## [PGP-I FM]: CML vs. SML

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

which describes expected return from efficient portfolios.

Later in the class we extended this idea of expected return as comprising reward for waiting , and reward for bearing risk to write the *Security Market Line (SML) *from the *Capital Asset Pricing Model (CAPM)* as:

where,

where the only, but the key, difference was that instead of using standard deviation as a measure of risk we used – 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 , 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, and :

.

If we let , and use the fact that , it immediately follows that

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.:

Dividing both sides by gives:

Using the intuition that 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:

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

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

Or, more succinctly, using the summation symbol:

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.**

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