1. Assume that an individual can either invest all of his resources in one of th
ID: 2734029 • Letter: 1
Question
1. Assume that an individual can either invest all of his resources in one of the two securities, A or B; or, alternatively, he can diversify his investment between the two. The distributions of the returns are as follows:
Security A Security B
Return Probability Return Probability
-10 1/2 -20 1/2
50 1/2 60 1/2
Assume that the correlation between the returns from the two securities is zero, and answer the following questions:
1) Calculate each security's expected return, variance and standard deviation.
2) Calculate the probability distribution of the returns on a mixed portfolio comprised of equal proportions of securities A and B, i.e. calculate all possible returns on this portfolio and the probability of each one.[1]
3) Also calculate the portfolio's expected return, variance and standard deviation.
4) Calculate the expected return and the variance of a mixed portfolio comprised of 75% of security A and 25% of security B.
Please show formulas
[1] In the case of the two independently distributed returns the joint probability that the return on A is x% and the return on B is y% at the same time is the product of marginal probabilities. That is
Explanation / Answer
Given, Security A Security B Return(%age) Probability Return Probability -10 0.5 -20 0.5 50 0.5 60 0.5 Answer 1) Expected Return = Return * Probability + Return * Probability Secutity A =-10%*.5 + 50%*.5 =20% Security B =-20%*.5 + 60%*.5 =20% Standard Deviation = probability(Given Return - Expected Return)2 Variance = Standard Deviation 2 Security A Given Return(X) Expected Return(Y) (X-Y) (X-Y)2 Prob. Prob(X-Y)2 -10 20 -30 900 0.5 450 50 20 30 900 0.5 450 probability(Given Return - Expected Return)2 900 Standard Deviation = 30 Variance =302 =900 Security B Given Return(X) Expected Return(Y) (X-Y) (X-Y)2 Prob. Prob(X-Y)2 -20 20 -40 1600 0.5 800 60 20 40 1600 0.5 800 probability(Given Return - Expected Return)2 1600 Standard Deviation = 40 Variance =402 =1600 Answer 2) Probability Distribution for return on security A and B Probability Distribution = Weight of security * Probability of Return State Return on Security A Probability Distribution 1 Return on Security A -10% 0.25 2 Return on Security A 50% 0.25 3 Return on Security B -20% 0.25 4 Return on Security B 60% 0.25 Answer 3) Portfolio Expected Return = Expected Return (A) * Weight (A) + Expected Return (B) * Weight (B) =20%*.5 + 20%*.5 =20% Standard Deviation of Portfolio(Where correlation is 0) = Std. Deviation(A)2*Weight(A)2 + Std. Deviation(B)2*Weight(B)2 =302 * .52 + 402 + .52 =25 Variance = Standard Deviation 2 Therefore, Variance = 625 Answer 4) Calculation of expected return and the variance of a mixed portfolio comprised of 75% of security A and 25% of security B Portfolio Expected Return = Expected Return (A) * Weight (A) + Expected Return (B) * Weight (B) =20%*.75 + 20%*.25 =20% Standard Deviation of Portfolio(Where correlation is 0) = Std. Deviation(A)2*Weight(A)2 + Std. Deviation(B)2*Weight(B)2 =302 * .752 + 402 + .252 =24.62 Variance = Standard Deviation 2 Therefore, Variance = 606.25
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