Question

The market is expected to yield 9.4% and the risk free rate is 1.2%. You currently hold a portfolio with a beta of 0.7 worth $24,400. You want to invest in another portfolio with a beta of 1.8. How much will you have to invest in the new risky asset so that the resulting portfolio will have an expected return of 11.4%? Note: You are adding funds to the new asset and to the overall portfolio. answer in $ not in percentage.

Answer is 23,864.90 but I am getting 20,328.14

Answer 1

Expected return on Market = R_M = 9.4%

Risk-free rate = R_F = 1.2%

First portfolio

Amount invested in the first portfolio = V₁ = $24400

Beta of the portfolio = β₁ = 0.7

Expected return on the first portfolio can be calculated using the CAPM equation:

Expected return on the first portfolio = E[R₁] = R_F + β₁*(R_M-R_F) = 1.2% + 0.7*(9.4%-1.2%) = 6.94%

Risky-asset

Now, suppose that the amount invested in the risky-asset = V₂

Beta of the risky-asset = β₂ = 1.8

Expected return on the risky-asset = E[R₂] = R_F + β₁*(R_M-R_F) = 1.2% + 1.8*(9.4%-1.2%) = 15.96%

Resulting portfolio

Total amount invested in the resulting portfolio = 24400 + V₂

Weight of the first portfolio in the resulting portfolio = W₁ = 24400/(24400+V₂)

Weight of the risky-asset in the resulting portfolio = W₂ = V₂/(24400+V₂)

Now, it is given that the expected return of the resulting portfolio = E[R] = 11.4%

Expected return of the resulting portfolio is calculated using the below formula:

E[R] = W₁*E[R₁] + W₂*E[R₂]

E[R₁] = 6.94%, E[R₂] = 15.96%

11.4% = [24400*6.94% + V₂*15.96%]/(24400+V₂)

11.4%*(24400+V₂) = 1693.36 + 0.1596*V₂

2781.6 + 0.114*V₂ = 1693.36 + 0.1596*V₂

2781.6 - 1693.36 = 0.1596*V₂ - 0.114*V₂

0.0456*V₂ = 1088.24

V₂ = 1088.24/0.0456 = 23864.9122807018 ~ 23864.9

Amount invested in new risky asset = V₂ = 23864.9

Answer -> 23864.9