How are CAPM and the Security Market Line related?

It’s crunch time for the CFA crowd, and being done with the process definitely gives Memorial Day weekend a special sweetness for me. I did spend some time teaching Portfolio Management to Level II candidates recently, and part of the discussion covered differences between the Security Market Line (SML) and the Capital Market Line that comes out of the Capital Asset Pricing Model (CAPM). The SML and the implications of CAPM are so similar that it can be extremely difficult to understand the difference between the two. Preparing the discussion made the issue clearer for me and I thought I might share the insights on this blog for the CFA crunch crowd.

Why the confusion?

There are two main reasons for the confusion between the SML, the CML, and CAPM. The first reason is that analysts frequently use what they call “CAPM” to estimate the cost of equity, when in fact they are actually using the SML. The CFA curriculum reinforces this perception by teaching “CAPM” as one of the ways to estimate required returns. Nonetheless, the estimation of returns as…

E(R) or K_e = RFR + Beta*(R_Market – RFR)

…is an application of the SML, not CAPM or the CML. One can argue that CAPM provides the theoretical framework that lets an analyst use the SML in this way, so the CFA curriculum isn’t technically incorrect by saying you are using CAPM. However, it does contribute to confusion between CAPM and the SML, because the main conclusion of CAPM is that all investors should simply hold portfolios that lie along the CML. The fact that individual security returns are best estimated with the SML is an implication of CAPM, but it is not the main result.

The second reason for the confusion is that the principal chart used to show CAPM looks almost identical to the main graphic used to explain the SML. Both graphs plot a measure of risk on the x-axis, returns on the y-axis, and then show a line that intercepts the y-axis at the Risk-Free Rate (RFR) and passes through the return on the market (R_Mkt).

When looking at these charts, one easy way to tell if you are looking at an SML or a CML/CAPM chart is to look at the x-axis. If the axis is labeled with Beta as a measure of risk, you are looking at the SML, if it is labeled with standard deviation (or possibly variance), you are looking at the CML and CAPM. In addition, efficient frontiers never show up on SML charts; they are only relevant on CML/CAPM charts.

The SML is about being paid for market risk

The SML is a more general model than CAPM. It says that investors get paid for taking on market risk, and that the quantity of paid risk is best measured by the Beta coefficient to the market return. That’s pretty much it. In contrast, CAPM is more specific than the SML in asserting that market risk is the *only* paid risk.

The theory behind the SML is that if one is paid to take market risk, it should be priced similarly for all securities according to their levels of market risk. If not, investors will gravitate to the securities that pay more for their level of market risk, and shy away from those that pay less. The process bids up the price of the higher-paying risk, lowering the expected return by forcing investors to pay more for returns, at the same time lowering the price of lesser-paid securities, raising those returns until all assets pay the same for their level of market risk.

One can argue that the process of paying returns for accepting degrees of market risk is simply a process of markets reaching equilibrium, or one can say that markets do this because arbitrageurs take advantage of mispriced market risk and equalize all market risk as a result. Both processes lead to the same single price for market risk, which helps explain why the SML is a fairly generalizable conclusion, and using the SML equation is often considered the best way to estimate expected or required returns.

Empirically, one observes that an extraordinary amount of variation in security returns is in fact correlated to the market, and so the idea that most if not all risky securities contain at least some market risk is empirically plausible.

Why that risk should be a paid risk is another question. Traditionally, most finance theory says that market risk needs to be a paid risk, or else no one would ever have an incentive to invest in the market. This may play a role, but to me it comes across as tautological, and plenty of people may simply invest in the market as a result of wishful thinking. I prefer and have argued elsewhere that markets should pay a return on average because we can expect that – over the long term – business, scientific, and technological innovation will allow us to produce more economic value over time, and those who provide capital to the economy are first in line to receive the profits generated by these innovations.

I do think that the need to have a positive expected return for taking on risk does place a lower limit on what the long term market return ought to be, because volatility does eat away at a portfolio over time and most investors will need to be compensated at least enough to avoid losing money. But other than that, the need for positive expected return does not tell us much more about how much the market should pay over time. By contrast, linking market returns to technological and business innovation can, even if the estimate is not necessarily precise.

The SML is more general than CAPM because all it says is that investors are paid to assume market risk. True, CAPM also concludes this, but the SML’s claim about paid market risk is compatible with other pricing theories besides CAPM. These alternate theories include, for example, the many formulations of the Arbitrage Pricing Theory (APT) that include a market factor among other paid risk factors.

CAPM is about the optimal efficiency of the market portfolio

What makes CAPM special is that it concludes that the market portfolio is not only on the efficient frontier, but is also the efficient portfolio with the highest possible Sharpe ratio. Since it has the highest possible Sharpe ratio, it is therefore the tangent portfolio to a line running through the RFR on the y-axis. Why? Because the slope of any line that runs through the RFR at the y-axis and some point on on the risk-return chart is equal to that point’s Sharpe ratio, and therefore – looking at a CAPM chart – the tangent point on the efficient frontier *must* therefore have the highest Sharpe ratio, or it would intersect the efficient frontier at more than one point.

In short, if

A) The Market portfolio has the highest Sharpe ratio, and
B) The highest Sharpe ratio portfolio is the tangent point on the frontier, then
C) The market portfolio is the tangent point on the efficient frontier.

A CAPM graphic typically shows a curved efficient frontier, with a point on it representing the “market portfolio”, and a tangent line running through the RFR on the y-axis and passing through the market portfolio on the efficient frontier. The line running through RFR on the y-axis and the market portfolio on the risk-return chart is the Capital Market Line, or CML. The CML represents the expected risk-return performance of all portfolios that are some combination of the market portfolio and either borrowing or lending at the risk-free rate.

If you look at a CAPM chart, you will see that the expected return for portfolios on the CML is always larger than the return for portfolios on the efficient frontier, except at the tangent point, where the returns are exactly the same. In addition, the CML allows an investor to achieve portfolios with lower risk levels than would be achievable without having a risk-free asset.

The conclusion is that investors should hold only portfolios that lie on the CML, which means their portfolios should only consist of some combination of the market portfolio and either lending or borrowing at the risk-free rate. Exactly how much they hold of each asset depends on how much risk an investor is willing to take, or alternately, how much return they demand.

But beware, the formula taught with CAPM, namely that:

E(R) = RFR + (Market Sharpe Ratio) * StDev(portfolio)

is not used to price assets in general. This formula tells you the expected return only for portfolios that lie on the CML. It does not tell you the expected return of individual assets or of any other portfolio except for combinations of the market portfolio and the RFR. If you accept CAPM’s assumptions, you can interpret the CML as telling you the highest expected returns possible for a given level of risk. Achieving those levels does admittedly result in a relatively boring looking portfolio, which no-doubt has stimulated much research into alternative models.

When looking for the expected return of individual assets, though, you still use the SML, and – if you think there are other paid risks – you would add additional regression factors to account for those. If you use the CAPM framework, which is what people are doing when they say they use CAPM to find the required return of an individual security, you are asserting that there are no other paid risk factors, and that the SML captures the only relevant risk for pricing returns.

Why is the market portfolio efficient under CAPM?

Why does the market portfolio have the highest possible Sharpe ratio? This is where CAPM leverages its assumption that market risk is the only risk that receives a (positive) expected return. If this is true, then any departure from the market portfolio adds asset-specific risk to the portfolio, without increasing its expected return (technically, without increasing its expected return beyond that which could be achieved simply by changing the proportions of RFR and the market portfolio). When computing Sharpe ratio of this alternate portfolio, the extra unpaid risk increases the denominator without increasing the numerator, bringing the Sharpe ratio down. As a result, the market portfolio has the highest Sharpe ratio.

What is special about the market portfolio is that it is “fully diversified,” which basically means that all of the unpaid risks (i.e. diversifiable risks) are perfectly balanced against each other to add up to zero. When you depart from the market portfolio, you essentially upset this perfect balancing, and start to introduce unpaid risks into your portfolio, and that means that you are inefficiently accepting risk.

Something tautological about all this?

If you think that there is something a bit tautological about saying that diversifiable risks must cancel out in the market portfolio because the market risk is determined by comparing its returns to “the market” (i.e. comparing its returns with itself), then you are justified. It *is* a bit strange to conclude that no other exposures should be paid simply because the market portfolio is perfectly correlated with the market (i.e. itself).

However, this seemingly circular argument is a consequence of the assumption that no other risks are paid risks, and the mathematics that follow. If you are willing to assume that there are other types of paid risks, then your pricing models will become substantially more rich and interesting. Perhaps you will be too.

In CAPM’s defense, however, I should add that empirically, market risk does appear to constitute by far the largest single component of most security returns, and especially so in the equity space. Therefore, even if CAPM is not strictly “true,” it is definitely defensible to think of CAPM as a good “first approximation” for coming up with expected / required returns.

A final summing up for the exam takers

If you are studying for the exams and having trouble understanding the distinction between CAPM and the SML, try to remember that when they say you are using CAPM to compute a security’s required return, you are actually using the SML. CAPM simply provides the set of theoretical justifications that tell you why you can feel comfortable using the SML to do this. If – in your mind – you can separate CAPM as a theory about how investors should construct their portfolios versus CAPM as a background justification for computing individual security returns with the SML, you will find that your mind is better able to absorb the other aspects of the CAPM model.

PS: If I get to it, I will try to insert a graphic or two to help illustrate these points, but my blog software does not integrate graphics well with blogger and so it is an intricate process.

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