Stock X and Stock Z both have an expected return of 10%. The standard deviation
ID: 2766693 • Letter: S
Question
Stock X and Stock Z both have an expected return of 10%. The standard deviation of the expected return is 8% for Stock X, and 12% for Stock Z. Assume that these are the only two stocks available in a hypothetical world.
A. Assume that the correlation between the returns of the two stocks is +1.
• What is the expected return and standard deviation of a portfolio containing 50% X and 50% Z?
• What is the optimal amount of Stock Z for an investor to hold in a portfolio (if the correlation is +1)?
B. Assume that the correlation between the returns of the two stocks is +0.1.
• What is the expected return and standard deviation of a portfolio containing 50% X and 50% Z?
• What is the optimal amount of Stock Z for an investor to hold in a portfolio (if the correlation is +0.1)?
C. Assume that the correlation between the returns of the two stocks is –1.
• What is the expected return and standard deviation of a portfolio containing 50% X and 50% Z?
• What is the optimal amount of Stock Z for an investor to hold in a portfolio (if the correlation is –1)?
D. If the Capital Asset Pricing Model is true, explain why it is possible for the two stocks to have the same expected return even though the standard deviation of the expected return is 8% for Stock X, and 12% for Stock Z. (What must be true about the types of risks facing the two companies for the given numbers to make sense?).
Explanation / Answer
Expected return rX = rZ = 10%
Standard Deviation of X SDX = 8%
Standard Deviation of Y SDZ = 12%
Answer A
Correlation coefficient between two stocks = 1.0
Weights in the portfolio = 50% of X and 50% of Z
Expected Return of Portfolio rP = 0.5 * 10% + 0.5 * 10% = 10%
Correlation of (X,Z) = Cov (X,Z)/(SDX * SDZ)
Cov (X,Z) = Correlation of (X,Z)*SDX*SDZ
Substituting values from above
Cov (X,Z) = 1.0 * 0.08 * 0.12 = 0.0096
Expected variance of portfolio of X, Z = Wt1^2*SDX^2+Wt2^2* SDZ^2+2*Wt1*Wt2*Cov (X,Z)
= 0.5^2 * 0.08^2+0.5^2*0.12^2+2*0.5*0.5*0.0096
= 0.25 *0.0064 + 0.25*0.0144+2*0.5*0.5*0.0096
= 0.0016 + 0.0036 + 0.0048
= 0.01
Expected Standard Deviation of Portfolio = Variance ^0.5 = 0.01^0.5 = 0.10 or 10%
Optimal amount of stock Z to be held in a portfolio is where the standard deviation of the portfolio is minimum
WZ = (SDZ^2 – Cor(X,Z)*SDX*SDZ) / (SDX^2+SDZ^2-2*Cor(X,Z)*SDX*SDZ)
= (0.12^2 – 1.0*0.08*0.12) / (0.08^2+0.12^2 – 2*1.0*0.08*0.12)
= (0.0144 – 0.0096)/(0.0064 + 0.0144 – 0.0192)
= 0.0048/0.0016
= 3 or 300%
As no other indications about short selling are available, since the SD of X is lower than then SD of Z, it is not possible to create a portfolio with a correlation of 1 and standard deviation lower than 8% without selling Stock X short. Hence it can be taken that the optimal holding of stock Z in a portfolio with a correlation as 1.0 as Zero.
Answer B
Correlation between Stock X and Z = 0.1
Expected return of the portfolio = 0.5 * 10% + 0.5 * 10% = 10%
Variance of Portfolio = 0.5^2*0.08^2+0.5^2*0.12^2+2*0.5*0.5*0.1*0.08*0.12
= 0.25 * 0.0064 + 0.25 * 0.0144 + 2*0.5*0.5*0.1*0.08*0.12
= 0.0016 + 0.0036 + 0.00048
= 0.000568
Portfolio Standard Deviation = 0.000568^0.5 = 0.075365 or 7.54% (rounded off)
Optimal amount of stock Z to be held in a portfolio is where the standard deviation of the portfolio is minimum
WZ = (SDZ^2 – Cor(X,Z)*SDX*SDZ) / (SDX^2+SDZ^2-2*Cor(X,Z)*SDX*SDZ)
= (0.12^2 – 0.1*0.08*0.12) / (0.08^2+0.12^2 – 2*0.1*0.08*0.12)
= (0.0144 – 0.00096)/(0.0064 + 0.0144 – 0.00192)
= 0.01344/0.01888
= 0.7118644 or 71.19% (rounded off)
Hence 71.19% the amount of stock to be held by an investor in the portfolio.
Answer C
Correlation coefficient between two stocks = -1.0
Weights in the portfolio = 50% of X and 50% of Z
Expected Return of Portfolio rP = 0.5 * 10% + 0.5 * 10% = 10%
Correlation of (X,Z) = Cov (X,Z)/(SDX * SDZ)
Cov (X,Z) = Correlation of (X,Z)*SDX*SDZ
Substituting values from above
Cov (X,Z) = -1.0 * 0.08 * 0.12 = - 0.0096
Expected variance of portfolio of X, Z = Wt1^2*SDX^2+Wt2^2* SDZ^2+2*Wt1*Wt2*Cov (X,Z)
= 0.5^2 * 0.08^2+0.5^2*0.12^2+2*0.5*0.5*-0.0096
= 0.25 *0.0064 + 0.25*0.0144+2*0.5*0.5*-0.0096
= 0.0016 + 0.0036 - 0.0048
= 0.0004
Expected Standard Deviation of Portfolio = Variance ^0.5 = 0.0004^0.5 = 0.02 or 2%
Optimal amount of stock Z to be held in a portfolio is where the standard deviation of the portfolio is minimum
WZ = (SDZ^2 – Cor(X,Z)*SDX*SDZ) / (SDX^2+SDZ^2-2*Cor(X,Z)*SDX*SDZ)
= (0.12^2 – (-1.0)*0.08*0.12) / (0.08^2+0.12^2 – 2*-1.0*0.08*0.12)
= (0.0144+0.0096)/(0.0064+0.0144+0.0192)
= 0.024/0.04 = 0.6 or 60%
Hence optimal holding of stock Z in portfolio with minimum standard deviation is 60%
Answer D
As per the Capital Asset Pricing Model, the expected return of a stock is dependent on the risk free rate, the expected market return and beta which is a measure of systematic risk of the stock. Standard Deviation on the other hand measures the total risk of the stock which consists of both systematic and unsystematic risks. Unsystematic risk the factor which is specific is a particular stock or industry which is diversified away by diversification.
The formula is as follows
Re = Rf + Beta * (Rm – Rf) where Re is expected return, Rm and Rf are market return and risk-free rate
Hence it is possible for two stocks with different standard deviations to have same expected returns based on their beta measures.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.