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| In finance, '''diversification''' means reducing [[financial risk|risk]] by [[investment|investing]] in a variety of [[asset]]s. If the asset values do not move up and down in perfect synchrony, a diversified [[Portfolio (finance)|portfolio]] will have less risk than the [[weighted mean|weighted average]] risk of its constituent assets, and often less risk than the least risky of its constituent.<ref>{{cite book
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| | last = Sullivan
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| | first = Arthur
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| | authorlink = Arthur O' Sullivan
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| | coauthors = Steven M. Sheffrin
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| | title = Economics: Principles in action
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| | publisher = Pearson Prentice Hall
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| | year = 2003
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| | location = Upper Saddle River, New Jersey 07458
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| | pages = 273
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| | url = http://books.google.com/books?vid=ISBN0131334832
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| | doi =
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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 relies on the lack of a tight positive relationship among the assets' returns, and works even when [[correlation and dependence|correlations]] are near zero or somewhat positive. Hedging relies on negative correlation among assets, or [[short (finance)|shorting]] assets with positive correlation.
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| ==Examples==
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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, the probability of any one basket being dropped notwithstanding. There is more risk of losing one egg (assuming at least one basket has a higher probability of being dropped than the original basket), 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, especially 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>{{fr icon}} {{Cite web |url=http://www.canadianeuropean.com/yahoo_site_admin/assets/docs/FONDS_DASIE-PACIFIQUE_REVUE_AF_APR_09.95131642.pdf |accessdate=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) |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}} }}</ref><ref>{{en icon}} {{Cite web |url=http://www.insead.edu/facultyresearch/centres/global_private_equity_initiative/publications/documents/INSEAD_AsiaPEReport_long.pdf |accessdate=2011-06-15 |title=see Michael Prahl, "Asian Private Equity – Will it Deliver on its Promise?", INSEAD Global Private Equity Initiative (GPEI) |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref>
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| ==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. ''Ex post'', 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 ''ex post'' 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 ''ex post'' 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]. II. Portfolios of Assets. Retrieved on November 20, 2008.</ref>
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| ==Maximum diversification==
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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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| 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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| 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 value or future economic footprint. "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>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>
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| ==Effect of diversification on variance==
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| One simple measure of [[financial risk]] is [[variance]]. 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>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 necessarily 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,"''Journal of Financial and Quantitative Analysis'' 2, March 1967, 1-13.</ref> Then the portfolio return's variance equals <math>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 risky assets to a portfolio,<ref>Samuelson, Paul, "Risk and uncertainty: A fallacy of large numbers," ''Scientia'' 98, 1963, 108-113.</ref><ref>Ross, Stephen, "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>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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| ==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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| Note that there is also the risk of [[overdiversifying]] to the point that your performance will suffer and you will end up paying mostly for fees.
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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
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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 = 978-0-15-504266-7}}</ref>
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| ==An empirical example relating diversification to risk reduction==
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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" border="1"
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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
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| |-
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| | width="90" | 1 || width="150" | 49.24% || width="200" | 1.00
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| | 2 || 37.36 || 0.76
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| | 4 || 29.69 || 0.60
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| | 6 || 26.64 || 0.54
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| | 8 || 24.98 || 0.51
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| | 10 || 23.93 || 0.49
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| | 20 || 21.68 || 0.44
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| | 30 || 20.87 || 0.42
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| | 40 || 20.46 || 0.42
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| | 50 || 20.20 || 0.41
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| | 400 || 19.29 || 0.39
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| | 500 || 19.27 || 0.39
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| | 1000 || 19.21 || 0.39
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| |}
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| ==Corporate diversification strategies==
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| 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.
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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.
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| ==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
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| | title = Life Application Study Bible: New Living Translation
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| | publisher = Tyndale House Publishers, Inc.
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| | year = 1996
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| | location = Wheaton, Illinois
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| | page = 1024
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| | isbn = 0-8423-3267-7}}</ref>
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| :But divide your investments among many places, | |
| :for you do not know what risks might lie ahead.<ref>[http://www.youversion.com/bible/nlt/eccl/11/2 Ecclesiastes 11:2 NLT]</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]]).
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| Diversification is mentioned in Shakespeare<ref>''The Only Guide to a Winning Investment Strategy You'll Ever Need''</ref> (''[[Merchant of Venice]]''):
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| :My ventures are not in one bottom trusted,
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| :Nor to one place; nor is my whole estate
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| :Upon the fortune of this present year:
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| :Therefore, my merchandise makes me not sad.
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| The modern understanding of diversification dates back to the work of [[Harry Markowitz]]<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> in the 1950s.
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| ==Diversification with an equally-weighted portfolio==
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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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| :<math> \mathbb{E}[R_P] = \sum^{n}_{i=1}x_i\mathbb{E}[R_i] </math>
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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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| The variance of the portfolio return is given by:
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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>. In an equally-weighted portfolio, <math> x_i = x_j = \frac{1}{n} , \forall i, j </math>.
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| The portfolio variance then becomes:
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| :<math> \sigma^2_P = n \frac{1}{n^2} \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>. Simplifying we obtain
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| :<math> \sigma^{2}_{P} = \frac{1}{n} \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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| ==Cointegration and correlation in finance==
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| Within the framework of the financial industry, when representing relationships between assets, correlation is typically used. However, academics have long since questioned this
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| method due to the plethora of issues that plague it. Indeed, it is thought that cointegration is a natural replacement in some of the cases as it is able to represent the physical reality of these assets better. However, despite this general academic consensus, financial practitioners refuse to accept cointegration as a better tool, or even, the lesser of two evils.<ref name="wilmott.com">{{cite journal|authors=Mahdavi Damghani B., Welch D., O'Malley C., Knights S.|title=The Misleading Value of Measured Correlation |journal=Wilmott Magazine|year=2012|url=http://wilmott.com/pdfs/MVMC_CSL_05_2012.pdf}}</ref> This interesting bias has led to the creation of the mathematical model referred to as [[Cointelation]] which is a hybrid model between [[correlation]] and [[cointegration]].<ref name="wilmott.com"/><ref name="wilmottM.com">{{cite journal|authors=Mahdavi Damghani B.|title=The Non-Misleading Value of Inferred Correlation: An Introduction to the Cointelation Model|journal=Wilmott Magazine|year=2013|url=http://onlinelibrary.wiley.com/doi/10.1002/wilm.10252/abstract}}</ref>
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| ==See also==
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| [[File:Asset allocation.gif|230px|right|thumbnail|'''[[b:Asset Allocation|Asset Allocation on Wikibook]]''']]
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| *[[Central limit theorem]]
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| *[[Cointelation]]
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| *[[Coherent risk measure]]
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| *[[Dollar cost averaging]]
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| *[[Financial correlation]]
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| *[[List of finance topics]]
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| *[[Modern portfolio theory]]
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| *[[Systematic risk]]
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| ==References==
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| {{clear}}
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| {{reflist|colwidth=35em}}
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| ==External links== | |
| * [http://www.stanford.edu/~wfsharpe/mia/mia.htm Macro-Investment Analysis], [[William Forsyth Sharpe|Prof. William F. Sharpe]], [[Stanford University]]
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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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| * [http://www.macroaxis.com/invest/marketRiskAndReturn Synthetically diversified portfolio based on mean-variance optimization], Risk-adjusted returns analysis for public companies
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| {{Financial risk}}
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| {{DEFAULTSORT:Diversification (Finance)}}
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| [[Category:Econometrics]]
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| [[Category:Financial risk]]
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