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HADM 2222 Fall 2009, Prof. Q. Ma Homework assignment #5 [Due 10 a.m. Wednesday November 4, 2009, Statler 435 drop box] 1. Consider the following information State Probability X Z_ Boom .25 15% 10% Normal .60 10% 9% Recession .15 5% 10% What is the standard deviation of a portfolio with an investment of $6000 in X and $4000 in Z? Solution: Portfolio weights are 60% and 40% for assets X and Z, respectively. Rp is (13%, 9.6%, 7%) in (boom, normal, recession). For example, Rp for Boom = 60% * 15% + 40% * 10%= 13%. E(Rp)= .25(13%)+.60(9.6%) + .15(7%) = 10.06%. Variance=.00036924; standard deviation = 1.92%. 2. You have $10,000 to invest in a stock portfolio. Your choices are Stock X with an expected return of 15 percent and Stock Y with an expected return of 10 percent. If your goal is to create a portfolio with an expected return of 12.2 percent, how much money will you invest in Stock X? In Stock Y? Solution: Here we are given the expected return of the portfolio and the expected return of each asset in the portfolio, and are asked to find the weight of each asset. We can use the equation for the expected return of a portfolio to solve this problem. Since the total weight of a portfolio must equal 1 (100%), the weight of Stock Y must be one minus the weight of Stock X. Mathematically speaking, this means: E(Rp) = .122 = .15wX + .10(1 wX) We can now solve this equation for the weight of Stock X as: .122 = .15wX + .10 .10wX; wX = 0.44. So, the dollar amount invested in Stock X is the weight of Stock X times the total portfolio value, or: Investment in X = 0.44($10,000) = $4,400; and the dollar amount invested in Stock Y is: Investment in Y = (1 0.44)($10,000) = $5,600 3. Based on the information below, calculate the expected return and standard deviation for the two stocks: State of Probability of Rate of Return if State Occurs Economy State of Economy Stock A Stock B Recession .15 .06 -.20 Normal .60 .07 .13 Boom .25 .11 .33 Solution: E(RA) = .15(.06) + .60(.07) + .25(.11) = .0785 or 7.85% E(RB) = .15(.2) + .60(.13) + .25(.33) = .1305 or 13.05% !A2 =.15(.06 .0785)2 + .60(.07 .0785)2 + .25(.11 .0785)2 = .00034;!A = (.00034)1/2 = 1.85% !B2 =.15(.2 .1305)2 .60(.13 + .1305)2 + .25(.33 .1305)2 = .02633; !B = (.02633)1/2 =16.23%
Prof. Q. Ma, HADM 2222 Fall 2009
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4. What are the portfolio weights for a portfolio that has 100 shares of Stock A that sell for $40 per share and 130 shares of Stock B that sell for $22 per share? Solution: The portfolio weight of an asset is total investment in that asset divided by the total portfolio value. First, we will find the portfolio value, which is: Total value = 100($40) + 130($22) = $6,860. The portfolio weight for each stock is: WeightA = 100($40)/$6,860 = .5831; WeightB = 130($22)/$6,860 = .4169 5. Consider the following information about Stocks I and II: State of Economy Probability of Rate of Return if State Occurs State of Economy Stock I Stock II Recession .25 .09 -.30 Normal .50 .42 .12 Irrational exuberance .25 .26 .44 What are the expected returns and standard deviations of the two stocks? Solution: E(RI) = .25(.09) + .50(.42) + .25(.26) = .2975 or 29.75% !I2 = .25(.09 .2975)2 + .50(.42 .2975)2 + .25(.26 .2975)2 = .01862; !I = (.01862)1/2 = 13.65% E(RII) = .25(.30) + .50(.12) + .25(.44) = .0950 !II2 = .25(.30 .0950)2 + .50(.12 .0950)2 + .25(.44 .0950)2 = .06908; !II = (.06908)1/2 = 26.28% 6. Consider the following information: State of Probability of State of Economy Economy Boom .20 Good .40 Poor .30 Bust .10 Rate of Return if State Occurs Stock A Stock B Stock C .30 .45 .33 .12 .10 .15 .01 -.15 -.05 -.06 -.30 -.09
a. Your portfolio is invested 30 percent each in A and C, and 40 percent in B. What is the expected return of the portfolio? Solution: Boom: E(Rp) = .30(.3) + .40(.45) + .30(.33) = .3690 or 36.90% Good: E(Rp) = .30(.12) + .40(.10) + .30(.15) = .1210 or 12.10% Poor: E(Rp) = .30(.01) + .40(.15) + .30(.05) = .0720 or 7.20% Bust: E(Rp) = .30(.06) + .40(.30) + .30(.09) = .1650 or 16.50% And the expected return of the portfolio is: E(Rp) = .20(.3690) + .40(.1210) + .30(.0720) + .10(.1650) = .0841 or 8.41% b. What is the standard deviation of the portfolio? Solution: !p2 = .20(.3690 .0841)2 + .40(.1210 .0841)2 + .30(.0720 .0841)2 + .10(.1650 .0841)2 = .03029; !p = (.03029)1/2 = .1741 or 17.41%
Prof. Q. Ma, HADM 2222 Fall 2009
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