Capital Asset Pricing Model

Efficient Frontier

The first assumption is that an investor's preference for a portfolio is entirely determined by two things: its expected rate of return and its expected volatility. This matches the assumptions we had for individual assets in the Markowitz model. Now, however, instead of using variance as our measure of volatility, we will use its square-root: standard deviation.

We can consider all such portfolios by plotting them on a Cartesian plane:

Obviously, returns are good while volatility (risk) is bad, so this results in an efficient frontier of portfolios. No investor has any reason to choose portfolios inside this frontier.

Now let's assume there exists a risk-free interest rate. Suppose I choose to invest x% of my assets in a portfolio and the remainder in the risk-free asset. My expected return and volatility (standard deviation) are simply the weighted sum of the risk and return of those two options, where the weights are the percent of my money I put in to each option.

In other words, I can draw a line between the risk-free interest rate and a portfolio, and then I can choose any point between the two to represent my investments. If we additionally assume that I can borrow at this risk-free interest rate, then I can also choose any point on that line even past the original portfolio.

It turns out there is exactly one portfolio such that the above line doesn't cross the Efficient Frontier: the Ideal Market Portfolio. This line through this point is called the "Capital Market Line":

Note, though, that for every point inside the original efficient frontier, there exists a point on the line that has at least as much return and no more risk. If we assume the market is efficient, this has a startling conclusion: all investors' portfolios will end up on this line. Moreover, since all investors Ideal Market Portfolio is identical, it follows that this portfolio must be the set of all assets weighted by market capitalization.

This all suggests rational investors only have one choice: how much leverage they take on - a question we addressed fairly in depth previously.

Sharpe Ratio Aside

The Sharpe ratio Sharpe ratio is probably the single most common number used to evaluate an investment's returns. For investment $i$, it is defined as

Why use this formula? It can be justified within the CAPM model as providing a leverage-adjusted and risk-adjusted measure of returns.

As noted above, using 10% more leverage increases both the expected returns above the risk-free rate and the expected risk (standard deviation) by 10%. Therefore, the ratio of these two metrics (i.e. the Sharpe ratio) will be unaffected by leverage. Put another way, every point on the Capital Market Line (CML) has the same Sharpe ratio.

Likewise, every point to the right of the CML will have a lower Sharpe ratio, and every point to the left will have a positive Sharpe ratio. In this way, the Sharpe ratio provides a leverage-adjusted way to measure how an investment performs relative to its risk.

What makes the Sharpe ratio unique relative to other leverage-adjusted measures is that it makes no assumptions concerning your risk tolerance. Indeed, this agnosticism regarding the degree of risk tolerance is a feature of the CAPM model more generally and is one of the reasons it is seen as so elegant and useful.

Individual Securities

It's important to note that we have not proven that all individual securities lie along this line, just all efficient portfolios (collections of securities rational people buy). In particular, pretty much all stocks should lie to the right of the line, because a portfolio consisting of exactly one stock is almost certainly a bad idea. However, by the same logic, no stock should lie to the left of the line.

Two Types of Risk

Securities (and portfolios of securities) have at least two kinds of risk: idiosyncratic risk and systematic risk. The former represents the risk inherent to the specific asset while the latter represents risk that affects the entire market. Other kinds of risk obviously exist, but let's focus on these two.

Idiosyncratic risk is specific to one asset. If we assume no single asset makes up a significant portion of the overall market, idiosyncratic risk can be eliminated via portfolio diversification without any loss of expected returns.

Systematic risk, on the other hand, cannot be eliminated unless you only want to earn the risk-free rate. The proof for this is that if you could eliminate systematic risk and earn more than the risk-free rate, then that portfolio would be above the Capital Market Line, which is (as shown above) impossible.

If we suppose that these are the only two types of risk, it follows that an individual security's expected return is entirely determined by its expected systematic risk. In particular, it then follows that, in fact, every individual security lies on this systematic-risk-capital-market-line (called the "security market line" Security market line). (TODO: why does this follow?)

Single-Index Model

In investor-speak, systemic risk is measured with "beta", which is defined as the slope of the line that relates the overall market's return to the specific stock's return. For instance, if ISBM typically goes up 1.5% on days when the S&P 500 goes up 1%, then ISBM has a beta of 1.5.

According to CAPM, the return on stock $i$ at time $t$ should be given by

where $R_t$ is the return of the overall market at time $t$. This model is called the single-index model Single-index model and its fundamental perspective is that a stock's returns can be summarized with three variables:

• $\epsilon_i$ - its degree of idiosyncratic risk
• $\beta_i$ - its degree of (compensated) systematic risk
• $\alpha_i$ - its returns beyond what should be expected for its level of risk

In an efficient market, $\alpha_i = 0$ for all stocks.

Criticism

Much criticism has been levied against CAPM - just see the Wikipedia page Capital asset pricing model. In Wikipedia and all the ways its failed empirical tests Fama. For these reasons, you definitely shouldn't take it literally and arguably shouldn't take it seriously.

Nevertheless, it remains a ubiquitous lens through which experts view the market and constantly shows up in the literature, so it's worth knowing about. By separating returns between $\alpha$ and $\beta$, this lens also makes it more difficult for fund managers to claim to be good investors when they are, in fact, merely in favor of leverage.