Question

The two risky assets you can invest in are Exxon and BP. Exxon has a mean return of 8% and a standard deviation of 10%. BP has mean return of 10 percent and standard deviation of 15 percent. The correlation between the two is 0.25. The tangency portfolio has weight of 55% in Exxon.

The risk free asset has return of 3.0 percent. What is the expected return and standard deviation of the tangency portfolio?

You desire an expected return of 25%. What will be the standard deviation of your portfolio? What fraction of your portfolio will be invested in Exxon?

Answer 1

Tangency Portfolio is the optimally risky portfolio as it is the point of tangency of the CAL to the efficient frontier of risky assets.

Exxon Weight = 0.55 and BP Weight = 0.45

Let the expected return and standard deviation of the tangency portfolio be Rt and St respectively.

Rt = 0.55 x 8 + 0.45 x 10 = 8.9 %

Correlation = 0.25

St = [{0.1 x 0.55}^(2) + {0.15 x 0.45}^(2) + 2 x 0.55 x 0.45 x 0.1 x 0.15 x 0.25]^(1/2) = 9.715 %

Risk-Free Rate = Rf = 3 %

Expected Return = E(r) = 25 %

Let the fraction of investment in risky asset be y

Therefore, E(r) = Rf + y x (Rt - Rf) = 3 + y x (8.9 - 3)

25 = 3 + 5.9y

y = (22/5.9) = 3.73

A value greater than 1 implies that this is a leverage portfolio, one where portfolio investments have been made by borrowing.

Standard deviation of Portfolio = St x y = 9.715 x 3.73 = 36.237 %

Fraction Invested in Exxon = (0.55 x 3.73) / 4.73 = .4337