CAPM, SML, CML

[Cipher2014]

Hi David,

As I read the chapters on CAPM, I'm a bit confused as to what's the relationship between SML, CML and CAPM. The book seem to associate CAPM exclusively with SML, but CML is also in the same chapter.

Could you kindly explain a that's the association between CML to CAPM. Is CML also derived from the CAPM model or they are unrelated?

Thanks,
Ying

 

[David Harper CFA FRM]

Hi Ying,

• SML manifests the CAPM, such that in practice--i.e., E[excess return] = priceRisk*quantityRisk--they are used interchangebly. SML is the line; CAPM is the broader theory and set of unrealistic assumptions that produces the SML but includes *ideas* like the all-important Equilibrium; SML/CAPM has systemic risk (beta) on the X-axis
• CML is the efficient frontier after the riskfree asset has been added to the minimum variance portfolios (the curvy line), of which the most important risky (i.e., all risk assets) is the Market portfolio (b/c it has the highest sharpe ratio). CML has total risk (volatility) on the X-axis.

My favorite conceptual distinction, which i use in the PQ, is this: the CML is a line of efficient portfolios and only efficient portfolios. The SML/CAPM is a line for ANY portfolio or asset, including inefficient: it will take that asset/portfolio and, because it cares only about the systemic risk, give you back the expected return. But your asset/portfolio is not necessarily on the CML; maybe it is below it. Here is that part of Elton that helped me clarify this distinction, FWIW:

The straight line depicted in Figure 13.2 is usually referred to as the capital market line. All investors will end up with portfolios somewhere along the capital market line and all efficient portfolios would lie along the capital market line. However, not all securities or portfolios lie along the capital market line. In fact, from the derivation of the efficient fron-tier, we know that all portfolios of risky and riskless assets, except those that are efficient, lie below the capital market line. By looking at the capital market line, we can learn some-thing - Elton , p 285

In terms of the exam, I frankly think CAPM is overassigned (i.e., more theory than they will test). I *think* it is possible that, from the exam standpoint, you can treat SML ~ CAPM such that you are only really dealing with two lines: 1. SML/CAPM and 2. CML, and the more likely testable concept is simply the application of the CAPM formula.

Hope the helps, thanks, David

 

[Cipher2014]

I think what got me confused is the naming convention: "Capital Asset Pricing Model" is associated with the "Security Market Line" where as "Capital Market Line", dispite of having "Capital" in its name is actually not part of CAPM.

Thanks for the detailed explanation David!

 

[BioNerd]

This is the clearest explaination I've seen so far. really helpful. thanks David!

 

[hawayi_vgo]

Got that!

How about Efficient frontier? How does the efficient frontier relate to CML, SML, risky asset and riskless assets on mean variance framework?

Specifically the relationship between efficient frontier and CML.

Thank you

 

[ShaktiRathore]

Hi,
Efficient frontier is basically the same as CML. Except that the CML line beyond the market portfolio(which is the tang-ency point of CML with the mean variance curve) represents this efficient frontier . Earlier through mean variance optimization we obtain the efficient frontier as the upper portion of mean variance curve that is above the tang-ency point. But once the riskfree asset is added to the market portfolio the efficient frontier becomes the straight line which is nothing but the CML.
CML: CML passes thru(0,Rf) and (sigma(m),E(Rm))
E(Rp)-Rf=slope*(sigma(p)-0)
=>E(Rp)-Rf=(E(Rm)-Rf/sigma(m))(sigma(p))
=>E(Rp)=Rf+(E(Rm)-Rf)[(sigma(p))/sigma(m)] which represents CAPM to be approximate
variance(p)=variance(systemetic risk)+variance of unsystemtic risk
=>variance(p)=variance(systemetic risk)+[beta*sigma(m)]^2
Now portfolio has two components of systematic risk and unsystematic risk we diversify away all unsystematic risk which is one of the assumption of capm hence variance(unsystemetic risk)=0.
=>variance(p)=[beta*sigma(m)]^2 =>sigma(p)=[beta*sigma(m)]
=> our CAPM becomes E(Rp)=Rf+(E(Rm)-Rf)[beta*sigma(m)/sigma(m)] =Rf+(E(Rm)-Rf)[beta] which represents CAPM to be approximate to be exact.
and from this CAPm we obtain the SML which is nothing but the straight line representing this CAPM equation.

thanks

 

[Backwardation]

Hi,

I have also questions regarding this topic, maybe you can comment on them:

- I would basically not say that the efficient frontier is the same as the CML, since the efficient frontiert is a curve (concave), starting a the minimum variance portfolio. The CML in contrast is derived by adding the risk-free asset (lending and borrowing) -> by doing so, the efficient frontier becomes straight line. Hence, I would rather argue, that the CML is the efficient frontiert including the possibility of risk-free lending and investing.

- On the other hand, I would say that the CML and the SML are the same (assuming a well diversified portfolio). The SML is just the term of the CML referred to in the CAPM model.

Am I right concerning this statemtns?

 

[ShaktiRathore]

CML is the expected return on portfolio versus risk of portfolio line representing the max. sharpe ratio. Here the investor hold the risky market portfolio which has max. sharpe ratio and may not be fully diversified to eliminate unsystematic risk as it cares with only the sharpe ratio(takes whole risk systematic as well as unsystematic) maximization of portfolio holding Risk free asset and the market portfolio.
IN CAPM we assume that investor is well diversified and can lend/borrow at risk free rate and such unrealistic assumptions which eliminates the systematic risk ans what remains is just the systematic risk. So security hold by a well diversified investor will have only the systematic risk because the unsystematic risk would already have been diversified away. So investor just prices the systematic risk, and this CAPM equation is represented by the SML.

thanks

 

[David Harper CFA FRM]

On the other hand, I would say that the CML and the SML are the same (assuming a well diversified portfolio). The SML is just the term of the CML referred to in the CAPM model.

I would not agree with the above. My favorite distinction between SML and CML is: CML is only efficient portfolios, but the SML includes any portfolio, including non-efficient portfolio

A portfolio can be on the SML but not on the CML: "the CML is a line of efficient portfolios and only efficient portfolios. The SML/CAPM is a line for ANY portfolio or asset, including inefficient: it will take that asset/portfolio and, because it cares only about the systemic risk, give you back the expected return. But your asset/portfolio is not necessarily on the CML"
see http://forum.bionicturtle.com/threads/capm-sml-cml.5347/#post-14867

thanks,

 

[Anova]

Hi;

Why do we use standard deviation to measure risk in CAL and CML? Standard deviation measures total risk -- both systematic and unsystematic. If we are dealing with efficient portfolios, unsystematic risks should not be considered.

Since CAL and CML include unsystematic risk, won't they overstating the expected return? What use is CAL and CML if they are using the wrong risk measure?

 

[ShaktiRathore]

Hi
The CAL is the plot of portfolios expected returns Vs the portfolios standard deviation(both systematic and unsystematic). CML is a special CAL where the risky portfolio is the market portfolio. CAPM makes further assumption for CML that portfolio is well diversified in that case only the systematic risk remains while the unsystematic risk diversifies away to zero , hence CML now reduces to a SML with the securities return on y axis and beta(systematic risk on X axis).
CAL and CML does not consider that portfolios are well diversified they correctly consider the whole risk both systematic and unsystematic(they correctly price risk). But when portfolio are well diversified then only we can consider only systematic risk. SO there is no point of overstating the expected return which depends only on systematic risk.

thanks

 

[Anova]

Thank you Shakti!

 

[afterworkguinness]

Hello,
The notes on the various non standard forms of the CAPM make reference to an equilibrium, can you elaborate ? Thanks

 

[David Harper CFA FRM]

Hi afterwork,

"Equilibrium" is deep at the heart of the theory, by which I mean to suggest I am not highly qualified to summarize it safely

I will tell you how I interpret it very simply, at the risk of exaggeration. In CAPM, equilibrium implies that in the efficient market buyers and sellers will trade the asset such that its price will adjust to equal its expected (forecast) price discounted to account for its risk (beta). For example, if stock will have a (consensus) expected price of $10 at the end of the single period, then under CAPM assumptions, all investors will "conspire" to produce a price today of $10*exp[-(Rf + beta*ERP)]. In this way, the rigid (unrealistic) CAPM assumptions create an artificial environment under which conditions imply prices adjust, and the prices adjust to inform a CONSENSUS expected return. This is why, in my videos, I always say I think the assumptions of investors' HOMOGENOUS EXPECTATIONS (i.e., that all investors have the same identical view on asset means and variances) is the most critical: first, because it's unrealistic; second, because it is the key to the "equilibrium" the produces an expected return (again, a function of a market-cleared price) which is only a function of beta (systematic risk).

In CAPM, equilibrium is just an instance (example or class) of a more general equilibrium concept which, I think, can be viewed as models that have a self-contained mechanism to "clear" a price (or return); e.g., in economics, supply and demand "clears" to find an "equilibrium" price/quantity due to tensions ... there is a balance in the model.

The new Tuckman (P2 FRM on fixed income) differentiates between equilibrium interest rate models versus arbitrage-free rate models, where the equilibrium models have a "self-contained" method to producing the interest rate evolution which does not necessarily match the observed interest rate term structure.

The notes of course summarize Elton's Modern Portfolio Theory and Investment Analysis, and the actual text devotes much space to equilibrium theory. I would naively reduce most of it to something like: equilibrium in CAPM and non-standard CAPM is when we can maintain, under the *unrealistic* assumptions, a model that "clears" (via risk/return tensions or imbalances) into a set of consensus and "balanced" returns (again, prices that inform returns!) as a function of risk.

But, I sort of prefer the clarity of the following from Amenc Chapter 4, selected mine:

"[4.1.1.2] Equilibrium Theory: Up until now we have only considered the case of an isolated investor. By now assuming that all investors have the same expectations concerning assets, they all then have the same return, variance and covariance values and construct the same efficient frontier of risky assets. In the presence of a risk-free asset, the reasoning employed for one investor is applied to all investors. The latter therefore all choose to divide their investment between the risk-free asset and the same risky asset portfolio M.

Now, for the market to be at equilibrium, all the available assets must be held in portfolios. The risky asset portfolioM, in which all investors choose to have a share, must therefore contain all the assets traded on the market in proportion to their stock market capitalisation. This portfolio is therefore the market portfolio. This result comes from Fama (1970).
In the presence of a risky asset, the efficient frontier that is common to all investors is the straight line of the following equation [CML line]. This line links the risk and return of efficient portfolios linearly. It is known as the capital market line. These results, associated with the notion of equilibrium, will now allow us to establish a relationship for individual securities.

[4.1.2.4] Market efficiency and market equilibrium: An equilibrium model can only exist in the context of market efficiency. Studying market efficiency enables the way in which prices of financial assets evolve towards their equilibrium value to be analysed. Let us first of all define market efficiency and its different forms
... There are several degrees of market efficiency. Efficiency is said to be weak if the information only includes past prices; efficiency is semi-strong if the information also includes public
information; efficiency is strong if all information, public and private, is included in the present prices of assets. Markets tend to respect the weak or semi-strong form of efficiency, but the CAPM’s assumption of perfect markets refers in fact to the strong form.
The demonstration of the CAPM is based on the efficiency of the market portfolio at equilibrium. This efficiency is a consequence of the assumption that all investors make the same forecasts concerning the assets. They all construct the same efficient frontier of risky assets and choose to invest only in the efficient portfolios on this frontier. Since the market is the aggregation of the individual investors’ portfolios, i.e. a set of efficient portfolios, the market portfolio is efficient. In the absence of this assumption of homogeneous investor forecasts, we are no longer assured of the efficiency of the market portfolio, and consequently of the validity of the equilibrium model. The theory of market efficiency is therefore closely linked to that of the CAPM. It is not possible to test the validity of one without the other. This problem constitutes an important point in Roll’s criticism of the model. "

 

[afterworkguinness]

Thanks for your detailed reply. I'm not clear on how the CAPM relates to the markets clearing at a given price when the CAPM tells us expected return of an asset.

 

[David Harper CFA FRM]

Right, Stulz' example is an expected cash flow at the end of the (single period) year; e.g., say we all expect a single cash flow of $10 at then end of one year, or for that matter, we all (market participants with homogenous) expectations expect the future spot price, E(St), to equal $10. (Stulz really simplifies by assuming only one cash flow at the end of one period, such that it will equal the future price).

The idea is that today's spot price would adjust to ensure the CAPM-expected return; e.g., S(0) = $10*exp(-kT), where k = CAPM discount rate. If k = 8%, S(0) = 10*exp(-8%*1) = $9.23 where, by definition, this current price is set so that the expected growth rate (the return) is 8%. If the asset has higher risk, the price is discounted to a lower level, in order to produce a higher expected return as compensation for the risk. In this way, a lower (higher) current spot price is a direct (theoretical) function of a higher (lower) discount rate, itself a function of the asset's risk. I hope that helps, thanks,

 

[afterworkguinness]

Thanks David, I understand it when you put it that way.