Risk-free allocations for target return or beta

EXERCISE 01

The stock has beta 1.2 and expected return 16%. The risk-free asset returns 5%.

E[R_P]=0.5(0.16)+0.5(0.05)=10.5\%

Expected return = 10.5%.

w=0.75/1.2=0.625

62.5% in the stock; 37.5% risk-free.

w=\frac{0.08-0.05}{0.16-0.05}=0.272727

\beta_P=\frac{0.08-0.05}{0.16-0.05}(1.2)\approx0.32727

Portfolio beta = 0.32727.

EXERCISE 02

A €1,000 portfolio contains €400 of A (beta 1.3), an unknown amount of B (beta 0.7), and the balance in a risk-free asset. Target beta is 0.90.

\beta_P=w_A\beta_A+w_B\beta_B

0.90=0.40(1.3)+\frac{B}{1000}(0.7)

B=1000\frac{0.90-0.40(1.3)}{0.7}

B = €542.86; risk-free balance = €57.14.

EXERCISE 03

Risk-free rate = 2.500%; market risk premium = 5.000%.

\beta_P=\sum_iw_i\beta_i,\quad E[R_P]=r_f+\beta_PMRP

\beta_P=(0.318182)(0.7)+(0.181818)(0.85)+(0.163636)(1.1)+(0.336364)(1.4)\approx 1.028182

E[R_P]=0.025+(1.028182)(0.05)\approx 0.076409

Beta = 1.02818; expected return = 7.641%.

w_P=\frac{r_{target}-r_f}{E[R_P]-r_f}

w_P=\frac{0.09-0.025}{0.076409-0.025}\approx 1.264368

w_f=1-w_P

Portfolio weight = 126.437%; risk-free weight = -26.437%.

The negative risk-free position means borrowing.