Answer-first summary for fast verification
Answer: It measures how the changes in the surprise factor k will affect the rate of return of security i.
## 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.
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A risk analyst at a bank is explaining to an intern the use of the Arbitrage Pricing Theory (APT) in estimating the expected return of a security. The risk analyst uses the following APT formula in the discussion:
Which of the following is a correct interpretation of ?
A
It is a coefficient measuring the effect of changes in the rate of return of security k on the expected value of factor l.
B
It measures the difference between the observed and expected values of factor k.
C
It measures the idiosyncratic random shock to the price of security i which has a mean of zero.
D
It measures how the changes in the surprise factor k will affect the rate of return of security i.