Talk:Capital asset pricing model

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I am concerned about a recent edit. There is an article showing that there is a math error in the derivation of the CAPM that was deleted as an irrelevant criticism. How could a math error in the proof be irrelevant on a model filled with empirical failures? The error is that the CAPM violates the rules of general summation. This is not a trivial or irrelevant argument. Suggestions? Winning a Nobel prize cannot be a reason to hide its problems. The possibility that the left hand side does not equal the right hand side is serious. Wikipedia isn't the place to argue this out, but a lot of people use Wikipedia and the problems are serious. Laypersons need to be adequately warned. It doesn't pass muster empirically or mathematically. It needs to go the way of luminiferous ether.

Response: Dear David, I will refer to you by name because the user "entpdave" who added the (subsequently removed) section here also shows up as a user on the INET website, which David Harris, the author of the criticizing article, is associated with[1]. Further, the Facebook page of the arguably same David Harris has a URL also featuring "entp.dave"[2]. While somebody could have faked the Wikipedia username to mimic David Harris' credentials, it seems as very unlikely.

Therefore, before I get to the factual criticism of your paper titled "Why heavy tails?", let me suggest that it is rather unethical to promote your own work here while pretending to be a third person, referring to your own article as "There is an article showing ...", rather than "I wrote an article showing...". That's just for starters.

And now to the factual criticism of your own work, which, I believe, can be made using arguments from Section 3.1 of your paper[3], section titled "Intuition behind the proof". In this section, you commit a standard logical fallacy. You make a set of assumptions different from those in standard (and numerous) derivations of the CAPM model, then you show that your set of assumptions leads to ill-defined objects of interest, which leads you to conclude that the CAPM model must be wrong.

But of course, there is nothing wrong with the CAPM model. It is your own assumptions that are poorly chosen. What you argue in Section 3.1 is the following. Assume that the value of an investment at time is , where is normally distributed. Then the gross return conditional on time t information is normally distributed. You are already running into some troubles here because your normality assumption implies that the value of the investment is negative with positive probability at time but that can still be worked around.

The critical problem arises when you try to express the distribution of conditional on time time t-1 information in your equations (5) and (6). The problem now is that conditional on time t-1 information, not only the numerator of the return but also its denominator can be negative with a strictly positive probability. Therefore, you have a strictly positive probability of an event that your investment has a negative value at time t, but a positive value at time t+1. This of course violates no-arbitrage and does not make any economic sense.

From this point on, you can still try to express the distribution of such a ratio, but it is economically completely meaningless. The Cauchy distribution[4] that is derived as the distribution of the ratio takes on both positive and negative values, and the negative values are generated by combinations of a negative numerator and a positive denominator, and vice versa (and this vice versa is economic garbage).

It simply does not make sense to even start talking about CAPM in such an environment. And in no way do your assumptions constitute a good model of returns in financial markets.

Your analysis is thus a clear example of "garbage assumptions in, garbage results out". But it simply does not disprove anything about CAPM.

To be constructive, let me suggest to you a modification of your model. A much better assumption (one that is also much closer to the empirical distribution of returns) is that the logarithm of the return is normally distributed. Make this assumption and you will see that your whole analysis goes through without a glitch. Your return will be lognormally distributed conditional both on information at time t, as well as information at time t-1. Problem solved. CAPM survives. — Preceding unsigned comment added by (talk) 06:48, 6 August 2014 (UTC)


I think the last point of the shortcomings should be explained better, or be removed:

Because CAPM prices a stock in terms of all stocks and bonds, it is really an arbitrage pricing model which throws no light on how a firm's beta gets determined.

I understand all other points, but do not know what is meant by the last point... 16:21, 16 March 2007 (UTC)

I think two of the shortcomings are not quite right.

"The model assumes that all investors are risk averse. Some investors (e.g., some day traders), however, can not be considered to be risk averse."

A central idea of the model is that a day trader and a retiree can use the same pricing model, because they can assemble a portfolio that matches their different risk objectives and they can observe the market price for risk.
All the model assumes is that 1) higher returns are preferred to lower returns and 2) lower risk is preferred to higher risk.
In other words, if you offer a day trader two assets -- one highly risky asset and one lower risk asset -- with the same expected return, the day trader will pick the lower risk asset.
One can argue that the model assumes that investor behavior is rational (and that that is a shortcoming).

--->The above is not said clearly. What he/she means to say is that: (i) for a given risk level, an investor will always prefer a higher return, and (ii) for a given expected return level, an investor will always prefer lower risk.

"The model assumes that all investors create mean-variance optimized portfolios. However, there are many investors who don't know what a mean-variance optimized portfolio is."

Again, the only thing the model assumes is that 1) higher returns are preferred to lower returns and 2) lower risk is preferred to higher risk.
Variance is used as a measure of observed risk. There is no assumption that investors are using the model.
Like gravity, you don't need to know : for it to work.

If no objections, I will remove these shortcomings. Thoughts? -Chris vLS 20:52, 10 Dec 2004 (UTC)

I agree with you, especially about the day traders. Whoever wrote this sentence confused willingness to accept risk with an actual DESIRE for risk. Evel Knievel was not a day trader. Furthermore, its been 2 weeks since you asked this question and nobody has sought to defend these passages yet, so ... let's roll! --Christofurio 00:43, Dec 22, 2004 (UTC)

Done! Thanks for the encouragement! --Chris vLS 16:01, 22 Dec 2004 (UTC)

Hi there. Although your point is well taken, I'm not sure that you should have removed the part of about investors preference for mean-variance optimized portfolios. That actually IS one of the assumptions of the model. This isn't physics, it's not a model that the market must follow. It's a model that assumes that investors always choose the market portfolio, together with the risk-free asset. This is inherent in the definition of the market portfolio, that it's defined by investor choices, is completely diversified, and has no non-systematic risk. The market portfolio has a beta of exactly 1, and CAPM assumes that investors can choose more or less risk by buying or selling the portfolio and borrowing or lending the risk-free asset. This is known as the separation theorum of capm: that every investor can participate with their own risk preference by buying just a market index fund and the risk free asset.

--Tristan Reid 10 Feb 2005 (UTC)

Actually, let me amend what I said: investors don't prefer mean-variance optimized portfolios, investors choices in aggregate (as contributing parts of the market portfolio) are mean-variance optimized. --Tristan Reid 09:46, Apr 06, 2005 (UTC)

--->Tristan, I'm not sure what you mean when you talk of aggregation. All investors are mean-variance optimisers in that they will prefer, for a given risk level, MAXIMUM returns. that is all. This is THE fundamental assumption of the CAPM.-Ben

Hi Ben. Not all markets are accessible to all people (such as 144a assets, or restrictions on foreign markets). The given risk level is only defined by the covariance of the assets in the portfolio, but some investors are also concerned with other risks, e.g. liquidity. Investors can add these different constraints on the mean-variance objective function without breaking CAPM. That's what I mean by 'in aggregate'. If you model that everyone is chasing the same assets, the liquity constraint will only be satisfied if investors either hold non-zero weighted chunks of the entire market, or that they hold enough of the market to be locally diversified. By considering all investors together, both of these problems are solved and CAPM is satisfied. It's very much like a Nash equilibrium. Tristanreid 21:55, 8 February 2006 (UTC)

Risk free rate[edit]

There appears to be a contradiction in the article regarding the risk free rate. At one point it is referred to as "such as interest arising from government bonds", later on as "cash (either borrowed or invested)". There are online references for optimal asset allocation, e.g. [5], which suggest (subject to an individual's different short-term needs for cash) that the optimal allocation of the non-risky part of an individual's assets should be about three quarters bonds and a quarter cash. What I am hoping is that someone has done some research to reconcile these.Dan88888 (talk) 13:39, 8 May 2012 (UTC)

Typically the risk free rate is the interest rate from government bonds (when the government is economically stable, for instance US T-Bills are usually used whereas Greek bonds would not be right now). I believe this is mentioned in risk-free bond. As for saying it is "cash", that is because in (simpler) financial math models it is assumed you can always borrow or invest at the risk free rate (with any time maturity). So from that point of view these are thought of as the same number in theory. In practice you won't be able to borrow at the risk-free rate because you have a risk of default so a premium is included. As for how to optimally allocate the riskless assets, I don't think that has a part in this page as that goes much beyond the CAPM model. I hope this helps. Zfeinst (talk) 14:13, 8 May 2012 (UTC)
OK, I found a piece of original research I could cite (which explains my point better than I could!). So I was bold an edited the article. Do please let me know if you disagree with the edit. — Preceding unsigned comment added by Dan88888 (talkcontribs) 16:22, 8 May 2012 (UTC)

Think it is time to delete this section. My question was answered we decided not to edit the page further. In fact I did delete, but had my edit rolled back. Please reply here if you think this section continues to have merit. Dan88888 (talk) 14:04, 23 June 2013 (UTC)


The CAPM graph is wrong for negative betas.[edit]

The CAPM graph looks wrong for negative betas. You cannot get an expected return less than the risk-free interest rate.

> YEs you can -if you buy an asset with a negative beta (i.e. an asset that moves against the market portfolio most of the time). THe benefit is having an asset with a negative beta. but then again you get a lower return than a risk free asset.


I agree with you, the graph isn't really showing the main points of CAPM. Just for the fun of discussion, though: You can buy insurance (or an option) that reduces your risk below the risk-free interest rate. Also, what if you just don't invest at all? Isn't that a return below the risk-free rate? And if you consider the US Treasury bond to be risk free, how much risk does an inflation-protected US Treasury bond have? Even less? TIPS have smaller coupons than TSY bonds. Tristanreid 20:08, 20 September 2005 (UTC)

Gold gives you lower than risk free rates of return because it has a negative beta (one of the few real assets that behaves this ways). As mentioned above, insurance gives you a negative return and a negative beta. There's no reason the graph can't extend even farther than shown into negative rates of return. Kjm 20:58, 30 December 2005 (UTC)

The perfect example of a negative beta, less than riskfree rate asset is your homeowner's insurance which is priced to yield negative absolue returns yets freely trades.

For Dummies?[edit]

Isn't the expression "for dummies" in this kind of usage copyrighted? And do we really need this section? If its explanations are better than those of the rest of the article, they should be put there, not in an addendum. --Christofurio 15:50, 5 February 2006 (UTC)

Article removed from Wikipedia:Good articles[edit]

This article was formerly listed as a good article, but was removed from the listing because the intro is just a bit impenetrable. If you could just explain a few terms a bit more I think it would certainly qualify. Imagine your reader is intelligent but totally ignorant! Worldtraveller 00:56, 24 February 2006 (UTC)

Should article include empirical estimation of CAPM?[edit]

Hello -- all --- reading this article offers a pretty good intro CAPM. Congrats to all. It explains the structure and assumptions of the model. Some extenstion suggestions to improve this excellent article and make it more comprehensive. Should this article include a brief review of the mail empirical issues in estimating CAPM? Or rather, should a new article, different from this entry, focus on the emprical issues of "correctly" estimating CAPM? Finally, should International CAPM (a la Solnik) be included in this entry or anchor a new one? Psw2xx

Formula is insufficient for the model[edit]

The formula is now given for the expected rates, as is done in most introductory texts, but this is not a sufficient assumption for the model. It should be in the form of a linear regression equation Y = a + bX + e, where Y is the excess return of the portfolio and X is the excess return of the benchmark. In the original, strict form of the CAPM the equation is Y = bX + e. The difference with the formula in the article is that the regression equation applies to each individual return, albeit with an error e, not only for the expected returns. The regression equation is necessary to estimate alpha and beta. It is logically impossible to do that from the formula with the expectations alone. JulesEllis 01:20, 18 January 2007 (UTC)

Formula is OK but tests should be mentioned[edit]

The most widely used tests such as the Sharpe Ratio, Traynor Ratio and Jensen Alpha should be incorporated within this topic as they ALL measure the efficiency of a portfolio manager using the CAPM as a reference model. The regression equation suggested before is infact the basis for Jensen's Alpha. Abh1984

I agree with that too. JulesEllis 05:13, 23 January 2007 (UTC)

What about location-scale?[edit]

It states that returns are assumed to be normally distributed. However, recent developments have shown that location-scale returns with constant skewness across location-scale choices are sufficient. In fact, this skewness restriction can be removed under some assumptions on the utility function. In fact, location-scale returns are not necessary. This can be generalized to spherical distributions. (see articles by Grootveld/Hallerbach (1999) and Bawa (1975)) Shouldn't there be a note about some of this? --TedPavlic 20:15, 8 May 2007 (UTC)

Formula: Arithmetic vs. Geometric average?[edit]

In the two notes about measuring the expected market rate of return and the risk-free rate of return arithmetic average is mentioned. Would the geometric average not give a better calculation? -- LJG59 (talk) 13:06, 27 August 2008 (UTC)

I don't think geometric for a one period model like CAPM. —Preceding unsigned comment added by (talk) 04:46, 3 December 2009 (UTC)

Link Sharpe/Mossin[edit]

Mossin (1966) did not write his article independently of Sharpe (1964), but based on it. Cf. the references of his article. — Preceding unsigned comment added by NecSpe (talkcontribs) 00:02, 9 February 2012 (UTC)

Asset pricing correction?[edit]

Perhaps I'm misunderstanding the model, but I believe "undervalued" and "overvalued" should be reversed in the following sentence from the Asset Pricing section: "If the observed price is higher than the CAPM valuation, then the asset is undervalued (and overvalued when the estimated price is below the CAPM valuation)."

If observed price is HIGHER than the price the model suggests for the given beta, doesn't that indicate the asset is OVER-valued? In terms of expected returns, the model associates an expected return with the given beta, which may be used to discount future cash flows to derive an expected price. An observed price HIGHER than the expected price implies a future return lower than the CAPM-generated expected return, placing the asset below the Security Market Line, which indicates the asset is OVER-valued. This matches what is written in the Security Market Line section.

If I am wrong, perhaps someone can add language to clarify this matter, because it seems (at least to me) the Security Market Line and Asset Pricing sections contradict each other.

- Mike — Preceding unsigned comment added by (talk) 16:47, 25 February 2012 (UTC)

Black CAPM[edit]

The article currently seems to be discussing Sharpe-Lintner CAPM, but it appears common in the literature to refer to an extention of it by Fischer Black as CAPM (at least from some papers I saw by Fama, et al). An extension to the article, or at least clarification, seems warranted. --C S (talk) 03:36, 26 September 2013 (UTC)


I am willing to add some references to the CAPM variants developed in insurance. For instance, there are demand-supply (econ) approaches described by, e.g., Borch (1986) [Risk theory and serendipity. Insurance: Mathematics and Economics 5, 103 - 112], and also computational ones (math), e.g., Furman and Zitikis (2017). [Beyond the Pearson correlation: Heavy-tailed risks, weighted Gini correlations, and a Gini-type weighted insurance pricing model. ASTIN Bulletin: the Journal of the International Actuarial Association 47(3), 919 - 942.] Would there be any objections to inserting a few references? — Preceding unsigned comment added by Efurman (talkcontribs) 12:45, 7 October 2017 (UTC)

Bear in mind that this is an encyclopedia. So, if you want to add citations, you need to show that the citations are relevant and significant. Significance comes from others, secondary sources, who quote your paper. For example, if there are numerous notable reviews (in peer reviewed journals) of CAPM variants that says that your approach is an important variant, then, perhaps, it can be included. Merely noting the existence of the paper, even if in peer reviewed journals, is not enough. Best. --regentspark (comment) 13:03, 7 October 2017 (UTC)

Many thanks for your quick reply. I agree with your note of `recognition by others'. Please note however that such recognition can come from distinct sources, e.g., citations - if the paper has been long enough around/also depending on how numerous the relevant scientific community is, or, alternatively, interest of practitioners (when you cite barclay wealth in your present article, I suppose this is what you pursue). The paper on insurance variant of the CAPM I am suggesting, has drawn interest of the Casualty Actuarial Society - one of the main bodies governing the profession. A version of the paper was published in their CAS E-Forum in addition to the pee-reviewed ASTIN Bulletin (btw, considered a top journal in insurance). So, isn't this sort of recognition sufficient? — Preceding unsigned comment added by Efurman (talkcontribs) 14:15, 7 October 2017 (UTC)

I would say no. The mere fact that a paper has been published is insufficient for inclusion. If that were sufficient, there would likely be many hundreds of candidate peer review papers that could be included here and that's obviously not going to work. It would be preferable to see a survey paper, in a good peer reviewed journal, that states that this paper makes a significant contribution to the field. --regentspark (comment) 14:59, 7 October 2017 (UTC)

Then why are you citing a research by Barclays Wealth, which is - I suspect - not even peer reviewed. If you remove Barclays Wealth's citation, then your `heavy-tailedness' part of the Problems section - however important and indeed a shortcoming of the CAPM - misses references. As to your note about a review paper, here you go: in which Section 6 reviews an insurance variant of the CAPM - nothing about heavy-tailedness, I admit. This review paper is published in North American Actuarial Journal - a top-tier peer-reviewed journal in Actuarial Science. The paper itself was sponsored by the Society of Actuaries - one of the main bodies that governs the profession. The paper is well--cited (for a small actuarial community). If you feel that this discussion has to involve more participants, please let's do it. — Preceding unsigned comment added by Efurman (talkcontribs) 17:45, 8 October 2017 (UTC)

Well written, but violates Wikipedia's policy on verifiability[edit]

Greetings Wikipedians! I commend all the contributors for their efforts. But sadly, this article lacks inline citations to reliable, verifiable sources. As such, it contains multiple violations of Wikipedia's policy on verifiability, which states: "Even if you are sure something is true, it must be verifiable before you can add it....The burden to demonstrate verifiability lies with the editor who adds or restores material, and it is satisfied by providing an inline citation to a reliable source that directly supports the contribution." The policy is set forth here.

In my view, this means we'd rather have a sparse article in which every important statement (including all these equations) is supported by an inline citation to a reliable source. There are no citations - none at all - in the following sections:

  • Formula;
  • Modified beta;
  • Security market line;
  • Asset-specific required return;
  • Risk and diversification; and
  • Efficient frontier.

I hope someone will step forward to remedy this problem, which crops up in a distressing number of articles about financial mathematics. Unsourced material is subject to being removed. My qualifications for this subject are set forth in my user profile. Cordially, BuzzWeiser196 (talk) 14:10, 7 April 2021 (UTC)