Workshop 7 Answers PDF

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Finance 261 Workshop 7 Answers 1. Distinguish among the following performance measures: a) Sharpe ratio b) Treynor measure c) Jensen’s alpha i) Describe how each of the three performance measures is calculated. ii) State whether each measure assumes that the relevant risk is systematic, unsystematic, or total. Explain how each measure relates excess return and the relevant risk. Answer: i) The Sharpe ratio is calculated by dividing the portfolio risk premium (actual portfolio return minus risk-free return) by the portfolio standard deviation. The Treynor ratio is calculated by dividing the portfolio risk premium (actual portfolio return minus risk-free return) by the portfolio beta. Jensen’s alpha is calculated by subtracting the market risk premium, adjusted for risk by the portfolio’s beta, from the actual portfolio excess return (risk premium). It is the difference in return earned by the portfolio compared to the return implied by the CAPM or SML. ii) The Sharpe ratio assumes that the relevant risk is total risk, and it measures excess return per unit of total risk. The Treynor measure assumes that the relevant risk is systematic risk, and it measures excess return per unit of systematic risk. Jensen’s alpha assumes that the relevant risk is systematic risk, and it measures excess return at a given level of systematic risk.

2. The finance committee of an endowment has decided to shift all of its investment in an index fund to one of two professionally managed portfolios. Upon examination of past performance, a committee member proposes to choose the portfolio that achieved a greater alpha value. a) Do you agree? Why or why not? No. Alpha alone does not determine which portfolio has a larger Sharpe ratio. Sharpe measure is the primary factor, since it tells us the real return per unit of risk. We only invest if the Sharpe measure is higher. The standard deviation of an investment and its correlation with the benchmark are also important. Thus, positive alpha is not a sufficient condition for a managed portfolio to offer a higher Sharpe measure than the passive benchmark

b) Could a positive alpha be associated with inferior performance? Explain. Yes. It is possible for a positive alpha to exist, but the Sharpe measure declines in which case, we would experience inferior performance.

3. Could portfolio A show a higher Sharpe ratio than that of B and at the same time a lower M2 measure? Explain. No. The M-squared is an equivalent representation of the Sharpe measure, with the added difference of providing a risk-adjusted measure of performance that can be easily interpreted as a differential return relative to a benchmark. Thus, it provides in a different format the same information as the Sharpe measure.

4. Consider the two (excess return) index-model regression results for stocks A and B. The risk-free rate over the period was 6%, and the market’s average return was 14%. Performance is measured using an index model regression on excess returns.

Index model regression estimates R-square SD of excess returns

Stock A 1%+1.2(rM-rf)

Stock B 2%+0.8(rM-rf)

0.576 21.6%

0.436 24.9%

Stock A 1% 0.4907

Stock B 2% 0.3373

8.833

10.5

Calculate: a. Jensen’s alpha b. Sharpe ratio c. Treynor index Answer: Alpha=regression intercept Sharpe ratio =(r-rf)/� Treynor measure=(r-rf)/β Stock A: E(r-rf)= 1%+1.2(rM-rf)=1+1.2(14-6)=10.6 Sharpe ratio: E(r-rf)/� =10.6/21.6=0.4907 Treynor measure: = E(r-rf)/β=10.6/1.2=8.833

Stock B:

E(r-rf)= 2%+0.8(rM-rf)=2+0.8(14-6)=8.4 Sharpe ratio: E(r-rf)/�=8.4/24.9=0.3373 Treynor measure: = E(r-rf)/β=8.4/0.8=10.5

5. Consider the following data for a particular sample period:

Average return Beta Standard deviation Tracking error (nonsystematic risk, �e)

Portfolio P

Market M

35% 1.20 42% 18%

28% 1 30% 0

a) Calculate the following performance measures for portfolio P and the market: Sharpe ratio, Jensen’s alpha, and Treynor index. The T-bill rate during the period was 6%. Answer: Sharpe:

( ´r −´r f ) σ

SRP = (35-6)/42=0.69 SRM = (28-6)/30=0.733

Alpha:

´r −[ ´r f +β ×( r´ M −´r f ) ]

aP=35-[6+1.2*(28-6)]=2.6 aM=0 (by definition)

Treynor:

( ´r −´r f ) β

TIP = (35-6)/1.2 =24.2 TIM = (28-6)/1 = 22

b) Calculate the M2 measure. Answer: To match 30% standard deviation with P and risk-free, the weight on P is 30/42.

Hence, the adjusted portfolio is formed by mixing bills and portfolio P with weights 30/42=0.714 in P and 1-0.714=0.286 in bills. The return on this portfolio is (0.714*35%) + (0.286*6%) =26.7%. This is 1.3% less than the market return. M2 measure = -1.3 Notice that the negative M2 coincides with P’s Sharpe ratio lower than that of M.

c) Calculate the information ratio. Answer: Information ratio = Alpha / Residual standard deviation IRP = 2.6 / 18 = 0.1444 IRM = 0 (by definition)

6.

You obtained the following results when you estimated the security characteristic lines for stocks A and B: Stock A: (RA – Rf) = αA + βA (RM - Rf) +eA Stock B: (RB – Rf) = αB + βB (RM - Rf) +eB Where RA and RB are returns on Stock A and B, RM is return on the market, Rf is the risk-free rate that has been constant throughout the sample period, and e A and eB are zero-mean error terms that are not correlated with anything.

Average return Return Standard deviation Alpha Beta R2

Stock A

Stock B

19% 40% 0.00 1.40 0.49

13% 30% 0.00 0.80 ?

a. What should the (constant) risk-free rate and the average market return have been? 19 - Rf = 1.4×(E[RM] - Rf) 13 - Rf = 0.8×(E[RM] - Rf) E[RM] = 15% Rf = 5%

b. What is Stock B’s R2?

0.49 = 1.42× σM2 / 0.42 σM = 20% 0.82×202 / 302 = 0.2844

c. Compute Stock B’s M2 measure. 5 + (13-5) × 20 / 30 – 15 = -4.67%

d. What is the correlation coefficient between Stock A and B? 1.4×0.8×202 / (40×30) = 0.3733

e. Assuming that the estimates are true descriptions of Stock A and B’s return generating processes, calculate the smallest standard deviation one can get by forming a portfolio of Stock A and B. MV weight on A = (302-1.4×0.8×202)/ (402+302-2×1.4×0.8×202) = 0.2818 MV = 0.28182×402+0.71822×302+2×0.2818×0.7182×1.4×0.8×202 = 772.63 Minimum SD = 27.80%...