Explanation

Using the one-factor APT model:

$E(R_p) = R_f + \beta_p \times \lambda$

For Portfolio X: $`$8`% = R_f + 0.8\lambda$$

For Portfolio Y: $`$16`% = R_f + 2.4\lambda$$

Subtract the first equation from the second: $(16\% - 8\%) = (R_f + 2.4\lambda) - (R_f + 0.8\lambda)$ $`$8`% = 1.6\lambda$$\lambda = 5%$$

Now substitute back into Portfolio X's equation: $`$8\% = R_f + 0.8 \times 5\%$$ $$8`% = R_f + 4%$$R_f = 4%$$

Therefore, the risk-free rate is 4.0%.