Diversification (finance): Difference between revisions
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{{Short description|Risk reduction technique}} |
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{{Finance sidebar}} |
{{Finance sidebar}} |
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In finance, '''diversification''' |
In [[finance]], '''diversification''' is the process of allocating capital in a way that reduces the exposure to any one particular asset or risk. A common path towards diversification is to reduce [[financial risk|risk]] or [[volatility (finance)|volatility]] by [[investment|investing]] in a variety of [[asset]]s. If asset prices do not change in perfect synchrony, a diversified [[Portfolio (finance)|portfolio]] will have less variance than the [[weighted mean|weighted average]] variance of its constituent assets, and often less volatility than the least volatile of its constituents.<ref>{{cite book |
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| last1 = O'Sullivan |
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| first1 = Arthur |
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| author-link = Arthur O'Sullivan (economist) |
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| first2 = Steven M. | last2 = Sheffrin | author-link2=Steven M. Sheffrin |
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| title = Economics: Principles in |
| title = Economics: Principles in Action |
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| publisher = Pearson Prentice Hall |
| publisher = Pearson Prentice Hall |
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| year = 2003 |
| year = 2003 |
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| location = Upper Saddle River, New Jersey |
| location = Upper Saddle River, New Jersey |
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| page = 273 |
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| isbn = 0-13-063085-3}}</ref> |
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| url = http://pearsonschool.mobi/index.cfm?locator=PSZ16o&filter_161=&filter_423=&filter_422=&filter_424=&filter_281=&filter_425=&programFilterTypeList=161%2C423%2C422%2C424%2C281%2C425&PMDbSiteid=2781&PMDbSolutionid=6724&PMDbSubSolutionid=&PMDbCategoryid=815&PMDbSubcategoryid=24843&&PMDbProgramID=23061 |
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| doi = |
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| id = |
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| isbn = 0-13-063085-3}}</ref>. Therefore, any [[risk-averse]] investor will diversify to at least some extent, with more risk-averse investors diversifying more completely than less risk-averse investors. |
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Diversification is one of two general techniques for reducing investment risk. The other is [[hedge (finance)|hedging]] |
Diversification is one of two general techniques for reducing investment risk. The other is [[hedge (finance)|hedging]]. |
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==Examples== |
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It is important to remember that diversification only works because investment in each individual asset is reduced. If someone starts with $10,000 in one [[stock]] and then puts $10,000 in another stock, they would have more risk, not less. Diversification would require the sale of $5,000 of the first stock to be put into the second. There would then be less risk. Hedging, by contrast, reduces risk without selling any of the original position<ref>{{cite book |
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| last = Fama |
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| first = Eugene F. |
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| authorlink = Eugene Fama |
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| coauthors = Merton H. Miller |
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| title = The Theory of Finance |
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| publisher = Holt Rinehart & Winston |
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| date = June 1972 |
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| isbn = 978-0155042667}}</ref>. |
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The simplest example of diversification is provided by the proverb "'''Don't put all your eggs in one basket'''". Dropping the basket will break all the eggs. Placing each egg in a different basket is more diversified. There is more risk of losing one egg, but less risk of losing all of them. On the other hand, having a lot of baskets may increase costs. |
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The risk reduction from diversification does not mean anyone else has to take more risk. If person A owns $10,000 of one stock and person B owns $10,000 of another, both A and B will reduce their risk if they exchange $5,000 of the two stocks, so each now has a more diversified portfolio<ref>{{cite book |
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| last = Sharpe |
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| first = William |
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| authorlink = William Forsyth Sharpe |
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| coauthors = Gordon J. Alexander, Jeffrey W. Bailey |
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| title = Investments |
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| publisher = Prentice Hall |
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| date = October 30, 1998 |
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| isbn = 978-0130101303}}</ref>. |
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==Examples== |
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In finance, an example of an undiversified portfolio is to hold only one stock. This is risky; it is not unusual for a single stock to go down 50% in one year. It is less common for a portfolio of 20 stocks to go down that much, especially if they are selected at random. If the stocks are selected from a variety of industries, company sizes and asset types it is even less likely to experience a 50% drop since it will mitigate any trends in that industry, company class, or asset type. |
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The simplest example of diversification is provided by the proverb "don't put all your eggs in one basket". Dropping the basket will break all the eggs. Placing each egg in a different basket is more diversified. There is more risk of losing one egg, but less risk of losing all of them. |
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In finance, an example of an undiversified portfolio is to hold only one stock. This is risky; it is not unusual for a single stock to go down 50% in one year. It is much less common for a portfolio of 20 stocks to go down that much, even if they are selected at random. If the stocks are selected from a variety of industries, company sizes and types (such as some [[growth stock]]s and some [[value investing|value stocks]]) it is still less likely. |
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Since the mid-1970s, it has also been argued that geographic diversification would generate superior risk-adjusted returns for large [[institutional investor]]s by reducing overall portfolio risk while capturing some of the higher rates of return offered by the [[emerging markets]] of Asia and Latin America.<ref>{{in lang|fr}} {{Cite web |url=http://www.canadianeuropean.com/yahoo_site_admin/assets/docs/FONDS_DASIE-PACIFIQUE_REVUE_AF_APR_09.95131642.pdf |access-date=2009-04-02 |title=see M. Nicolas J. Firzli, "Asia-Pacific Funds as Diversification Tools for Institutional Investors", Revue Analyse Financière/The French Society of Financial Analysts (SFAF) |url-status=dead |archive-url=https://web.archive.org/web/20100508172054/http://canadianeuropean.com/yahoo_site_admin/assets/docs/FONDS_DASIE-PACIFIQUE_REVUE_AF_APR_09.95131642.pdf |archive-date=2010-05-08 }}</ref><ref>{{in lang|en}} {{Cite web |url=http://www.insead.edu/facultyresearch/centres/global_private_equity_initiative/publications/documents/INSEAD_AsiaPEReport_long.pdf |access-date=2011-06-15 |title=see Michael Prahl, "Asian Private Equity – Will it Deliver on its Promise?", INSEAD Global Private Equity Initiative (GPEI) }}</ref> |
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Further diversification can be obtained by investing in stocks from different countries, and in different asset classes such as [[bond (finance)|bonds]], [[real estate]] and commodities like [[heating oil]] or [[gold]]<ref>{{cite book |
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| last = Campbell |
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| first = John Y. Campbell |
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| authorlink = Merton Miller |
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| coauthors = Andrew W. Lo, A. Craig MacKinlay |
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| title = The Econometrics of Financial Markets |
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| publisher = Princeton University Press |
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| date = December 9, 1996 |
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| isbn = 978-0691043012}}</ref>. |
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==Return expectations while diversifying== |
==Return expectations while diversifying== |
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If the prior [[expected value|expectations]] of the returns on all assets in the portfolio are identical, the [[expected return]] on a diversified portfolio will be identical to that on an undiversified portfolio. |
If the prior [[expected value|expectations]] of the returns on all assets in the portfolio are identical, the [[expected return]] on a diversified portfolio will be identical to that on an undiversified portfolio. Some assets will do better than others; but since one does not know in advance which assets will perform better, this fact cannot be exploited in advance. The return on a diversified portfolio can never exceed that of the top-performing investment, and indeed will always be lower than the highest return (unless all returns are identical). Conversely, the diversified portfolio's return will always be higher than that of the worst-performing investment. So by diversifying, one loses the chance of having invested solely in the single asset that comes out best, but one also avoids having invested solely in the asset that comes out worst. That is the role of diversification: it narrows the range of possible outcomes. Diversification need not either help or hurt expected returns, unless the alternative non-diversified portfolio has a higher expected return.<ref>Goetzmann, William N. [http://viking.som.yale.edu/will/finman540/classnotes/class2.html An Introduction to Investment Theory] {{Webarchive|url=https://web.archive.org/web/20170331163028/http://viking.som.yale.edu/will/finman540/classnotes/class2.html |date=2017-03-31 }}. II. Portfolios of Assets. Retrieved on November 20, 2008.</ref> |
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==Amount of diversification== |
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There is no magic number of stocks that is diversified versus not. Sometimes quoted is 30, although it can be as low as 10, provided they are carefully chosen. This is based on a result from John Evans and Stephen Archer.<ref>[http://financialfuse.com/investment-guide-beginners-introduction/ Investment Guide Beginners Introduction]</ref> Similarly, a 1985 book reported that most value from diversification comes from the first 15 or 20 different stocks in a portfolio.<ref>{{cite book |author=James Lorie |author2=Peter Dodd |author3=Mary Kimpton |title=The Stock Market: Theories and Evidence |url=https://archive.org/details/stockmarkettheor0000lori |url-access=registration |edition=2nd |year=1985 |page=[https://archive.org/details/stockmarkettheor0000lori/page/85 85]|publisher=Dow Jones-Irwin |isbn=9780870946189 }}</ref> More stocks give lower price volatility. |
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Given the advantages of diversification, many experts recommend maximum diversification, also known as “buying the [[market portfolio]].” Unfortunately, identifying that portfolio is not straightforward. |
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Given the advantages of diversification, many experts {{who|date=March 2014}} recommend maximum diversification, also known as "buying the [[market portfolio]]". Identifying that portfolio is not straightforward. |
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The earliest definition comes from the [[capital asset pricing model]] which argues the maximum diversification comes from buying a ''pro rata'' share of all available [[asset]]s. This is the idea underlying [[index fund]]s. |
The earliest definition comes from the [[capital asset pricing model]] which argues the maximum diversification comes from buying a ''pro rata'' share of all available [[asset]]s. This is the idea underlying [[index fund]]s. |
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Diversification has no maximum so long as more assets are available.<ref>{{cite journal | url=https://www.jstor.org/stable/2330969 | jstor=2330969 | doi=10.2307/2330969 | title=How Many Stocks Make a Diversified Portfolio? | last1=Statman | first1=Meir | journal=The Journal of Financial and Quantitative Analysis | date=1987 | volume=22 | issue=3 | pages=353–363 }}</ref> Every equally weighted, uncorrelated asset added to a portfolio can add to that portfolio's measured diversification. When assets are not uniformly uncorrelated, a weighting approach that puts assets in proportion to their relative correlation can maximize the available diversification. |
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One objection to that is it means avoiding investments like [[futures contract|futures]] that exist in zero net supply. Another is that the portfolio is determined by what securities come to market, rather than underlying economic value. Finally, buying ''pro rata'' shares means that the portfolio overweights any assets that are overvalued, and underweights any assets that are undervalued. This line of argument leads to portfolios that are weighted according to some definition of “economic footprint,” such as total underlying assets or annual cash flow.<ref>Wagner, Hans [http://www.investopedia.com/articles/mutualfund/07/fw_index.asp Fundamentally Weighted Index Investing]. Retrieved on June 20, 2010.</ref> |
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"Risk parity" is an alternative idea. This weights assets in inverse proportion to risk, so the portfolio has equal risk in all asset classes. This is justified both on theoretical grounds, and with the pragmatic argument that future risk is much easier to forecast than either future market price or future economic footprint.<ref>Asness, Cliff; [[David Kabiller]] and Michael Mendelson [http://www.iimagazine.com/Popups/PrintArticle.aspx?ArticleID=2486929 Using Derivatives and Leverage To Improve Portfolio Performance, Institutional Investor, May 13, 2010]. Retrieved on June 21, 2010.</ref> "Correlation parity" is an extension of risk parity, and is the solution whereby each asset in a portfolio has an equal correlation with the portfolio, and is therefore the "most diversified portfolio". Risk parity is the special case of correlation parity when all pair-wise correlations are equal.<ref>Schoen, Robert [https://www.putnam.com/literature/pdf/SU830.pdf Parity Strategies and Maximum Diversification, Putnam Investments, June, 2013] {{Webarchive|url=https://web.archive.org/web/20150402181913/https://www.putnam.com/literature/pdf/SU830.pdf |date=2015-04-02 }}.</ref> |
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==Effect of diversification on variance== |
==Effect of diversification on variance== |
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One simple measure of [[financial risk]] is [[variance]] |
One simple measure of [[financial risk]] is [[variance]] of the [[rate of return|return]] on the portfolio. Diversification can lower the variance of a portfolio's return below what it would be if the entire portfolio were invested in the asset with the lowest variance of return, even if the assets' returns are uncorrelated. For example, let asset X have stochastic return <math>x</math> and asset Y have stochastic return <math>y</math>, with respective return variances <math>\sigma^{2}_x</math> and <math>\sigma^{2}_y</math>. If the fraction <math>q</math> of a one-unit (e.g. one-million-dollar) portfolio is placed in asset X and the fraction <math>1-q</math> is placed in Y, the stochastic portfolio return is <math>qx+(1-q)y</math>. If <math>x</math> and <math>y</math> are uncorrelated, the variance of portfolio return is <math>\text{var}(qx+(1-q)y)=q^{2}\sigma^{2}_x+(1-q)^{2}\sigma^{2}_y</math>. The variance-minimizing value of <math>q</math> is <math>q=\sigma^{2}_y/[\sigma^{2}_x+\sigma^{2}_y]</math>, which is strictly between <math>0</math> and <math>1</math>. Using this value of <math>q</math> in the expression for the variance of portfolio return gives the latter as <math>\sigma^{2}_x\sigma^{2}_y/[\sigma^{2}_x+\sigma^{2}_y]</math>, which is less than what it would be at either of the undiversified values <math>q=1</math> and <math>q=0</math> (which respectively give portfolio return variance of <math>\sigma^{2}_x</math> and <math>\sigma^{2}_y</math>). Note that the favorable effect of diversification on portfolio variance would be enhanced if <math>x</math> and <math>y</math> were negatively correlated but diminished (though not eliminated) if they were positively correlated. |
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In general, the presence of more assets in a portfolio leads to greater diversification benefits, as can be seen by considering portfolio variance as a function of <math>n</math>, the number of assets. For example, if all assets' returns are mutually uncorrelated and have identical variances <math>\sigma^{2}_x</math>, portfolio variance is minimized by holding all assets in the equal proportions <math>1/n</math><ref>Samuelson, Paul, "General Proof that Diversification Pays |
In general, the presence of more assets in a portfolio leads to greater diversification benefits, as can be seen by considering portfolio variance as a function of <math>n</math>, the number of assets. For example, if all assets' returns are mutually uncorrelated and have identical variances <math>\sigma^{2}_x</math>, portfolio variance is minimized by holding all assets in the equal proportions <math>1/n</math>.<ref>Samuelson, Paul, [https://www.gwern.net/docs/statistics/decision/1967-samuelson.pdf "General Proof that Diversification Pays"], ''Journal of Financial and Quantitative Analysis'' 2, March 1967, 1-13.</ref> Then the portfolio return's variance equals <math>\text{var}[(1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n}]</math> = <math>n(1/n^{2})\sigma^{2}_{x}</math> = <math>\sigma^{2}_{x}/n</math>, which is monotonically decreasing in <math>n</math>. |
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The latter analysis can be adapted to show why ''adding'' uncorrelated |
The latter analysis can be adapted to show why ''adding'' uncorrelated volatile assets to a portfolio,<ref>Samuelson, Paul, [https://www.casact.org/pubs/forum/94sforum/94sf049.pdf "Risk and uncertainty: A fallacy of large numbers"], ''Scientia'' 98, 1963, 108-113.</ref><ref>Ross, Stephen, [https://www.gwern.net/docs/statistics/decision/1999-ross.pdf "Adding risks: Samuelson's fallacy of large numbers revisited"] ''[[Journal of Financial and Quantitative Analysis]]'' 34, September 1999, 323-339.</ref> thereby increasing the portfolio's size, is not diversification, which involves subdividing the portfolio among many smaller investments. In the case of adding investments, the portfolio's return is <math>x_1+x_2+ \dots +x_n</math> instead of <math>(1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n},</math> and the variance of the portfolio return if the assets are uncorrelated is <math>\text{var}[x_1+x_2+\dots +x_n] = \sigma^{2}_{x} + \sigma^{2}_{x}+ \dots + \sigma^{2}_{x} = n\sigma^{2}_{x},</math> which is ''increasing'' in ''n'' rather than decreasing. Thus, for example, when an insurance company adds more and more uncorrelated policies to its portfolio, this expansion does not itself represent diversification—the diversification occurs in the spreading of the insurance company's risks over a large number of part-owners of the company. |
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==Diversification with correlated returns via an equally weighted portfolio== |
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==Diversifiable and non-diversifable risk== |
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The expected return on a portfolio is a weighted average of the expected returns on each individual asset: |
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The [[Capital Asset Pricing Model]] introduced the concepts of diversifiable and non-diversifiable risk. Synonyms for diversifiable risk are idiosyncratic risk and security-specific risk. Synonyms for non-diversifiable risk are [[systematic risk]], [[beta (finance)|beta]] risk and [[market risk]]. |
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:<math> \mathbb{E}[R_P] = \sum^{n}_{i=1}x_i\mathbb{E}[R_i] </math> |
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If one buys all the stocks in the [[S&P 500]] one is obviously exposed only to movements in that [[index (economics)|index]]. If one buys a single stock in the S&P 500, one is exposed both to index movements and movements in the stock relative to the index. The first risk is called “non-diversifiable,” because it exists however many S&P 500 stocks are bought. The second risk is called “diversifiable,” because it can be reduced it by diversifying among stocks, and it can be eliminated completely by buying all the stocks in the index. |
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where <math> x_i </math> is the proportion of the investor's total invested wealth in asset <math> i </math>. |
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Of course, there's nothing special about the S&P 500; the same argument can apply to any index, up to and including the [[market portfolio]] of all assets. |
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The variance of the portfolio return is given by: |
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The Capital Asset Pricing Model argues that investors should only be compensated for non-diversifiable risk. Other financial models allow for multiple sources of non-diversifiable risk, but also insist that diversifiable risk should not carry any extra expected return. Still other models do not accept this contention<ref>.{{cite book |
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:<math> \underbrace{\text{Var}(R_P)}_{\equiv \sigma^{2}_{P}} = \mathbb{E}[R_P - \mathbb{E}[R_P]]^2 .</math> |
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Inserting in the expression for <math> \mathbb{E}[R_P] </math>: |
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:<math> \sigma^{2}_{P} = \mathbb{E}\left[\sum^{n}_{i=1}x_i R_i - \sum^{n}_{i=1}x_i\mathbb{E}[R_i]\right]^2 . </math> |
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Rearranging: |
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:<math> \sigma^{2}_{P} = \mathbb{E}\left[\sum^{n}_{i=1}x_i(R_i - \mathbb{E}[R_i])\right]^2 </math> |
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:<math> \sigma^{2}_{P} = \mathbb{E}\left[\sum^{n}_{i=1} \sum^{n}_{j=1} x_i x_j(R_i - \mathbb{E}[R_i])(R_j - \mathbb{E}[R_j])\right] </math> |
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:<math>\sigma_{P}^{2}=\mathbb{E}\left[\sum_{i=1}^{n}x_{i}^{2}(R_{i}-\mathbb{E}[R_{i}])^{2}+\sum_{i=1}^{n}\sum_{j=1,i\neq j}^{n}x_{i}x_{j}(R_{i}-\mathbb{E}[R_{i}])(R_{j}-\mathbb{E}[R_{j}])\right]</math> |
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:<math> \sigma_{P}^{2}=\sum_{i=1}^{n}x_{i}^{2}\underbrace{\mathbb{E}\left[R_{i}-\mathbb{E}[R_{i}]\right]^{2}}_{\equiv\sigma_{i}^{2}}+\sum_{i=1}^{n}\sum_{j=1,i\neq j}^{n}x_{i}x_{j}\underbrace{\mathbb{E}\left[(R_{i}-\mathbb{E}[R_{i}])(R_{j}-\mathbb{E}[R_{j}])\right]}_{\equiv\sigma_{ij}}</math> |
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:<math> \sigma^{2}_{P} = \sum^{n}_{i=1} x^{2}_{i} \sigma^{2}_{i} + \sum^{n}_{i=1} \sum^{n}_{j=1, i \neq j} x_i x_j \sigma_{ij} </math> |
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where <math> \sigma^{2}_{i} </math> is the variance on asset <math> i </math> and <math> \sigma_{ij} </math> is the covariance between assets <math> i </math> and <math> j </math>. |
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In an equally weighted portfolio, <math> x_i = x_j = \frac{1}{n} , \forall i, j </math>. The portfolio variance then becomes: |
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:<math> \sigma^2_P = \frac{1}{n^2} \ {n} \bar{\sigma}^{2}_{i} + n(n-1) \frac{1}{n} \frac{1}{n} \bar{\sigma}_{ij} </math> |
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where <math>\bar{\sigma}_{ij}</math> is the average of the covariances <math>\sigma_{ij} </math> for <math>i\neq j</math> and <math>\bar{\sigma}^2_i</math> is the average of the variances. Simplifying, we obtain |
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:<math> \sigma^{2}_{P} = \frac{1}{n} \bar{\sigma}^{2}_{i} + \frac{n-1}{n} \bar{\sigma}_{ij} .</math> |
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As the number of assets grows we get the asymptotic formula: |
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:<math> \lim_{n \rightarrow \infty} \sigma^2_P = \bar{\sigma}_{ij}. </math> |
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Thus, in an equally weighted portfolio, the portfolio variance tends to the average of covariances between securities as the number of securities becomes arbitrarily large. |
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==Diversifiable and non-diversifiable risk== |
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The [[capital asset pricing model]] introduced the concepts of diversifiable and non-diversifiable risk. Synonyms for diversifiable risk are idiosyncratic risk, unsystematic risk, and security-specific risk. Synonyms for non-diversifiable risk are [[systematic risk]], [[beta (finance)|beta]] risk and [[market risk]]. |
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If one buys all the stocks in the [[S&P 500]] one is obviously exposed only to movements in that [[index (economics)|index]]. If one buys a single stock in the S&P 500, one is exposed both to index movements and movements in the stock based on its underlying company. The first risk is called "non-diversifiable", because it exists however many S&P 500 stocks are bought. The second risk is called "diversifiable", because it can be reduced by diversifying among stocks. |
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In the presence of per-asset investment fees, there is also the possibility of [[overdiversifying]] to the point that the portfolio's performance will suffer because the fees outweigh the gains from diversification. |
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The capital asset pricing model argues that investors should only be compensated for non-diversifiable risk. Other financial models allow for multiple sources of non-diversifiable risk, but also insist that diversifiable risk should not carry any extra expected return. Still other models do not accept this contention.<ref>.{{cite book |
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| last = Fama |
| last = Fama |
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| first = Eugene F. |
| first = Eugene F. |
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| author-link = Eugene Fama |
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|author2=Merton H. Miller |
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| title = The Theory of Finance |
| title = The Theory of Finance |
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| url = https://archive.org/details/theoryoffinance00fama |
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| url-access = registration |
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| publisher = Holt Rinehart & Winston |
| publisher = Holt Rinehart & Winston |
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| date = June 1972 |
| date = June 1972 |
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| isbn = 978- |
| isbn = 978-0-15-504266-7}}</ref> |
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== An empirical example relating diversification to risk reduction == |
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In 1977 [[Ed Elton|Edwin Elton]] and [[Martin J. Gruber|Martin Gruber]]<ref>E. J. Elton and M. J. Gruber, "Risk Reduction and Portfolio Size: An Analytic Solution," ''[[Journal of Business]]'' 50 (October 1977), pp. 415–437</ref> worked out an empirical example of the gains from diversification. Their approach was to consider a population of 3,290 securities available for possible inclusion in a portfolio, and to consider the average risk over all possible randomly chosen ''n''-asset portfolios with equal amounts held in each included asset, for various values of ''n''. Their results are summarized in the following table. |
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==An empirical example relating diversification to risk reduction== |
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The result for ''n''=30 is close to ''n''=1,000, and even four stocks provide most of the reduction in risk compared with one stock. |
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In 1977 Elton and Gruber<ref>E. J. Elton and M. J. Gruber, "Risk Reduction and Portfolio Size: An Analytic Solution," ''[[Journal of Business]]'' 50 (October 1977), pp. 415-37</ref> worked out an empirical example of the gains from diversification. Their approach was to consider a population of 3290 securities available for possible inclusion in a portfolio, and to consider the average risk over all possible randomly chosen ''n''-asset portfolios with equal amounts held in each included asset, for various values of ''n''. Their results are summarized in the following table. It can be seen that most of the gains from diversification come for n≤30. |
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{| class="wikitable |
{| class="wikitable" |
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! Number of Stocks in Portfolio !! Average Standard Deviation of Annual Portfolio Returns !! Ratio of Portfolio Standard Deviation to Standard Deviation of a Single Stock |
! Number of Stocks in Portfolio !! Average Standard Deviation of Annual Portfolio Returns !! Ratio of Portfolio Standard Deviation to Standard Deviation of a Single Stock |
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|- |
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| width="90" | 1 || width="150" | 49.24% || width="200" | 1.00 |
| width = "90" | 1 || width="150" | 49.24% || width = "200" | 1.00 |
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| 2 || 37.36 || 0.76 |
| 2 || 37.36 || 0.76 |
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| 500 || 19.27 || 0.39 |
| 500 || 19.27 || 0.39 |
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| 1,000 || 19.21 || 0.39 |
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Non-incremental diversification is a strategy followed by conglomerates, where the individual business lines have little to do with one another, yet the company is attaining diversification from exogenous risk factors to stabilize and provide opportunity for active management of diverse resources. |
Non-incremental diversification is a strategy followed by conglomerates, where the individual business lines have little to do with one another, yet the company is attaining diversification from exogenous risk factors to stabilize and provide opportunity for active management of diverse resources. |
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==Fallacy of time diversification== |
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The argument is often made that time reduces variance in a portfolio: a "time diversification". A common belief is younger investors should avoid bonds and emphasize stocks, due to the belief investors will have time to recover from any downturns. Yet this belief has flaws, as John Norstad explains: |
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{{Blockquote|text=This kind of statement makes the implicit assumption that given enough time good returns will cancel out any possible bad returns. While the basic argument that the standard deviations of the annualized returns decrease as the time horizon increases is true, it is also misleading, and it fatally misses the point, because for an investor concerned with the value of his portfolio at the end of a period of time, it is the total return that matters, not the annualized return. Because of the effects of compounding, the standard deviation of the total return actually increases with time horizon. Thus, if we use the traditional measure of uncertainty as the standard deviation of return over the time period in question, uncertainty increases with time.<ref>{{Cite web |url=http://www.norstad.org/finance/risk-and-time.html |date=2012-12-22 |title=Risk and Time |author=John Norstad |url-status=dead |archive-url=https://web.archive.org/web/20170911171611/http://www.norstad.org/finance/risk-and-time.html |archive-date=2017-09-11 |access-date=2019-05-30 }}</ref>}} |
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Three notable contributions to the literature on the fallacy of time diversification have been from [[Paul Samuelson]],<ref>{{cite journal |last1=Samuelson |first1=Paul |date=1963 |title=Risk and Uncertainty: A Fallacy of Large Numbers |journal=Scientia |volume=98 |issue=4 |pages=108–113}}</ref> [[Zvi Bodie]],<ref>{{cite journal |last1=Bodie |first1=Zvi |date=May–June 1995 |title=On the Risk of Stocks in the Long Run |journal=Financial Analysts Journal |volume=51 |issue=3 |pages=18–22|doi=10.2469/faj.v51.n3.1901 }}</ref> and Mark Kritzman.<ref>{{cite journal |last1=Kritzman |first1=Mark |date=October 2005 |title=A New Twist on Time Diversification |journal=InvestmentNews}}</ref> |
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==History== |
==History== |
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Diversification is mentioned in the [[Bible]], in the book of [[Ecclesiastes]] which was written in approximately 935 B.C.<ref>{{cite book |
Diversification is mentioned in the [[Bible]], in the book of [[Ecclesiastes]] which was written in approximately 935 B.C.:<ref>{{cite book |
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| title = Life Application Study Bible: New Living Translation |
| title = Life Application Study Bible: New Living Translation |
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| publisher = Tyndale House Publishers, Inc. |
| publisher = Tyndale House Publishers, Inc. |
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| year = 1996 |
| year = 1996 |
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| location = Wheaton, Illinois |
| location = Wheaton, Illinois |
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| page = 1024 |
| page = [https://archive.org/details/isbn_9780842332675/page/1024 1024] |
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| isbn = 0-8423-3267-7 |
| isbn = 0-8423-3267-7 |
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| url-access = registration |
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| url = https://archive.org/details/isbn_9780842332675/page/1024 |
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}}</ref> |
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:But divide your investments among many places, |
:But divide your investments among many places, |
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:for you do not know what risks might lie ahead.<ref> |
:for you do not know what risks might lie ahead.<ref>{{Cite web |url=http://www.youversion.com/bible/nlt/eccl/11/2 |title=Ecclesiastes 11:2 NLT |access-date=2010-01-09 |archive-url=https://web.archive.org/web/20110718144514/http://www.youversion.com/bible/nlt/eccl/11/2 |archive-date=2011-07-18 |url-status=dead }}</ref> |
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Diversification is also mentioned in the [[Talmud]]. The formula given there is to split one's assets into thirds: one third in business (buying and selling things), one third kept liquid (e.g. gold coins), and one third in land ([[real estate]]). |
Diversification is also mentioned in the [[Talmud]]. The formula given there is to split one's assets into thirds: one third in business (buying and selling things), one third kept liquid (e.g. gold coins), and one third in land ([[real estate]]). This strategy of splitting wealth equally among available options is now known as "naive diversification", "Talmudic diversification" or "1/n diversification", a concept which has earned renewed attention since the year 2000 due to research showing it may offer advantages in some scenarios.<ref>{{Cite journal |last1=Duchin |first1=Ran |last2=Levy |first2=Haim |date=2009 |title=Markowitz Versus the Talmudic Portfolio Diversification Strategies |url=http://pm-research.com/lookup/doi/10.3905/JPM.2009.35.2.071 |journal=The Journal of Portfolio Management |language=en |volume=35 |issue=2 |pages=71–74 |doi=10.3905/JPM.2009.35.2.071}}</ref><ref>Prince C Nwakanma1, Monday Aberiate Gbanador. Talmud and Markowitz Diversification Strategies: Evidence from the Nigerian Stock Market. Accounting and Finance Research Vol. 3, No. 2; 2014</ref> |
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Diversification is mentioned in Shakespeare<ref>''The Only Guide to a Winning Investment Strategy You'll Ever Need''</ref> |
Diversification is mentioned in Shakespeare's ''[[Merchant of Venice]]'' (ca. 1599):<ref>''The Only Guide to a Winning Investment Strategy You'll Ever Need''</ref> |
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:My ventures are not in one bottom trusted, |
:My ventures are not in one bottom trusted, |
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:Nor to one place; nor is my whole estate |
:Nor to one place; nor is my whole estate |
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Line 150: | Line 179: | ||
:Therefore, my merchandise makes me not sad. |
:Therefore, my merchandise makes me not sad. |
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Modern understanding of diversification dates back to the influential work of economist [[Harry Markowitz]] in the 1950s,<ref>{{Cite journal | doi = 10.2307/2975974 | last1 = Markowitz | first1 = Harry M. | year = 1952 | title = Portfolio Selection | jstor = 2975974| journal = Journal of Finance | volume = 7 | issue = 1| pages = 77–91 }}</ref> whose work pioneered [[modern portfolio theory]] (see [[Markowitz model]]). |
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An earlier precedent for diversification was economist [[John Maynard Keynes]], who managed the endowment of [[King's College, Cambridge]] from the 1920s to his 1946 death with a stock-selection strategy similar to what was later called [[value investing]].<ref>Chambers, David and Dimson, Elroy, John Maynard Keynes, Investment Innovator (June 30, 2013). Journal of Economic Perspectives, 2013, Vol 27, No 3, pages 1–18, Available at SSRN: https://ssrn.com/abstract=2287262 or http://dx.doi.org/10.2139/ssrn.2287262</ref> While diversification in the modern sense was "not easily available in Keynes's day"<ref>M. Lawlor (2016). The Economics of Keynes in Historical Context: An Intellectual History of the General Theory, Palgrave Macmillan UK, {{ISBN|9780230288775}}, p. 316</ref> and Keynes typically held a small number of assets compared to later investment theories, he nonetheless is recognized as a pioneer of financial diversification. Keynes came to recognize the importance, "if possible", he wrote, of holding assets with "opposed risks [...] since they are likely to move in opposite directions when there are general fluctuations"<ref>[[Kenneth L. Fisher]] (2007). 100 Minds That Made the Market. Wiley, {{ISBN|9780470139516}}</ref> Keynes was a pioneer of "international diversification" due to substantial holdings in non-U.K. stocks, up to 75%, and avoiding [[Equity home bias puzzle|home bias]] at a time when university endowments in the U.S. and U.K. were invested almost entirely in domestic assets.<ref>David Chambers, Elroy Dimson, Justin Foo (2015). [https://www.nber.org/chapters/c12860 Keynes, King's, and Endowment Asset Management], in ''How the Financial Crisis and Great Recession Affected Higher Education'' (2015), Jeffrey R. Brown and Caroline M. Hoxby, editors (p. 127–150). Conference held September 27–28, 2012.</ref> |
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==See also== |
==See also== |
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[[ |
[[File:Asset allocation.png|230px|right|thumbnail|'''[[b:Asset Allocation|Asset Allocation on Wikibook]]''']] |
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*[[Modern portfolio theory]] |
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*[[Central limit theorem]] |
*[[Central limit theorem]] |
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*[[Coherent risk measure]] |
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*[[Dollar cost averaging]] |
*[[Dollar cost averaging]] |
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*[[Equity repositioning]] |
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*[[Financial correlation]] |
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*[[Outline of finance]] |
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*[[Modern portfolio theory]] |
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*[[Systematic risk]] |
*[[Systematic risk]] |
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*[[List of finance topics]] |
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==References== |
==References== |
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{{reflist|colwidth=30em}} |
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{{clear}} |
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{{reflist|colwidth=35em}} |
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==External links== |
==External links== |
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* [http://www.stanford.edu/~wfsharpe/mia/mia.htm Macro-Investment Analysis], |
* [http://www.stanford.edu/~wfsharpe/mia/mia.htm Macro-Investment Analysis], Prof. [[William F. Sharpe]], [[Stanford University]] |
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* [https://web.archive.org/web/20080805124142/http://viking.som.yale.edu/will/finman540/classnotes/notes.html An Introduction to Investment Theory], Prof. William N. Goetzmann, [[Yale School of Management]] |
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* [http://www.riskcog.com/portfolio.jsp Portfolio Diversifier], Dynamically-generated diversified portfolios |
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* [http://www.assetcorrelation.com/ Asset Correlations], Dynamically-generated correlation matrices for the major asset classes |
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* [http://viking.som.yale.edu/will/finman540/classnotes/notes.html An Introduction to Investment Theory], Prof. William N. Goetzmann, [[Yale School of Management]] |
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* [http://www.barclayhedge.com/research/educational-articles/managed-futures-articles/managed-futures-overview.html Overview of Managed Futures] |
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{{Financial risk}} |
{{Financial risk}} |
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{{Authority control}} |
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{{DEFAULTSORT:Diversification (Finance)}} |
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Latest revision as of 21:35, 10 July 2024
Part of a series on |
Finance |
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In finance, diversification is the process of allocating capital in a way that reduces the exposure to any one particular asset or risk. A common path towards diversification is to reduce risk or volatility by investing in a variety of assets. If asset prices do not change in perfect synchrony, a diversified portfolio will have less variance than the weighted average variance of its constituent assets, and often less volatility than the least volatile of its constituents.[1]
Diversification is one of two general techniques for reducing investment risk. The other is hedging.
Examples
[edit]The simplest example of diversification is provided by the proverb "Don't put all your eggs in one basket". Dropping the basket will break all the eggs. Placing each egg in a different basket is more diversified. There is more risk of losing one egg, but less risk of losing all of them. On the other hand, having a lot of baskets may increase costs.
In finance, an example of an undiversified portfolio is to hold only one stock. This is risky; it is not unusual for a single stock to go down 50% in one year. It is less common for a portfolio of 20 stocks to go down that much, especially if they are selected at random. If the stocks are selected from a variety of industries, company sizes and asset types it is even less likely to experience a 50% drop since it will mitigate any trends in that industry, company class, or asset type.
Since the mid-1970s, it has also been argued that geographic diversification would generate superior risk-adjusted returns for large institutional investors by reducing overall portfolio risk while capturing some of the higher rates of return offered by the emerging markets of Asia and Latin America.[2][3]
Return expectations while diversifying
[edit]If the prior expectations of the returns on all assets in the portfolio are identical, the expected return on a diversified portfolio will be identical to that on an undiversified portfolio. Some assets will do better than others; but since one does not know in advance which assets will perform better, this fact cannot be exploited in advance. The return on a diversified portfolio can never exceed that of the top-performing investment, and indeed will always be lower than the highest return (unless all returns are identical). Conversely, the diversified portfolio's return will always be higher than that of the worst-performing investment. So by diversifying, one loses the chance of having invested solely in the single asset that comes out best, but one also avoids having invested solely in the asset that comes out worst. That is the role of diversification: it narrows the range of possible outcomes. Diversification need not either help or hurt expected returns, unless the alternative non-diversified portfolio has a higher expected return.[4]
Amount of diversification
[edit]There is no magic number of stocks that is diversified versus not. Sometimes quoted is 30, although it can be as low as 10, provided they are carefully chosen. This is based on a result from John Evans and Stephen Archer.[5] Similarly, a 1985 book reported that most value from diversification comes from the first 15 or 20 different stocks in a portfolio.[6] More stocks give lower price volatility.
Given the advantages of diversification, many experts [who?] recommend maximum diversification, also known as "buying the market portfolio". Identifying that portfolio is not straightforward. The earliest definition comes from the capital asset pricing model which argues the maximum diversification comes from buying a pro rata share of all available assets. This is the idea underlying index funds.
Diversification has no maximum so long as more assets are available.[7] Every equally weighted, uncorrelated asset added to a portfolio can add to that portfolio's measured diversification. When assets are not uniformly uncorrelated, a weighting approach that puts assets in proportion to their relative correlation can maximize the available diversification.
"Risk parity" is an alternative idea. This weights assets in inverse proportion to risk, so the portfolio has equal risk in all asset classes. This is justified both on theoretical grounds, and with the pragmatic argument that future risk is much easier to forecast than either future market price or future economic footprint.[8] "Correlation parity" is an extension of risk parity, and is the solution whereby each asset in a portfolio has an equal correlation with the portfolio, and is therefore the "most diversified portfolio". Risk parity is the special case of correlation parity when all pair-wise correlations are equal.[9]
Effect of diversification on variance
[edit]One simple measure of financial risk is variance of the return on the portfolio. Diversification can lower the variance of a portfolio's return below what it would be if the entire portfolio were invested in the asset with the lowest variance of return, even if the assets' returns are uncorrelated. For example, let asset X have stochastic return and asset Y have stochastic return , with respective return variances and . If the fraction of a one-unit (e.g. one-million-dollar) portfolio is placed in asset X and the fraction is placed in Y, the stochastic portfolio return is . If and are uncorrelated, the variance of portfolio return is . The variance-minimizing value of is , which is strictly between and . Using this value of in the expression for the variance of portfolio return gives the latter as , which is less than what it would be at either of the undiversified values and (which respectively give portfolio return variance of and ). Note that the favorable effect of diversification on portfolio variance would be enhanced if and were negatively correlated but diminished (though not eliminated) if they were positively correlated.
In general, the presence of more assets in a portfolio leads to greater diversification benefits, as can be seen by considering portfolio variance as a function of , the number of assets. For example, if all assets' returns are mutually uncorrelated and have identical variances , portfolio variance is minimized by holding all assets in the equal proportions .[10] Then the portfolio return's variance equals = = , which is monotonically decreasing in .
The latter analysis can be adapted to show why adding uncorrelated volatile assets to a portfolio,[11][12] thereby increasing the portfolio's size, is not diversification, which involves subdividing the portfolio among many smaller investments. In the case of adding investments, the portfolio's return is instead of and the variance of the portfolio return if the assets are uncorrelated is which is increasing in n rather than decreasing. Thus, for example, when an insurance company adds more and more uncorrelated policies to its portfolio, this expansion does not itself represent diversification—the diversification occurs in the spreading of the insurance company's risks over a large number of part-owners of the company.
Diversification with correlated returns via an equally weighted portfolio
[edit]The expected return on a portfolio is a weighted average of the expected returns on each individual asset:
where is the proportion of the investor's total invested wealth in asset .
The variance of the portfolio return is given by:
Inserting in the expression for :
Rearranging:
where is the variance on asset and is the covariance between assets and .
In an equally weighted portfolio, . The portfolio variance then becomes:
where is the average of the covariances for and is the average of the variances. Simplifying, we obtain
As the number of assets grows we get the asymptotic formula:
Thus, in an equally weighted portfolio, the portfolio variance tends to the average of covariances between securities as the number of securities becomes arbitrarily large.
Diversifiable and non-diversifiable risk
[edit]The capital asset pricing model introduced the concepts of diversifiable and non-diversifiable risk. Synonyms for diversifiable risk are idiosyncratic risk, unsystematic risk, and security-specific risk. Synonyms for non-diversifiable risk are systematic risk, beta risk and market risk.
If one buys all the stocks in the S&P 500 one is obviously exposed only to movements in that index. If one buys a single stock in the S&P 500, one is exposed both to index movements and movements in the stock based on its underlying company. The first risk is called "non-diversifiable", because it exists however many S&P 500 stocks are bought. The second risk is called "diversifiable", because it can be reduced by diversifying among stocks.
In the presence of per-asset investment fees, there is also the possibility of overdiversifying to the point that the portfolio's performance will suffer because the fees outweigh the gains from diversification.
The capital asset pricing model argues that investors should only be compensated for non-diversifiable risk. Other financial models allow for multiple sources of non-diversifiable risk, but also insist that diversifiable risk should not carry any extra expected return. Still other models do not accept this contention.[13]
An empirical example relating diversification to risk reduction
[edit]In 1977 Edwin Elton and Martin Gruber[14] worked out an empirical example of the gains from diversification. Their approach was to consider a population of 3,290 securities available for possible inclusion in a portfolio, and to consider the average risk over all possible randomly chosen n-asset portfolios with equal amounts held in each included asset, for various values of n. Their results are summarized in the following table.
The result for n=30 is close to n=1,000, and even four stocks provide most of the reduction in risk compared with one stock.
Number of Stocks in Portfolio | Average Standard Deviation of Annual Portfolio Returns | Ratio of Portfolio Standard Deviation to Standard Deviation of a Single Stock |
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1 | 49.24% | 1.00 |
2 | 37.36 | 0.76 |
4 | 29.69 | 0.60 |
6 | 26.64 | 0.54 |
8 | 24.98 | 0.51 |
10 | 23.93 | 0.49 |
20 | 21.68 | 0.44 |
30 | 20.87 | 0.42 |
40 | 20.46 | 0.42 |
50 | 20.20 | 0.41 |
400 | 19.29 | 0.39 |
500 | 19.27 | 0.39 |
1,000 | 19.21 | 0.39 |
Corporate diversification strategies
[edit]In corporate portfolio models, diversification is thought of as being vertical or horizontal. Horizontal diversification is thought of as expanding a product line or acquiring related companies. Vertical diversification is synonymous with integrating the supply chain or amalgamating distributions channels.
Non-incremental diversification is a strategy followed by conglomerates, where the individual business lines have little to do with one another, yet the company is attaining diversification from exogenous risk factors to stabilize and provide opportunity for active management of diverse resources.
Fallacy of time diversification
[edit]The argument is often made that time reduces variance in a portfolio: a "time diversification". A common belief is younger investors should avoid bonds and emphasize stocks, due to the belief investors will have time to recover from any downturns. Yet this belief has flaws, as John Norstad explains:
This kind of statement makes the implicit assumption that given enough time good returns will cancel out any possible bad returns. While the basic argument that the standard deviations of the annualized returns decrease as the time horizon increases is true, it is also misleading, and it fatally misses the point, because for an investor concerned with the value of his portfolio at the end of a period of time, it is the total return that matters, not the annualized return. Because of the effects of compounding, the standard deviation of the total return actually increases with time horizon. Thus, if we use the traditional measure of uncertainty as the standard deviation of return over the time period in question, uncertainty increases with time.[15]
Three notable contributions to the literature on the fallacy of time diversification have been from Paul Samuelson,[16] Zvi Bodie,[17] and Mark Kritzman.[18]
History
[edit]Diversification is mentioned in the Bible, in the book of Ecclesiastes which was written in approximately 935 B.C.:[19]
- But divide your investments among many places,
- for you do not know what risks might lie ahead.[20]
Diversification is also mentioned in the Talmud. The formula given there is to split one's assets into thirds: one third in business (buying and selling things), one third kept liquid (e.g. gold coins), and one third in land (real estate). This strategy of splitting wealth equally among available options is now known as "naive diversification", "Talmudic diversification" or "1/n diversification", a concept which has earned renewed attention since the year 2000 due to research showing it may offer advantages in some scenarios.[21][22]
Diversification is mentioned in Shakespeare's Merchant of Venice (ca. 1599):[23]
- My ventures are not in one bottom trusted,
- Nor to one place; nor is my whole estate
- Upon the fortune of this present year:
- Therefore, my merchandise makes me not sad.
Modern understanding of diversification dates back to the influential work of economist Harry Markowitz in the 1950s,[24] whose work pioneered modern portfolio theory (see Markowitz model).
An earlier precedent for diversification was economist John Maynard Keynes, who managed the endowment of King's College, Cambridge from the 1920s to his 1946 death with a stock-selection strategy similar to what was later called value investing.[25] While diversification in the modern sense was "not easily available in Keynes's day"[26] and Keynes typically held a small number of assets compared to later investment theories, he nonetheless is recognized as a pioneer of financial diversification. Keynes came to recognize the importance, "if possible", he wrote, of holding assets with "opposed risks [...] since they are likely to move in opposite directions when there are general fluctuations"[27] Keynes was a pioneer of "international diversification" due to substantial holdings in non-U.K. stocks, up to 75%, and avoiding home bias at a time when university endowments in the U.S. and U.K. were invested almost entirely in domestic assets.[28]
See also
[edit]- Central limit theorem
- Coherent risk measure
- Dollar cost averaging
- Equity repositioning
- Financial correlation
- Outline of finance
- Modern portfolio theory
- Systematic risk
References
[edit]- ^ O'Sullivan, Arthur; Sheffrin, Steven M. (2003). Economics: Principles in Action. Upper Saddle River, New Jersey: Pearson Prentice Hall. p. 273. ISBN 0-13-063085-3.
- ^ (in French) "see M. Nicolas J. Firzli, "Asia-Pacific Funds as Diversification Tools for Institutional Investors", Revue Analyse Financière/The French Society of Financial Analysts (SFAF)" (PDF). Archived from the original (PDF) on 2010-05-08. Retrieved 2009-04-02.
- ^ (in English) "see Michael Prahl, "Asian Private Equity – Will it Deliver on its Promise?", INSEAD Global Private Equity Initiative (GPEI)" (PDF). Retrieved 2011-06-15.
- ^ Goetzmann, William N. An Introduction to Investment Theory Archived 2017-03-31 at the Wayback Machine. II. Portfolios of Assets. Retrieved on November 20, 2008.
- ^ Investment Guide Beginners Introduction
- ^ James Lorie; Peter Dodd; Mary Kimpton (1985). The Stock Market: Theories and Evidence (2nd ed.). Dow Jones-Irwin. p. 85. ISBN 9780870946189.
- ^ Statman, Meir (1987). "How Many Stocks Make a Diversified Portfolio?". The Journal of Financial and Quantitative Analysis. 22 (3): 353–363. doi:10.2307/2330969. JSTOR 2330969.
- ^ Asness, Cliff; David Kabiller and Michael Mendelson Using Derivatives and Leverage To Improve Portfolio Performance, Institutional Investor, May 13, 2010. Retrieved on June 21, 2010.
- ^ Schoen, Robert Parity Strategies and Maximum Diversification, Putnam Investments, June, 2013 Archived 2015-04-02 at the Wayback Machine.
- ^ Samuelson, Paul, "General Proof that Diversification Pays", Journal of Financial and Quantitative Analysis 2, March 1967, 1-13.
- ^ Samuelson, Paul, "Risk and uncertainty: A fallacy of large numbers", Scientia 98, 1963, 108-113.
- ^ Ross, Stephen, "Adding risks: Samuelson's fallacy of large numbers revisited" Journal of Financial and Quantitative Analysis 34, September 1999, 323-339.
- ^ .Fama, Eugene F.; Merton H. Miller (June 1972). The Theory of Finance. Holt Rinehart & Winston. ISBN 978-0-15-504266-7.
- ^ E. J. Elton and M. J. Gruber, "Risk Reduction and Portfolio Size: An Analytic Solution," Journal of Business 50 (October 1977), pp. 415–437
- ^ John Norstad (2012-12-22). "Risk and Time". Archived from the original on 2017-09-11. Retrieved 2019-05-30.
- ^ Samuelson, Paul (1963). "Risk and Uncertainty: A Fallacy of Large Numbers". Scientia. 98 (4): 108–113.
- ^ Bodie, Zvi (May–June 1995). "On the Risk of Stocks in the Long Run". Financial Analysts Journal. 51 (3): 18–22. doi:10.2469/faj.v51.n3.1901.
- ^ Kritzman, Mark (October 2005). "A New Twist on Time Diversification". InvestmentNews.
- ^ Life Application Study Bible: New Living Translation. Wheaton, Illinois: Tyndale House Publishers, Inc. 1996. p. 1024. ISBN 0-8423-3267-7.
- ^ "Ecclesiastes 11:2 NLT". Archived from the original on 2011-07-18. Retrieved 2010-01-09.
- ^ Duchin, Ran; Levy, Haim (2009). "Markowitz Versus the Talmudic Portfolio Diversification Strategies". The Journal of Portfolio Management. 35 (2): 71–74. doi:10.3905/JPM.2009.35.2.071.
- ^ Prince C Nwakanma1, Monday Aberiate Gbanador. Talmud and Markowitz Diversification Strategies: Evidence from the Nigerian Stock Market. Accounting and Finance Research Vol. 3, No. 2; 2014
- ^ The Only Guide to a Winning Investment Strategy You'll Ever Need
- ^ Markowitz, Harry M. (1952). "Portfolio Selection". Journal of Finance. 7 (1): 77–91. doi:10.2307/2975974. JSTOR 2975974.
- ^ Chambers, David and Dimson, Elroy, John Maynard Keynes, Investment Innovator (June 30, 2013). Journal of Economic Perspectives, 2013, Vol 27, No 3, pages 1–18, Available at SSRN: https://ssrn.com/abstract=2287262 or http://dx.doi.org/10.2139/ssrn.2287262
- ^ M. Lawlor (2016). The Economics of Keynes in Historical Context: An Intellectual History of the General Theory, Palgrave Macmillan UK, ISBN 9780230288775, p. 316
- ^ Kenneth L. Fisher (2007). 100 Minds That Made the Market. Wiley, ISBN 9780470139516
- ^ David Chambers, Elroy Dimson, Justin Foo (2015). Keynes, King's, and Endowment Asset Management, in How the Financial Crisis and Great Recession Affected Higher Education (2015), Jeffrey R. Brown and Caroline M. Hoxby, editors (p. 127–150). Conference held September 27–28, 2012.
External links
[edit]- Macro-Investment Analysis, Prof. William F. Sharpe, Stanford University
- An Introduction to Investment Theory, Prof. William N. Goetzmann, Yale School of Management