Question

Which statements are true of the number of inputs necessary in the single-factor index model?

a. there are 3n+2 total estimates
b. there are a total of (n^2 + 3n)/2 estimates
c. there are n estimates of alpha
d. there are n(n-1)/2 different covariances

Answer 1

True statements regarding the single-factor index model are that there are n estimates of alpha and n(n-1)/2 different covariances between the returns of different assets.

The model involves estimating both the risk specific to each security and the risk shared across securities.

Step-by-step explanation:

The single-factor index model in finance is a model that relates the returns of a security to a single common factor and the security's unique factors. When considering the model, it is important to assess the number of inputs or estimates that are necessary to calculate the expected returns. Here is the breakdown of the estimates required:

• There are n estimates of alpha (α), which represent the expected return of each asset above the return predicted by the model. Since there are n assets, there are n alphas.
• The number of variances is also n, as each asset has its own variance.
• Regarding covariances, there are n(n-1)/2 different covariances because covariance is calculated between pairs of different assets.
• There are additional estimates including the market risk premium and the risk-free rate, adding to two more estimates.

So the correct statements from the provided choices are:

• c. There are nn estimates of alpha.
• d. There are n(n-1)/2 different covariances.