Diversification (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 In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environme ...
or volatility by
investing Investment is the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance, the purpose of investing is ...
in a variety of
asset In financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) that can be used to produce positive economic value. Assets represent value of ownership that c ...
s. If asset prices do not change in perfect synchrony, a diversified
portfolio Portfolio may refer to: Objects * Portfolio (briefcase), a type of briefcase Collections * Portfolio (finance), a collection of assets held by an institution or a private individual * Artist's portfolio, a sample of an artist's work or a c ...
will have less variance than the
weighted average The weighted arithmetic mean is similar to an ordinary arithmetic mean In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are con ...
variance 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 is hedging.

# Examples

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 investor An institutional investor is an entity which pools money to purchase Security (finance), securities, real property, and other investment assets or originate loans. Institutional investors include commercial banks, central banks, credit unions, ...
s by reducing overall portfolio risk while capturing some of the higher rates of return offered by the
emerging markets An emerging market (or an emerging country or an emerging economy) is a market that has some characteristics of a developed market, but does not fully meet its standards. This includes markets that may become developed markets in the future or were ...
of Asia and Latin America.

# Return expectations while diversifying

If the prior expectations of the returns on all assets in the portfolio are identical, the
expected return The expected return (or expected gain) on a financial investment is the expected value of its return (finance), return (of the profit on the investment). It is a measure of the center of the distribution of the random variable that is the 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.

# Amount of diversification

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 experts recommend maximum diversification, also known as "buying the
market portfolio Market portfolio is a portfolio consisting of a weighted sum of every asset in the market, with weights in the proportions that they exist in the market, with the necessary assumption that these assets are infinitely divisible. Richard Roll's crit ...
". Unfortunately, identifying that portfolio is not straightforward. The earliest definition comes from the
capital asset pricing model In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a Diversification (finance), well-diversified Portfolio (f ...
which argues the maximum diversification comes from buying a ''pro rata'' share of all available
asset In financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) that can be used to produce positive economic value. Assets represent value of ownership that c ...
s. This is the idea underlying
index fund An index fund (also index tracker) is a mutual fund A mutual fund is a professionally managed investment fund that pools money from many investors to purchase Security (finance), securities. The term is typically used in the United States, Canada, ...
s. 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.

# Effect of diversification on variance

One simple measure of
financial risk Financial risk is any of various types of risk associated with finance, financing, including financial transactions that include company loans in risk of default (finance), default. Often it is understood to include only downside risk, meaning t ...
is
variance In probability theory and statistics, variance is the expected value, expectation of the squared Deviation (statistics), deviation of a random variable from its population mean or sample mean. Variance is a measure of statistical dispersion, di ...
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 $x$ and asset Y have stochastic return $y$, with respective return variances $\sigma^_x$ and $\sigma^_y$. If the fraction $q$ of a one-unit (e.g. one-million-dollar) portfolio is placed in asset X and the fraction $1-q$ is placed in Y, the stochastic portfolio return is $qx+\left(1-q\right)y$. If $x$ and $y$ are uncorrelated, the variance of portfolio return is $\text\left(qx+\left(1-q\right)y\right)=q^\sigma^_x+\left(1-q\right)^\sigma^_y$. The variance-minimizing value of $q$ is

# Diversification with correlated returns via an equally weighted portfolio

The expected return on a portfolio is a weighted average of the expected returns on each individual asset: : where $x_i$ is the proportion of the investor's total invested wealth in asset $i$. The variance of the portfolio return is given by: : Inserting in the expression for : :$\sigma^_ = \mathbb\left\left[\sum^_x_i R_i - \sum^_x_i\mathbb\left[R_i\right]\right\right]^2 .$ Rearranging: :$\sigma^_ = \mathbb\left\left[\sum^_x_i\left(R_i - \mathbb\left[R_i\right]\right)\right\right]^2$ : : :$\sigma_^=\sum_^x_^\underbrace_+\sum_^\sum_^x_x_\underbrace_$ :$\sigma^_ = \sum^_ x^_ \sigma^_ + \sum^_ \sum^_ x_i x_j \sigma_$ where $\sigma^_$ is the variance on asset $i$ and $\sigma_$ is the covariance between assets $i$ and $j$. In an equally weighted portfolio, $x_i = x_j = \frac , \forall i, j$. The portfolio variance then becomes: :$\sigma^2_P = \frac \ \bar^_ + n\left(n-1\right) \frac \frac \bar_$ where $\bar_$ is the average of the covariances $\sigma_$ for $i\neq j$ and $\bar^2_i$ is the average of the variances. Simplifying, we obtain :$\sigma^_ = \frac \bar^_ + \frac \bar_ .$ As the number of assets grows we get the asymptotic formula: :$\lim_ \sigma^2_P = \bar_.$ 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

The
capital asset pricing model In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a Diversification (finance), well-diversified Portfolio (f ...
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 In finance and economics, systematic risk (in economics often called aggregate risk or undiversifiable risk) is vulnerability to events which affect aggregate outcomes such as broad Financial market, market returns, total economy-wide resource ho ...
,
beta Beta (, ; uppercase , lowercase , or cursive Greek, cursive ; grc, βῆτα, bē̂ta or ell, βήτα, víta) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Modern Greek, it represents th ...
risk and
market risk Market risk is the risk of losses in positions arising from movements in market variables like prices and volatility. There is no unique classification as each classification may refer to different aspects of market risk. Nevertheless, the most ...
. If one buys all the stocks in the
S&P 500 The Standard and Poor's 500, or simply the S&P 500, is a stock market index tracking the stock performance of 500 large companies listed on stock exchanges in the United States. It is one of the most commonly followed equity indices. As of D ...
one is obviously exposed only to movements in that
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
. 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.

# An empirical example relating diversification to risk reduction

In 1977 Edwin Elton and Martin Gruber 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.

# Corporate diversification strategies

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

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: A paper by Vanguard Investment Counseling & Research explores the collected research on this topic further, in general supporting Norstad's conclusion, but allowing for the counteracted effects of inflation risk and human capital:

# History

Diversification is mentioned in the
Bible The Bible (from Koine Greek , , 'the books') is a collection of religious texts or scriptures that are held to be sacredness, sacred in Christianity, Judaism, Samaritanism, and many other religions. The Bible is an anthologya compilation of ...
, in the book of
Ecclesiastes Ecclesiastes (; hbo, קֹהֶלֶת, Qōheleṯ, grc, Ἐκκλησιαστής, Ekklēsiastēs) is one of the Ketuvim ("Writings") of the Hebrew Bible and part of the Wisdom literature of the Christianity, Christian Old Testament. The titl ...
which was written in approximately 935 B.C.: :But divide your investments among many places, :for you do not know what risks might lie ahead. Diversification is also mentioned in the
Talmud The Talmud (; he, , Talmūḏ) is the central text of Rabbinic Judaism and the primary source of Jewish religious law (''halakha'') and Jewish theology. Until the advent of modernity, in nearly all Jewish communities, the Talmud was the center ...
. 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 Real estate is property consisting of land and the buildings on it, along with its natural resources such as crops, minerals or water; immovable property of this nature; an interest vested in this (also) an item of real property, (more general ...
). 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's ''
Merchant of Venice ''The Merchant of Venice'' is a play by William Shakespeare William Shakespeare ( 26 April 1564 – 23 April 1616) was an English playwright, poet and actor. He is widely regarded as the greatest writer in the English language and the ...
'' (ca. 1599): :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 Harry Max Markowitz (born August 24, 1927) is an American economist who received the 1989 John von Neumann Theory Prize and the 1990 Nobel Memorial Prize in Economic Sciences. Markowitz is a professor of finance at the Rady School of Management ...
in the 1950s, whose work pioneered
modern portfolio theory Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of Diversificatio ...
(see Markowitz model). An earlier precedent for diversification was economist
John Maynard Keynes John Maynard Keynes, 1st Baron Keynes, ( ; 5 June 1883 – 21 April 1946), was an English economist whose ideas fundamentally changed the theory and practice of macroeconomics and the economic policies of governments. Originally trained in m ...
, who managed the endowment of
King's College, Cambridge King's College is a List of colleges of the University of Cambridge, constituent college of the University of Cambridge. Formally The King's College of Our Lady and Saint Nicholas in Cambridge, the college lies beside the River Cam and faces ...
from the 1920s to his 1946 death with a stock-selection strategy similar to what was later called
value investing Value investing is an investment investor profile, paradigm that involves buying security (finance), securities that appear underpriced by some form of fundamental analysis. The various forms of value investing derive from the investment philoso ...
. 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 avoiding home bias at a time when university endowments in the U.S. and U.K. were invested almost entirely in domestic assets.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.

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Central limit theorem In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
* Coherent risk measure * Dollar cost averaging * Equity repositioning * Financial correlation *
List of finance topics The following Outline (list), outline is provided as an overview of and topical guide to finance: Finance – addresses the ways in which individuals and organizations raise and allocate monetary factors of production, resources over time, ta ...
*
Modern portfolio theory Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of Diversificatio ...
*
Systematic risk In finance and economics, systematic risk (in economics often called aggregate risk or undiversifiable risk) is vulnerability to events which affect aggregate outcomes such as broad Financial market, market returns, total economy-wide resource ho ...