Bob has a $50,000 stock portfolio with a beta of 1.2,an expected return of 10.8%,and a standard deviation of 25%.Becky also has a $50,000 portfolio,but it has a beta of 0.8,an expected return of 9.2%,and a standard deviation that is also 25%.The correlation coefficient,r,between Bob's and Becky's portfolios is zero.If Bob and Becky marry and combine their portfolios,which of the following best describes their combined $100,000 portfolio?
A) The combined portfolio's expected return will be less than the simple weighted average of the expected returns of the two individual portfolios,10.0%.
B) The combined portfolio's beta will be equal to a simple weighted average of the betas of the two individual portfolios,1.0;its expected return will be equal to a simple weighted average of the expected returns of the two individual portfolios,10.0%;and its standard deviation will be less than the simple average of the two portfolios' standard deviations,25%.
C) The combined portfolio's expected return will be greater than the simple weighted average of the expected returns of the two individual portfolios,10.0%.
D) The combined portfolio's standard deviation will be greater than the simple average of the two portfolios' standard deviations,25%.
E) The combined portfolio's standard deviation will be equal to a simple average of the two portfolios' standard deviations,25%.

Question

Quizplus · Accepted Answer

When Bob and Becky combine portfolios, their combined beta will be the weighted average of their individual betas: 
Combined beta = (Bob's beta x Bob's portfolio value + Becky's beta x Becky's portfolio value) /Total portfolio value
Combined beta = (1.2 x $50,000 + 0.8 x $50,000) / $100,000 = 1.0
The expected return of the combined portfolio will also be the weighted average of the individual expected returns:
Combined expected return = (Bob's expected return x Bob's portfolio value + Becky's expected return x Becky's portfolio value) / Total portfolio value
Combined expected return = (10.8% x $50,000 + 9.2% x $50,000) / $100,000 = 10.0%
The combined portfolio's standard deviation can be calculated using the formula:
Combined standard deviation = sqrt[(Bob's proportion of portfolio x Bob's standard deviation)^2 + (Becky's proportion of portfolio x Becky's standard deviation)^2 + 2 x Bob's proportion of portfolio x Becky's proportion of portfolio x Bob-Becky covariance x Bob's standard deviation x Becky's standard deviation]
Since the correlation coefficient is 0, the covariance is also 0. Therefore, the formula simplifies to:
Combined standard deviation = sqrt[(0.5 x 0.25)^2 + (0.5 x 0.25)^2] = 0.1768 or approximately 17.7%
This is less than the simple average of the two portfolios' standard deviations, which is 25%. Thus, option B is correct.

Bob Has a $50,000 Stock Portfolio with a Beta of 1.2,an