Question

Suppose that among the many stocks in the market there are two securities, A and B, with the following characteristics: A has mean return of 8% and return standard deviation σ = 0.4 and B has mean return of 13% and return standard deviation σ = 0.6. If the correlation between these two is ρ =−1, and if it is possible to borrow and lend at the risk-free rate, rf, then the equilibrium risk-free rate must be: (Hint: the minimum variance portfolio constructed using A and B has zero variance)

Answer 1

Let w be the proportion in Stock A and 1-w in Stock B

For perfectly negatively correlated stocks, portfolio standard deviation=w1*s1-w2*s2

Risk free rate has zero standard deviation

Hence,
w*40%-(1-w)*60%=0
=>w=0.60
and 1-w=0.40

The portfolio has zero standard deviation i.e., zero risk and hence the returns of the portfolio must equal the risk free rate, to prevent arbitrage

Hence,
Expected returns=w1*R1+w2*R2=0.60*8%+0.40*13%=10.0000%

Hence, risk free rate is 10.000%