Looking backward (ex post, or after the fact), alpha is the average of the realized residual returns.

where,
r_p(t): portfolio excess return.
r_B(t): benchmark excess return.

The estimates of β_P and α_Pobtained from the regression are the realized or historical β and α.

Looking forward (ex ante, or before the event), alpha is a forecast of residual return. This is the focus of the reading.

where θ_n is residual return on stock n.

Alpha has the portfolio property. For example, the alpha of the portfolio which has two stocks can be estimated as follows:

where h_P(1) and h_P(2) are holdings for stock 1 and 2, respectively.

By definition, the benchmark portfolio, risk-free portfolio and cash portfolio all have a residual return of zero. That is, alphas are benchmark-neutral.

Practice Question 1

Which portfolio has a zero residual return?

I. benchmark portfolio.
II. risk-free portfolio.
III. cash portfolio.

Correct Answer: I, II and III

Practice Question 2

Alpha, on ex ante basis, reflects:

A. average of excess returns.
B. forecast of residual return.
C. historical expected return.

Correct Answer: B

Practice Question 3

If we have a two-stock holding with holdings h_P(1) in stock 1 (with α₁) and h_P(2) in stock 2 (with α₂), the alpha of the portfolio will be:

A. h_P(1) α₁ + h_P(2) α₂.
B. α₁ x α₂.
C. cannot be calculated based on the data given.

Correct Answer: A

This is because alpha has the portfolio property.