Question

3% 1. Here are the probability distributions for three investment project returns: Up (prob = Down (prob = 0.4) 0.6) 5% -2% -2. 5% 14% -8% a. Without calculation, which two assets (X&Y, Y&Z, or X&Z) can reach the goal of diversification, which not? Explain. b. Calculate the expected return and standard deviation of each project and explain which one a rational investor would choose. c. Given your answer in part a., if you can only split your investment by half, which two projects would you invest in, and what becomes the expected return and standard deviation of your portfolio?

Answer 1

Part (a)

The behavior of X and Z are similar under the two situation. They both have positive return in up state and negative return in downstate. Because of this a combination of X and Z will not achieve diversification. X & Y and Y & Z have complementary behavior. Hence, diversification can be achieved by two assets, X & Y and Y & Z.

Part (b)

Please see the table below. Please be guided by the second row to understand the mathematics. The columns highlighted in yellow contain your answer. Figures in parenthesis, if any, mean negative values. All financials are in $. Adjacent cells in blue contain the formula in excel I have used to get the final output.

N R Asset Up return Std deviation Excel formula Ru Pd Re = PuRu + PdRd 5% Down return. Probability of Up Probability of Down Rd Pu -2% 0.4 0.6 3% 0.4 0.6 -8% 0.4 0.6 0.80% 3.43% Y 0.80% 2.69% -2.50% 14% '=SQRT(Q96*(096-596)^2+R96*(P96-596)^2) '=SQRT(Q97*(097-597)^2+R97*(P97-597)^2) "=SQRT(Q98*(098-598)^2+R98*(P98-998)^2) 0.80% 10.78%

Expected return of all the three assets are same. Hence, the rational investor will choose the asset with the least standard deviation, given the same expected return. Hence, the choice will be asset / project Y.

Part (c)

Since expected return is same, I will choose the projects with least standard deviation. Hence, I will choose a combination of X & Y.

Please see the table below.

Yellow colored cells contain your answer. Adjacent cells in blue has excel formula.

P Down return Rd 105 Asset 106 107 X 108 Y 109 Portfolio 0.5X + 0.5Y 110 Probability 111 Expected return of portfolio Up return Ru 5% -2.50% 1.25% -2% 0.4 0.80% 3% 0.50% 0.6 '=Q109*Q110+R109*R110 '=SQRT(Q110*(Q109- Q111)^2+R110*(R109- Q111)^2) 0.37% Standard deviation 112