BM Ch9 optimal portfolio practice exam question

The market has only two risky securities, with expected returns, standard deviations, and market values of

    Expected Return    Standard Deviation    Market Value
Stock Y     9.19%    30.40%    45.59 million
Stock Z    13.80%    67.20%    18.97 million


The correlation of stocks Y and Z is 54.62%. Risk-free bonds yield 4.74%. An investor who can borrow or lend at the risk-free rate forms an optimal portfolio of risk-free bonds and risky securities.

Question 8.1: Expected return of market portfolio of risky assets

What is the expected return of the market portfolio of risky securities?

Answer 8.1: The expected return of the market portfolio of risky securities is a weighted average by market value:

    ( (9.19%) × 45.59 + (13.80%) × 18.97 ) / (45.59 + 18.97) = 10.54%


Question 8.2: Variance of market portfolio of risky assets

What is the variance of the market portfolio of risky securities?

Answer 8.2: The variance of the market portfolio of risky securities is

    ( (30.40%)2 × 45.592 + (67.20%)2 × 18.972 + 2 × (54.62%) × (30.40%) × 45.59 × (67.20%) × 18.97) / (45.59 + 18.97)2 = 13.14%



Question 8.3: Standard deviation of market portfolio of risky assets

What is the standard deviation of the market portfolio of risky securities?


Answer 8.3: The standard deviation of the market portfolio of risky securities is the square root of the variance

    (13.14%)0.5 = 36.25%


Question 8.4: Composition of optimal portfolio

What is the composition of the optimal portfolio with a standard deviation of 42.7%?

Answer 8.4: The optimal portfolio is a combination of risk-free bonds with a standard deviation of zero and the market portfolio of risky securities that have a standard deviation of 36.25%. For a standard deviation of 42.7%, we solve

36.25% × Z + 0 × (1 – Z) = 42.7%
➾ Z = (42.7%) / (36.25%) = 117.79%
➾ (1 – Z) = 1 – 117.79% = -17.79%

For an optimal portfolio with a standard deviation of 42.7% and a market value of 100, the investor sells 17.79 of risk-free bonds with a standard deviation of zero and buys 117.79 of the market portfolio of risky securities with a standard deviation of 36.25%. This investor wants a portfolio even riskier than the market portfolio.


Question 8.5: Expected return of optimal portfolio

What is the expected return on an optimal portfolio with a standard deviation of 42.7%?

The expected return on the optimal portfolio with a standard deviation of 42.7% is a weighted average of its two parts:

    117.79% × 10.54% + -17.79% × 4.74% = 11.57%


Question 8.6: Composition of optimal portfolio

What is the composition of the optimal portfolio with an expected return of 6.24%?

Answer 8.6: The optimal portfolio is a combination of risk-free bonds with an expected return of 4.74% and the market portfolio of risky securities that have an expected return of 10.54%. For an expected return of 6.24%, we solve

10.54% × Z + 4.74% × (1 – Z) = 6.24%

➾ Z = (6.24% – 4.74% ) / (10.54% – 4.74%) = 25.86%

➾ (1 – Z) = 1 – 25.86% = 74.14%

For an optimal portfolio with an expected return of 6.24% and a market value of 100, the investor

●    buys 25.86 of the market portfolio of risky securities with an expected return of 10.54% and
●    buys 74.14 of risk-free bonds with an expected return of 4.74%.


Question 8.7: Standard deviation of optimal portfolio

What is the standard deviation on an optimal portfolio with an expected return of 6.24%?


Answer 8.7: The risk-free bonds have a standard deviation of zero and are not correlated with the market portfolio of risky securities, which have a standard deviation of 36.25%. This optimal portfolio has a standard deviation of 25.86% × 36.25% = 9.37%.