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Cml and cal 2 4 6 8 10 12 14 16 18 01 2 standard

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CML and CAL 0 2 4 6 8 10 12 14 16 18 01 0 2 0 Standard Deviation Ex 3 0 pected Retrun CAL: Slope = 0.3571 CML: Slope = 0.20 6-6

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21. a. With 70% of his money invested in my fund’s portfolio, the client’s expected return is 15% per year and standard deviation is 19.6% per year. If he shifts that money to the passive portfolio (which has an expected return of 13% and standard deviation of 25%), his overall expected return becomes: E(r C ) = r f + 0.7[E(r M ) r f ] = 8 + [0.7 × (13 – 8)] = 11.5% The standard deviation of the complete portfolio using the passive portfolio would be: σ C = 0.7 × σ M = 0.7 × 25% = 17.5% Therefore, the shift entails a decrease in mean from 14% to 11.5% and a decrease in standard deviation from 19.6% to 17.5%. Since both mean return and standard deviation decrease, it is not yet clear whether the move is beneficial. The disadvantage of the shift is that, if the client is willing to accept a mean return on his total portfolio of 11.5%, he can achieve it with a lower standard deviation using my fund rather than the passive portfolio. To achieve a target mean of 11.5%, we first write the mean of the complete portfolio as a function of the proportion invested in my fund ( y ): E(r C ) = 8 + y(18 8) = 8 + 10y Our target is: E(r C ) = 11.5%. Therefore, the proportion that must be invested in my fund is determined as follows: 11.5 = 8 + 10y 35 . 0 10 8 5 . 11 y = = The standard deviation of this portfolio would be: σ C = y × 28% = 0.35 × 28% = 9.8% Thus, by using my portfolio, the same 11.5% expected return can be achieved with a standard deviation of only 9.8% as opposed to the standard deviation of 17.5% using the passive portfolio. b. The fee would reduce the reward-to-variability ratio, i.e., the slope of the CAL. The client will be indifferent between my fund and the passive portfolio if the slope of the after-fee CAL and the CML are equal. Let f denote the fee: Slope of CAL with fee 28 f 10 28 f 8 18 = = Slope of CML (which requires no fee) 20 . 0 25 8 13 = = Setting these slopes equal we have: 20 . 0 28 f 10 = 10 f = 28 × 0.20 = 5.6 f = 10 5.6 = 4.4% per year 6-7
22. a. The formula for the optimal proportion to invest in the passive portfolio is: 2 M f M A σ r ) E(r y* = Substitute the following: E(r M ) = 13%; r f = 8%; σ M = 25%; A = 3.5: 0.2286 0.25 3.5 0.08 0.13 y* 2 = × = b. The answer here is the same as the answer to Problem 21(b). The fee that you can charge a client is the same regardless of the asset allocation mix of the client’s portfolio. You can charge a fee that will equate the reward-to- variability ratio of your portfolio to that of your competition. 23. a. If the period 1926 - 2005 is assumed to be representative of future expected performance, then we use the following data to compute the fraction allocated to equity: A = 4, E(r M ) r f = 8.39%, σ M = 20.54% (we use the standard deviation of the risk premium from Table 6.8). Then y * is given by: 0.4972 0.2054 4 0.0839 A σ r ) E(r y* 2 2 M f M = × = = That is, 49.72% of the portfolio should be allocated to equity and 50.28% should be allocated to T-bills. b. If the period 1986 - 2005 is assumed to be representative of future expected performance, then we use the following data to compute the fraction allocated to equity: A = 4, E(r M ) r f = 8.60%, σ M = 16.24% and y* is given by: 0.8152 0.1624 4 0.0860 A σ r ) E(r y* 2 2 M f M = × = = Therefore, 81.52% of the complete portfolio should be allocated to equity and 18.48% should be allocated to T-bills.

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CML and CAL 2 4 6 8 10 12 14 16 18 01 2 Standard Deviation...

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