Diversification (finance)

Infinance,diversificationis 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 reduceriskorvolatilitybyinvestingin a variety ofassets. If asset prices do not change in perfect synchrony, a diversifiedportfoliowill have less variance than theweighted averagevariance of its constituent assets, and often less volatility than the least volatile of its constituents.
Diversification is one of two general techniques for reducing investment risk. The other ishedging.
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 largeinstitutional investorsby reducing overall portfolio risk while capturing some of the higher rates of return offered by theemerging marketsof Asia and Latin America.
If the priorexpectationsof the returns on all assets in the portfolio are identical, theexpected returnon 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.
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.Similarly, a 1985 book reported that most value from diversification comes from the first 15 or 20 different stocks in a portfolio.More stocks give lower price volatility.
Given the advantages of diversification, many expertsrecommend maximum diversification, also known as "buying themarket portfolio". Identifying that portfolio is not straightforward.
The earliest definition comes from thecapital asset pricing modelwhich argues the maximum diversification comes from buying apro ratashare of all availableassets. This is the idea underlyingindex funds.
Diversification has no maximum so long as more assets are available.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."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.
One simple measure offinancial riskisvarianceof thereturnon 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 returnx{\displaystyle x}and asset Y have stochastic returny{\displaystyle y}, with respective return variancesσx2{\displaystyle \sigma _{x}^{2}}andσy2{\displaystyle \sigma _{y}^{2}}.  If the fractionq{\displaystyle q}of a one-unit (e.g. one-million-dollar) portfolio is placed in asset X and the fraction1−q{\displaystyle 1-q}is placed in Y, the stochastic portfolio return isqx+(1−q)y{\displaystyle qx+(1-q)y}.  Ifx{\displaystyle x}andy{\displaystyle y}are uncorrelated, the variance of portfolio return isvar(qx+(1−q)y)=q2σx2+(1−q)2σy2{\displaystyle {\text{var}}(qx+(1-q)y)=q^{2}\sigma _{x}^{2}+(1-q)^{2}\sigma _{y}^{2}}.  The variance-minimizing value ofq{\displaystyle q}isq=σy2/[σx2+σy2]{\displaystyle q=\sigma _{y}^{2}/[\sigma _{x}^{2}+\sigma _{y}^{2}]}, which is strictly between0{\displaystyle 0}and1{\displaystyle 1}.  Using this value ofq{\displaystyle q}in the expression for the variance of portfolio return gives the latter asσx2σy2/[σx2+σy2]{\displaystyle \sigma _{x}^{2}\sigma _{y}^{2}/[\sigma _{x}^{2}+\sigma _{y}^{2}]}, which is less than what it would be at either of the undiversified valuesq=1{\displaystyle q=1}andq=0{\displaystyle q=0}(which respectively give portfolio return variance ofσx2{\displaystyle \sigma _{x}^{2}}andσy2{\displaystyle \sigma _{y}^{2}}).  Note that the favorable effect of diversification on portfolio variance would be enhanced ifx{\displaystyle x}andy{\displaystyle y}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 ofn{\displaystyle n}, the number of assets.  For example, if all assets' returns are mutually uncorrelated and have identical variancesσx2{\displaystyle \sigma _{x}^{2}}, portfolio variance is minimized by holding all assets in the equal proportions1/n{\displaystyle 1/n}.Then the portfolio return's variance equalsvar[(1/n)x1+(1/n)x2+...+(1/n)xn]{\displaystyle {\text{var}}[(1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n}]}=n(1/n2)σx2{\displaystyle n(1/n^{2})\sigma _{x}^{2}}=σx2/n{\displaystyle \sigma _{x}^{2}/n}, which is monotonically decreasing inn{\displaystyle n}.
The latter analysis can be adapted to show whyaddinguncorrelated volatile assets to a portfolio,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 isx1+x2+⋯+xn{\displaystyle x_{1}+x_{2}+\dots +x_{n}}instead of(1/n)x1+(1/n)x2+...+(1/n)xn,{\displaystyle (1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n},}and the variance of the portfolio return if the assets are uncorrelated isvar[x1+x2+⋯+xn]=σx2+σx2+⋯+σx2=nσx2,{\displaystyle {\text{var}}[x_{1}+x_{2}+\dots +x_{n}]=\sigma _{x}^{2}+\sigma _{x}^{2}+\dots +\sigma _{x}^{2}=n\sigma _{x}^{2},}which isincreasinginnrather 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.
The expected return on a portfolio is a weighted average of the expected returns on each individual asset:
wherexi{\displaystyle x_{i}}is the proportion of the investor's total invested wealth in asseti{\displaystyle i}.
The variance of the portfolio return is given by:
Inserting in the expression forE[RP]{\displaystyle \mathbb {E} [R_{P}]}:
Rearranging:
whereσi2{\displaystyle \sigma _{i}^{2}}is the variance on asseti{\displaystyle i}andσij{\displaystyle \sigma _{ij}}is the covariance between assetsi{\displaystyle i}andj{\displaystyle j}.
In an equally weighted portfolio,xi=xj=1n,∀i,j{\displaystyle x_{i}=x_{j}={\frac {1}{n}},\forall i,j}. The portfolio variance then becomes:
whereσ¯ij{\displaystyle {\bar {\sigma }}_{ij}}is the average of the covariancesσij{\displaystyle \sigma _{ij}}fori≠j{\displaystyle i\neq j}andσ¯i2{\displaystyle {\bar {\sigma }}_{i}^{2}}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.
Thecapital asset pricing modelintroduced 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 aresystematic risk,betarisk andmarket risk.
If one buys all the stocks in theS&P 500one is obviously exposed only to movements in thatindex. 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 ofoverdiversifyingto 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.
In 1977Edwin EltonandMartin Gruberworked 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 chosenn-asset portfolios with equal amounts held in each included asset, for various values ofn. Their results are summarized in the following table.
The result forn=30 is close ton=1,000, and even four stocks provide most of the reduction in risk compared with one stock.
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.
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:
Three notable contributions to the literature on the fallacy of time diversification have been fromPaul Samuelson,Zvi Bodie,and Mark Kritzman.
Diversification is mentioned in theBible, in the book ofEcclesiasteswhich was written in approximately 935 B.C.:
Diversification is also mentioned in theTalmud. 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.
Diversification is mentioned in Shakespeare'sMerchant of Venice(ca. 1599):
Modern understanding of diversification dates back to the influential work of economistHarry Markowitzin the 1950s,whose work pioneeredmodern portfolio theory(seeMarkowitz model).
An earlier precedent for diversification was economistJohn Maynard Keynes, who managed the endowment ofKing's College, Cambridgefrom the 1920s to his 1946 death with a stock-selection strategy similar to what was later calledvalue investing.While diversification in the modern sense was "not easily available in Keynes's day"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"Keynes was a pioneer of "international diversification" due to substantial holdings in non-U.K. stocks, up to 75%, and avoidinghome biasat a time when university endowments in the U.S. and U.K. were invested almost entirely in domestic assets.