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Diversification (finance)
Encyclopedia
In finance, diversification means reducing risk
by investing
in a variety of asset
s. If the asset values do not move up and down in perfect synchrony, a diversified portfolio
will have less risk than the weighted average
risk of its constituent assets, and often less risk than the least risky of its constituents. 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.
Diversification is one of two general techniques for reducing investment risk. The other is hedging
. Diversification relies on the lack of a tight positive relationship among the assets' returns, and works even when correlations are near zero or somewhat positive. Hedging relies on negative correlation among assets, or shorting assets with positive correlation.
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.
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.
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 stocks
) it is still less likely.
Further diversification can be obtained by investing in stocks from different countries, and in different asset classes such as bonds
, real estate
, private equity
, infrastructure
and commodities such as heating oil
or gold
.
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.
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. But risk averse investors may find it beneficial to diversify into assets with lower expected returns, thereby lowering the expected return on the portfolio, when the risk-reduction benefit of doing so exceeds the cost in terms of diminished expected return.
.” 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 asset
s. This is the idea underlying index fund
s.
One objection to that is it means avoiding investments like 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.
“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.
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
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 necessarily 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 
. Then the portfolio return's variance equals 
= 
= 
, which is monotonically decreasing in 
.
The latter analysis can be adapted to show why adding uncorrelated risky 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
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.
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 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 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.
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.
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
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.
, in the book of Ecclesiastes
which was written in approximately 935 B.C.:
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):
The modern understanding of diversification dates back to the work of Harry Markowitz
in the 1950s.
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:
And simplifying:
Now, taking the number of assets,
, to the limit gives:
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.
Financial risk
Financial risk an umbrella term for multiple types of risk associated with financing, including financial transactions that include company loans in risk of default. Risk is a term often used to imply downside risk, meaning the uncertainty of a return and the potential for financial loss...
by investing
Investment
Investment has different meanings in finance and economics. Finance investment is putting money into something with the expectation of gain, that upon thorough analysis, has a high degree of security for the principal amount, as well as security of return, within an expected period of time...
in a variety of asset
Asset
In financial accounting, assets are economic resources. Anything tangible or intangible that is capable of being owned or controlled to produce value and that is held to have positive economic value is considered an asset...
s. If the asset values do not move up and down in perfect synchrony, a diversified portfolio
Portfolio (finance)
Portfolio is a financial term denoting a collection of investments held by an investment company, hedge fund, financial institution or individual.-Definition:The term portfolio refers to any collection of financial assets such as stocks, bonds and cash...
will have less risk than the weighted average
Weighted mean
The weighted mean is similar to an arithmetic mean , where instead of each of the data points contributing equally to the final average, some data points contribute more than others...
risk of its constituent assets, and often less risk than the least risky of its constituents. 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.
Diversification is one of two general techniques for reducing investment risk. The other is hedging
Hedge (finance)
A hedge is an investment position intended to offset potential losses that may be incurred by a companion investment.A hedge can be constructed from many types of financial instruments, including stocks, exchange-traded funds, insurance, forward contracts, swaps, options, many types of...
. Diversification relies on the lack of a tight positive relationship among the assets' returns, and works even when correlations are near zero or somewhat positive. Hedging relies on negative correlation among assets, or shorting assets with positive correlation.
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
Stock
The capital stock of a business entity represents the original capital paid into or invested in the business by its founders. It serves as a security for the creditors of a business since it cannot be withdrawn to the detriment of the creditors...
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.
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.
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.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
Growth stock
In finance, a growth stock is a stockof a company that generates substantial and sustainable positive cash flow and whose revenues and earnings are expected to increase at a faster rate than the average company within the same industry...
s and some value stocks
Value investing
Value investing is an investment paradigm that derives from the ideas on investment and speculation that Ben Graham and David Dodd began teaching at Columbia Business School in 1928 and subsequently developed in their 1934 text Security Analysis...
) it is still less likely.
Further diversification can be obtained by investing in stocks from different countries, and in different asset classes such as bonds
Bond (finance)
In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...
, real estate
Real estate
In general use, esp. North American, 'real estate' is taken to mean "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; an item of real property; buildings or...
, private equity
Private equity
Private equity, in finance, is an asset class consisting of equity securities in operating companies that are not publicly traded on a stock exchange....
, infrastructure
Infrastructure
Infrastructure is basic physical and organizational structures needed for the operation of a society or enterprise, or the services and facilities necessary for an economy to function...
and commodities such as heating oil
Heating oil
Heating oil, or oil heat, is a low viscosity, flammable liquid petroleum product used as a fuel for furnaces or boilers in buildings. Home heating oil is often abbreviated as HHO...
or gold
Gold
Gold is a chemical element with the symbol Au and an atomic number of 79. Gold is a dense, soft, shiny, malleable and ductile metal. Pure gold has a bright yellow color and luster traditionally considered attractive, which it maintains without oxidizing in air or water. Chemically, gold is a...
.
Since the mid-1970s, it has also been argued that geographic diversification would generate superior risk-adjusted returns for large institutional investor
Institutional investor
Institutional investors are organizations which pool large sums of money and invest those sums in securities, real property and other investment assets...
s by reducing overall portfolio risk while capturing some of the higher rates of return offered by the emerging markets
Emerging markets
Emerging markets are nations with social or business activity in the process of rapid growth and industrialization. Based on data from 2006, there are around 28 emerging markets in the world . The economies of China and India are considered to be the largest...
of Asia and Latin America.
Return expectations while diversifying
If the prior expectationsExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of the returns on all assets in the portfolio are identical, the expected return
Expected return
The expected return is the weighted-average outcome in gambling, probability theory, economics or finance.It isthe average of a probability distribution of possible returns, calculated by using the following formula:...
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. But risk averse investors may find it beneficial to diversify into assets with lower expected returns, thereby lowering the expected return on the portfolio, when the risk-reduction benefit of doing so exceeds the cost in terms of diminished expected return.
Maximum diversification
Given the advantages of diversification, many experts recommend maximum diversification, also known as “buying the market portfolioMarket 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....
.” Unfortunately, identifying that portfolio is not straightforward.
The earliest definition comes from the capital asset pricing model
Capital asset pricing model
In finance, the capital asset pricing model is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk...
which argues the maximum diversification comes from buying a pro rata share of all available asset
Asset
In financial accounting, assets are economic resources. Anything tangible or intangible that is capable of being owned or controlled to produce value and that is held to have positive economic value is considered an asset...
s. This is the idea underlying index fund
Index fund
An index fund or index tracker is a collective investment scheme that aims to replicate the movements of an index of a specific financial market, or a set of rules of ownership that are held constant, regardless of market conditions.-Tracking:Tracking can be achieved by trying to hold all of the...
s.
One objection to that is it means avoiding investments like futures
Futures contract
In finance, a futures contract is a standardized contract between two parties to exchange a specified asset of standardized quantity and quality for a price agreed today with delivery occurring at a specified future date, the delivery date. The contracts are traded on a futures exchange...
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.
“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.
Effect of diversification on variance
One simple measure of financial riskFinancial risk
Financial risk an umbrella term for multiple types of risk associated with financing, including financial transactions that include company loans in risk of default. Risk is a term often used to imply downside risk, meaning the uncertainty of a return and the potential for financial loss...
is variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
. 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 necessarily 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 . Then the portfolio return's variance equals = = , which is monotonically decreasing in .The latter analysis can be adapted to show why adding uncorrelated risky 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
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.Diversifiable and non-diversifiable risk
The Capital Asset Pricing ModelCapital asset pricing model
In finance, the capital asset pricing model is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk...
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
Systematic risk
In finance, systematic risk, sometimes called market risk, aggregate risk, or undiversifiable risk, is the risk associated with aggregate market returns....
, beta risk and market risk
Market risk
Market risk is the risk that the value of a portfolio, either an investment portfolio or a trading portfolio, will decrease due to the change in value of the market risk factors. The four standard market risk factors are stock prices, interest rates, foreign exchange rates, and commodity prices...
.
If one buys all the stocks in the S&P 500
S&P 500
The S&P 500 is a free-float capitalization-weighted index published since 1957 of the prices of 500 large-cap common stocks actively traded in the United States. The stocks included in the S&P 500 are those of large publicly held companies that trade on either of the two largest American stock...
one is obviously exposed only to movements in that index
Index (economics)
In economics and finance, an index is a statistical measure of changes in a representative group of individual data points. These data may be derived from any number of sources, including company performance, prices, productivity, and employment. Economic indices track economic health from...
. 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.
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
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....
of all assets.
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 Elton and Gruber 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.| 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 |
| 1000 | 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.
History
Diversification is mentioned in the BibleBible
The Bible refers to any one of the collections of the primary religious texts of Judaism and Christianity. There is no common version of the Bible, as the individual books , their contents and their order vary among denominations...
, in the book of Ecclesiastes
Ecclesiastes
The Book of Ecclesiastes, called , is a book of the Hebrew Bible. The English name derives from the Greek translation of the Hebrew title.The main speaker in the book, identified by the name or title Qoheleth , introduces himself as "son of David, king in Jerusalem." The work consists of personal...
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
Talmud
The Talmud is a central text of mainstream Judaism. It takes the form of a record of rabbinic discussions pertaining to Jewish law, ethics, philosophy, customs and history....
. 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
In general use, esp. North American, 'real estate' is taken to mean "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; an item of real property; buildings or...
).
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 Harry Markowitz
Harry Markowitz
Harry Max Markowitz is an American economist and a recipient of the John von Neumann Theory Prize and the Nobel Memorial Prize in Economic Sciences....
in the 1950s.
Diversification with an equally-weighted portfolio
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:
And simplifying:
Now, taking the number of assets,
, to the limit gives: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.
See also
- Modern portfolio theoryModern portfolio theoryModern portfolio theory is a theory of investment which attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets...
- Central limit theoremCentral limit theoremIn probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
- Dollar cost averagingDollar cost averagingDollar cost averaging is an investment strategy that may be used with any currency. It takes the form of investing equal monetary amounts regularly and periodically over specific time periods in a particular investment or portfolio. By doing so, more shares are purchased when prices are low and...
- Systematic riskSystematic riskIn finance, systematic risk, sometimes called market risk, aggregate risk, or undiversifiable risk, is the risk associated with aggregate market returns....
- List of finance topics
External links
- Macro-Investment Analysis, Prof. William F. SharpeWilliam Forsyth SharpeWilliam Forsyth Sharpe is the STANCO 25 Professor of Finance, Emeritus at Stanford University's Graduate School of Business and the winner of the 1990 Nobel Memorial Prize in Economic Sciences....
, Stanford UniversityStanford UniversityThe Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San... - Portfolio Diversifier, Dynamically-generated diversified portfolios
- Asset Correlations, Dynamically-generated correlation matrices for the major asset classes
- An Introduction to Investment Theory, Prof. William N. Goetzmann, Yale School of ManagementYale School of ManagementThe Yale School of Management is the graduate business school of Yale University and is located on Hillhouse Avenue in New Haven, Connecticut, United States. The School offers Master of Business Administration and Ph.D. degree programs. As of January 2011, 454 students were enrolled in its MBA...
- Overview of Managed Futures