# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Capital Market Line

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At this stage, having introduced the new straight line efficient set, we are all but there to our final destination. So, let’s step back a bit and try and understand the larger picture.

In the beginning was the efficient frontier. Markowitz gave us that. Efficient frontier describes the maximum possible expected return for any given amount of risk from the portfolio of available assets. Or alternatively, the minimum amount of risk that one must live with for any given amount of expected return.

As a first step, we moved from individual assets to portfolios that lay on the efficient frontier. When we did that implicitly the x-axis (labeled as risk/standard deviation) then became the risk of the portfolio (and not the risk of the individual assets). There should be no cause for this confusion, but no harm emphasizing it nonetheless – because we would always lie on the frontier the right risk/standard deviation to consider is the risk/standard deviation of the portfolio and not the individual asset.

Moral of the Story 1: When we consider the efficient frontier the relevant quantities to consider are portfolio risk and portfolio expected return.

Then, of course, Tobin came along and introduced a risk free asset in the Markowitz world, and he said we could ignore all other points on the frontier except the tangency one – because everybody would hold some proportion of only the tangency portfolio $\mbox{M}$ (as all other points even on the envelope are now inferior), and the line connecting the return from the risk free asset $R_f$ and the tangency portfolio offers the best possible combinations of portfolio risk and expected return. Remember, the operative word here is portfolio.

This gave us our revised efficient set as:

(Click on the figure to zoom.)

The equation of the new efficient set immediately follows (it’s a linear line with intercept at $R_f$ and slope $\displaystyle \frac{E[R_M] - r_f}{\sigma_M}$)  as:

$\boxed{\displaystyle E[R_p] = R_f + \frac{(E[R_M] - R_f)}{\sigma_M}\sigma_P}$

What is the tangency portfolio $M$?

Having said that all investors should hold the tangency portfolio $M$, the next thing to understand is the meaning of this tangency portfolio. By saying that all investors should hold $M$, what we are essentially saying is that investors would demand only combinations of portfolio $M$ and the risk-free asset. (Holding any other risky portfolio other than $M$ is inefficient.) This is the demand side of the problem. What is the supply side? The supply side is just all the assets that exist in the market.

And by now you would know enough of microeconomics to understand that equilibrium requires that demand be same as supply. That is, assets demanded in the portfolio $M$ must exactly equal the supply of each asset in the market. And the supply of each asset in the market is given by its market capitalization. So, in equilibrium all assets must be held in $M$ in exactly the same proportion as their market capitalization. That is, in percentage terms weight of assets in the total market capitalization and in the portfolio $M$ must be the same.

Consider the case where you run your Markowitz optimizer and find that that weight of a particular asset, say $\omega_i$ is $0$. Is that possible? Mathematically, of course, yes. But what about economically? Let’s try and understand this.

Saying that the weight of an asset $\omega_i$ in the Markowitz portfolio $M$ is $0$ is saying that no investor wants to hold the $i^{th}$ asset. If no investor wants to hold that asset, but the asset exists in the market then we have a state of disequilibrium. And what happens in a state of disequilibrium? Prices adjust. So, if no wants to hold an asset, its price will drop. Once the price starts to drop its expected return:

$E[r_i] = \displaystyle \frac{E[P_i] - P_0}{P_0}$

will rise. As the price starts to fall, and expected return starts to rise, investors would start to find this asset more attractive. As its expected return $E[r_i]$ rises even more, then when you re-run your Markowitz optimizer again, you’ll find that this asset has a non-zero weight in the tangency portfolio $M$. That is, all assets that exist in the market must be held. This brings us to another important lesson:

Moral of the Story 2: The tangency portfolio $M$ is nothing but the market itself!

As another example consider a situation where the Markowitz optimizer prescribes a weight of $2.1\%$ for an asset whose market capitalization is $2\%$. What happens in that case? Well, now you know how to think about such disequilibrium situations. This is the case where the asset has more demand than supply. When demand is more than supply, prices rise. As price rises, the expected return will fall. As expected return falls, the Markowitz optimizer will prescribe a lower weight to this asset and in equilibrium the price and the market capitalization of the asset would adjust to make the demand exactly equal supply. That is:

When one imposes equilibrium, the line passing through the tangency portfolio has a specific name and it is called the Capital Market Line.

Note that at this stage, when we impose economic equilibrium, we have to necessarily assume that everybody has the same information – things don’t quite work the same way otherwise. And this brings us to the last moral of the story for today:

Moral of the Story 3: All efficient portfolios lie on the Capital Market Line.

Again, as in the case of the efficient frontier, the relevant quantities in the Capital Market Line are the expected return and risk of efficient portfolios. All individual stocks and other inefficient portfolios, however, would be anywhere below the efficient frontier, as say in the shaded portion of the graph below (from your book; think of $S$ in the plot below as the equilibrium market portfolio):

CML with the Efficient Frontier (Click to zoom; Source: Brealey-Myers, 9th Ed.)