Elton, Gruber, Brown and Goetzmann Modern …

1 Elton, Gruber, Brown and Goetzmann 1 Modern portfolio theory and investment analysis selected solutions to Text Problems Elton, Gruber, Brown and Goetzmann Modern portfolio theory and investment analysis selected solutions to Text Problems Chapter 1: Problem 1 A. Opportunity Set With one dollar, you can buy 500 red hots and no rock candies (point A), or 100 rock candies and no red hots (point B), or any combination of red hots and rock candies (any point along the opportunity set line AB). Algebraically, if X = quantity of red hots and Y = quantity of rock candies, then: +YX That is, the money spent on candies, where red hots sell for cents a piece and rock candy sells for 1 cent a piece, cannot exceed 100 cents ($ ).

2 Solving the above equation for X gives: YX5500 = which is the equation of a straight line, with an intercept of 500 and a slope of 5. Elton, Gruber, Brown and Goetzmann 2 Modern portfolio theory and investment analysis selected solutions to Text Problems B. Indifference Map Below is one indifference map. The indifference curves up and to the right indicate greater happiness, since these curves indicate more consumption from both candies. Each curve is negatively sloped, indicating a preference of more to less, and each curve is convex, indicating that the rate of exchange of red hots for rock candies decreases as more and more rock candies are consumed.

3 Note that the exact slopes of the indifference curves in the indifference map will depend an the individual s utility function and may differ among students. Elton, Gruber, Brown and Goetzmann 3 Modern portfolio theory and investment analysis selected solutions to Text Problems Chapter 1: Problem 3 If you consume nothing at time 1 and instead invest all of your time61 income at a riskless rate of 10%, then at time 2 you will be able to consume all of your time62 income plus the proceeds earned from your investment : $5,000 + $5,000 ( ) = $10,500. If you consume nothing at time 2 and instead borrow at time 1 the present value of your time62 income at a riskless rate of 10%, then at time 1 you will be able to consume all of the borrowed proceeds plus your time61 income: $5,000 + $5,000 ( ) = $9, All other possible optimal consumption patterns of time 1 and time 2 consumption appear on a straight line (the opportunity set) with an intercept of $10,500 and a slope of.

4 C2 = $5,000 + ($5,000 C1) ( ) = $10,500 Elton, Gruber, Brown and Goetzmann 4 Modern portfolio theory and investment analysis selected solutions to Text Problems Chapter 1: Problem 5 With Job 1 you can consume $30 + $50 ( ) = $ at time 2 and nothing at time 1, $50 + $30 ( ) = $ at time 1 and nothing at time 2, or any consumption pattern of time 1 and time 2 consumption shown along the line AB: C2 = $ With Job 2 you can consume $40 + $40 ( ) = $ at time 2 and nothing at time 1, $40 + $40 ( ) = $ at time 1 and nothing at time 2, or any consumption pattern of time 1 and time 2 consumption shown along the line CD: C2 = $ The individual should select Job 1, since the opportunity set associated with it (line AB) dominates the opportunity set of Job 2 (line CD).

5 Elton, Gruber, Brown and Goetzmann 5 Modern portfolio theory and investment analysis selected solutions to Text Problems Chapter 1: Problem 9 Let X = the number of pizza slices, and let Y = the number of hamburgers. Then, if pizza slices are $2 each, hamburgers are $ each, and you have $10, your opportunity set is given algebraically by $2X + $ = $10 Solving the above equation for X gives X = 5 , which is the equation for a straight line with an intercept of 5 and a slope of Graphically, the opportunity set appears as follows: Assuming you like both pizza and hamburgers, your indifference curves will be negatively sloped, and you will be better off on an indifference curve to the right of another indifference curve.

6 Assuming diminishing marginal rate of substitution between pizza slices and hamburgers (the lower the number of hamburgers you have, the more pizza slices you need to give up one more burger without changing your level of satisfaction), your indifference curves will also be convex. Elton, Gruber, Brown and Goetzmann 6 Modern portfolio theory and investment analysis selected solutions to Text Problems A typical family of indifference curves appears below. Although you would rather be on an indifference curve as far to the right as possible, you are constrained by your $10 budget to be on an indifference curve that is on or to the left of the opportunity set.

7 Therefore, your optimal choice is the combination of pizza slices and hamburgers that is represented by the point where your indifference curve is just tangent to the opportunity set (point A below). Elton, Gruber, Brown and Goetzmann 7 Modern portfolio theory and investment analysis selected solutions to Text Problems Chapter 4: Problem 1 A. Expected return is the sum of each outcome times its associated probability. Expected return of Asset 1 = =1R16% + 12% + 8% = 12% 2R = 6%; 3R = 14%; 4R = 12% Standard deviation of return is the square root of the sum of the squares of each outcome minus the mean times the associated probability.

8 Standard deviation of Asset 1 = 1 = [(16% 12%)2 + (12% 12%)2 + (8% 12%)2 ]1/2 = 81/2 = 2 = 21/2 = ; 3 = 181/2 = ; 4 = = B. Covariance of return between Assets 1 and 2 = 12 = (16 12) (4 6) + (12 12) (6 6) + (8 12) (8 6) = 4 The variance/covariance matrix for all pairs of assets is: 1 2 3 4 1 8 4 12 0 2 4 2 6 0 3 12 6 18 0 4 0 0 0 Correlation of return between Assets 1 and 2 = = = . The correlation matrix for all pairs of assets is: 1 2 3 4 1 1 1 1 0 2 1 1 1 0 3 1 1 1 0 4 0 0 0 1 Elton, Gruber, Brown and Goetzmann 8 Modern portfolio theory and investment analysis selected solutions to Text Problems C.

9 portfolio Expected Return A 1/2 12% + 1/2 6% = 9% B 13% C 12% D 10% E 13% F 1/3 12% + 1/3 6% + 1/3 14% = G H I 1/4 12% + 1/4 6% + 1/4 14% + 1/4 12% = 11% portfolio Variance A (1/2)2 8 + (1/2)2 2 + 2 1/2 1/2 ( 4) = B C D 2 E 7 F (1/3)2 8 + (1/3)2 2 + (1/3)2 18 + 2 1/3 1/3 ( 4) + 2 1/3 1/3 12 + 2 1/3 1/3 ( 6) = G 2 H I (1/4)2 8 + (1/4)2 2 + (1/4)2 18 + (1/4)2 + 2 1/4 1/4 ( 4) + 2 1/4 1/4 12 + 2 1/4 1/4 0 + 2 1/4 1/4 ( 6) + 2 1/4 1/4 0 + 2 1/4 1/4 0 = Elton, Gruber, Brown and Goetzmann 9 Modern portfolio theory and investment analysis selected solutions to Text Problems Chapter 5: Problem 1 From Problem 1 of Chapter 4, we know that: R1 = 12% R2 = 6% R3 = 14% R4 = 12% 21 = 8 22 = 2 23 = 18 24 = 1 = 2 = 3 = 4 = 12 = 4 13 = 12 14 = 0 23 = 6 24 = 0 34 = 0 12 = 1 13 = 1 14 = 0 23 = 24 = 0 34 = 0 In this problem, we will examine 26asset portfolios consisting of the following pairs of securities.

10 Pair Securities A 1 and 2 B 1 and 3 C 1 and 4 D 2 and 3 E 2 and 4 F 3 and 4 A. Short Selling Not Allowed (Note that the answers to part are integrated with the answers to parts , and below.) We want to find the weights, the standard deviation and the expected return of the minimum6risk porfolio, also known as the global minimum variance (GMV) portfolio , of a pair of assets when short sales are not allowed. Elton, Gruber, Brown and Goetzmann 10 Modern portfolio theory and investment analysis selected solutions to Text Problems We further know that the compostion of the GMV portfolio of any two assets i and j is: ijjiijjGMViX 2222 + = GMViGMVjXX =1 Pair A (assets 1 and 2): Applying the above GMV weight formula to Pair A yields the following weights: 31186)4)(2(28)4(2212222112221== + = + = GMVX (or ) 32311112= = =GMVGMVXX (or ) This in turn gives the following for the GMV portfolio of Pair A.