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100-439: The primary benefit of this arrangement is to diversify a large stock position without triggering a " taxable event ". Note that the tax is not avoided, just deferred. Deferring taxes avoids tax drag, as the money lost to taxes remains invested in the market, letting the portfolio compound from a larger base, which could create a significant advantage with time. When the diversified holdings are eventually sold, tax will be due on

200-521: A b x f ( x ) d x = ∫ a b x x 2 + π 2 d x = 1 2 ln ⁡ b 2 + π 2 a 2 + π 2 . {\displaystyle \int _{a}^{b}xf(x)\,dx=\int _{a}^{b}{\frac {x}{x^{2}+\pi ^{2}}}\,dx={\frac {1}{2}}\ln {\frac {b^{2}+\pi ^{2}}{a^{2}+\pi ^{2}}}.} The limit of this expression as

300-401: A weighted average of the x i values, with weights given by their probabilities p i . In the special case that all possible outcomes are equiprobable (that is, p 1 = ⋅⋅⋅ = p k ), the weighted average is given by the standard average . In the general case, the expected value takes into account the fact that some outcomes are more likely than others. Informally,

400-421: A → −∞ and b → ∞ does not exist: if the limits are taken so that a = − b , then the limit is zero, while if the constraint 2 a = − b is taken, then the limit is ln(2) . To avoid such ambiguities, in mathematical textbooks it is common to require that the given integral converges absolutely , with E[ X ] left undefined otherwise. However, measure-theoretic notions as given below can be used to give

500-435: 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 ". Identifying that portfolio is not straightforward. The earliest definition comes from the capital asset pricing model which argues

600-486: A broader range of investors. Exchange funds became popular after Eaton Vance obtained a private ruling from the IRS in 1975 allowing their use. The U.S. Securities and Exchange Commission has investigated the use of these arrangements with reference to the potential for market abuse by directors not disclosing their effective divestment in stocks for which they are privy to sensitive market information. In addition, there

700-413: A function of n {\displaystyle n} , the number of assets. For example, if all assets' returns are mutually uncorrelated and have identical variances σ x 2 {\displaystyle \sigma _{x}^{2}} , portfolio variance is minimized by holding all assets in the equal proportions 1 / n {\displaystyle 1/n} . Then

800-488: A multidimensional random variable, i.e. a random vector X . It is defined component by component, as E[ X ] i = E[ X i ] . Similarly, one may define the expected value of a random matrix X with components X ij by E[ X ] ij = E[ X ij ] . Consider a random variable X with a finite list x 1 , ..., x k of possible outcomes, each of which (respectively) has probability p 1 , ..., p k of occurring. The expectation of X

900-570: A real number μ {\displaystyle \mu } if and only if the two surfaces in the x {\displaystyle x} - y {\displaystyle y} -plane, described by x ≤ μ , 0 ≤ y ≤ F ( x ) or x ≥ μ , F ( x ) ≤ y ≤ 1 {\displaystyle x\leq \mu ,\;\,0\leq y\leq F(x)\quad {\text{or}}\quad x\geq \mu ,\;\,F(x)\leq y\leq 1} respectively, have

1000-459: A small circle of mutual scientific friends in Paris about it. In Dutch mathematician Christiaan Huygens' book, he considered the problem of points, and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens published his treatise in 1657, (see Huygens (1657) ) " De ratiociniis in ludo aleæ " on probability theory just after visiting Paris. The book extended

1100-479: 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

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1200-616: A systematic definition of E[ X ] for more general random variables X . All definitions of the expected value may be expressed in the language of measure theory . In general, if X is a real-valued random variable defined on a probability space (Ω, Σ, P) , then the expected value of X , denoted by E[ X ] , is defined as the Lebesgue integral E ⁡ [ X ] = ∫ Ω X d P . {\displaystyle \operatorname {E} [X]=\int _{\Omega }X\,d\operatorname {P} .} Despite

1300-399: A time when university endowments in the U.S. and U.K. were invested almost entirely in domestic assets. Expected value In probability theory , the expected value (also called expectation , expectancy , expectation operator , mathematical expectation , mean , expectation value , or first moment ) is a generalization of the weighted average . Informally, the expected value

1400-507: A value in any given open interval is given by the integral of f over that interval. The expectation of X is then given by the integral E ⁡ [ X ] = ∫ − ∞ ∞ x f ( x ) d x . {\displaystyle \operatorname {E} [X]=\int _{-\infty }^{\infty }xf(x)\,dx.} A general and mathematically precise formulation of this definition uses measure theory and Lebesgue integration , and

1500-469: A variety of stylizations: the expectation operator can be stylized as E (upright), E (italic), or E {\displaystyle \mathbb {E} } (in blackboard bold ), while a variety of bracket notations (such as E( X ) , E[ X ] , and E X ) are all used. Another popular notation is μ X . ⟨ X ⟩ , ⟨ X ⟩ av , and X ¯ {\displaystyle {\overline {X}}} are commonly used in physics. M( X )

1600-426: Is q = σ y 2 / [ σ x 2 + σ y 2 ] {\displaystyle q=\sigma _{y}^{2}/[\sigma _{x}^{2}+\sigma _{y}^{2}]} , which is strictly between 0 {\displaystyle 0} and 1 {\displaystyle 1} . Using this value of q {\displaystyle q} in

1700-404: 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. The expected return on a portfolio is a weighted average of

1800-640: Is a Borel function ), we can use this inversion formula to obtain E ⁡ [ g ( X ) ] = 1 2 π ∫ R g ( x ) [ ∫ R e − i t x φ X ( t ) d t ] d x . {\displaystyle \operatorname {E} [g(X)]={\frac {1}{2\pi }}\int _{\mathbb {R} }g(x)\left[\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt\right]dx.} If E ⁡ [ g ( X ) ] {\displaystyle \operatorname {E} [g(X)]}

1900-546: Is any random variable with finite expectation, then Markov's inequality may be applied to the random variable | X −E[ X ]| to obtain Chebyshev's inequality P ⁡ ( | X − E [ X ] | ≥ a ) ≤ Var ⁡ [ X ] a 2 , {\displaystyle \operatorname {P} (|X-{\text{E}}[X]|\geq a)\leq {\frac {\operatorname {Var} [X]}{a^{2}}},} where Var

2000-462: Is as in the previous example. A number of convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below. The probability density function f X {\displaystyle f_{X}} of a scalar random variable X {\displaystyle X} is related to its characteristic function φ X {\displaystyle \varphi _{X}} by

2100-635: Is called the probability density function of X (relative to Lebesgue measure). According to the change-of-variables formula for Lebesgue integration, combined with the law of the unconscious statistician , it follows that E ⁡ [ X ] ≡ ∫ Ω X d P = ∫ R x f ( x ) d x {\displaystyle \operatorname {E} [X]\equiv \int _{\Omega }X\,d\operatorname {P} =\int _{\mathbb {R} }xf(x)\,dx} for any absolutely continuous random variable X . The above discussion of continuous random variables

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2200-408: Is defined as E ⁡ [ X ] = x 1 p 1 + x 2 p 2 + ⋯ + x k p k . {\displaystyle \operatorname {E} [X]=x_{1}p_{1}+x_{2}p_{2}+\cdots +x_{k}p_{k}.} Since the probabilities must satisfy p 1 + ⋅⋅⋅ + p k = 1 , it is natural to interpret E[ X ] as

2300-407: Is easily obtained by setting Y 0 = X 1 {\displaystyle Y_{0}=X_{1}} and Y n = X n + 1 − X n {\displaystyle Y_{n}=X_{n+1}-X_{n}} for n ≥ 1 , {\displaystyle n\geq 1,} where X n {\displaystyle X_{n}}

2400-564: Is equivalent to the representation E ⁡ [ X ] = ∫ 0 ∞ ( 1 − F ( x ) ) d x − ∫ − ∞ 0 F ( x ) d x , {\displaystyle \operatorname {E} [X]=\int _{0}^{\infty }{\bigl (}1-F(x){\bigr )}\,dx-\int _{-\infty }^{0}F(x)\,dx,} also with convergent integrals. Expected values as defined above are automatically finite numbers. However, in many cases it

2500-496: 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 investors by reducing overall portfolio risk while capturing some of the higher rates of return offered by the emerging markets of Asia and Latin America. If

2600-436: Is finite if and only if E[ X ] and E[ X ] are both finite. Due to the formula | X | = X + X , this is the case if and only if E| X | is finite, and this is equivalent to the absolute convergence conditions in the definitions above. As such, the present considerations do not define finite expected values in any cases not previously considered; they are only useful for infinite expectations. The following table gives

2700-626: Is finite, changing the order of integration, we get, in accordance with Fubini–Tonelli theorem , E ⁡ [ g ( X ) ] = 1 2 π ∫ R G ( t ) φ X ( t ) d t , {\displaystyle \operatorname {E} [g(X)]={\frac {1}{2\pi }}\int _{\mathbb {R} }G(t)\varphi _{X}(t)\,dt,} where G ( t ) = ∫ R g ( x ) e − i t x d x {\displaystyle G(t)=\int _{\mathbb {R} }g(x)e^{-itx}\,dx}

2800-1055: Is fundamental to be able to consider expected values of ±∞ . This is intuitive, for example, in the case of the St. Petersburg paradox , in which one considers a random variable with possible outcomes x i = 2 , with associated probabilities p i = 2 , for i ranging over all positive integers. According to the summation formula in the case of random variables with countably many outcomes, one has E ⁡ [ X ] = ∑ i = 1 ∞ x i p i = 2 ⋅ 1 2 + 4 ⋅ 1 4 + 8 ⋅ 1 8 + 16 ⋅ 1 16 + ⋯ = 1 + 1 + 1 + 1 + ⋯ . {\displaystyle \operatorname {E} [X]=\sum _{i=1}^{\infty }x_{i}\,p_{i}=2\cdot {\frac {1}{2}}+4\cdot {\frac {1}{4}}+8\cdot {\frac {1}{8}}+16\cdot {\frac {1}{16}}+\cdots =1+1+1+1+\cdots .} It

2900-560: Is general public policy disagreement whether tax revenue that is generated from exchange funds and other like-kind exchanges should be deferred or avoided. Many holders of appreciated positions may elect to hold the concentrated position and borrow against it rather than sell and pay the associated capital gains tax, which results in deadweight loss to the economy. Proponents argue that exchange funds help with this significant deadweight loss as holders of appreciated stock can diversify and liquidate their positions, re-injecting this capital into

3000-795: Is monotonically decreasing in n {\displaystyle 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 + ⋯ + x n {\displaystyle x_{1}+x_{2}+\dots +x_{n}} instead of ( 1 / n ) x 1 + ( 1 / n ) x 2 + . . . + ( 1 / n ) x n , {\displaystyle (1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n},} and

3100-472: Is natural to say that the expected value equals +∞ . There is a rigorous mathematical theory underlying such ideas, which is often taken as part of the definition of the Lebesgue integral. The first fundamental observation is that, whichever of the above definitions are followed, any nonnegative random variable whatsoever can be given an unambiguous expected value; whenever absolute convergence fails, then

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3200-482: 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 (ca. 1599): Modern understanding of diversification dates back to the influential work of economist Harry Markowitz in

3300-642: Is otherwise available. For example, in the case of an unweighted dice, Chebyshev's inequality says that odds of rolling between 1 and 6 is at least 53%; in reality, the odds are of course 100%. The Kolmogorov inequality extends the Chebyshev inequality to the context of sums of random variables. The following three inequalities are of fundamental importance in the field of mathematical analysis and its applications to probability theory. The Hölder and Minkowski inequalities can be extended to general measure spaces , and are often given in that context. By contrast,

3400-604: Is the Fourier transform of g ( x ) . {\displaystyle g(x).} The expression for E ⁡ [ g ( X ) ] {\displaystyle \operatorname {E} [g(X)]} also follows directly from the Plancherel theorem . The expectation of a random variable plays an important role in a variety of contexts. In statistics , where one seeks estimates for unknown parameters based on available data gained from samples ,

3500-416: Is the mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would "expect" to get in reality. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In

3600-482: Is the variance . These inequalities are significant for their nearly complete lack of conditional assumptions. For example, for any random variable with finite expectation, the Chebyshev inequality implies that there is at least a 75% probability of an outcome being within two standard deviations of the expected value. However, in special cases the Markov and Chebyshev inequalities often give much weaker information than

3700-459: Is the average of the covariances σ i j {\displaystyle \sigma _{ij}} for i ≠ j {\displaystyle i\neq j} and σ ¯ i 2 {\displaystyle {\bar {\sigma }}_{i}^{2}} is the average of the variances. Simplifying, we obtain As the number of assets grows we get

3800-421: 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 or volatility by investing in a variety of assets . If asset prices do not change in perfect synchrony, a diversified portfolio will have less variance than the weighted average variance of its constituent assets, and often less volatility than

3900-455: 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,

4000-421: 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 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

4100-653: Is the variance on asset i {\displaystyle i} and σ i j {\displaystyle \sigma _{ij}} is the covariance between assets i {\displaystyle i} and j {\displaystyle j} . In an equally weighted portfolio, x i = x j = 1 n , ∀ i , j {\displaystyle x_{i}=x_{j}={\frac {1}{n}},\forall i,j} . The portfolio variance then becomes: where σ ¯ i j {\displaystyle {\bar {\sigma }}_{ij}}

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4200-1673: Is then natural to define: E ⁡ [ X ] = { E ⁡ [ X + ] − E ⁡ [ X − ] if  E ⁡ [ X + ] < ∞  and  E ⁡ [ X − ] < ∞ ; + ∞ if  E ⁡ [ X + ] = ∞  and  E ⁡ [ X − ] < ∞ ; − ∞ if  E ⁡ [ X + ] < ∞  and  E ⁡ [ X − ] = ∞ ; undefined if  E ⁡ [ X + ] = ∞  and  E ⁡ [ X − ] = ∞ . {\displaystyle \operatorname {E} [X]={\begin{cases}\operatorname {E} [X^{+}]-\operatorname {E} [X^{-}]&{\text{if }}\operatorname {E} [X^{+}]<\infty {\text{ and }}\operatorname {E} [X^{-}]<\infty ;\\+\infty &{\text{if }}\operatorname {E} [X^{+}]=\infty {\text{ and }}\operatorname {E} [X^{-}]<\infty ;\\-\infty &{\text{if }}\operatorname {E} [X^{+}]<\infty {\text{ and }}\operatorname {E} [X^{-}]=\infty ;\\{\text{undefined}}&{\text{if }}\operatorname {E} [X^{+}]=\infty {\text{ and }}\operatorname {E} [X^{-}]=\infty .\end{cases}}} According to this definition, E[ X ] exists and

4300-514: Is thus a special case of the general Lebesgue theory, due to the fact that every piecewise-continuous function is measurable. The expected value of any real-valued random variable X {\displaystyle X} can also be defined on the graph of its cumulative distribution function F {\displaystyle F} by a nearby equality of areas. In fact, E ⁡ [ X ] = μ {\displaystyle \operatorname {E} [X]=\mu } with

4400-466: Is used in Russian-language literature. As discussed above, there are several context-dependent ways of defining the expected value. The simplest and original definition deals with the case of finitely many possible outcomes, such as in the flip of a coin. With the theory of infinite series, this can be extended to the case of countably many possible outcomes. It is also very common to consider

4500-403: Is worth just such a Sum, as wou'd procure in the same Chance and Expectation at a fair Lay. ... If I expect a or b, and have an equal chance of gaining them, my Expectation is worth (a+b)/2. More than a hundred years later, in 1814, Pierre-Simon Laplace published his tract " Théorie analytique des probabilités ", where the concept of expected value was defined explicitly: ... this advantage in

4600-408: The sample mean serves as an estimate for the expectation, and is itself a random variable. In such settings, the sample mean is considered to meet the desirable criterion for a "good" estimator in being unbiased ; that is, the expected value of the estimate is equal to the true value of the underlying parameter. For a different example, in decision theory , an agent making an optimal choice in

4700-503: 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

4800-531: The Jensen inequality is special to the case of probability spaces. In general, it is not the case that E ⁡ [ X n ] → E ⁡ [ X ] {\displaystyle \operatorname {E} [X_{n}]\to \operatorname {E} [X]} even if X n → X {\displaystyle X_{n}\to X} pointwise. Thus, one cannot interchange limits and expectation, without additional conditions on

4900-398: The Lebesgue theory of expectation is identical to the summation formulas given above. However, the Lebesgue theory clarifies the scope of the theory of probability density functions. A random variable X is said to be absolutely continuous if any of the following conditions are satisfied: These conditions are all equivalent, although this is nontrivial to establish. In this definition, f

5000-514: 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. 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

5100-418: 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

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5200-425: The belief investors will have time to recover from any downturns. Yet this belief has flaws, as John Norstad explains: This kind of statement makes the implicit assumption that given enough time good returns will cancel out any possible bad returns. While the basic argument that the standard deviations of the annualized returns decrease as the time horizon increases is true, it is also misleading, and it fatally misses

5300-433: The case of a continuum of possible outcomes, the expectation is defined by integration . In the axiomatic foundation for probability provided by measure theory , the expectation is given by Lebesgue integration . The expected value of a random variable X is often denoted by E( X ) , E[ X ] , or E X , with E also often stylized as E {\displaystyle \mathbb {E} } or E . The idea of

5400-507: The concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem (e.g., for three or more players), and can be seen as the first successful attempt at laying down the foundations of the theory of probability . In the foreword to his treatise, Huygens wrote: It should be said, also, that for some time some of the best mathematicians of France have occupied themselves with this kind of calculus so that no one should attribute to me

5500-462: The corresponding theory of absolutely continuous random variables is described in the next section. The density functions of many common distributions are piecewise continuous , and as such the theory is often developed in this restricted setting. For such functions, it is sufficient to only consider the standard Riemann integration . Sometimes continuous random variables are defined as those corresponding to this special class of densities, although

5600-431: The difference between the sales price and the original cost basis of the contributed stock. Historically, exchange funds have been offered primarily by two major investment firms, specifically for their ultra-wealthy clients. Morgan Stanley (through Eaton Vance ) is a prominent provider of these funds. Goldman Sachs offers exchange funds as well. Newer entrants like Cache offer exchange funds that are more accessible to

5700-490: The distinct case of random variables dictated by (piecewise-)continuous probability density functions , as these arise in many natural contexts. All of these specific definitions may be viewed as special cases of the general definition based upon the mathematical tools of measure theory and Lebesgue integration , which provide these different contexts with an axiomatic foundation and common language. Any definition of expected value may be extended to define an expected value of

5800-432: The economy. Opponents argue that exchange funds only serve a narrow slice of the population. For example, public figures like politician Mitt Romney and businessman Eli Broad have been identified as using exchange funds to reduce their tax obligations. Regulatory filings indicate that it is a frequently used strategy by high-ranking corporate executives. Diversification (finance) In finance , diversification

5900-528: 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 {\displaystyle x} and asset Y have stochastic return y {\displaystyle y} , with respective return variances σ x 2 {\displaystyle \sigma _{x}^{2}} and σ y 2 {\displaystyle \sigma _{y}^{2}} . If

6000-527: The expectation of a random variable with a countably infinite set of possible outcomes is defined analogously as the weighted average of all possible outcomes, where the weights are given by the probabilities of realizing each given value. This is to say that E ⁡ [ X ] = ∑ i = 1 ∞ x i p i , {\displaystyle \operatorname {E} [X]=\sum _{i=1}^{\infty }x_{i}\,p_{i},} where x 1 , x 2 , ... are

6100-519: The expected returns on each individual asset: where x i {\displaystyle x_{i}} is the proportion of the investor's total invested wealth in asset i {\displaystyle i} . The variance of the portfolio return is given by: Inserting in the expression for E [ R P ] {\displaystyle \mathbb {E} [R_{P}]} : Rearranging: where σ i 2 {\displaystyle \sigma _{i}^{2}}

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6200-492: The expected value can be defined as +∞ . The second fundamental observation is that any random variable can be written as the difference of two nonnegative random variables. Given a random variable X , one defines the positive and negative parts by X = max( X , 0) and X = −min( X , 0) . These are nonnegative random variables, and it can be directly checked that X = X − X . Since E[ X ] and E[ X ] are both then defined as either nonnegative numbers or +∞ , it

6300-494: The expected value operator is not σ {\displaystyle \sigma } -additive, i.e. E ⁡ [ ∑ n = 0 ∞ Y n ] ≠ ∑ n = 0 ∞ E ⁡ [ Y n ] . {\displaystyle \operatorname {E} \left[\sum _{n=0}^{\infty }Y_{n}\right]\neq \sum _{n=0}^{\infty }\operatorname {E} [Y_{n}].} An example

6400-573: The expected value originated in the middle of the 17th century from the study of the so-called problem of points , which seeks to divide the stakes in a fair way between two players, who have to end their game before it is properly finished. This problem had been debated for centuries. Many conflicting proposals and solutions had been suggested over the years when it was posed to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré in 1654. Méré claimed that this problem could not be solved and that it showed just how flawed mathematics

6500-488: The expected values of some commonly occurring probability distributions . The third column gives the expected values both in the form immediately given by the definition, as well as in the simplified form obtained by computation therefrom. The details of these computations, which are not always straightforward, can be found in the indicated references. The basic properties below (and their names in bold) replicate or follow immediately from those of Lebesgue integral . Note that

6600-785: The expression for the variance of portfolio return gives the latter as σ x 2 σ y 2 / [ σ x 2 + σ y 2 ] {\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 values q = 1 {\displaystyle q=1} and q = 0 {\displaystyle q=0} (which respectively give portfolio return variance of σ x 2 {\displaystyle \sigma _{x}^{2}} and σ y 2 {\displaystyle \sigma _{y}^{2}} ). Note that

6700-432: The favorable effect of diversification on portfolio variance would be enhanced if x {\displaystyle x} and y {\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

6800-485: The fraction q {\displaystyle q} of a one-unit (e.g. one-million-dollar) portfolio is placed in asset X and the fraction 1 − q {\displaystyle 1-q} is placed in Y, the stochastic portfolio return is q x + ( 1 − q ) y {\displaystyle qx+(1-q)y} . If x {\displaystyle x} and y {\displaystyle y} are uncorrelated,

6900-444: 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

7000-401: The honour of the first invention. This does not belong to me. But these savants, although they put each other to the test by proposing to each other many questions difficult to solve, have hidden their methods. I have had therefore to examine and go deeply for myself into this matter by beginning with the elements, and it is impossible for me for this reason to affirm that I have even started from

7100-1193: The indicator function of the event A . {\displaystyle A.} Then, it follows that X n → 0 {\displaystyle X_{n}\to 0} pointwise. But, E ⁡ [ X n ] = n ⋅ Pr ( U ∈ [ 0 , 1 n ] ) = n ⋅ 1 n = 1 {\displaystyle \operatorname {E} [X_{n}]=n\cdot \Pr \left(U\in \left[0,{\tfrac {1}{n}}\right]\right)=n\cdot {\tfrac {1}{n}}=1} for each n . {\displaystyle n.} Hence, lim n → ∞ E ⁡ [ X n ] = 1 ≠ 0 = E ⁡ [ lim n → ∞ X n ] . {\displaystyle \lim _{n\to \infty }\operatorname {E} [X_{n}]=1\neq 0=\operatorname {E} \left[\lim _{n\to \infty }X_{n}\right].} Analogously, for general sequence of random variables { Y n : n ≥ 0 } , {\displaystyle \{Y_{n}:n\geq 0\},}

7200-403: 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

7300-399: The infinite sum is a finite number independent of the ordering of summands. In the alternative case that the infinite sum does not converge absolutely, one says the random variable does not have finite expectation. Now consider a random variable X which has a probability density function given by a function f on the real number line . This means that the probability of X taking on

7400-552: The inversion formula: f X ( x ) = 1 2 π ∫ R e − i t x φ X ( t ) d t . {\displaystyle f_{X}(x)={\frac {1}{2\pi }}\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt.} For the expected value of g ( X ) {\displaystyle g(X)} (where g : R → R {\displaystyle g:{\mathbb {R} }\to {\mathbb {R} }}

7500-440: The least volatile of its constituents. Diversification is one of two general techniques for reducing investment risk. The other is hedging . 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

7600-447: The letters "a.s." stand for " almost surely "—a central property of the Lebesgue integral. Basically, one says that an inequality like X ≥ 0 {\displaystyle X\geq 0} is true almost surely, when the probability measure attributes zero-mass to the complementary event { X < 0 } . {\displaystyle \left\{X<0\right\}.} Concentration inequalities control

7700-447: The likelihood of a random variable taking on large values. Markov's inequality is among the best-known and simplest to prove: for a nonnegative random variable X and any positive number a , it states that P ⁡ ( X ≥ a ) ≤ E ⁡ [ X ] a . {\displaystyle \operatorname {P} (X\geq a)\leq {\frac {\operatorname {E} [X]}{a}}.} If X

7800-659: The literature on the fallacy of time diversification have been from Paul Samuelson , Zvi Bodie , and Mark Kritzman. Diversification is mentioned in the Bible , 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 ). This strategy of splitting wealth equally among available options

7900-464: 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

8000-432: The newly abstract situation, this definition is extremely similar in nature to the very simplest definition of expected values, given above, as certain weighted averages. This is because, in measure theory, the value of the Lebesgue integral of X is defined via weighted averages of approximations of X which take on finitely many values. Moreover, if given a random variable with finitely or countably many possible values,

8100-423: 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

8200-483: The point, because for an investor concerned with the value of his portfolio at the end of a period of time, it is the total return that matters, not the annualized return. Because of the effects of compounding, the standard deviation of the total return actually increases with time horizon. Thus, if we use the traditional measure of uncertainty as the standard deviation of return over the time period in question, uncertainty increases with time. Three notable contributions to

8300-584: The portfolio return's variance equals var [ ( 1 / n ) x 1 + ( 1 / n ) x 2 + . . . + ( 1 / n ) x n ] {\displaystyle {\text{var}}[(1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n}]} = n ( 1 / n 2 ) σ x 2 {\displaystyle n(1/n^{2})\sigma _{x}^{2}} = σ x 2 / n {\displaystyle \sigma _{x}^{2}/n} , which

8400-502: The possible outcomes of the random variable X and p 1 , p 2 , ... are their corresponding probabilities. In many non-mathematical textbooks, this is presented as the full definition of expected values in this context. However, there are some subtleties with infinite summation, so the above formula is not suitable as a mathematical definition. In particular, the Riemann series theorem of mathematical analysis illustrates that

8500-718: 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. In 1977 Edwin Elton and Martin Gruber worked out an empirical example of

8600-410: The prior 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. 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

8700-672: The random variables. To see this, let U {\displaystyle U} be a random variable distributed uniformly on [ 0 , 1 ] . {\displaystyle [0,1].} For n ≥ 1 , {\displaystyle n\geq 1,} define a sequence of random variables X n = n ⋅ 1 { U ∈ ( 0 , 1 n ) } , {\displaystyle X_{n}=n\cdot \mathbf {1} \left\{U\in \left(0,{\tfrac {1}{n}}\right)\right\},} with 1 { A } {\displaystyle \mathbf {1} \{A\}} being

8800-439: 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

8900-438: The same finite area, i.e. if ∫ − ∞ μ F ( x ) d x = ∫ μ ∞ ( 1 − F ( x ) ) d x {\displaystyle \int _{-\infty }^{\mu }F(x)\,dx=\int _{\mu }^{\infty }{\big (}1-F(x){\big )}\,dx} and both improper Riemann integrals converge. Finally, this

9000-449: The same fundamental principle. The principle is that the value of a future gain should be directly proportional to the chance of getting it. This principle seemed to have come naturally to both of them. They were very pleased by the fact that they had found essentially the same solution, and this in turn made them absolutely convinced that they had solved the problem conclusively; however, they did not publish their findings. They only informed

9100-418: The same principle. But finally I have found that my answers in many cases do not differ from theirs. In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the expectations of random variables . Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes: That any one Chance or Expectation to win any thing

9200-552: 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

9300-514: The sum hoped for. We will call this advantage mathematical hope. The use of the letter E to denote "expected value" goes back to W. A. Whitworth in 1901. The symbol has since become popular for English writers. In German, E stands for Erwartungswert , in Spanish for esperanza matemática , and in French for espérance mathématique. When "E" is used to denote "expected value", authors use

9400-464: The term is used differently by various authors. Analogously to the countably-infinite case above, there are subtleties with this expression due to the infinite region of integration. Such subtleties can be seen concretely if the distribution of X is given by the Cauchy distribution Cauchy(0, π) , so that f ( x ) = ( x + π ) . It is straightforward to compute in this case that ∫

9500-411: The theory of chance is the product of the sum hoped for by the probability of obtaining it; it is the partial sum which ought to result when we do not wish to run the risks of the event in supposing that the division is made proportional to the probabilities. This division is the only equitable one when all strange circumstances are eliminated; because an equal degree of probability gives an equal right for

9600-428: 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

9700-411: The value of certain infinite sums involving positive and negative summands depends on the order in which the summands are given. Since the outcomes of a random variable have no naturally given order, this creates a difficulty in defining expected value precisely. For this reason, many mathematical textbooks only consider the case that the infinite sum given above converges absolutely , which implies that

9800-435: The variance of portfolio return is var ( q x + ( 1 − q ) y ) = q 2 σ x 2 + ( 1 − q ) 2 σ y 2 {\displaystyle {\text{var}}(qx+(1-q)y)=q^{2}\sigma _{x}^{2}+(1-q)^{2}\sigma _{y}^{2}} . The variance-minimizing value of q {\displaystyle q}

9900-518: The variance of the portfolio return if the assets are uncorrelated is var [ x 1 + x 2 + ⋯ + x n ] = σ x 2 + σ x 2 + ⋯ + σ x 2 = n σ x 2 , {\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

10000-434: Was when it came to its application to the real world. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all. He began to discuss the problem in the famous series of letters to Pierre de Fermat . Soon enough, they both independently came up with a solution. They solved the problem in different computational ways, but their results were identical because their computations were based on

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