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**Subject 1. The definition of alpha**

Looking backward (

r

r

**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_{P}obtained 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.

#### Practice Question 2

Alpha, on ex ante basis, reflects:
A. average of excess returns.

B. forecast of residual return.

C. historical expected return.

#### Practice Question 3

If we have a two-stock holding with holdings h_{P}(1) in stock 1 (with α

_{1}) and h

_{P}(2) in stock 2 (with α

_{2}), the alpha of the portfolio will be:

A. h

_{P}(1) α

_{1}+ h

_{P}(2) α

_{2}.

B. α

_{1}x α

_{2}.

C. cannot be calculated based on the data given.Correct Answer: A

This is because alpha has the portfolio property.

### Study notes from a previous year's CFA exam:

a. define the terms "alpha" and "information ratio" in both their ex post and exante senses;