# bkmsol_ch27 - CHAPTER 27 THE THEORY OF ACTIVE PORTFOLIO...

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CHAPTER 27: THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT 1. a.Define R = r – r f Note that we compute the estimates of standard deviation using 4 degrees of freedom (i.e., we divide the sum of the squared deviations from the mean by 4 despite the fact that we have 5 observations), since deviations are taken from the sample mean, not the theoretical population mean. E(R B ) = 11.16% σ B = 21.24% E(R U ) = 8.42% σ U = 14.85% Risk neutral investors would prefer the Bull Fund because its performance suggests a higher mean. b. Using the reward-to-volatility (Sharpe) measure: 5254 . 0 24 . 21 16 . 11 ) R ( E r ) r ( E S B B B f B B = = σ = σ - = 5670 . 0 85 . 14 42 . 8 ) R ( E r ) r ( E S U U U f U U = = σ = σ - = The data suggest that the Unicorn Fund dominates for a risk averse investor. c. The decision rule for the proportion to be invested in the risky asset is given by the following formula: 2 f A 01 . 0 ) R ( E r ) r ( E y σ = σ - = This value of y maximizes a mean-variance utility function of the form: U = E(r) – 0.005A σ 2 For utility functions of this form, Sharpe’s measure is the appropriate criterion for the selection of optimal risky portfolios. An investor with A = 3 would invest the following fraction in Unicorn: 2727 . 1 85 . 14 3 01 . 0 42 . 8 y 2 U = × × = Note that the investor seeks to borrow in order to invest in Unicorn. In that case, his portfolio risk premium and standard deviation would be: E(r P ) – r f = 1.2727 × 8.42% = 10.72% σ P = 1.2727 × 14.85% = 18.90% 27-1

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The investor’s utility level would be: U(P) = r f + 10.72 – (0.005 × 3 × 18.90 2 ) = r f + 5.36 If borrowing is not allowed, investing 100% in Unicorn would lead to: E(r P ) – r f = 8.42% σ P = 14.85% U(P) = r f + 8.42 – (0.005 × 3 × 14.85 2 ) = r f + 5.11 Note that, if Bull must be chosen, then: 8246 . 0 24 . 21 3 01 . 0 16 . 11 y 2 B = × × = E(r P ) – r f = 0.8246 × 11.16% = 9.20% σ P = 0.8246 × 21.24% = 17.51% U(P) = r f + 9.20 – (0.005 × 3 × 17.51 2 ) = r f + 4.60 Thus, even with a borrowing restriction, Unicorn (with the lower mean) is still superior to Bull. 2. Write the Black-Scholes formula from Chapter 21 as: C = S 0 N(d 1 ) – PV(X)N(d 2 ) In this application, where we express the value of timing per dollar of assets, we use S 0 = 1.0 for the value of the stock. The present value of the exercise price is also equal to 1. Note that: T T ) 2 / r ( ) X / S ln( d 2 0 1 σ σ + + = and T d d 1 2 σ - = When S 0 = PV(X) and T = 1, the formula for d 1 reduces to: d 1 = σ/2 The formula for d 2 becomes: d 2 = – σ /2 Therefore: C = N( σ/2 ) – N(– σ/2 ) Finally, recall that: N(–x) = 1 – N(x) Therefore, we can write the value of the call as: C = N( σ/2 ) – [1 – N( σ/2 )] = 2N( σ /2) – 1 Since σ = 0.055, the value of the option is: C = 2N(0.0275) – 1 Interpolating from the standard normal table in Chapter 21: 27-2
0220 . 0 0220 . 1 1 ) 5080 . 0 5160 . 0 ( 0200 . 0 0075 . 0 5080 . 0 2 C = = - - × + = Hence the added value of a perfect timing strategy is 2.2% per month. 3. a.Using the relative frequencies to estimate the conditional probabilities P 1 and P 2 for timers A and B, we find: Timer A Timer B P 1 78/135 = 0.58 86/135 = 0.64 P 2 57/92 = 0.62 50/92 = 0.54 P* = P 1 + P 2 - 1 0.20 0.18 The data suggest that timer A is the better forecaster.

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