Solutions to Practice Questions (Diversification)

Transcription

1 Simon School of Business University of Rochester FIN 402 Capital Budgeting & Corporate Objectives Prof. Ron Kaniel Solutions to Practice Questions (Diversification) 1. These practice questions are a suplement to the problem sets, and are intended for those of you who want more practice. They are Optional, and are not part of the required material. 2. It is recommended that you look at these problems only after you fully understand how to solve the problem sets, the examples we covered in class, and the ones in the lecture notes. 3. Please note that I have collected these exmples from previous teaching material I have had. As such, while in most cases the notation will match the one used in class, the match is not 100%. 4. Some of these questions are easier than the ones you are expected to know how to solve, while others are above the level of knowledge you are expected to show on quizes and the final. ENJOY!

2 FIN 402 Solutions to Practice Questions 2 1. (a) The expected return of the portfolio is the weighted average of the returns of the components of the portfolio: ER p = w 1 (r 1 ) + w 2 (r 2 ) = 0.3(0.12) + 0.7(0.18) = 16.2% (b) The variance of a portfolio is given by the following equation: σ 2 p = w 2 1 (σ2 1 ) + w2 2 (σ2 2 ) + 2w 1w 2 σ 1 σ 2 ρ 12 = (0.09) (0.25) 2 + 2(0.3)(0.7)(0.09)(0.25)(0.2) = σ p = ( ) 1 2 = 18.23% 2. (a) The expected returns are the sum of the possible returns multiplied by the associated probabilities: ER A =.15(.2) +.2(.5) +.6(.3) = 25% ER B =.2(.2) +.3(.5) +.4(.3) = 31% The variances are the sums of the squared deviations from the expected returns multiplied by their probabilities: σ 2 A =.2(.15.25) 2 +.5(.2.25) 2 +.3(.6.25) 2 =.2(.16)+.5(.0025)+.3(.1225) =.0700 σb 2 =.2(.2.31) 2 +.5(.3.31) 2 +.3(.4.31) 2 =.2(.0121)+.5(.0001)+.3(.0081) =.0049 The standard deviations are thus: σa 2 =.0700 = 26.46% σb 2 =.0049 = 7% (b) The portfolio weights are $15, , 000 =.75 and $5, , 000 =.25. The expected return is thus: ER p =.75ER A +.25ER B =.75(.25)+.25(.31) = 26.5% Note we could have calculated the portfolio return in each state and calculated as we did before.

3 FIN 402 Solutions to Practice Questions 3 The portfolio s variance is: σ 2 p =.2((.75(.15)+.25(.2)).265) 2 +.5((.75(.2)+.25(.3)).265) 2 +.2((.75(.6)+.25(.4)).265) 2 =.2( ) 2 +.5( ) 2 +.3( ) 2 =.0466 So the standard deviation of the portfolio is: σ 2 p =.0466 = 21.59% 3. (a) Janet holds $80(100) = $8, 000 of Macrosoft and $40(300) = $12, 000 of Intelligence. $8, 000 Therefore, her portfolio weighting in Macrosoft (w M ) is 8, , 000 =.4 and her weighting in Intelligence (w I ) is = 0.6. The expected return on this portfolio is thus: ER p = w M (R M ) + w I (R I ) = 0.4(0.15) + 0.6(0.20) = 18% The variance of this portfolio is given by: σ 2 p = w 2 M(σ 2 M) + w 2 I(σ 2 I) + 2w M w I σ M σ I ρ MI = (0.08) (0.2) 2 + 2(0.4)(0.6)(0.08)(0.20)(0.38) = σ p = ( ) 1 2 = 13.54% (b) Jane now holds only $4,000 of Intelligence, so her new portfolio weights are w M = and w I = The expected return of the new portfolio is: ER p = w M (R M ) + w I (R I ) = 0.667(0.15) (0.20) = 16.66% The variance of this portfolio is given by: σ 2 p = w 2 M(σ 2 M) + w 2 I(σ 2 I) + 2w M w I σ M σ I ρ MI = (0.08) (0.2) 2 + 2(0.667)(0.333)(0.08)(0.20)(0.38) = σ p = ( ) 1 2 = 9.99% 4. (a) The expected return of a portfolio p of three securities is calculated as follows: r p E( r p ) = w i r i = 0.3(15%) + 0.4(20%) + 0.3(35%) = 23%.

4 FIN 402 Solutions to Practice Questions 4 The variance of the same portfolio is calculated as follows: σ 2 p Var( r p) = = wi 2 σi j=1 w i w j σ ij = 2 j=i+1 wi 2σ2 i + w i w j ρ ij σ i σ j w i w j σ ij = (0.3) 2 (0.2) 2 + (0.4) 2 (0.4) 2 + (0.3) 2 (0.7) 2 [ ] + 2 (0.3)(0.4)(0.5)(0.2)(0.4) + (0.3)(0.3)(0.7)(0.2)(0.7) + (0.4)(0.3)(0.9)(0.4)(0.7) = Finally the standard deviation of this portfolio is simply the square root of its variance, that is σ p = = (b) Let w i, i = 1, 2, 3, denote the fraction of our wealth that we will put in each of the three available stocks. Since we want to invest all of our wealth in these three stocks, these weights must satisfy w 1 + w 2 + w 3 = 1. (1) Further, we want to form a portfolio that will have an expected return of 30%. Using the expression for the expected return of our portfolio above, we want j=1 j i 0.3 = 0.15w w w 3. (2) Finally, we want this portfolio to have a variance of So, using the expression for the variance of our portfolio above, we set 0.35 = w 2 1(0.2) 2 + w 2 2(0.4) 2 + w 2 3(0.7) 2 + 2w 1 w 2 (0.5)(0.2)(0.4)+ 2w 1 w 3 (0.7)(0.2)(0.7)+ 2w 2 w 3 (0.9)(0.4)(0.7). (3) We can then solve (1), (2), and (2) for w 1, w 2, and w 3. Although this is all you needed to write to answer this question, you can check that the two portfolios that would satisfy our specifications are given by w 1 = , w 2 = , w 3 = , and w 1 = , w 2 = , w 3 = (a) We are looking for the portfolio (w 1, 1 w 1 ) which minimizes the portfolio s variance, σ 2 p = w 2 1σ (1 w 1 ) 2 σ w 1 (1 w 1 )ρ 12 σ 1 σ 2. There are three ways one can find the portfolio which minimizes the above expression for the variance. The first is simply by trial and error one can pick different values of w 1 till the minimum is found. The second is by plotting the portofolio variance as a function of w 1. In that case one plots the function w 2 1(0.3) 2 + (1 w 1 ) 2 (0.2) 2 + 2w 1 (1 w 1 )(0.2)(0.3)(0.2)

5 FIN 402 Solutions to Practice Questions 5 and then looks were the minimum is attained. The third is by using some basic calculus. For the first two alternatives the procedure for finding the minimum is straight forward. For the third alternative, differentiating the variance with respect to w 1 results in dσ 2 p dw 1 = 2w 1 σ 2 1 2(1 w 1 )σ (1 2w 1 )ρ 12 σ 1 σ 2. We can then set this last expression equal to zero, and solve for w 1 : 1 w 1 = σ 2 2 ρ 12σ 1 σ 2 σ σ2 2 2ρ 12σ 1 σ 2 = (0.2) 2 (0.2)(0.3)(0.2) (0.3) 2 + (0.2) 2 2(0.2)(0.3)(0.2) = Therefore, the minimum variance portfolio consists in investing 26.4% of the portfolio in asset 1, and % = 73.6% in asset 2. The standard deviation of this portfolio is equal to σ p = (0.264) 2 (0.3) 2 + (0.736) 2 (0.2) 2 + 2(0.264)(0.736)(0.2)(0.3)(0.2) = , and its expected return is given by r p = 0.264(0.1) (0.3) = (b) All the portfolios with a variance smaller than or equal to 0.25 (i.e. with a standard deviation smaller than or equal to 0.25 = 0.5) lie on the thick line in the following figure: r p Obviously, the portfolio (w 1, 1 w 1 ) that we are looking for is the one on top of that thick line. This portfolio solves p 0.25 = w 2 1σ (1 w 1 ) 2 σ w 1 (1 w 1 )ρ 12 σ 1 σ = w 2 1(0.3) 2 + (1 w 1 ) 2 (0.2) 2 + 2w 1 (1 w 1 )(0.2)(0.3)(0.2) 0 = 0.106w w Solving this equation by trial and error or equivalently using the formula for the solution of a quadratic equation yields two solutions: w 1 = ± (0.056) 2 4(0.106)( 0.21) 2(0.106) = or It is easily verified that the second derivative is positive, so that we indeed found a minimum.

6 FIN 402 Solutions to Practice Questions 6 The reason we get two different solutions is that there are two portfolios with a variance of 0.25; in fact, this is quite obvious from the figure above. Since the portfolio we are interested in has the higher expected return, we only need to calculate r p with w 1 = and w 1 = We find r p = and r p = respectively, so that w 1 = is the portfolio that we want. For every dollar invested in this portfolio, $(1 w 1 ) = $ is invested in asset 2, and $ comes from (short-)selling asset 1.