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 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%.

Question

Quizplus · Accepted Answer

The combined portfolio's beta will be the simple weighted average of the betas of the two individual portfolios:

Weighted average beta = (50,000/100,000) x 1.2 + (50,000/100,000) x 0.8 = 1.0

The combined portfolio's expected return will also be the simple weighted average of the expected returns of the two individual portfolios:

Weighted average expected return = (50,000/100,000) x 10.8% + (50,000/100,000) x 9.2% = 10.0%

However, the combined portfolio's standard deviation will be less than the simple average of the two portfolios' standard deviations due to their zero correlation coefficient. The formula for the standard deviation of the combined portfolio is:

Standard deviation = sqrt[(50,000/100,000)^2 x (0.25)^2 + (50,000/100,000)^2 x (0.25)^2 + 2 x (50,000/100,000) x (50,000/100,000) x 0.8 x 1.2 x 0.25 x 0.25 x 0] = 0.20, or 20%

Since 20% is less than the simple average of the two portfolios' standard deviations (25%), choice B is the correct answer.

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