Question

A mutual fund manager has a $90.0 million portfolio with a beta of 1.25. The risk-free rate is 3.50%, and the market risk premium is 6.00%. The manager expects to receive an additional $30.0 million which she plans to invest in a number of stocks. After investing the additional funds, she wants to reduce the portfolio’s risk level so that once the additional funds are invested the portfolio’s required return will be 11.75%. What must the average beta of the new stocks added to the portfolio be to achieve the desired required rate of return?

Answer 1

Investment in Old Portfolio = $90.0 million
Investment in New Stock = $30.0 million

Investment in New Portfolio = Investment in Old Portfolio + Investment in New Stock
Investment in New Portfolio = $90.0 million + $30.0 million
Investment in New Portfolio = $120.0 million

Weight of Old Portfolio = Investment in Old Portfolio / Investment in New Portfolio
Weight of Old Portfolio = $90.0 million / $120.0 million
Weight of Old Portfolio = 0.75

Weight of New Stock = Investment in New Stock / Investment in New Portfolio
Weight of New Stock = $30.0 million / $120.0 million
Weight of New Stock = 0.25

Required Return of New Portfolio = Risk-free Rate + Beta of New Portfolio * Market Risk Premium
11.75% = 3.50% + Beta of New Portfolio * 6.00%
8.25% = Beta of New Portfolio * 6.00%
Beta of New Portfolio = 1.375

Beta of New Portfolio = Weight of Old Portfolio * Beta of Old Portfolio + Weight of New Stock * Beta of New Stock
1.375 = 0.75 * 1.25 + 0.25 * Beta of New Stock
1.375 = 0.9375 + 0.25 * Beta of New Stock
0.4375 = 0.25 * Beta of New Stock
Beta of New Stock = 1.75

So, the average beta of the new stocks must be 1.75