# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM]: CML vs. SML

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:

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

where,

\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.