Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance


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We earlier wrote the Capital Market Line (CML) as:

\displaystyle E[R_p] = \underbrace{R_f }_{\mbox{Reward for Waiting}}+ \underbrace{\frac{E[R_m] - R_f}{\sigma_M}}_{\mbox{Reward per unit of Risk}} \sigma_p

which describes expected return from efficient portfolios.

Later in the class we extended this idea of expected return as comprising reward for waiting R_f, and reward for bearing risk (\frac{E[R_M] - R_f}{\sigma_M})\sigma_p to write the Security Market Line (SML) from the Capital Asset Pricing Model (CAPM) as:

\boxed{E[R_i] = R_f + (E[R_M] - R_f) \beta_i}


\begin{aligned} \beta_i &= \frac{\mbox{Cov}(R_i, R_M)}{\sigma_M^2} \\&= \frac{\sigma_{iM}}{\sigma_M^2} \end{aligned}

where the only, but the key, difference was that instead of using standard deviation as a measure of risk we used \beta_i – sensitivity of change in return from a stock to change in the market – as the measure of risk.

Another important distinction between CML and SML is that while the former can be used only for efficient/optimal portfolios, the latter can be used all assets, inefficient portfolios as well as efficient portfolios.

That is CML is like a subset of SML, or alternatively SML subsumes CML.

Before we can prove this result, however, we need another result – that market beta is a weighted average of beta of individual securities. In this part of the post we establish the relationship for \beta, and in the next part we prove that SML implies CML.

Market beta is a weighted average of beta of individual securities

Recall from your basic probability theory that for any three random variables, I, J and M:

\begin{aligned} Cov[\omega_1 I + \omega_2 J, M] &= \omega_1 Cov[I, M] + \omega_2 Cov[J, M] \\ &= \omega_1 \sigma_{IM} + \omega_2 \sigma_{JM} \end{aligned}.

If we let M = \omega_1 I + \omega_2 J, and use the fact that Cov[M, M] = \sigma^2_M, it immediately follows that

\begin{aligned}\sigma^2_M &= \omega_1 \sigma_{IM} + \omega_2 \sigma_{JM} \\ \Rightarrow 1 &= \omega_1 \frac{\sigma_{IM}}{\sigma_M^2} + \omega_2 \frac{\sigma_{JM}}{\sigma_M^2}  \\ \mbox{or  } 1 &=\omega_1 \beta_1 + \omega_2 \beta_2 \end{aligned}

Our proof below is just a generalization of this result.

Taking as starting point the result that market variance is a weighted average of covariance of individual assets with the market (previous post), i.e.:

\begin{aligned} \sigma^2_M &= \sum_{i=1}^N \omega_i \sigma_{iM} \\ & = \omega_1\sigma_{1M} + \omega_2\sigma_{2M} + \omega_3\sigma_{3M} + ... + \omega_N\sigma_{NM} \end{aligned}

Dividing both sides by \sigma^2_M gives:

\begin{aligned} 1 &= \omega_1\frac{\sigma_{1M}}{\sigma^2_M} + \omega_2\frac{\sigma_{2M}}{\sigma^2_M}+ \omega_3\frac{\sigma_{3M}}{\sigma^2_M}+ ... + \omega_N\frac{\sigma_{NM}}{\sigma^2_M} \end{aligned}

Using the intuition that \beta is the sensitivity of change in a stock’s return to change in market return, that is, it can be interpreted as the regression coefficient, we have the result that:

\displaystyle \beta_i = \frac{\sigma_{iM}}{\sigma^2_M}

We can now substitute this formula for \beta in the previous equation and write:

\begin{aligned} 1 &= \omega_1\frac{\sigma_{1M}}{\sigma^2_M} + \omega_2\frac{\sigma_{2M}}{\sigma^2_M}+ \omega_3\frac{\sigma_{3M}}{\sigma^2_M}+ ... + \omega_N\frac{\sigma_{NM}}{\sigma^2_M} \\ & = \omega_1\beta_1 + \omega_2\beta_2 + \omega_3\beta_3 + ... \omega_N\beta_N \end{aligned}

By definition market \beta_M is 1 (this is trivial, beta of the market represent change in the market when market changes), we have our required result that:

\begin{aligned} \beta_M & = \omega_1\beta_1 + \omega_2\beta_2 + \omega_3\beta_3 + ... \omega_N\beta_N = 1 \end{aligned}

Or, more succinctly, using the summation symbol:

\boxed{\displaystyle \beta_M = \sum_{i=1}^N \omega_i \beta_i = 1}

This result can be used to show that SML implies CML – which brings us to our last lesson for this module:

Moral of the Story 5: In the CAPM world, while CML describes expected return from efficient portfolios SML describes expected return from individual stocks, inefficient portfolios as well as efficient portfolios. SML implies CML.


Written by Vineet

September 9, 2016 at 9:07 pm

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