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M.Sc. Finance
M.Sc. International Banking and Finance
and
M.Sc. International Accounting and Finance
Portfolio Theory: Tutorial Four
SOLUTIONS
Q1. A portfolio manager buys 30,000 shares in a company today at £9.00 per share. At the end of each the next three years he sells 10,000 shares. The share price increases to £10.00 per share at the end of year one, to £13.00 by the end of year two and falls back to £12.50 by the end of year three. The shares do not pay any dividends over the three year period.
a) Calculate the time weighted geometric and arithmetic average returns on the portfolio. Return in period one (10 – 9)/9 = 0. Return in period two (13 – 10)/10 = 0. Return in period three (12.50 – 13)/13 = -0.
Arithmetric average = (11.0% + 30.0% - 3.85%)/3 = 12.4%
Geometric average = 3 √(1.111)(1.300)(0.9615) – 1 = 0. b) Calculate the dollar weighted average return on the portfolio Dollar Weighted Return 0 = -370,000 + 100,000 (1 + i)-1^ + 130,000 (1 + i) -2^ + 125,000 (1 + i)- i = 0.1445 14.
Q2. You are given the following information on the average returns produced by three investment funds over the last four years. The risk free rate is 6 per cent, the average return on the market portfolio is 15 per cent and the standard deviation of this return was 20 per cent.
Fund Average Annual Return Standard Deviation Beta A 19% 25% 0. B 21% 24% 1. C 25% 32% 1.
a) Calculate the Sharpe, Treynor and Jensen indices and comment on the rankings produced.
J (^) A = 19 – (6 + 0.80[9]) = 5.80%
J (^) B = 21 – (6 + 1.40[9]) = 2.40%
J (^) C = 25 – (6 + 1.80[9]) = 2.80%
Market return per unit of risk = (15 – 6)/20 = 0. Market return per unit of beta risk is 9 per cent (15 – 6) = 9
Sharpe ranking B, C, A Treynor ranking A, B, C Sharpe ranking A, C, B
b) Explain why the Treynor and Jensen measures, both based on the capital asset pricing model, can provide different rankings of portfolios (illustrate this on a diagrammatic basis). The Treynor measure provides a return per unit of beta risk whereas the Jensen measure focuses on the absolute value of the abnormal return [Jensen’s alpha] - See slides attached.
Market: = (20 – 6)/21 = 0. 14 x 1.3125 – 6 x 0.3125 = 16. 16.5 – 6 = 10.5 (less than the market premium)
Q4.
A fund manager has been given a target beta risk of 1.2. For the periods under consideration the risk free rate is 6 per cent and the average return on the market portfolio is 18 per cent with a standard deviation of 20 per cent. The average return on the fund is 32 per cent, with a standard deviation of 38 per cent, while the beta of the portfolio is 1.5. Use Fama’s decomposition of risk to assess the portfolio’s performance, undertaking the relevant calculations and drawing an appropriate diagram to demonstrate the outcome. Expected return on a portfolio with a beta of 1.2 is 20.4 = 6 + (18 – 6) 1.20. Expected return on a portfolio with a beta of 1.5 is 24.0 = 6 + (18 – 6) 1.
A portfolio based on the market portfolio and lending or borrowing (at the risk free rate) with a standard deviation of 38 per cent would have a beta of
βp = ρpm SD(R (^) p)/SD(Rm ) = 1.0 x 38/20 = 1. SD(Rp ) = wm SD(R (^) m ) SD(Rp ) =38 = wm 20 wm = 1. βp = wm βm = 1.90 x 1.0 = 1.
The expected return on a portfolio with a beta of 1.90 would be 28.8 = 6 + (18-6)1. Net selectivity = Rp – E(Rp|given adjusted beta) = 32 – 28. 80 = 3.
Returns 32.0 32
Net Selectivity { 3.
Diversification Risk
Manager Risk
βm = 1.0 β (^) pT^ = 1.2 β (^) p = 1.50 β (^) A = 1. Beta The manager’s excess return in relation to the risk of the portfolio is 3.2 per cent and is attributed to the superior performance of securities included in the portfolio.
