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# Definition of excess return in Study Notes CAPM and APT

#### QuantFFM

##### Member
Hi all,
just one point regarding the use of the term "excess return" which confuses me:

In the Notes for CAPM_Elton, page 5: Excess return is defined as RsubM minus RsubF (I understand)

In the Notes for APT_Bodie, page 8: Excess return on portfolio is defined as E(RsubA) + BetasubA times F

I would have expectetd that the excess return is BetasubA times F because Beta times F is the portion of the revised expected return over the old expected return and thus the excess return.

I would be thankfull for an explanation.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @QuantFFM Great observation with respect to an inconsistency. Let me first say that I have a mistake on page 8 of Bodie. Rather than read "The excess return on the portfolio is therefore: E(R_a) = β(A) +*F = 10% + 1.0*F" it should read "The excess return on the portfolio is therefore: E(R_A) = β(A) +*F = 10% + 1.0*F" That is, this means to refer to a gross return not an excess return. However, I do not believe we can infer the excess return from the given assumptions (because we cannot solve for two variables, risk-free rate and equity risk premium, in one function, simultaneously).

The (default) definition of excess return is "return in excess of the risk-free rate." Let me explain the aspect that can be confusing (to me, at least). Amenc's single-factor model is given by:
• R(p) = E[R(p)] + β(p)*F + e(p); i.e., 10.6. Where if the portfolio is well-diversified, then e(p) ~ 0 and R(p) = E[R(p)] + β(p)*F; in the illustrated example, R(A) = E[R(A)] + β(A)*F where E[R(A)] = 10.0% and β(A) = 1.0,
such that E[R(A)] = 10.0% + 1.0*F
• But E[R(A)] = Risk-free + β(A,F)*RP(F); for example, in the single-factor SML/CAPM, E[R(A)] = Risk-free + β(A,F)*(E[R(M)] - Riskfree), so if we plug that into Bodie's single-factor model, it expands to:
• Because RP(F) = F, R(A) = E[R(A)] + β(A)*F --> R(A) = [Rf + β(A,F)*F] + β(A,F)*F; I believe this is the key to understanding Bodie's "transition" from expected return to realized return in the chapter (which confused me for a while). Please note this is not ambiguous with respect to excess return: we can subtract Rf from the right side and we will have the realized excess return given by R(A) - Rf = β(A,F)*F + β(A,F)*F. However, this is still confusing because (F) appears twice, and this is where I think his notation confused me at least. I think a better notation to represent the expanded concept, which is consistent with his text, is something like:
• R(A) = [Rf + β(A,F)*E(F)] + β(A,F)*ΔF. So here we have the realized gross return of A, R(A), as the sum of two components:
• The expected gross return given by the sum of the risk-free rate, sensitivity to factor (F) which I denote β(A,F) because it is the sensitivity of A with respect to F, and the risk premium associated with one unit of exposure to factor F, i.e., [Rf + β(A,F)*E(F)]. And, again, that's the expected (gross, not net) return of A such that if we stopped here, we'd have only E[R(A)] = [Rf + β(A,F)*E(F)].
• But the additional component is the same beta sensitivity multiplied by the surprise (unexpected change) in the factor, β(A,F)*ΔF. In this way, the realized excess return is given by R(A) - Rf = β(A,F)*[E(F)+ΔF]; ie, the beta sensitivity multiplied by the realized factor. I hope that clarifies why the "excess" that you quoted in Bodie is mistake (sorry) and yet why I don't think we can solve for the excess in this example (because 10% expected return allows for several different variations of risk-free rate and risk premium; e..g, if Rf = 1%, then ERP = 9%; if Rf = 2%, then ERP = 8%). Thanks!