SML, CML and CAPM

[sudeepdoon]

Hi David,

First of all let me take the opportunity to let you know that the spread sheets and the videos you had uploaded had helped a lot to make the understanding better.

I still had a few questions in my mind and would like to throw them to you

• The SML line is defined to be the line that shows the relation between the returns and the beta for a particular security, then why is that in all example I see; Asset A and Asset B, and then we talk about correlation and covariance..
• The CAPM stands for Capital Asset Pricing Model; which gives me the impression that it can be used for any Asset in general; but the equation that we derive has a term of beta in it. I am under the impression that beta is used for Stocks, so how is that CAPM can be used for any asset

Thanks,
Sudeep

 

[David Harper CFA FRM]

Hi Sudeep,

Thanks for finding them helpful. Good questions...

1. You are right about the SML but it plots securities or *portfolios* of risky securities; so if you consider Asset A, Asset A itself lies on the SML. But now add Asset B, so you have a portfolio of [A+B], this is a "risky portfolio" [no riskless asset]. This risky portfolio [A+B] has a beta which is simply a (linear) combination; e.g., 50% of A and 50% of B implies portfolio beta of [50%* beta of A] + [50% * beta of B]. This risky portfolio of [A+B] has itself a beta, and lies on the SML. So, i did not mean to confuse by generating the market portfolio (the optimal mix of risky assets), but i think if you see that portfolio beta is a linear combination of component (assets), you'll see it doesn't matter if we plot security or portfolio beta against return on the SML.

2. Yes, understood, this is common misunderstanding. Beta is not stock specific; you may have noticed I try to stress the generic form of Beta = cov()/variance because it has many applications; e.g., Stulz has a cash flow beta (project cash flow, company cash flow) = covariance (project, company)/variance(company); even the marginal value at risk is a form of beta. I recommend not imbue specific meaning into a generic beta; e.g., in Grinold, he even employs two forms of the stock beta (levered, unlevered). Rather, here is how I would view beta: it is cousin to correlation coefficient as both "standardize" *covariance* into something more usable. Correlation translates covariance into unitless format; beta translates covariance into a sensitivity (vis a vis the denominator asset). If you let beta be generic, you will see it all over the FRM. In the CAPM specifically, the risky assets are beyond stocks, encompassing all risky assets (e.g., alternative, commodities). It is literally any asset which is not riskless; this is a plank of the equilibrium theory. So, the beta here is quite qeneric: standardized covariance between *any* risky asset and the (optimal) market portfolio.

Hope that helps...David

append: you might find this 9-minute video helpful ("beta is one idea with many faces"):
http://www.bionicturtle.com/learn/article/beta_is_one_idea_with_many_faces_9_min_briefcast/
...shows beta is also the hedge ratio (from Hull)

 

[frm_daniel]

Hi David,
I do not understand on your reply to Sudeep's question item 1. The equity risk premium ERP is the market excess return of the return of market less risk free rate.
The Beta of security (say stock A) is defined as the cov (market, security A)/variance (market.) What does the market return refer ? Does it refer to particular set of securities other than security A. But then the Beta given should be the risky portfolio of [security A+other securities] which is the porfolio beta. Why it is named as beta for the security A. What does "equity" risk premium mean? It is a porfolio of equity's(other than security A) return in excess of risk free rate? Shouldn't it be named as market risk premium?

Also, when we say beta of security A is equal to 1.2, does it mean the security A has a change in the proportion in the market portfolio such that the beta of security A (cov of security A with market/ variance of market) has increased from 1 to 1.2. Sorry I am not clear about the concept and may have asked a stupid question.
Daniel

 

[David Harper CFA FRM]

Hi Daniel,

Great questions, for sure. I only recently came into a better (still imperfect) understanding after the Grinold was assigned to FRM; so, if you have time, early Grinold chapters--e.g., Chapter 1--are recommended.

Most abstractly, I think it's helpful to keep in mind that beta is a generic sister of correlation: beta of X with respect to Y = cov(x,y)/var(y) = correlation(x,y) * volatilty(x)/volatility (y) = correlation (x,y) * cross-volatility. In other words, beta is volatility-adjusted correlation and thefore can refer to ANY linear relationship. For example, we can compute asset beta with respect to portfolio (where the portfolio contains the asset, such that we expect positive beta w.r.t portfolio). Even better, beta = minimum variance hedge ratio (Hull) because, after all, both are the slope of the regression line, where in that case, it just happens to be about the relationship between forward and spot (!). But it's not too much to say the M.V. hedge ratio is the "beta" of the (change in) spot w.r.t. forward.

So, then we have CAPM (ex post; predictive)/market model (ex ante) where both are "merely" regressions. In the case of CAPM, it's a highly theoretical regression of asset/portfolio excess return (y-axis) against the total market excess return. As ex post predictor, it is ~ Gujarati's population regression function (PRF) without error: we don't really see it, and the excess return is a predicted: excess return = beta*ERP.

So in the fanciful CAPM world, ERP = excess return of the entire portfolio of all risky assets (it occurs to me that maybe ERP, which i was taught, is inferior to MRP b/c it's all encompassing...). And, the beta in CAPM is therefore intended to be: beta (asset/portfolio excess return, excess market return).

... but in practice we don't have access to "all risky assets," so we do something like an ex-post regression: asset to market index returns. And the slope of that line is a market model-based beta that we then go and plug into the ex ante CAPM.

In regard to "Why it is named as beta for the security A." Please note that once we have a ERP (or excess market return), then we are just applying the SML to predict the return based on any portfolio's beta. (my point in #1 above). It may be worth a meditation on the fact that the SML (i.e., excess return versus beta; which therefore has slope of ERP!) is just an expression of the CAPM: as a one-factor regression, it says any predicted excess return is only due to beta and nothing else.

So, what i mean is: we first may compute Asset A's beta by regressing its excess returns against a proxy for the total market. Then, once we have Asset A's beta, we can "plug it into" the SML, or we may combine with other assets into a portfolio beta (beta is conveniently a weighted average!) and use SML on the portfolio.

Re: meaning of "beta of security A is equal to 1.2"
you'll note that I have come to think it is best to be specific here; i.e., beta of security with respect to what? because we ought to specify the common factor...
...but okay, if we are talking CAPM, then IMO the most intuitive interpretation is simply as the slope of the regression line (excess security/portfolio return versus excess market return) where beta of 1.2 then implies "a +1% increase in excess market returns implies a +1.2% increase in portfolio return." ... and even here, i think it's helpful to use beta = correlation * cross-volatility; e.g,. the +1.2% could be due to 1.0 correlation and 1.2X higher volatility; or 0.5 correlation and 2.4X higher volatility.

hope this helps, David

 

[frm_daniel]

Thanks for your detail reply. It helps.I find studying FRM is not easy if you go to drill it down to understand each AIM indepth.

Daniel

 

[David Harper CFA FRM]

Daniel, sure thing. I could not agree more. The additional challenge, unfortunately and frankly, is that the deep dive in some areas will not necessarily "payoff" in the form of a test question; the FRM casts such a wide net that invariably a dutiful preparation (i.e., a literal following of the AIMs) necessarily over-prepares and meets with the exam's historical tracking error (AIMs versus test questions). I would like to think a deep deep overall correlates to exam success, in particular in the sense that it reinforces common principles, but still I would like to see GARP reduce what i have dubbed the exam's tracking error (even with the L1 + L2, the ratio of AIM-implied concepts is multiples of, far exceeds, the number of exam questions such that some pockets are rarely tested) ...David

 

[sucheta_isi]

David,

I have some questions about Jensen's Alpha.

1. When it is given that a portfolio has underperformed its benchmark then will ALPHA be always negative? Or can we conclude anything about ALPHA?

2. Suppose a portfolio has a very high ALPHA. Now how will it behave with market if it has positive/negative Beta?

Thanks,
Sucheta

 

[David Harper CFA FRM]

Hi Sucheta -

1. A question worth a meditation. We cannot conclude. Performance below benchmark is negative ACTIVE return. But the point of (FRM Assigned) Grinold Ch 17 is that active returns are determined by active systemic (common factor) exposures in addition to alpha; e.g., if portfolio beta = 0.8, may under perform S&P benchmark (assume beta = 1.0) and then add some positive skill (alpha) which mitigates but could leave the active return negative or positive. So, you'd need to parse out the performance that is explained by beta exposures

2. Jensen's alpha implies CAPM: CAPM gives you the ex ante expected return (without skill): excess return = beta*ERP. Pure CAPM says: higher beta = higher return. The alpha justs adds to that.

David

 

[sucheta_isi]

David,

Thanks for your reply. But now I need some more clarifications. Let me be more specific:

1. ACTIVE return and ALPHA are different?

2.A portfolio has outperformed(underperformed) the benchmark portfolio DOES NOT always mean that the given portfolio has positive(negative) ALPHA. BUT we can conclude the other way round i.e. positive(negative) ALPHA always imply that the portfolio has outperformed (underperformed) the benchmark portfolio. Am I right or wrong?

 

[sucheta_isi]

David,

To be more specific my question is :

If it is stated that "A portfolio has underperformed its benchmark by 5%" then can we conclude anything about the ALPHA?


I am sorry for asking questions on this again and again.

Thanks,
Sucheta

 

[David Harper CFA FRM]

Sucheta,

1. They are different: active return includes alpha plus "alternative beta" (i.e., skillful overweighting to common factors if they "surprise" on the upside; underweight if on the downside). So, active = portfoio return - benchmark, and active = alpha + alternative/active beta contribution.
2. No, we cannot conclude
Re: "A portfolio has underperformed its benchmark by 5%” then can we conclude anything about the ALPHA:" no, we cannot conclude

e.g., assume riskfree rate = 4%, equity premium = 5% and benchmark is S&P 1500 with beta = 1.0

benchmark return = 4% + 5%*1 = 9%;
now assume portfolio return = 8%
... that could be -1% alpha
... or could be benchmark timing that gave portfolio an effective beta = 0.8 with 0 alpha
... or could be worse benchmark timing that gave beta = 0.6 with +1% alpha

alpha is a true residual, therefore is dubious to say it is the only skill as skill also comes from over- (and under-) weighting exposure to common factors

David

 

[David Harper CFA FRM]

Sucheta,

I thought you might enjoy this great 2-part article @ allaboutalpha: http://allaboutalpha.com/blog/2010/04/07/the-five-faces-of-alpha-ii-true-alpha-reveals-itself/
... he goes deep (I don't quite agree, or maybe i just don't comprehend why, that insurance beta is a true beta) but notice the "exotic beta:" it is not alpha that contributes to active (portfolio - benchmark) return - David