Charlie and Lucinda each have $50,000 invested in stock portfolios. Charlie's has a beta of 1.2, an expected return of 10.8%, and a standard deviation of 25%. Lucinda's 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 Charlie's and Lucinda's portfolios is zero. If Charlie and Lucinda marry and combine their portfolios, which of the following best describes their combined $100,000 portfolioσ
A) 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%.
B) 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%.
C) the combined portfolio's standard deviation will be greater than the simple average of the two portfolios' standard deviations, 25%.
D) the combined portfolio's standard deviation will be equal to a simple average of the two portfolios' standard deviations, 25%.
E) 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%.

Question

Quizplus · Accepted Answer

Since the correlation coefficient, r, between Charlie's and Lucinda's portfolios is zero, the diversification benefits of combining the two portfolios will be maximized, resulting in a combined portfolio that has a standard deviation less than the simple average of the two individual portfolios' standard deviations.

Using the formulas for combining two portfolios:
Weighted Average Beta = (Weight of Portfolio A * Beta of Portfolio A) + (Weight of Portfolio B * Beta of Portfolio B)
= (50,000/100,000 * 1.2) + (50,000/100,000 * 0.8) = 1.0
Weighted Average Expected Return = (Weight of Portfolio A * Expected Return of Portfolio A) + (Weight of Portfolio B * Expected Return of Portfolio B)
= (50,000/100,000 * 10.8%) + (50,000/100,000 * 9.2%) = 10.0%
Combined Portfolio Standard Deviation = sqrt[(Weight of Portfolio A)^2 * (Standard Deviation of Portfolio A)^2 + (Weight of Portfolio B)^2 * (Standard Deviation of Portfolio B)^2 + 2 * (Weight of Portfolio A) * (Weight of Portfolio B) * (Standard Deviation of Portfolio A) * (Standard Deviation of Portfolio B) * (Correlation Coefficient)]
= sqrt[(0.5)^2 * (0.25)^2 + (0.5)^2 * (0.25)^2 + 2 * (0.5) * (0.5) * (0.25) * (0.25) * (0)]
= 0.176 < 0.25 (the simple average of the two individual portfolios' standard deviations)

Therefore, the combined portfolio's beta will be a simple weighted average of the betas of the two individual portfolios, 1.0; its expected return will be 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%.

Charlie and Lucinda Each Have $50,000 Invested in Stock Portfolios