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Problems on Portfolio Standard Deviation

Coursework Instructions:

You need to do the question (e) (f) (g)in the document. Thank you !

Problem 1 Suppose you are given the following information for two stocks, A and B  
a) Calculate the expected return of each stock Expected return on stock A = 0.1*60% + 0.2*20% + 0.5*10% + 0.2*(-25%) = 0.1Expected return on stock B = 0.1*5% + 0.2*25% + 0.5*5% + 0.2*0% = 0.08
b) Calculate the standard deviation (SD) of each stock Std. deviation of Stock A = 22.69%, Std. deviation of stock B = 8.72%
c) Calculate the weights and expected return for a portfolio that has $600 invested in stock A and $1400 invested in stock B Weight of A = 600/(600+1400) = 600/2000 = 0.3Weight of B = 1-0.3 = 0.7
d) Calculate the portfolio standard deviations assuming that the correlation between the returns of A and B is:i. -1 ii. 0 iii. 0.6iv. 1 
Case Correlation Weight of A Weight of B Std. deviation of A Std. deviation of B Std. deviation of the portfolioi -1 0.3 0.7 22.69% 8.72% 0.71%ii 0 0.3 0.7 22.69% 8.72% 9.14%iii 0.6 0.3 0.7 22.69% 8.72% 11.55%iv 1 0.3 0.7 22.69% 8.72% 12.91%
e) In the light of your answers in part (d) above, describe how a change in the degree of correlation affects the portfolio standard deviation? Which particular correlation gives the lowest portfolio standard deviation? 
f) Given correlation of -1, what portfolio weights will reduce the portfolio standard deviation to zero? Show calculation to support your answer. 
g) Calculate the weighted average of the standard deviation of individual stocks A and B; and show that each portfolio standard deviation is less than this weighted average except the portfolio standard deviation with correlation of 1.

Coursework Sample Content Preview:
e) In the light of your answers in part (d) above, describe how a change in the degree of correlation affects the portfolio standard deviation. Which particular correlation gives the lowest portfolio standard deviation?
Based on the calculation from part (D), the correlation between the returns of two assets in a portfolio has a significant impact on the portfolio standard deviation. When the correlation between the returns of Asset A and Asset B is negative (correlation = -1), the portfolio standard deviation is at its minimum, as the returns of the two assets tend to move in opposite directions and help to reduce the overall volatility of the portfolio.
On the other hand, when the correlation is positive (correlation = 1), the portfolio standard deviation is at its highest, as the returns of the two assets tend to move in the same direction, which results in higher overall volatility of the portfolio. When the correlation is between -1 and 1, the portfolio standard deviation increases as the correlation increases, and the returns of the two assets become more and more positively correlated. In other words, the higher the correlation between the two assets, the higher the portfolio standard deviation will be.
Therefore, to minimize the portfolio standard deviation, Asset A and Asset B should have a negative or low correlation, as they will help to diversify the portfolio and reduce overall volatility. The correlation that gives the lowest portfolio standard deviation is a correlation of -1, where the returns of the two assets move in opposite directions.
f) Given a correlation of -1, what portfolio weights will reduce the portfolio standard deviation to zero? Show calculations to support your answer.
Portfolio standard deviation (σ_p) = Sqrt (w_A^2 × σ_A^2 + w_B^2 × σ_B^2 + 2 × w_A × w_B × σ_A × σ_B × ρ_AB)
where:
σ_p is the standard deviation of the portfolio
w_A and w_B are the weights of Asset A and Asset B in the portfolio, respectively
σ_A and σ_B are the standard deviations of Asset A and Asset B, respectively
ρ_AB is the correlation between the returns of Asset A and Asset B.
Substituting the given values for the correlation of -1, Asset A standard deviation of 22.69%, and Asset B standar...
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