Stock X has an expected return of 8% and Stock Z has an expected return of 12%.
ID: 2715365 • Letter: S
Question
Stock X has an expected return of 8% and Stock Z has an expected return of 12%. The standard deviation of Stock X is 12% and the standard deviation of Stock Z is 8%. Assume that these are the only two stocks available in a hypothetical world.
If the correlation between the returns of the two stocks is +1:
What is the expected return and standard deviation of a portfolio containing:
100% Z
25% X and 75% Z
50% X and 50% Z
75% X and 25% Z
100% X
Will any investor include Stock X in his or her portfolio? Explain why or why not.
If the correlation between the returns of the two stocks is +0.3:
What is the expected return and standard deviation of a portfolio containing:
100% Z
25% X and 75% Z
50% X and 50% Z
75% X and 25% Z
100% X
Will any investor include Stock X in his or her portfolio (if the correlation is +0.3)? If so, what is the maximum amount of Stock X a rational investor might include in a portfolio? Support your answer numerically.
Explanation / Answer
The solution is for part C of two questions.
Use it as sample for remaining.
Xr = 8%, Zr = 12%, Xs = 12%, Zs = 8%, Wx = 50%, Wy = 50%
A. Cor (X,Z) = +1
Cor (X,Z) = Covar(X,Z)/(Xs * Zs)
Covar(X,Z) = Cor (X,Z) * Xs * Zs
= 1 * 12% * 8%
= 0.0096
Expected return of portfolio = weight(X) * return (X) + weight (Z) * return (Z)
= 0.5 * 8% + 0.5 * 12%
= 10%
Expected standard deviation = (Wx2 * Xs2 + Wz2 * Zs2 + 2Wx * Wz * Covar (X,Z))0.5
= (0.52 * 12%2 + 0.52 * 8%2 + 2 * 0.5 * 0.5 * 0.0096)0.5
= 0.5 * (12%2 + 8%2 + 2 * 0.0096)0.5
= 0.5 * 20%
= 10%
Now stock X has lower return of 8% and higher standard deviation of 12%. Higher standard deviation means that the stock X is more risky as compared to stock Z which has a standard deviation of 8%. Also, the correlation between the stocks is 1, which means that if one price of one stock rises, price of other stock will also rise.
Hence, there is no point of keeping stock X in the portfolio since it has a lower return and higher standard deviation. Now, if the portfolio consisted of only stock z,
Return of the portfolio = 12%, and standard deviation = 8% , which are better than the portfolio having X in it.
Hence, no investor would include stock X in the portfolio
B. Cor (X,Z) = 0.3
Cor (X,Z) = Covar(X,Z)/(Xs * Zs)
Covar(X,Z) = Cor (X,Z) * Xs * Zs
= 0.3* 12% * 8%
= 0.00288
Now expected return would not change from Part A, hence Expected return = 10%
Expected standard deviation = (Wx2 * Xs2 + Wz2 * Zs2 + 2Wx * Wz * Covar (X,Z))0.5
= (0.52 * 12%2 + 0.52 * 8%2 + 2 * 0.5 * 0.5 * 0.0029)0.5
= 0.5 * (12%2 + 8%2 + 2 * 0.0029)0.5
= 0.5 * 16.3%
= 8.15%
In this case, when correlation is 0.3, the standard deviation comes down to 8.15% from 10% in part A, hence if an investor includes stock X in the portfolio with stock Z, the standard deviation gets decreased.
Hence, the investor should include stock X in the portfolio.
Now the maximum amount of X would be the point at which the standard deviation of the portfolio becomes equal to 8. Since, after that amount, the standard deviation would be more than the standard deviation of portfolio consisting of stock Z only. Let Wx be X and Wz = (1-X)
Expected standard deviation = (Wx2 * Xs2 + Wz2 * Zs2 + 2Wx * Wz * Covar (X,Z))0.5
8%2 = (X2 * 12%2 + (1-X)2 * 8%2 + 2 * X* (1-X) * 0.0029)
64 = 144X2 + 64 + 64X2 - 128X + 58X -58X2
Solving for X , X= 7/15 = 0.47
Hence the maximum amount of stock X in the portfolio should be 47%
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.