on this measure. The 4.64 and 2.90 figures come from dividing each fund's total return by its tracking error rather than using active return. The lower-tracking-error statement ignores that Fund 2's active return is also much smaller, and the 220-basis-point outperformance statement ignores tracking error; neither provides a risk-adjusted comparison.
An IAR is evaluating an actively managed large-cap fund for a client's portfolio. Over the past three years, the fund's annualized return was 11%, while its benchmark index returned 9%. The fund's standard deviation was 16%, its tracking error was 4%, its beta relative to the benchmark was 1.05, and its R² relative to the benchmark was 0.95. The risk-free rate averaged 2%.
Which of the following statements about the fund's risk-adjusted performance is correct? Answer: The information ratio is 0.5: it measures active return per unit of tracking error and is the appropriate benchmark-relative measure.
Governing concept: information ratio, Sharpe ratio, and alpha. The information ratio measures active return per unit of active risk: (11% - 9%) / 4% = 0.5. For an actively managed fund being evaluated against a benchmark, the information ratio is the appropriate measure of the manager's ability to add value relative to the benchmark. The 2.0 figure inverts the calculation (4% / 2%). The Sharpe ratio is (11% - 2%) / 16% = 0.56, but it measures excess return per unit of total risk; it does not isolate the manager's value added beyond the benchmark, and a high R² does not make it the better measure for that purpose. The required return based on beta is 2% + 1.05 × (9% - 2%) = 9.35%, so alpha is 11% - 9.35% = 1.65%; it is the return above the required return based on beta, not the return in excess of the benchmark after adjusting for tracking error.
- Which of the following statements regarding risk-adjusted performance measures is correct?
Answer: A portfolio with a 12% return, a 2% risk-free rate, and a 16% standard deviation has a Sharpe ratio of 0.625.
Under standard performance measurement conventions, the Sharpe ratio is excess return divided by total risk: (12% - 2%) / 16% = 0.625. The Treynor ratio divides excess return by beta, not multiplies by it: 10% / 1.25 = 8.0%, so the 12.5% statement is wrong. Jensen's alpha is actual return minus the CAPM required return: 12% - [2% + 1.2 × (10% - 2%)] = 0.4%, not 2.4%. The Sortino ratio uses excess return over the target return divided by downside deviation: 10% / 8% = 1.25, not 1.50; the 1.50 figure puts total return in the numerator.
One year ago, a client invested $100,000 in a mutual fund. Six months ago, when the account was worth $120,000, the client added $60,000. The account is now worth $198,000. The fund's benchmark index returned 31% over the year. The client asks the IAR whether the fund manager outperformed the benchmark.
Which of the following statements correctly answers the client's question? Answer: The manager outperformed the index, because the time-weighted return of 32% exceeds the index's 31% and removes the effect of the client's contribution timing.
Governing concept: time-weighted vs. dollar-weighted return. For evaluating the fund manager, the time-weighted return is the appropriate measure because it removes the effect of the client's contribution timing. The account grew 20% in the first six months ($100,000 to $120,000) and 10% in the second six months ($180,000 to $198,000), so the time-weighted return is (1.20 × 1.10) − 1 = 32%, which exceeds the benchmark's 31%. The manager therefore outperformed. The 38% figure divides the $38,000 gain by the original $100,000 investment only; it ignores the $60,000 added six months ago and therefore is not a valid measure of the manager's performance. The 15% figure is the arithmetic average of the two semiannual returns; it ignores compounding and cannot be compared with the annual benchmark return. The dollar-weighted return of about 29.7% is the internal rate of return on the client's actual cash flows; it reflects the client's experience, which was hurt by contributing money before the weaker second half, not the manager's skill.
- A client's portfolio returned 13.4% over the past year. The portfolio's beta is 1.15, the risk-free rate was 2.5%, and the return on the market was 10.0%. Based on the capital asset pricing model, what is the portfolio's alpha?
Answer: 2.28%
Alpha = actual return − CAPM expected return, where expected return = risk-free rate + beta × (market return − risk-free rate). Expected return = 2.5% + 1.15 × (10.0% − 2.5%) = 2.5% + 8.625% = 11.125%. Alpha = 13.4% − 11.125% = 2.28%. The 1.90% figure comes from using beta times the full 10.0% market return (11.5%) as the expected return, with no risk-free rate added. The 3.40% figure comes from using the market return itself (10.0%) as the expected return, ignoring beta and the risk-free rate. The −0.60% figure comes from substituting the full market return into the risk premium term: expected return = 2.5% + (1.15 × 10.0%) = 14.0%; alpha = 13.4% − 14.0% = −0.60%.
- Which of the following statements regarding the selection and use of a performance benchmark is correct?
Answer: A benchmark should be established before the evaluation period and reflect the portfolio's stated investment universe, using a broad market index or a customized blend of style indexes.
Under standard performance-measurement practice, a benchmark should be specified before the measurement period and should reflect the portfolio's stated mandate and opportunity set; a broad market index or a customized blend of published style indexes can serve this purpose. Selecting a benchmark after seeing the manager's actual returns is look-ahead bias and invalidates the comparison. A benchmark should be changed when the mandate or investment universe changes, not merely to make the comparison more favorable, and it need not be replaced on an annual schedule; doing so could undermine comparability. A customized blend of published style indexes is not inherently invalid.
An IAR is reviewing a mutual fund's performance for a client's annual statement. The fund produced annual returns of +30%, -20%, and +10% over the past three years. Its benchmark index returned exactly 5% in each of those three years.
Which of the following statements correctly compares the fund's performance with the benchmark's? Answer: The fund's geometric mean return of about 4.6% trails the benchmark's 5.0% annual return, so the manager underperformed on a compounded basis.
Governing concept: arithmetic vs. geometric mean return. The geometric mean is the annualized return that compounds to the fund's actual terminal value: (1.30×0.80×1.10)1/3 - 1 = 1.1441/3 - 1 ≈ 4.6%, whereas the arithmetic mean is (30% - 20% + 10%) / 3 = 6.7%. Because the benchmark compounded at 5% in each year, the fund's annualized return of about 4.6% trails the benchmark, so the manager underperformed on a compounded basis. The arithmetic mean overstates compound growth whenever returns vary from period to period. The statement that the geometric mean understates performance is backwards: the geometric mean is the actual compound annual growth rate, not a conservative estimate.
