R10.P1.T1.BODIE_CH10_SINGLE_FACTOR_MODEL_vs_CAPM

[gargi.adhikari]

In reference to R10.P1.T1.BODIE_CH10_SINGLE_FACTOR_MODEL_vs_CAPM :-
The CAPM Pricing Model is often referred to as the Single Factor Model.
But the Single Factor Model is :-
Ri = E(Ri) + Beta*F(Macro-Factor) + Non-Systemic-Firm-Specific-Risk

Whereas,
For the CAPM:-
Ri = Rf (Risk-Free-Rate) + Beta*(Rm-Rf)

We do not have the Rf factored into the Single Factor Model....so how is the CAPM = the Single Factor Model ....?

 

[David Harper CFA FRM]

HI @gargi.adhikari

The risk-free rate is embedded in the E[R(i)]. If we parse the equity (aka, market) risk premium, as the common factor, into an expected value plus an unexpected change, where ERP = ERP + ΔERP, then CAPM can be given by:

• R(i) = Rf + β*(ERP + ΔERP) = Rf + β*ERP + β*ΔERP -->
• R(i) = E[R(i)] + β*ΔERP

The CAPM is a specific model with a very particular common factor, ERP, or really the excess return on the most efficient (highest sharpe ratio) portfolio that investors hold (my recent youtube series includes a video devoted to CAPM here http://trtl.bz/2hxHdM6). It is an example of a single-factor model because it says that (exposure to) only a single factor (the market's expected excess premium) explains a security's excess return. The single-index model uses a market index (eg, S&P 500) as proxy for the CAPM because the CAPM's single factor is too theoretical to actually access. I hope that explains.

The above math (parsing the unexpected component is related to an post I have here (although it's all in source Bodie, nothing too original) https://forum.bionicturtle.com/thre...ultifactor-models-continued.10115/#post-51969 i.e.,

Thank you @Eltanariel ! @lavi5h These questions follow Bodie carefully in order to correctly reflect Bodie's theory, but there is a step in the chapter that (IMO) can be confusing, related to your question. Please note that the answer to 705.3 can be expressed, instead of the displayed two-step solution that solves first for the un-revised expected return and then adds the revision, simply solves for the revised Expected Return of the stock with:

• revised E(r) = Rf + β1*RP1 + β2*RP2 + β2*RP2 = 2.0% + 0.80*3.0% + -0.50*1.0%+ 1.30*4.0% = 2.0% + 0.80*(2.0% + 1.0%) + -0.50*(1.0% + 0%) + 1.30*(3.0% + 1.0%) = 9.10%; i.e., actual versus expected rate of change is a way of parsing a single risk premium into into an un-revised (expected) and its revised ("with surprise") components

My point is that all we really have here is a mulifactor SML, by which I mean a linear combination (summation) of the product of a sensitivity (ie, β) multiplied by its risk premium:

• E(r) = Rf + β1*RP1 + β2*RP2 + β3*RP3

... But the risk premiums are parsed into their expected and actual values, so that we have:

• E(r) = Rf + β1*[expected RP1 + ΔRP1] + β2*[expected RP2 + ΔRP2] + β3*[expected RP3 + ΔRP3]

This relates to the theory of the APT model and allows for Bodie to says that "each factor has zero expected value because each measures the surprise in the systematic variable rather then the level of the variable." So, if we just isolate on the interest rate factor in 705.3, the stock's (un-revised) expected return of 7.0% already includes the contribution of β2*RP2 = -0.50*1.0%. The actual contribution of the interest rate factor needs to add any unexpected change, so we need to add -0.50*ΔRP2 = -.50*(1% - 1%). Just as, similarly, with respect to the first factor, inflation, the stock's (un-revised) expected return of 7.0% already includes the contribution of (0.80*2.0%) such that adding the surprise must add 0.80*(2.0% - 1.0%) because including the surprise reflects the fact that the revised risk premium is really 3.0%: 0.80*3.0% = (0.80*2.0%) + 0.80 * (2.0% - 1.0%). I hope that provides the intuition, thanks!

 

[gargi.adhikari]

@David Harper CFA FRM Perfectoo!!! Thank you so much !!

 

[tusharkango]

Hi,

I was going through this reading and i cannot understand the explanation for Q7 of end of chapter questions. I was hoping if someone could clarify as to how are the variance and standard deviation is calculated for N=20, 50 and 100.

Thanks.

 

[David Harper CFA FRM]

Hi @tusharkango The question is copied below. This assumes the single-index model ("market index as a common factor") such that a security's return is a function of only two components, systematic and firm-specific (aka, idiosyncratic), where E[R(i)] = β(i, F)*F + e, where (e) is firm-specific, such that the variance of this formula is equal to σ^2(Ri) = β(i, F)*σ^2(F) + σ^2(e). This just applies variance properties: var(aX + Y) = a^2*var(X) + var(Y) + 2*a*cov(X,Y) except that cov(X,Y) is zero, so the last term drops. As you can see from the solution explanation, the systematic risk is assumed to cancel out in the "market neutral" portfolio which leaves only the firm-specific component, σ^2(e). That is, each security has variance given by, σ^2(i) = β(i,F)*σ^2(F) + σ^2(e) but since the β(i,F)s will cancel out, the standard deviation solution reduces to finding the portfolio standard deviation (for N = 20, 50, 100) when each security has variance given by σ^2(e) = 30%^2 = 0.090.

Then the solution step is instructive. It relies on variance properties, in particular variance(aX) = a^2*variance(X). Similar to Bodie's given solution, we can first infer that, in the case of n = 20, each position's value = $2.0 million / 20 = $100,000. Each position's dollar volatility = 30%*$100,000 = $30,000 so that its dollar variance = $30,000^2 and the dollar variance of 20 positions = 20*$30,000^2 = $18.0 billion with standard deviation = sqrt(18.0 bb) = $134,164. Similarly, for example, for n = 100 positions, each position's dollar volatility = 30%*($2.0 mm / 100) = $6,000 and the dollar variance of all 100 positions = 100*$6,000^2 = $3.6 BB with standard deviation = sqrt(3.6 bb) = $60,000.

The key assumption here is the independence (non correlation) between the firm-specific returns. If we are more comfortable with the variance mechanics, we might realize that we can scale volatility (per the square root rule under i.i.d.) so that the percentage volatility of n = 20 identical positions is given by 30%*sqrt(20) = 134.2%. And this can efficiently be multiplied by the per-position dollar amount of $100,000 to get the same $134,164. I hope that's helpful!

 

[tusharkango]

Oh yes thanks a lot @David Harper CFA FRM . That did clear everything up.