Zero-volatility portfolios and consistency of assumptions

EXERCISE 01

For the first calculation, A returns 8% with volatility 10%; B returns 12% with volatility 18%; their correlation is −1. Separately, the exercise asks for CAPM implications with market risk premium 5% and beta(A) = 0.4.

A.Find the zero-volatility weights and return.
B.Calculate the rates implied by the separate CAPM inputs.
C.Are the two sets of assumptions jointly consistent?

A.

w_A=\frac{0.18}{0.10+0.18}=\frac9{14},\quad w_B=\frac5{14}

\sigma_P=\left|\frac9{14}(0.10)-\frac5{14}(0.18)\right|=0

E[R_P]=\frac9{14}(0.08)+\frac5{14}(0.12)\approx9.42857\%

Zero-volatility portfolio return = 9.42857%.

B.

r_f=8\%-0.4(5\%)=6\%,\quad E[R_M]=11\%

\beta_B=(12\%-6\%)/5\%=1.2

C.

No: the zero-risk combination would earn 9.42857%, not the implied 6% risk-free rate. Also, the source’s positive correlations of both assets with the market (0.8 and 0.7) cannot hold together with their perfect negative correlation. The two computations above are conditional calculations, not a jointly feasible CAPM economy.