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# P2.T9.21.10. Time- versus dollar-weighted returns and risk-adjusted measures

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Learning outcomes: Differentiate between the time-weighted and dollar-weighted returns of a portfolio and describe their appropriate uses. Describe risk-adjusted performance measures, such as Sharpe’s measure, Treynor’s measure, Jensen’s measure (Jensen’s alpha), and the information ratio, and Identify the circumstances under which the use of each measure is most relevant. Describe the uses for the Modigliani-squared and Treynor’s measure in comparing two portfolios and the graphical representation of these measures.

Questions:

21.10.1. Bertha initially purchases a share (at Time Zero) for $100.00. At the end of the year, while the price has dropped to$80.00, the share pays a $5.00 dividend (which is 5.0% of the price at the beginning of the year). As the stock has dropped, Bertha purchases another share at this time (Time 1). At the end of the second year (Time 2), Bertha receives$8.00 in dividends (i.e., $4.00 for each share, based again on 5.0% of the stock's price at the beginning of the year). At this time (Time 2), the stock price recovers to$92.00 and Bertha buys a third share. Finally, at the end of the third year when the price reaches $016.00, Berth collects$13.80 in dividends (i.e., 3 * 5.0% * $92.00 =$13.80) and sells all three shares for \$318.00. The stock price path and Bertha's trades are displayed below. That holding period returns are -15.00%, +20.00% and +20.22%; e.g., (106.00 + 4.60 - 92.00)/92.00 = 20.22%. Further, the net cash flows for each period are shown in the bottom row. What are, respectively, the time-weighted and dollar-weighted returns over the three-year period?

a. 7.0% (time) and 12.5% (dollar)
b. 11.0% (time) and 15.0% (dollar)
c. 13.3% (time) and 17.9% (dollar)
d. 15.0% (time) and 21.4% (dollar)

21.10.2. While the riskfree rate was 3.0%, the Market's (gross) return was 11.0% with a volatility of 20.0%. During the same period, Donald's portfolio generated an average return of 15.0% but with a volatility of 40.0% which was double the Market's volatility! His portfolio's correlation to the Market, ρ(P,M) was 0.80 and its residual risk (standard deviation), σ(.), was 32.0%. How did Donald's portfolio compare to the Market; i.e., which of the following statements is TRUE?

a. We cannot answer without the portfolio's beta with respect to the Market, β(P,M).
b. Donald's Jensen's alpha (and information ratio) was negative; both his Sharpe ratio and Treynor's measure were below the Market's ratio/measure
c. Donald's Jensen's alpha (and information ratio) was positive; both his Sharpe ratio and Treynor's measure were above the Market's ratio/measure
d. Donald's Jensen's alpha (and information ratio) was zero/neutral; both his Sharpe ratio and Treynor's measure matched the Market's ratio/measure

21.10.3. Below are displayed the performance of two Portfolios, P & Q, in comparison to the market. The riskfree rate is 3.0%. In regard to Portfolio P as it compares to Portfolio Q, which of the following statements is TRUE?

a. Portfolio P has a higher alpha (than Q) but is equal to Portfolio Q in regard to both M^2 and T^2
b. Portfolio P has a higher alpha (than Q) but is inferior (to Q) in regard to both M^2 and T^2
c. Portfolio P has a lower alpha (than Q) but is superior (to Q) in regard to both M^2 and T^2
d. Portfolio P has a lower alpha (than Q) and is also inferior (to Portfolio Q) in regard to both M^2 and T^2