CAPM - Formulation for CAPM

[navjyotbirdy]

Hey guys,

Though I might sound stupid, but can you please put in more light to the statement that market portfolio must have a beta of 1. Though I understand that any individual investor as per CAPM will have identical portfolios but there respective portfolio will vary as per their risk preferences due to which the quantity of Beta can vary across all individual investor.

Here's a screenshot for reference:




Regards,
Navjyot

 

[David Harper CFA FRM]

Hi Navjyot (@navjyotbirdy) I think it's a good question. My sense is there may be an easy answer and a harder answer. The easy answer is that the market portfolio must have a beta of 1.0 by definition of CAPM's beta! We can think of generic beta as a re-scaled correlation between two assets: the beta of X with respect to Y, β(X,Y) = cov(X,Y)/variance(Y) = ρ(X,Y)*σ(X)/σ(Y). Mathematically, beta is just a relationship between two variables, although unlike correlation we want to be careful about which is the "with respect to" variable because it's not symmetrical, β(X,Y) <> β(Y,X).

In CAPM, the beta is defined as "with respect to the market portfolio:" β(X,M) where M is the Market Portfolio. By virtue of the many (ten!) CAPM assumptions, notably to me that homogeneity of expectations (i.e., the grossly unrealistic assumptions that all investors perceive the same risk/returns), it is demonstrated that all investors would hold this Market Portfolio for their risky allocation (then they can tailor their risk/reward simply by allocating more/less to the riskfree asset; aka, moving up/down the CML). So we arrive at a definition of beta that happens to be β(X,M), that is, with respect to the Market Portfolio because the theory importantly shows that this is the only relevant measure of risk. Each investor can have a different mix of risky Market portfolio versus riskless asset, which which increase/decrease their portfolio standard deviation (per the CML).

The beta of the market portfolio with respect to the market portfolio, M, is given by β(M,M) = COV(M,M)/variance(M) and since COV(M,M) = variance(M), it must be true that β(M,M) = variance(M)/variance(M). In much the same way that the correlation of a thing to itself must be 1.0 as we can see that β(M,M) = ρ(M,M)*σ(M)/σ(M) = 1.0.

In practice, we often proxy the market by using the S&P 500 or S&P 1500, and we can't expect the β(S&P1500, theoretical M) to be exactly 1.0. If the index is sufficiently broad, we can expect high correlation. It's a whole other topic, but I hope that's helpful regarding what i think is the easy answer! Thanks,

 

[navjyotbirdy]

Thanks David this makes perfect sense to me..

Navjyot Birdy