Explanation

In the APT formula: $R_i = E(Ri) + \beta_{i1}[I_1 - E(I_1)] + \cdots + \beta_{iK}[I_K - E(I_K)] + e_i$

• $R_i$ is the actual return of security i
• $E(R_i)$ is the expected return of security i
• $I_k$ is the actual value of factor k
• $E(I_k)$ is the expected value of factor k
• $[I_k - E(I_k)]$ is the surprise or unexpected component of factor k
• $\beta_{ik}$ is the sensitivity of security i's return to factor k
• $e_i$ is the idiosyncratic error term

Correct Interpretation of $\beta_{ik}$:

$\beta_{ik}$ measures how changes in the surprise component of factor k ($[I_k - E(I_k)]$) will affect the rate of return of security i. It represents the sensitivity or exposure of security i to factor k's unexpected movements.

Why other options are incorrect:

• A: Incorrect - $\beta_{ik}$ measures the effect of factor k on security i, not security k on factor l
• B: Incorrect - The difference $[I_k - E(I_k)]$ measures the surprise, not $\beta_{ik}$
• C: Incorrect - $e_i$ measures the idiosyncratic shock, not $\beta_{ik}$

Therefore, option D provides the correct interpretation of $\beta_{ik}$ in the APT framework.