Markowitz Mean-Variance Portfolio Theory

Markowitz Mean-Variance Portfolio Theory Exercise Solutions 1. (a) In this problem me, and so the expected return for any set of weights w is mw = μew = 1. Therefore, the efficient set must consist of a single point with expected return and variance equal to the minimum variance. (b) The minimum variance solution is were The associated variance is w² Σω = (ee)-1 = 1/02. Therefore the minimum variance point on the efficient frontier, and which in fact equals the entire frontier, is (8¹, μ). 2. There are two ways to interpret this problem: (A) The planner spends 1mil on the concert and spends more money on rain insurance. Here 0 ≤ u ≤ 3. (B) The planner spends u million dollars on insurance and (1-u) million dollars on the concert where 0 ≤ u ≤ 1 since 1mil is the entire budget for the concert. For the concert to proceed, the planner needs partners to come up with the remaining u million dollars. Solutions for both scenarios are provided. (A) Letr be the rate of return random variable. If we buy u million dollars of insurance, then r = expected payout. Therefore, r= (+1) 2-(+1) (+1) (1+) where is the 1+ , if no rain, if rain. 3-(+1) The probability of each of the two events, rain and no rain, is 1/2, consequently Hr = E(r) = = 13-(u+1) 12u(u+1) 2 (+1) + 2 (u+1) E(r²) - ² = 4u² - 124 +9 μ 4(+1) 2(u+1) We have do du 20-30 4(1+)3 so the minimum variance solution occurs at u 3/2 at which point the variance of the rate of return is zero yielding a guar- enteed rate of return of 1/5. That is, the planner should buy 1.5mil in insurance for a guarenteed payout of $3,000,000. (B) In this scenario we have two investment opportunities: (1) the concert and (2) rain insurance. Let r₁ be the rate of return on the concert investmment and r₂ be the rate of return on the rain insurance: 71 T2 = 1 =2 , if no rain, 0-1 =-1, if rain. 0-1 1 -1 21 , if no rain, if rain. = We spend the entire Imil on a portfolio consisting of r₁ and 72. and so this portfolio is. r=(1-u)rı+ ur2. Consequently, = (1-4)E(r1)+ UE(12) = (1-4)2 + 24(1 – 4)012 + 1203 1 =(1-4) σ 3 =(1-4)2-24(1-4)+u². To obtain the minimum variance portfolio solve 0 = do =(25-15) du to get u. The corresponding expected return and variance are 1/5 and 0, respectively. That is, if we purchase $600,000 and invest $400,000 in the concert, then we get a guarenteed payout of $1,200,000. Hence the minimum variance weights are yielding a minimum variance portfolio of r = +2. (b) A second efficient portfolio is given by the weights yielding the efficient portfolior (3r1-2r2+5r3). (c) The market portfolio in this setting has weights WM = 2-1(m-re) If desired, these weights can be combined with the weights (1,0,0,0)7 to eliminate the risk free component to get the efficient weights (0, 1, -1, 2) yielding the efficient portfolior(r1r2+213). 4. (a) The random vector2 = (M. 71,72,...,n) has by definition the covariance matrix Σ =[] 8 Σ Hence (ii) The case where the constraint µm²w is inactive has al- ready been studied above, so we assume that = mw. In this case we get (3-1)+81-3 where α = (-)(-e) [(1-2-3)-( - 2-18) – (m)] δ and 8 = (mm) (ee) – (em)². - (iii) Yes! The two funds are 1-2, Σ - 1- Σ e-m Wi = Σ = Σ+ -3 5. (α) μ = .07 + 10 (b) .64 (c) Borrow $1000 at the risk free rate and invest $2000 in the market portfolio. (d) $ 1182 6. (a) We have σε = (1 - α)σ + 20(1 – α)στη + α²², 2 and since om is the minimum variance portfolio we have do da 0 = a=0 = 2(στη -ση). Therefore, A1, 1.c.,0m = στη (b) Just solve 0 = cov (wz, w.) for a to get a = 7. Apply the CAPM formula to get P = μα 1000 1+ry+Br(m-1) 1.1.6.17.1) 1000 1.142 = 875.66. Hence according to the CAPM these shares are a bargin at $800.