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==Risk averse definition==
Isn't the definition of risk adverse that an investor with two options, A showing a risk-corrected higher expected return and B a lower risk-corrected expected return but a lower risk choose B (that is, trading off expected return for less risk)?? I would not consider choosing the less risky assets with the SAME expected return as risk adversion as instead is defined in the article.
[[User:A.lobianco|A.lobianco]] 09:16, 21 Oct 2013 (UTC)
==Graphics==
I added a picture. I have the xfig source, so it is possible to modify it if this is too clunky. [[User:CSTAR|CSTAR]] 04:15, 1 Jul 2004 (UTC)
Oops I noticed in the text that you're plotting risk on the Y-axis, whereas I plotted it on the X-axis. Also I'm callin the measn \mu (which is common ampng probabilists) Well I can fix this, if you think it's worth it. [[User:CSTAR|CSTAR]] 05:07, 1 Jul 2004 (UTC)
I think it would tie in closer with the article if the axes were labelled as per the text -- ie. return and risk -- ''as well as'' per the notation as above.
I'll change the graphic, although it seems that yuu changed the text to have risk on X axis.
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[[User:CSTAR|CSTAR]] 12:42, 2 Jul 2004 (UTC)
Thanks for the graphic. The arguement is (ultimately) built around standard deviation as opposed to variance, could the graph reflect that?
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==Shorting ==
Maybe you should something about ''shorting''? Or at least point to some place in Wikipedia where this is mentioned. [[User:CSTAR|CSTAR]] 18:28, 2 Jul 2004 (UTC)
:Great graphic. Will add discussion on shorting
MPT has nothing to do with shorting. It has everything to do with the balance of risk and return of assets classes and their relationship over time. Shorting stocks does not fall into that framework.
==Black-Scholes==
I corrected two incorrect references to the Black-Scholes model. That model does not imply or assume that stock prices follow a martingale or that the best forecast of tomorrows prices is the price today. <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:MathHisSci|MathHisSci]] ([[User talk:MathHisSci|talk]] • [[Special:Contributions/MathHisSci|contribs]]) 22:01, 29 October 2009 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
:It is my recollection that returns could be achieved above the Efficient Frontier, but constrained by the Capital Market Line, by leveraging, i. e. shorting the risk free asset. Also, couldn't being short an asset be seen as a synthetic asset that has an expected return with a (presumably high!) standard deviation. Since "shorts" would be expected to have low covariance with "long" assets "shorts" might be great assets to create efficient portfolios.[[User:X17bc8|X17bc8]] ([[User talk:X17bc8|talk]]) 17:49, 19 May 2009 (UTC)
== Risk ==
In order for the capital market line to be straight, risk should be standard deviation not (variance). I think. [[User:CSTAR|CSTAR]] 22:47, 14 Jul 2004 (UTC)
:That is true - I see it has been changed. [[User:Zain Ebrahim111|Zain Ebrahim]] ([[User talk:Zain Ebrahim111|talk]]) 22:21, 17 May 2008 (UTC)
== distribution about the mean ==
I see no discussion of what happens when there is an asymmetric distribution about the mean, which is always the case in financial models where the asset value is bounded on the low side by ZERO.
*Exactly. This all seems to reply on the Gauss-Markov assumption (<b>returns are normally distributed => wake up, they aren't</b>, just use HRH on Bloomberg for a few indices or stocks). As I understand it, the CAPM can outside of the Gauss Markov assumptions (significantly beyond the scope of the definition given on this page). I do not know of any 'Omega metric' pages on wikipedia (which the derivation comes off), and feel I am probably not the best person to write one, though I would be happy to have a stab and link it in somewhere?
*I shall write something on Omega metrics, and stick a reference in the page (Omega being a generalised treatment for asymmetric returns, though is it non-parametric - CAPM can still be derived from it).
: Financial models don't a normal distribution of prices. They assume a log-normal distribution of prices, aka a normal distriubtion of returns. --[[Special:Contributions/206.248.177.127|206.248.177.127]] ([[User talk:206.248.177.127|talk]]) 19:35, 2 January 2011 (UTC)
== Expected beta ==
''Note that the theory uses an historical parameter, volatility, as a proxy for risk while return is an expectation on the future.'' Is this an accurate description of the theory or of practice? I would think that the theory says that expected return and expected volatility should be used. I don't the the theory prescribes using historical beta. In practice people use historical return and historical beta, then adjust for their expectations. Since it is very hard to develop numerical estimates of beta, it is usually expected return and historical beta.
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: Ideally, one should use some kind of time series analysis to estimate these parameters. If one makes an assumption about the nature of the underlying time series process, then there is some rational justification for making estimates of the various parameters characterizing the process. I don't have a clue in fact how the nuts-and-bolts financial analysts actually compute these numbers and whether they would take exception to the characterization given in the article, but I think the person that put that material in ([[User:fintor|fintor]]) is knowledgeable about practices in this area. [[User:CSTAR|CSTAR]] 22:40, 19 Nov 2004 (UTC)
:: Oh, I'd
::: I totally withdraw my comment. The text is correct. Nearly all forms I can find of the formula use E() to denote what is expected, and do not show E(beta). Hence, the description is correct. Sorry for the trouble. [[User:Chrisvls|Chris vLS]] 19:02, 30 Nov 2004 (UTC)
== Risk free asset ==
If you're going to change notation for rf, change it everywhere. [[User:CSTAR|CSTAR]] 18:40, 30 Nov 2004 (UTC)
== Distribution about the mean and other objections ==
Right now, there are some comments that come close to attacking the theory in the same paragraphs that explain it. While that's fine for the sophisitcate, for the novice I think it's confusing -- kind of like saying "Note that general relatively is not accounted for in F=ma" in the middle of the description of Newton's force formula. I think it would be better to create a new section, called "Shortcomings and challenges" or something like that and discuss where modern portfolio theory has gone since the CAPM.
----
Do you mean something like the following assertion?
: ''(Here again, the theory accepts in its assumptions that a parameter based on past data can be combined with a future expectation.)''
I don't know who put this in, but in the model as formulated this assertion is meaningless and should be removed. The MPT model described in this article is a probabilistic model, but the random variables have no time dependency whatsoever, and although such models are used in finance, this isn't one of them.[[User:CSTAR|CSTAR]] 21:25, 7 Dec 2004 (UTC)
:Yes and I agree. Here are the statements I see that could be moved:
::''"The investor is indifferent to other characteristics of the distribution of returns, such as its skew."''
::''"Note that the theory uses an historical parameter, volatility, as a proxy for risk while return is an expectation on the future."''
::''"(Here again, the theory accepts in its assumptions that a parameter based on past data can be combined with a future expectation.)"''
--[[User:Chrisvls|Chris vLS]] 21:17, 10 Dec 2004 (UTC)
----
:''"The investor is indifferent to other characteristics of the distribution of returns, such as its skew."''
This is a meaningful assertion about this model, so I don't see any reason to change it.
On the other hand, the article should say something intelligible about parameter estimation somewhere. I suppose this was the intent of whoever adding the assertion of the parameter is based on past performance. It is a truism that historical data is the only data available to estimate parameters of time series. But this requires a time series model.
I would wait to whoever wrote that part of the article to respond before deleting anything. [[User:CSTAR|CSTAR]] 22:41, 10 Dec 2004 (UTC)
----
Oh, I agree with the statement about skew. I am not looking to delete, just clarify.
All I'm saying is that we should consider moving and expanding the discussion of shortcomings into its own section, as is done in the article on the [[CAPM]]. --[[User:Chrisvls|Chris vLS]] 06:09, 12 Dec 2004 (UTC)
:Agreed: we need a seperate "shortcomings" section. I also think that the article could use an explicit "assumptions" section - I would think that the two should to some extent dovetail... [[user:Fintor | Fintor]] | [[User_talk:Fintor | talk]] |January 9 18:49 UTC
----
For those talking about distributions about the mean, please note that the CAPM works _whatever_ the distribution of returns it is derived from is - you could even have a classic lottery distribution of returns and it'll work, but you cannot use the variance-covariance method to work it out (so while it can be derived from raw data, standard deviation cannot be used as a probability).
Perhaps the derivation section (as it stands now, discussing parametric specification of CAPM) can be moved to a subsection, or 'appendix' under the discussion of what CAPM actually means, I am happy to add a section on how to derive CAPM for _any_ distribution of returns (whatever the skew, kurotsis, or any other partial moments of conditions of a distribution), its really pretty simple stuff if you know how: uses 'Omega Metrics', mentioned above.
To repeat, the result of the SML/CAPM do not depend features of the underlying distribution if the SML/CAPM have been derived correctly (unbiased) w.r.t. that distribution.
[[user:zhte415 | zhte415]] | [[User_talk:zhte415 | talk]] |January 16 01:37 GMT
== Practical guide ==
This article was a nice introduction for me, but for those who plan on ''using'' this theory in structuring a portfolio, it's not very practical. I was reading the same page on investopedia (http://www.investopedia.com/terms/m/modernportfoliotheory.asp) and they said there were four distinct steps:
#Security Valuation
#Asset Allocation
#Portfolio Optimization
#Performance Measurement
Is there any chance of getting a section describing in practical terms how this theory is used? --[[User:Jens Schriver|Jens Schriver]] 19:10, 11 January 2006 (UTC)
== Merge with Portfolio Theory, more APT needed here ==
The Article [[Portfolio theory]] is a stub on the same topic.
I suggest that it should be merged into this article, and will do so fairly quickly if there are no objections.
Also I think that an article named modern portfolio theory needs to have something more than just a reference to arbitrage pricing theory. [[User:Smallbones|Smallbones]] 09:49, 27 March 2006 (UTC)
:I don't see why the word "modern" is needed at all. There's nothing more modern about portfolio theory than other theories in finance and economics, or in science generally, and the term isn't used by specialists in the field. All theories change over time, but that's to be expected. So, I think the correct course is to merge this into [[Portfolio theory]][[User:John Quiggin|JQ]] 02:36, 28 May 2006 (UTC)
==Errors in the article ?==
Please check. In Diversification Section:
"An investor can reduce portfolio risk simply by holding instruments which are not perfectly correlated." Shouldn't it be "negatively correlated"
Also, "From the formulae above: if any two assets in the portfolio have a correlation of less than 1..." shouldn't it be "a correlation of less than 0"?
Agreed. Perfectly correlated assets implies that if one goes down, so does the next in the series. Reduce risk by holding varied, negatively correlated assets. [[User:66.9.159.226|66.9.159.226]] 22:30, 25 January 2007 (UTC)Steve
: For a two asset portfolio, the statement in the article is correct ''generally'', if one allows selling short (e.g. allowing the weights ''w'' for portfolio composition to be negative).
:Moreover, if κ is the correlation between asset 1 and 2 and
::<math> \sigma_1 > \kappa \sigma_2 </math>
::<math> \sigma_2 > \kappa \sigma_1 </math>
:then the minimum variance portfolio is achieved by positive weights for both assets; (i.e., no shorting).
:--[[User:CSTAR|CSTAR]] 02:53, 26 January 2007 (UTC)
Disagre!!!
"From the formulae above: if any two assets in the portfolio have a correlation of less than 1..." shouldn't it be "a correlation of less than 0"?
NO!!
The article was right, now it's wrong. "if all assets of a portfolio have a correlation of 0, the portfolio variance and hence volatility will be the weighted average of the individual instruments' volatilities..."
Volatility will be the weighted average of volatilities if correlation is 1 and not 0.
(σp = ωa.σa + ωb.σb ↔ σp^2 = ωa^2.σa^2 + ωb^2.σb^2 + 2.ωa.ωb.σa.σb ↔ ρab=1).[[User:Davivalle|Davivalle]] 21:34, 2 September 2007 (UTC)
:Please correct me if I am wrong!
:The interesting thing is that for perfect correlations <math>\rho=1</math> the portfolio ''volatility'' (i.e. standard deviation) is the weighted sum of the assets' volatilities - as you have said Davidalle. However, for uncorrelated assets <math>\rho=0</math> the portfolio ''variance'' is the weighted sum of the assets' variances. [[User:Tomeasy|Tomeasy]] ([[User talk:Tomeasy|talk]]) 00:47, 2 March 2008 (UTC)
in Mathematically section
"the where w is a vector of portfolio weights. Each wi>=0 "
IMHO the condition wi>=0 is true with 2 assets but wrong in general. With more than 2 assets efficient portfolios can have wi<0. With more than 2 assets the condition wi>=0 is in fact an additional constraint that prohibites short selling and as any additional constraint will reduce the efficient frontier. check out Investments 7th edtion Bodie Kane Markus pg 226 <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/137.111.130.201|137.111.130.201]] ([[User talk:137.111.130.201|talk]]) 00:45, 30 July 2008 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
== Criticism by Taleb ==
I was reading a recent FT [http://www.ft.com/cms/s/0/4eb6ae86-8166-11dc-a351-0000779fd2ac.html?nclick_check=1 article] by Taleb that was pretty critical of some of this, and I was wondering why there isn't any criticism in the article. Taleb's book, "The Black Swan" is fairly popular, should we bring in some of his (and others'?) criticisms? Some articles like [[CAPM]] have "shortcomings" sections, but I'm not sure that is enough. What would be a good way to bring in some criticism? [[User:Smmurphy|Smmurphy]]<sup>([[User talk:Smmurphy|Talk]])</sup> 19:29, 27 October 2007 (UTC)
Also [[Benoît Mandelbrot]]'s [http://www.yaleeconomicreview.com/issues/fall2005/fractalfinance The (Mis)Behavior of Markets] <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Fintor|Fintor]] ([[User talk:Fintor|talk]] • [[Special:Contributions/Fintor|contribs]]) 08:03, 30 October 2007 (UTC)</small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
I agree that a criticism section should be added--one of Taleb's criticisms is that modern portfolio theory is a fraud because it pretends that the data follows a Gaussian distribution when it doesn't, and thus events like the October 1987 crash occur far more frequently than the model predicts. The link to Mandelbrot's book is no good, but looking at the Amazon reviews of that book, it looks like an excellent source of material for a criticism section as well. [[User:Lippard|Lippard]] ([[User talk:Lippard|talk]]) 20:23, 22 May 2008 (UTC)
In a bloomberg interview on 7th Nov 2008, Taleb called Modern Portfolio Theory "hogwash" and said that academics like Scholes and Merton should be discredited. [http://media.bloomberg.com/bb/avfile/Economics/On_Economy/vCheV1KoaJAw.mp3] <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Vynbos|Vynbos]] ([[User talk:Vynbos|talk]] • [[Special:Contributions/Vynbos|contribs]]) 08:16, 13 November 2008 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
== Efficient Frontier Concavity ==
The article claims that the graph of the efficient frontier is convex. This seems wrong. Convex functions are U-shaped. [[Convex function]] . Shouldn't it say "concave"? A [[Concave function]] is frown-shaped, upside-down U. I don't want to change it myself because maybe the wording comes from a reliable source. [[User:Eyal0|Eyal0]] ([[User talk:Eyal0|talk]]) 06:22, 27 May 2009 (UTC)
:It could also be "upper convex" or "convex upward". I think the use of the word "convex" in this article corresponds to ordinary English usage (and probably standard usage in the field) rather than mathematical jargon. --[[User:Doradus|Doradus]] ([[User talk:P3d0|talk]]) 17:00, 16 October 2009 (UTC)
::Hello, not a mathematician here, so what I say may be incorrect. The article claims that the efficient frontier is a hyperbola and cites a source by saying, "See bottom of slide 6" (reference number 4). I checked the source; it isn't a deck slide - it is an entire course at MIT. I checked the 6th lesson, but I could not find anything about the hyperbolic shape of the frontier. However, I found another reference (see below, bottom of slide 5) where the professor calculates the equation of the efficient frontier. It looks to me like a hyperbola indeed - although the professor calls it parabola, so whether he is making a typo or I misunderstand the basics.
::Could you please verify the source? Thank you!
::Here is the slide I mentioned https://stanford.edu/~ashlearn/RLForFinanceBook/EfficientFrontier.pdf [[Special:Contributions/79.20.61.201|79.20.61.201]] ([[User talk:79.20.61.201|talk]]) 15:22, 4 July 2023 (UTC)
== Nice job ==
I just want to congratulate the authors of this article on doing a good job of explaining some very complex concepts in a concise manner. It may be confusing to the ordinary Encyclopedia reader, but then a topic this arcane can hardly avoid that. --[[User:Doradus|Doradus]] ([[User talk:P3d0|talk]]) 16:58, 16 October 2009 (UTC)
== Criticism section ==
I've added a "criticism" section. Currently it has two parts, assumptions and shortcomings, both largely cribbed from the CAPM article (with which there is much overlap, an area ripe for cleanup.) To this I've added all of the problems with MPT I could think of or find here in the talk page.
Much more work on the criticism section is needed. Your improvements would be very welcome! --[[User:Jonathanstray|Jonathan Stray]] ([[User talk:Jonathanstray|talk]]) 08:01, 8 November 2009 (UTC)
: Further, there is growing evidence that investors are not rational and markets are not efficient.[1][2] - I don't think this is true in general. As most people seem to know, it is actually very hard to make "above-average" money in financial markets, to add alpha or whatever you want to call it. If it was so inefficient as some people seem to espouse, it wouldn't be that hard. I don't understand why people bother having this discussion. uhh-hmm i will end there. --[[Special:Contributions/129.240.146.32|129.240.146.32]] ([[User talk:129.240.146.32|talk]]) 16:54, 11 November 2010 (UTC)
"An empirical proof of this is the price-hike that stocks typically experience once they are included in major indices like the S&P 500" - I completely disagree here. Is there a proof this is due to MPT portfolios? As far as I know the main effects are a) Index funds & index arbitrageurs having to rebalance their holdings, b) speculation that a) will drive the price up. <small><span class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:LaurentP|LaurentP]] ([[User talk:LaurentP|talk]] • [[Special:Contributions/LaurentP|contribs]]) 16:55, 9 December 2010 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
I think there needs to be a citation for each of the rebuttals to the assumptions in this section. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/131.181.251.131|131.181.251.131]] ([[User talk:131.181.251.131|talk]]) 04:12, 10 September 2012 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
"Connection with rational choice theory": The claim about monotonicity is nonsense. If P(X > Y) = 1, then, necessarily, E(X) > E(Y).
If Var(X) > Var(Y), then X and Y are incomparable under the partial ordering on (mean, variance); this does not mean that an agent would choose Y over X - the "may recommend" comment needs justification. <!-- Template:Unsigned IP --><small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/2604:2000:DE52:4B00:B09F:FAE9:3499:F403|2604:2000:DE52:4B00:B09F:FAE9:3499:F403]] ([[User talk:2604:2000:DE52:4B00:B09F:FAE9:3499:F403#top|talk]]) 15:41, 26 August 2018 (UTC)</small> <!--Autosigned by SineBot-->
I also suggest adding the goals-based investing critique of MPT, something along the lines of:
[[Goal-based investing]] (GBI) objects to MPT's assumption that investors are unbounded in their ability to employ leverage and [[Short (finance)|short-selling]] in their investment portfolios. In practice, investors are limited in their ability to sell short or buy securities using leverage (especially in tax-qualified accounts like a [[401(k)]] or [[Individual retirement account|IRA]]). GBI further critiques MPT on the grounds that it takes no account an investor's financial goals, rather MPT selects portfolios using a risk-tolerance metric which can be dubious to measure.<ref>{{Cite journal|last=Pan|first=Carrie H.|last2=Statman|first2=Meir|date=2012-08-01|title=Questionnaires of Risk Tolerance, Regret, Overconfidence, and Other Investor Propensities|url=https://papers.ssrn.com/abstract=2144481|language=en|___location=Rochester, NY}}</ref> In the context of achieving a future financial goal (i.e. a minimum wealth level within a specified period of time) with real-world constraints on short-selling and leverage, MPT portfolios are [[Stochastic dominance|first-order stochastically dominated]] by goals-based portfolios.<ref>{{Cite journal|last=Parker|first=Franklin J.|date=2020-02-06|title=A Goals-Based Theory of Utility|url=https://doi.org/10.1080/15427560.2020.1716359|journal=Journal of Behavioral Finance|volume=0|issue=0|pages=1–16|doi=10.1080/15427560.2020.1716359|issn=1542-7560}}</ref>
The citation to support the last sentence is a reference to my own peer-reviewed work, so the sentence could be left out if COI concerns are too great. [[User:Fjparker|Fjparker]] ([[User talk:Fjparker|talk]]) 18:38, 23 July 2020 (UTC)
{{reflist-talk}}
== Last sentence of intro means nothing to newcomer ==
"A study conducted by Myron Scholes, Michael Jenson, and Fischer Black in 1972 suggests that the relationship between return and beta might be flat or even negatively correlated."
What is beta? I don't think this should be in the introduction unless beta is defined there. Does flat mean no correlation? [[User:Surement|Surement]] ([[User talk:Surement|talk]]) 01:41, 1 March 2013 (UTC)
== CAL? ==
There's something called CAL in the graphic. Not defined anywhere in the article. <small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/75.76.168.93|75.76.168.93]] ([[User talk:75.76.168.93|talk]]) 19:06, 7 October 2015 (UTC)</small><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
:Hmm. That would be the Capital Allocation Line. You're right and it does need explanation somewhere. Will do that if no else gets there first. --[[User:RegentsPark|regentspark]] <small>([[User talk:RegentsPark|comment]])</small> 19:49, 7 October 2015 (UTC)
::Actually, it is explained in a different section on the page. [[Modern_portfolio_theory#Risk-free_asset_and_the_capital_allocation_line]]. I'll connect the two. A lot easier.--[[User:RegentsPark|regentspark]] <small>([[User talk:RegentsPark|comment]])</small> 19:51, 7 October 2015 (UTC)
==Assessment comment==
{{Substituted comment|length=222|lastedit=20090527061907|comment=Would it be appropriate to update this article for the current credit crisis and the implications it may imply regarding the validity of the model?[[User:X17bc8|X17bc8]] ([[User talk:X17bc8|talk]]) 18:37, 19 May 2009 (UTC)}}
Substituted at 00:15, 30 April 2016 (UTC)
== Dr. Platen's comment on this article ==
Dr. Platen has reviewed [https://en.wikipedia.org/w/index.php?title=Modern_portfolio_theory&oldid=725891889 this Wikipedia page], and provided us with the following comments to improve its quality:
{{quote|text=The article is reasonably well written. Currently I would not see any need to improve the content substantially. However, in two year's time I would like to come back on this question and potentially add some material that would extend the article.}}
We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.
Dr. Platen has published scholarly research which seems to be relevant to this Wikipedia article:
*'''Reference ''': Jan Baldeaux & Man Chung Fung & Katja Ignatieva & Eckhard Platen, 2014. "A Hybrid Model for Pricing and Hedging of Long Dated Bonds," Research Paper Series 343, Quantitative Finance Research Centre, University of Technology, Sydney.
[[User:ExpertIdeasBot|ExpertIdeasBot]] ([[User talk:ExpertIdeasBot|talk]]) 14:52, 24 June 2016 (UTC)
<br>
==Violates Wikipedia verifiability policy==
Greetings Wikipedians! I commend all the contributors for their efforts. But sadly, this article lacks inline citations to reliable, verifiable sources. There are no citations - none at all! - in the following sections:
* 1.1: Risk and expected return
* 1.2: Diversification
* 2.1: Systematic risk and specific risk
This violates [[Wikipedia:Verifiability|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." Readers need a path to follow to verify that what they are reading is accurate and accepted by authorities in the field - particularly when the article is this technical and contains so many equations in math-major notation, proofs, and so on.
I hope someone will step forward to remedy this problem. Unsourced material is subject to being removed. My modest qualifications for this subject, such as they are, are set forth in my user profile. Cordially,[[User:BuzzWeiser196|BuzzWeiser196]] ([[User talk:BuzzWeiser196|talk]]) 12:54, 11 April 2021 (UTC)<br>
== Standard deviation vs. variance, hyperbole vs. parabole ==
I am new to the topic but it seems to me that to have the Capital Market Line as a straight line, the x-axis should be standard deviation. This is the case in the graphic, but the text says: "The image shows expected return on the vertical axis, and the horizontal axis should be labeled variance instead of standard deviation (volatility)." while the graphic's legend says "Note that the horizontal axis should be labeled variance, not volatility."
I have the impression that these two sentences should be removed, but as I am no expert in the field, I prefer not to do it (I could be mistaken).
A related confusion is whether the efficient frontier is hyperbolic or parabolic. I have the impression that it is hyperbolic if plotted with the standard deviation, and parabolic if plotted with the variance. Here again, I am not sure, but all should be consistent in this article and in others, in particular [[Markowitz model]] and [[Efficient frontier]].
[[Special:Contributions/2001:67C:10EC:3C91:8000:0:0:106D|2001:67C:10EC:3C91:8000:0:0:106D]] ([[User talk:2001:67C:10EC:3C91:8000:0:0:106D|talk]]) 16:14, 13 November 2021 (UTC)
:I replaced parabola with hyperbola (you're right, obviously hyperbolic not parabolic) and removed the horizontal axis sentence (probably a remanent of other edits). --[[User:RegentsPark|RegentsPark]] <small>([[User talk:RegentsPark|comment]])</small> 16:27, 13 November 2021 (UTC)
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