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# P1.T1.20.9. Performance measures

#### Nicole Seaman

Staff member
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Learning objectives: Calculate, compare and interpret the following performance measures: the Sharpe performance index, the Treynor performance index, the Jensen performance index, the tracking error, information ratio, and Sortino ratio.

Questions:

20.9.1. The riskfree rate is 3.0% and the market portfolio's expected return is 10.0% (put another way, the market's excess expected return is 7.0%). Consider two portfolios:
• Portfolio A has a high volatility, σ(A) = 50.0% per annum, but its correlation to the market portfolio is only, ρ(A, M) = 0.30
• Portfolio B has a moderate volatility, σ(B) = 30.0% per annum, and its correlation to the market portfolio is, ρ(B, M) = 0.70
If both portfolios lie on the security market line (SML), and if Portfolio A has an expected return, ER(A) = 7.20%, then what is the expected return of Portfolio B?

a. 5.100%
b. 7.900%
c. 8.880%.
d. 9.540%

20.9.2. Let Rm(P) denote the monthly return of the portfolio and Rm(B) denote the monthly return of its benchmark. Over a three-year measurement period, the following statistics are calculated:
• The average monthly return for, respectively, the portfolio and the benchmark was Rm(P) = 8.50% and Rm(B) = 6.90%; therefore, on average, the portfolio outperformed its benchmark by +1.60%.
• The monthly standard deviation of the difference between the portfolio's and benchmark's return, σ[Rm(P) - Rm(B)] = 11.80%
• A regression of the portfolio's excess return against the benchmark's excess return produced the sample regression function, ERm(P) = -0.0140 + 1.35 * ERm(B); therefore, the regression intercept (aka, alpha) is -1.40%
• The standard error of the regression (SER), which approximates the volatility of alpha, σ(α), is 11.0%
Although the periodicity is monthly, an information ratio is generally annualized. In regard to annualized information ratios, which of the following statements is accurate?

a. The (both annualized) active information ratio is +0.136 and the residual information ratio is +1.426
b. The (both annualized) active information ratio is +0.470 and the residual information ratio is -0.441
c. The (both annualized) active information ratio is +0.636 and the residual information ratio is -0.127
d. The (both annualized) active information ratio is -0.250 and the residual information ratio is +0.889

20.9.3. During a 36-month period during which the risk-free rate was 2.0%, consider a comparison between the market portfolio and two fund managers, Betty and Peter:
• The market portfolio's excess return was +6.0% (its gross return was +8.0%) with a volatility of 24.0% per annum
• Peter's portfolio's excess return was +7.0% (his gross return was + 9.0%) with a volatility of 36.0% per annum and beta, β(P,M) = 0.750
• Betty's portfolio's excess return was +11.0% (her gross return was + 13.0%) with a volatility of 44.0% per annum and beta, β(B,M) = 1.50
If the capital asset pricing model (CAPM) is valid, then each of the following statements is true EXCEPT which is false?

a. If the investor is diversified (or the portfolio represents only a fraction of the investor's entire wealth), the Treynor (TPI) is a good measure and Peter's TPI is better than Betty's
b. If the investor is not diversified (or the portfolio represents the investor's entire wealth), the Sharpe (SPI) is a good measure and Betty's SPI is better than Peter's
c. Betty's portfolio already lies on the efficient frontier such that we do not expect her to sustain further improvement in the reward-to-variability ratio
d. If Betty's and Peter's portfolio belong to different peer groups, Jensen's alpha is a good measure for ranking them and Betty's (Jensen's) alpha is better than Peter's