Portfolio risk and return: suppose that the S&P500, with a beta of 1 has an expe
ID: 2726349 • Letter: P
Question
Portfolio risk and return: suppose that the S&P500, with a beta of 1 has an expected return of 10%, and T-bills provide a risk free return of 4%.
i) How would you construct a portfolio from these two assets with an expected return of 8%
ii) How would you construct a portfolio from these 2 assets with a beta of 0.4?
iii) Now consider a borrow and invest strategy in which you use 1 million of your own money and borrow anothing 1 million to invest 2 million in total in a market index fund. If the risk free rate is 4% (you can borrow at this rate) and the expected return on the market index fund is 12% what is the risk premium and expected return on the borrow and invest strategy?
Explanation / Answer
Beta of S&P 500 = 1.0
Expected return = 10% or 0.10
Risk-free return (T-bill rate) = 4% or 0.04
Beta of T-bills = 0
Answer (i)
Let X be the weight of S&P in the portfolio. Then (1-X) will be the weight of T-Bills in the portfolio
Target return of the portfolio = 8% or 0.08
Portfolio return can be calculated as
Portfolio return = Weight of asset A * Return of A + Weight of Asset B * Return of B
0.08 = X * 0.10 + (1-X) * 0.04
0.08 = 0.10 * X + 0.04 – 0.04 * X
0.08 – 0.04 = X * (0.10 – 0.04)
0.04 = 0.06 * X
X = 0.04/0.06 = 0.06667 or 66.67%
Weight of S&P500 in portfolio = 66.67%
Weight of T-Bills in portfolio = 100-66.67% = 33.33%
Beta of the portfolio = 0.6667 * 1.0 + 0.3333 * 0 = 0.6667
Answer (ii)
Let X be the weight of S&P 500 in the portfolio. Then 1-X will be the weight of T-Bills in the portfolio
Target Beta of the portfolio = 0.40
Portfolio Beta can be calculated as
Portfolio Beta = Weight of Asset A * Beta of A + Weight of Asset B * Beta of B
0.40 = X * 1.0 + (1-X) * 0
0.40 = 1.0 * X or X = 0.40
Weight of S&P 500 in portfolio = 0.4 or 40%
Weight of T-Bills in portfolio = 1- 0.40 =0.60 or 60%
Return on portfolio = 0.4 * 0.10 + 0.6 * 0.04 = 0.04 + 0.024 = 0.064 or6.40%
Answer (iii)
Risk Premium = Return from market – Return from risk free investment
Risk Premium = 12%-4% =8%
Expected return = 0.5*4%+0.5*12%
Expected return = 2%+6%
Expected return = 8%