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##### Subject 1. Return Concepts
Holding Period Return

It is the percentage by which the value of an investment has grown for a particular period. It is the sum of investment income and capital gains divided by the initial value.

HPRn = (Income + Pn - P0)/P0

Realized and Expected Return

If the investment is sold, then the realized return can be calculated. If not, then the investor can form an expectation concerning the investment income and final selling price, and thereby have an expected holding period return. The expected return is the average of probability distribution of possible returns. For example, if you knew a given investment had a 50% chance of earning a 10% return, a 25% chance of earning 20% and a 25% chance of earning -10%, the expected return would be equal to 7.5%: (0.5) (0.1) + (0.25) (0.2) + (0.25) (-0.1) = 0.075 = 7.5%.

Although this is what you expect the return to be, there is no guarantee that it will be the actual, realized return.

Required Return

It is the rate of return used by investors to decide whether an investment is attractive or not. It represents the opportunity cost for investing in the asset given the asset's riskiness. If the investor's expected return is higher (lower) than the required return, the asset will appear to be undervalued (overvalued).

Expected alpha = Expected return - Required return
Realized alpha = Actual holding period return - Contemporaneous required return

When an investor's intrinsic value estimate for an asset differs from its market price, the investor generally expects to earn the required return plus a return from the convergence of price to value. When the asset's intrinsic value equals price, however, the investors only expects to earn the required return.

Discount Rate

It is the rate required by investors for delaying consumption. The market required rate is typically used as the discount rate in finding the present values of expected future cash flows.

Internal Rate of Return

This is the discount rate that equates the present value of the asset's expected future cash flows to the asset's price. If we define cash flows as dividends and assume a stable dividend growth rate for the indefinite future, the intrinsic value of an asset is:

Year-ahead dividend / (Required return - Expected dividend growth rate)

Learning Outcome Statements

a. contrast realized holding period return, expected holding period return, required return, return from convergence of price to intrinsic value, discount rate, and internal rate of return;

CFA® 2023 Level I Curriculum, Volume 3, Module 21

User Comment
nedved19 Is
Expected alpha = ex ante alpha
and
Realized alpha = ex post alpha?
soorajiyer @nedved
ex ante alpha = expected return - required return

ex post alpha = historical HPR for the security - historical HPR for similar security.

Hope this helps!
bablig Ex post alpha is the difference between historical realized return and required return.
The diff between Ex Ante and Ex Post is that for the former we take expected return and for the latter we take realized return. But the comparison is with Expected return for both.