You manage an index fund that is an exact replica of the market index. The marke

Question

You manage an index fund that is an exact replica of the market index. The market expected annual rate of return is 16.5% with a standard deviation of 12.5%. Annual T-bill rate is 3.3%. A client of yours wants you to invest 75% of his portfolio in your fund, with the rest in a T- bill money market fund. What is the expected return and standard deviation of this client's portfolio? a. b. What are the Sharpe ratios of your fund and client's portfolio? c. Suppose your client prefers to invest in your fund a proportion y which maximizes the expected return on his portfolio subject to the constraint that the portfolio's standard deviation will not exceed 10%. i. ii. ili. What is the investment proportion, y? What is the expected rate of return on your client's portfolio? What is the degree of risk aversion (A) of this client?

Explanation / Answer

a. Expected Return of the portfolio = (Weightage in index fund X Expected annual return) + (Weightage in T-bill X Expected annual return)

                                                   = (0.75 x 0.165) + (0.25 x 0.033) = 0.12375 + 0.0082 =0.132

                                                     = 13.2%

Variance of the portfolio = [(Weightage in index fund)2 x (standard deviation index fund)2] + [(Weightage in T-bill)2 x (standard deviation T-bill)2] + [2x(Weightage in index fund)x (Weightage in T-bill)x(Covariance of index and T-Bill)]

                                             = [(Weightage in index fund)2 x (standard deviation index fund)2]

(The rest of the terms become zero as standard deviation of T-Bill is zero )

                                             = [(0.75 x 0.75) x (0.125 x 0.125) = [0.5625 x 0.015625] = 0.008789

Standard deviation of portfolio = ?(variance of portfolio) = ?(0.008789) = 0.0937 = 9.37%

b. Sharpe Ratio of Portfolio = (Portfolio Return – T-Bill Rate)/ Standard deviation of the portfolio

                                                  = (0.132 – 0.033)/0.0937 = 1.057

Sharpe Ratio of the Index Fund = (Index Fund Return - T-Bill rate)/Standard deviation of Index Fund

                                                         = (0.165 – 0.033)/0.125 = 1.056

c. Considering, Standard deviation of portfolio = ? [(Weightage in index fund)2 x (standard deviation index fund)2]

or,                      0.1 = ?(y*y) x (0.125*0.125)

or,                      (0.1*0.1) = y2 x (0.125*0.125)

or,                      0.01 = y2 x 0.015625

or,                      y2 = 0.01/0.015625 = 0.64

Therefore, weightage y = ?0.64 = 0.8

i) So, the client should invest 80% (y) of its wealth in the index fund

ii) Expected return on clients portfolio = (Weightage in index fund X Expected annual return) + (Weightage in T-bill X Expected annual return)

                                                             = (0.8 x 0.165) + (0.2 x 0.033) = 0.139 = 13.9%

iii) Utility function :

U = E(r) - [(1/2) * (A) * (Standard deviation of portfolio)2] Here A is the risk aversion parameter

now, 0.132 = 0.139 - [0.5 * A * (0.1*0.1)]

or,     0.132 = 0.139 - 0.005A

or,     0.005A = 0.007

or,     A = 1.4

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