Beta in CAPM

CFA General
#1

Hi,

When I was reviewing level 3 notes, a question came up in my mind which I couldn’t really answer. Would someone be able to help and point out the stupidity inside it?

In CAPM model, we have a concept of beta which really is the sensitivity of return of an asset class to the return of the market portofolio which is the ultimate well-diversified portofolio which should include traditional asset classes as well as alternative asset classes.

And the expected return of this asset class in question, according to the CAPM model, is the risk free rate plus the market portoflio return in excess of the risk free return amplified by beta.

The beta can be negative, 0 and positive.

Thus, from the above, beta is the non-diversifiable risk and the market portfolio has a beta of 1. The market portofolio should have included all the asset classes and inherited all the systematic risks from them. If we add a new asset to the market portofolio, then the beta of the market portofolio can not increase.

Hence, I think the beta can not be bigger than 1. It is the same as to say any asset can not have systematic risk bigger than it in the market portfolio. Unless the asset we try to price is outside the scope of the market portfolio, the market portfolio should have included the asset and hence has already inherited its systematic risk.

I know that my understanding is not correct but where it goes wrong?

much appreciated,

magictiger

#2

If there are individual stocks with betas less than 1, there must be ones with betas greater than 1 to even it out, if the individuals compose the whole, right?

#3

Hi,

Thanks for the reply.

I am not sure if I understand it because beta is the non diversifiable risk. If the “even out” can happen, then non diversifiable risk could be diversified away.

:frowning:

#4

I think CFA must have omitted some concept around this. It also says some asset class can have beta less than 0. In the sense of beta being the sensitivity return to market return, beta less than 0 can make sense. If we think beta as the non diversiable risk, then risk less than 0 does not make sense.

#5

Beta should be considered as deviation of an asset class (or single security) in relation to the market (calculated as covariance divided by variance of the market).

Higher beta means higher risk, hence higher return as expressed in CAPM.

#6

If an asset is more volatile than the market, it can (but does not have to) have a beta > 1.

If you had a levered version of the market portfolio - say you bought SPY on margin, are now are 2x levered, and you intend to rebalance to maintain the right leverage ratio - then your portfolio will have a beta of 2 to the market: when the market goes up 1% your portfolio will go up 2%, and when it goes down 1%, your portfolio will go down 2%. Your portfolio has 2x the market risk, and a beta of 2, even though it is “only” correlated to the market portfolio by +1.00.

That’s because your portfolio is more volatile than the market portfolio. It is riskier: if something bad happens to the market portfolio, it’s expected to be twice as bad for your portfolio. That’s what beta measures. The idea that beta can be used as a measure of nondiversifiable risk is actually a separate interpretation of what beta means that makes sense in the context of some asset pricing models like CAPM.

Now, assume that you have an asset that is more volatile than the market but highly correlated to it (i.e. much like the margin example above, except that the asset is just naturally more volatile than the market portfolio or the company has a high D/E ratio or something). That asset will likely have a beta > 1.

This, incidentally, is one of the reasons that lower volatility securities are thought to outperform higher volatility securities. Investors who can’t lever using margin will sometimes try to do it by buying higher beta stocks instead. This drives up demand for these stocks at the expense of lower volatility stocks, which raises the price, and consequently lowers the expected return. Voila, high beta stocks return less because there are more people who want them and who therefore bid up the price.

What’s special about the market portfolio (if you accept CAPM) is that it is the most EFFICIENT portfolio in terms of risk/reward. Beta is just a measure of how much you can expect something bad happening in the market to transfer to your asset or your portfolio, and then to get a risk-return figure for your asset/portfolio, you’d have to add in any ideosyncratic sources of risk too. The market portfolio can have a whole bunch of companies with betas of 3 or 4 or 5 as long as they are small percentages of the portfolio. The market portfolio’s beta is 1 by definition in CAPM, and it is a weighted average of all the betas of all the portfolio’s constitutents, so there will be some greater and some smaller.

EDIT: In theory, you can have a negative beta if your asset is negatively correlated to the market (it goes up when the market goes down). From a portfolio diversification perspective, these are very attractive assets, but they are not very common in the real world. That’s because the long-term consequence of a negative beta is that that the asset will slowly go to zero as the market grows (long term). If that’s true, most people will not want to invest in it. Diversification only makes sense after you’ve determined that the long term expected returns are positive (and greater than RFR). However, it is also true that beta can change depending on the data and - particularly - depending on the frequency of your data. Mildly negative betas are more likley to represent correlations close to zero which happen to have estimation error than something which is truly negative.

#7

You’re talking about correlation of _ prices _. When we talk about correlation in MPT, we mean correlation of _ returns _, not correlation of prices; they’re two very different beasts. An asset can have a negative beta and increase in value as the market rises. Your statement that “it goes up whenthe market goes down” and your conclusion about the long-term consequence of negative beta are, simply, wrong.

This, by the way, is the reason that I hate hearing finance types talk about the “correlation of assets” when they mean the “correlation of assets’ returns”; finance people do this all the time. The sloppiness of the language leads to sloppy thinking – drifting from correlation of returns to correlation of prices and back – and, consequently, to poor understanding.

#8

Magic tiger, I think I get what you’re saying. If beta of 1 is equal to the risk of the market portfolio, how can beta be greater than 1 if the market portfolio is really an aggregate of all the stocks in the market, and thus must inherit all of the stock’s systematic and unsystematic risks. So by that logic, the market portfolio should be the riskier than a stock. So my rebuttal to this is that the unsystematic risk that affects a stock can be almost trivial to the market portfolio. Think about holding a porfolio of two stocks versus the market portfolio. If one of the stocks that the both portfolios share tanks, which portfolio will this have a greater impact on?

A more analytical take on this could be done if you look at the composition of beta.

Beta = Cov(stock,market) / Var(market). So if you look at the beta of the market…

Beta = Cov(market,market) / Var(market). If you have ever taken stastistics, you should know that Cov(market,market) = Var(market). If you haven’t, just know that Variance is a specific case of Covariance when you take Cov(x,y) and x=y.

So than the Beta of the market will always = 1, just by definition. So it doesn’t matter if you add a stock to the market portfolio… take one out… add a million stocks, or have just one stock. As long as there is one stock in the portfolio (because variance of 0 stocks is 0 and 0/0 is undefined), than the beta of the market portfolio will always be 1.

So then, following this math thing.

Beta represents the variance of the expected returns of a particular stock, with respect to the expected returns of the market portfolio.

#9

Magician, I think you’ve misread my argument here. It’s true that it is correlation of returns rather than prices and that sometimes causes trouble, particularly since it’s common to say “assets are correllated/uncorrelated” instead of “asset returns are correlated/uncorrelated.”

In my example above, I clearly used returns rather than prices, so I don’t think you can say that I misunderstood price versus returns.

However, let’s look at an example that illustrates better what I was talking about.

What is the consequence of an asset with a beta of -0.5?

If we set the RFR and equity risk premium to long-term historical averages, that’s approximately 3.5% for RFR and 5% for the ERP. So by the index model or CAPM, you get a long-term expected return of

RFR + Beta * ERP = 3.5% + (-0.5) * 5% = 1%

Now it’s true that this is a positive return, but it is also a return that is LOWER than the RFR but still contains both market risk (and perhaps ideosyncratic risk), so it’s not a good investment on its own merits. There might be value for the asset as a diversifier in a portfolio, but the investment can never be claimed to be a good one on its own, without some new source of information.

Now, it’s been a while since I’ve done this analysis, and what I realize is that in the CURRENT interest rate environment, where short term rates are close to 0, it’s very easy to observe negative beta --> negative returns. Or if beta is more negative (I’ll use a more negative beta so as to avoid an argument about which end of the yield curve to use for the RFR):

3.5% + (-0.75) * 5% = -0.25%

Which is a negative expected return, and assets that deliver a negative expected return have ZERO value for portfolio diversification, because - even if they reduce risk, they reduce return even more. Negative-beta assets can be useful for hedging, but that is about eliminating unwanted risks, rather than improving diversification.

My main argument still holds, which is that highly negative betas are very uncommon and should be looked at with suspicion, and that’s because if beta gets too high, long-term returns will start looking negative and the price will slowly grind down to zero. However, it is true that the critical beta where that happens does depend on market conditions and is = - RFR/ERP, if you are using the index or CAPM SML. If you use short term treasuries right now, that beta is very close to 0. If you use long term treasury rates right now, the critical beta where there is no reason to own an asset at all is around (-3/5) = -0.6.

#10

That was my point: that the expected return can still be positive even when beta is negative. (Note: for simplicity, we’re ignoring any alpha here.) Therefore, your statement that when beta is negative “it goes up when the market goes down” is incorrect: they both have positive expected returns. And your conclusion that “the long-term consequence of a negative beta is that that the asset will slowly go to zero as the market grows” is also incorrect: they both grow.

To say that a negative beta implies that when the value of the market rises, the value of the asset falls (and vice-versa) suggests that beta depends on prices, not on returns.

(By the way, it’s interesting to go to various financial websites that have glossaries and look at their definitions of beta: about 80% of them are incorrect. And many are incorrect specifically because they say that beta is calculated based on prices, not on returns.)

#11

Not if Beta is too negative.

#12

True, but you wrote, “the consequence of _ a _ negative beta . . . .” Not “a sufficiently large negative beta”, but _ any _ negative beta. That’s my quibble.

#13

I’m tired of nit-picking here. Yes, I forgot to include that the RFR can permit a little bit of negative beta and it can be justified as a portfolio diversifier. At short time scales the RFR is often nearly zero, and so the “when the market goes up, the asset is expected to go down” is a reasonable interpretation negative beta, particularly in a low interest rate environment like the last 5 years. My mistake was forgetting that when you extend this out, the RFR can become significant, and this is why prices don’t necessarily ground to down to zero for all negative betas.

The main point - which is that you should not expect to find very many assets with strongly negative betas - remains. They are not good investments on their own merits (because they have risk and expected returns lower than the RFR) and can only be justified from a diversification perspective (or if there is alpha, but this will typically require some kind of beta-neutralization to capture).

And while we’re nit-picking, let me say that YOU ARE WRONG when you identified thinking that beta is a regression of prices rather than returns as the reason for the mistake. The mistake was forgetting that excess returns being negative doesn’t mean ordinary returns are negative unless the absolute value of the excess returns are larger than the RFR for the same period. I know that this is a trivial distinction about whether you are wrong or you are right, but since we’re bathing in technicalities here, it seems appropriate to say.

And yes, we aren’t talking about alpha, because OP posted the question in the context of CAPM.

#14

bathing… oh bchad. magician is a dirty boy and needs a technicality bath.

#15

Absolutely correct.

Fair enough. I’ve seen so many people who misunderstand beta (people who are smart enough that they shouldn’t), and explicitly state that it compares prices rather than that it compares returns, that I, too, reached the wrong conclusion.

My apologies.

#16

Getting back to the original poster (and as answered by Guitardude) - let me put a less mathematical (and maybe more intuitive) spin on it.

When you add a high-beta stock to the market portfolio, the Beta of the market will still be one BY DEFINITION. That’s because beta measures exposure to systematic risk RELATIVE TO THE BENCHMARK (i.e. THE MARKET). In this case, you’ve essentially changed the composition of the benchmark. But the beta of the market captures its exposure to systematic risk relative to itself (as Guitardude showed). By definition, this relationship must be unity.

However, the betas of all the other stock could (and likely will) change when you add a high-beta (or even a low-beta) stock to the market. Think of it as measuring a beta based on the S&P vs the Wilshire. Each index has a beta of 1 relative to itself, but individual stocks will have different betas measured using the two indexes.

#17

Buzzkill.

#18

LOL

(I could make part of 'em disappear.)

#19

WoooWoo, I am back. Nice, I like overwhelming replies!

Let me read through them all.

Thanks to all!!!

#20

No killing is allowed in this post, thanks