Մասնակից:Anahit Israyelyan/Ավազարկղ

Դիվերսիֆիկացիա, կապիտալի տեղաբաշխում այնպես, որ կախվածությունը մեկ ակտիվից, դրա ռիսկից նվազում է: Դիվերսիֆիկացիայի ամենատարածված եղանակներից է տարբեր ակտիվներում ներդրումներ կատարելը, որի արդյունքում նվազում է մեկ ակտիվի ռիսկից՝ գնային տատանումներից ամբողջ պորտֆելի կախվածությունը: Եթե պորտֆելում ներառվոծ ակտիվների գնային տատանումները ժամանակի մեջ կատարյալ կերպով չեն համընկնում, ապա պորտֆելի վարիացիան ավելի փոքր կլինի դրա առանձին ակտիվների վարիացիաների միջին կշռվածից, և հաճախ պորտֆելում ներարառված առանձին ակտիվների վարիացիաներից:

Դիվերսիֆիկացիան ներդրումային ռիսկի նվազեցման երկու հիմնական եղանակներից մեկն է․ մյուսը հեջավորումն է:

Օրինակներ

Ամենապարզ եղանակը դիվերսիֆիկացված պորտվել կառուցելու տարբեր ակտիվներում ներդրումներ կատարելն է՝ առանց առանձին ակտիվների առանձնահատկություններին ու դրանց կապին անդրադառնալու: Այս դեպքում մի ակտիվի գնի տատանումը ամբողջ պորտֆելի վրա էական ազդեցություն չի ունենում:

1970֊ականներից ի վեր քննարկումների առիթ է նաև այն տեսակետը, որ աշխարհագրական դիվերսիֆիկացիան կարող է էականորեն նվազեցնելով պորտֆելի ռիսկը ապահովելբարձր եկամտաբերություն:

Դիվերսիֆիկացիայի եղանակներից է նաև Մարկովիցի պորտֆելի կառուցումը:

Դիվերսիֆիկացման չափը

Չկա որևէ հստակ քանակ, թե որքան տեսակի ակտիվներում պետք է կատարվի ներդրում լավագույնս դիվերսիֆիկացված պորտֆել ունենալու համար: Այդ քանակը կարող է լինել 30 կամ 10 կամ ցանկացած այլ քանակ․ առավել կարևոր է ակտիվների ընտրությունը, քան քանակը: Հիմնականում տեսակետն այն է, որ դիվերսիֆիկացման հիմնական արդյունքը ստացվում է 15-ից 20 ակտիվում ներդրման արդյունքում, դրանից ավել տեսակներում ներդրումը ավելի քիչ չափով է ազդում ռիսկի նվազեցման չափի վրա:

Given the advantages of diversification, many experts recommend maximum diversification, also known as "buying the market portfolio". Unfortunately, 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. 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 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 $x$ and asset Y have stochastic return $y$, with respective return variances $\sigma _{x}^{2}$ and $\sigma _{y}^{2}$. 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 $qx+(1-q)y$. If $x$ and $y$ are uncorrelated, the variance of portfolio return is ${\text{var}}(qx+(1-q)y)=q^{2}\sigma _{x}^{2}+(1-q)^{2}\sigma _{y}^{2}$. The variance-minimizing value of $q$ is $q=\sigma _{y}^{2}/[\sigma _{x}^{2}+\sigma _{y}^{2}]$, which is strictly between $0$ and $1$. Using this value of $q$ in the expression for the variance of portfolio return gives the latter as $\sigma _{x}^{2}\sigma _{y}^{2}/[\sigma _{x}^{2}+\sigma _{y}^{2}]$, which is less than what it would be at either of the undiversified values $q=1$ and $q=0$ (which respectively give portfolio return variance of $\sigma _{x}^{2}$ and $\sigma _{y}^{2}$). Note that the favorable effect of diversification on portfolio variance would be enhanced if $x$ and $y$ 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 of $n$, the number of assets. For example, if all assets' returns are mutually uncorrelated and have identical variances $\sigma _{x}^{2}$, portfolio variance is minimized by holding all assets in the equal proportions $1/n$. Then the portfolio return's variance equals ${\text{var}}[(1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n}]$ = $n(1/n^{2})\sigma _{x}^{2}$ = $\sigma _{x}^{2}/n$, which is monotonically decreasing in $n$.

The latter analysis can be adapted to show why adding uncorrelated 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 is $x_{1}+x_{2}+\dots +x_{n}$ instead of $(1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n},$ and the variance of the portfolio return if the assets are uncorrelated is ${\text{var}}[x_{1}+x_{2}+\dots +x_{n}]=\sigma _{x}^{2}+\sigma _{x}^{2}+\dots +\sigma _{x}^{2}=n\sigma _{x}^{2},$ 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

The expected return on a portfolio is a weighted average of the expected returns on each individual asset:

$\mathbb {E} [R_{P}]=\sum _{i=1}^{n}x_{i}\mathbb {E} [R_{i}]$

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:

$\underbrace {{\text{Var}}(R_{P})} _{\equiv \sigma _{P}^{2}}=\mathbb {E} [R_{P}-\mathbb {E} [R_{P}]]^{2}.$

Inserting in the expression for $\mathbb {E} [R_{P}]$:

$\sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}x_{i}R_{i}-\sum _{i=1}^{n}x_{i}\mathbb {E} [R_{i}]\right]^{2}.$

Rearranging:

$\sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}x_{i}(R_{i}-\mathbb {E} [R_{i}])\right]^{2}$

$\sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}\sum _{j=1}^{n}x_{i}x_{j}(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])\right]$

$\sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}x_{i}^{2}(R_{i}-\mathbb {E} [R_{i}])^{2}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])\right]$

$\sigma _{P}^{2}=\sum _{i=1}^{n}x_{i}^{2}\underbrace {\mathbb {E} \left[R_{i}-\mathbb {E} [R_{i}]\right]^{2}} _{\equiv \sigma _{i}^{2}}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}\underbrace {\mathbb {E} \left[(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])\right]} _{\equiv \sigma _{ij}}$

$\sigma _{P}^{2}=\sum _{i=1}^{n}x_{i}^{2}\sigma _{i}^{2}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}\sigma _{ij}$

where $\sigma _{i}^{2}$ is the variance on asset $i$ and $\sigma _{ij}$ is the covariance between assets $i$ and $j$.

In an equally weighted portfolio, $x_{i}=x_{j}={\frac {1}{n}},\forall i,j$. The portfolio variance then becomes:

$\sigma _{P}^{2}={\frac {1}{n^{2}}}\ {n}{\bar {\sigma }}_{i}^{2}+n(n-1){\frac {1}{n}}{\frac {1}{n}}{\bar {\sigma }}_{ij}$

where ${\bar {\sigma }}_{ij}$ is the average of the covariances $\sigma _{ij}$ for $i\neq j$ and ${\bar {\sigma }}_{i}^{2}$ is the average of the variances. Simplifying, we obtain

$\sigma _{P}^{2}={\frac {1}{n}}{\bar {\sigma }}_{i}^{2}+{\frac {n-1}{n}}{\bar {\sigma }}_{ij}.$

As the number of assets grows we get the asymptotic formula:

$\lim _{n\rightarrow \infty }\sigma _{P}^{2}={\bar {\sigma }}_{ij}.$

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 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.

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.

Number of Stocks in Portfolio	Average Standard Deviation of Annual Portfolio Returns	Ratio of Portfolio Standard Deviation to Standard Deviation of a Single Stock
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

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 phrasing: "At your young age, you have enough time to recover from any dips in the market, so you can safely ignore bonds and go with an all stock retirement portfolio." As John Norstad explains:

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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:

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History

Diversification is mentioned in the Bible, in the book of Ecclesiastes 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 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 mentioned in Shakespeare (Merchant of Venice):

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.

The modern understanding of diversification dates back to the work of economist Harry Markowitz in the 1950s, 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. While diversification in the modern sense was "not easily available in Keynes's day" and Keynes typically held a small number of assets, 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.

See also

Asset Allocation on Wikibook

• Central limit theorem
• Coherent risk measure
• Dollar cost averaging
• Equity repositioning
• Financial correlation
• List of finance topics
• Modern portfolio theory
• Systematic risk

External links

• Macro-Investment Analysis, Prof. William F. Sharpe, Stanford University
• An Introduction to Investment Theory, Prof. William N. Goetzmann, Yale School of Management