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Jump to: navigation, search "Portfolio analysis" redirects here. For theorems about the mean-variance efficient frontier, see  Mutual fund separation theorem. For non-mean-variance portfolio analysis, see Marginal  conditional stochastic dominance.  dominance.  Modern portfolio theory (MPT) is a theory of finance of  finance which attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for risk for a given level of expected return, by carefully choosing the proportions of various assets.  assets.  Although MPT is widely used in practice in the financial industry and several of its [1] creators won a Nobel memorial prize  for the theory, in recent years the basic assumptions of MPT have been widely challenged by fields such as behavioral economics.  economics.  MPT is a mathematical formulation of the concept of diversification of  diversification in investing, with the aim of selecting a collection of investment assets that has collectively lower risk than any individual asset. That this is possible can be seen intuitively because different types of  [2] assets often change in value in opposite ways. ways .  For example, to the extent prices in the stock market move differently from prices in the bond market, a collection of both types of  assets can in theory face lower overall risk than either individually. But diversification lowers risk even if assets' returns are not negatively correlated — indeed, indeed, even if they are [3] positively correlated. correlated.   More technically, MPT models an asset's return as a normally distributed function (or more generally as an elliptically distributed random variable), variable), defines risk  risk as as the standard deviation of return, and models a portfolio as a weighted combination of assets, so that the return of a portfolio is the weighted combination of the assets' returns. By combining different assets whose returns are not perfectly positively correlated, MPT seeks to reduce the total variance of the portfolio return. MPT also assumes that investors are rational and markets are efficient.  efficient.   MPT was developed in the 1950s through the early 1970s and was considered an important advance in the mathematical modeling of finance. Since then, many theoretical and practical criticisms have been leveled against it. These include the fact that financial returns do not follow a Gaussian distribution or indeed any symmetric distribution, and that correlations between asset classes are not fixed but can vary depending on external events (especially in crises). Further, there is growing evidence that investors are not [4][5] rational and markets are not efficient. efficient .  

Contents [hide]  



1 Concept 

 

   





 



 



   

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2 History  model   3 Mathematical model o  3.1 Risk and expected return  o  3.2 Diversification  o  3.3 The efficient frontier with no risk-free asset  3.4 The two mutual fund theorem  o  o  3.5 The risk-free asset and the capital allocation line  4 Asset pricing using MPT MPT   o  4.1 Systematic risk and specific risk   o  4.2 Capital asset pricing model   Criticisms   5 Criticisms 5.1 Assumptions  o  o  5.2 MPT does not really model the market  5.3 The MPT does not take its own effect on asset prices into account   o  6 Extensions  Applications   7 Other Applications o  7.1 Applications to project portfolios and other "non-financial" assets   o  7.2 Application to other disciplines   8 Comparison with arbitrage pricing theory   9 See also  10 References  11 Further reading  12 External links 

[edit] edit] Concept The fundamental concept behind MPT is that the  the assets  assets in an investment investment  portfolio portfolio  should not be selected individually, each on their own merits. Rather, it is important to consider how each asset changes in price relative to how every other asset in the portfolio changes in price. Investing is a tradeoff between  between  risk  and expected  expected return. return. In general, assets with higher expected returns are riskier. For a given amount of risk, MPT describes how to select a portfolio with the highest possible expected return. Or, for a given expected return, MPT explains how to select a portfolio with the lowest possible risk (the targeted expected return cannot be more than the highest-returning available security, of course, unless negative [6] holdings of assets are possible.) possible.)   Therefore, MPT is a form of  diversification. diversification . Under certain  certain assumptions  assumptions and for specific quantitative  quantitative  definitions of risk and return, MPT explains how to find the best possible diversification strategy.

[edit] edit] History

 

[7]

[8]

Harry Markowitz  article  and a 1959 book .  Markowitz Markowitz introduced MPT in a 1952 article classifies it simply as "Portfolio Theory," because "There's nothing modern about it." See [6] also this this  survey of the history.

[edit] edit] Mathematical model i s  In somethe sense the have mathematical below.[6]   MPT, although the basic concepts behind model also beenderivation very influential. influential

This section develops the "classic" MPT model. There have been many  many  extensions  extensions  since.

[edit] edit] Risk and expected return MPT assumes that investors are risk averse, meaning that given two portfolios that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher expected returns must accept more risk. The exact trade-off will be the same for all investors, but different investors will evaluate the trade-off differently based on individual risk aversion characteristics. The implication is that a  a rational  rational investor will not invest in a portfolio if a second portfolio exists with a more favorable  favorable  risk-expected  –  i.e., if for that level of risk an alternative portfolio exists which has better return profile  profile  –  expected returns. Note that the theory uses standard deviation of return as a proxy for risk, which is valid if  distributed  or otherwise  otherwise elliptically distributed distributed.. There asset returns are  are jointly  jointly normally normally distributed are problems with this, however; see  see criticism. criticism.  Under the model:  

Portfolio return is the  the proportion-weighted combination  combination of the constituent assets' returns.    Portfolio volatility is a function of the  the correlations  correlations  ρij of the component assets, for all asset pairs (i , j   j )).. 





In general:  



Expected return: 

where is the return on the portfolio, is the return on asset i and the weighting of component component asset (that is, the share of asset i in the portfolio).  



Portfolio return variance: 

is

 

  coefficient  between the returns on assets i and j . where is the  the correlation coefficient Alternatively the expression can be written as: , where  



 j .  j  for i =

Portfolio return volatility (standard deviation):  

For a two asset portfolio:  

Portfolio return:

 

Portfolio variance:





For a three asset portfolio:    





Portfolio return: Portfolio variance: 

[edit] edit] Diversification An investor can reduce portfolio risk simply by holding combinations of instruments which correlated  (correlation coefficient  coefficient  ). In other are not perfectly positively  positively correlated words, investors can reduce their exposure to individual asset risk by holding a   diversified diversified   portfolio of assets. Diversification may allow for the same portfolio expected return with reduced risk. These ideas have been started with Markowitz and then reinforced by other economists and mathematicians such as Andrew Brennan who have expressed ideas in the limitation of variance through portfolio theory. If all the asset pairs have correlations of 0 — they they are perfectly uncorrelated — the the portfolio's return variance is the sum over all assets of the square of the fraction held in the asset times the asset's return variance (and the portfolio standard deviation is the square root of  this sum).

[edit] edit] The efficient frontier with no risk-free asset icient nt f rontie rontier  r    M ain article:  article:   Eff icie

 

  Efficient Frontier. The hyperbola is sometimes referred to as the 'Markowitz Bullet', and is the efficient frontier if no risk-free asset is available. With a risk-free asset, the straight line is the efficient frontier. As shown in this graph, every possible combination of the risky assets, without including any holdings of the risk-free asset, can be plotted in risk-expected return space, and the collection of  all such possible portfolios defines a region in this space. The left boundary of this region is a frontier in the absence of a risk-free hyperbola,,[9] and the upper edge of this region is the efficient frontier in hyperbola asset (sometimes called "the Markowitz bullet"). Combinations along this upper edge represent  portfolios (including no holdings of the risk-free asset) for which there is lowest risk for a given given level of expected return. Equivalently, Equivalentl y, a portfolio lying on the efficient frontier represents the combination offering the best possible expected return for given risk level. Matrices are preferred for calculations of the efficient frontier. In matrix form, for a given "risk  Matrices  tolerance"

, the efficient frontier is found by minimizing the following expression:

where

 



 



 



 



 



 



is a vector of portfolio weights and (The weights can be negative, which means investors can  can short  short a security.); p ortfolio; is the  the covariance matrix  matrix for the returns on the assets in the portfolio; is a "risk tolerance" factor, where 0 results in the portfolio p ortfolio with minimal risk and results in the portfolio infinitely far out on the frontier with both expected return and risk unbounded; and is a vector of expected returns. is the variance of portfolio return. is the expected return on the portfolio.

 

The above optimization finds the point po int on the frontier at which the inverse o off the slope of the frontier would be q if portfolio return variance instead of standard deviation were plotted horizontally. The frontier in its entirety is parametric on q. Many software packages, including  including MATLAB MATLAB,, Microsoft Excel Excel,, Mathematica Mathematica  and and  R , provide optimization  optimization routines suitable for the above problem. An alternative approach to specifying the efficient frontier is to do so parametrically on the expected portfolio return This version of the problem requires that we minimize

subject to

for parameter

. This problem is easily solved using a Lagrange multiplier . 

[edit] edit] The two mutual fund theorem [9]

One key result of the above abov e analysis is the  the two mutual fund theorem theorem..  This theorem states that any portfolio on the efficient frontier can be generated by holding a combination of any two given portfolios on the frontier; the latter two given portfolios p ortfolios are the "mutual funds" in the theorem's name. So in the absence of a risk-free asset, an investor can achieve any desired efficient portfolio even if all that is accessible is a pair of efficient mutual funds. f unds. If the location of the desired portfolio on the frontier is between the locations of the two mutual funds, both mutual funds will be held in positive quantities. qu antities. If the desired portfolio is outside the range spanned by the two mutual funds, then one of the mutual funds must be sold short (held in negative quantity) while the size of the investment in the other mutual fund must be greater than the amount available for investment (the excess ex cess being funded by the borrowing from the other  fund).

[edit] edit] The risk-free asset and the capital allocation line  Main article:  article: Capital allocation line  line  The risk-free asset is the (hypothetical) asset which pays a  a  risk-free rate rate.. In practice, short-term the y pay a government securities (such as US  US treasury bills) bills) are used as a risk-free asset, because they default  risk. The risk-free asset has zero fixed rate of interest and have exceptionally exceptionall y low  low default variance in returns (hence is risk-free); it is also uncorrelated with any other asset (by definition, since its variance is zero). As a result, when it is combined with any other asset or portfolio of  assets, the change in return is linearly related to the change in risk as the proportions in the combination vary. When a risk-free assettoisthe introduced, half-line shown in the figure is the new  efficient Sharpe ratio. frontier. It is tangent hyperbolathe at the pure risky portfolio with the highest highest  ratio. Its

 

horizontal intercept represents a portfolio with 100% of holdings in the risk-free asset; the tangency with the hyperbola represents a portfolio with no risk-free holdings and 100% of assets held in the portfolio occurring at the tangency tan gency point; points between those points are portfolios containing positive amounts of both the risky tangency portfolio and the risk-free asset; and leveraged   portfolios portfolios involving negative  points on the half-line beyond the tangency point are  are leveraged holdings of the risk-free asset (the latter has been sold short —   — in in other words, the investor has  borrowed at the risk-free rate) and an amount invested in the tangency portfolio equal to more than 100% of the investor's initial capital. This efficient half-line is called the  the  capital allocation line  (CAL), and its formula can be shown to be line

In this formula P  formula P is is the sub-portfolio of risky assets at the tangency with the Markowitz bullet, F  bullet,  F   is the risk-free asset, and C  C is is a combination of portfolios P  portfolios P and and F   F . By the diagram, the introduction of the risk-free asset as a possible component of the portfolio has improved the range of risk-expected return combinations available, because everywhere except at the tangency portfolio the half-line gives a higher expected return than the hyperbola does at every possible risk level. The fact that all points on the linear efficient locus can be achieved by a combination of holdings of the risk-free asset and the tangenc tangency y portfolio is known [9] tangenc y portfolio. as the  the one mutual fund theorem, theorem,  where the mutual fund referred to is the tangency

[edit edit]] Asset pricing using MPT The above analysis describes optimal behavior of an individual investor. Asset pricing theory  builds on this analysis in the following way. Since everyone holds the risky assets in identical identical  proportions to each other   — namely namely in the proportions given by b y the tangency portfolio — in in market equilibrium the risky assets' prices, and therefore their expected returns, will adjust so that the ratios in the tangency portfolio are the same as the ratios in which the risky assets are supplied to the market. Thus relative supplies will equal relative demands. MPT derives the required expected return for a correctly priced asset in this context.

edit]] Systematic risk and specific risk  [edit Specific risk is the risk associated with individual assets - within a portfolio these risks can be reduced through diversification (specific risks "cancel out"). Specific risk is also called diversifiable, unique, unsystematic, or idiosyncratic risk.  risk. Systematic risk  (a.k.a. portfolio risk or  market risk) refers to the risk common to all securities — except except for selling short as noted below, systematic risk cannot be diversified away (within one market). Within the market portfolio, asset specific risk will be diversified away to the extent possible. Systematic risk is therefore equated with the risk (standard deviation) of the market portfolio. Since a security will be purchased only onl y if it improves the risk-expected return characteristics of  the market portfolio, the relevant measure of the risk of a security is the risk it adds to the market

 

 portfolio, and not its risk in isolation. In this context, the volatility of the asset, and its correlation with the market portfolio, are historically observed and are therefore th erefore given. (There are several approaches to asset pricing that attempt to price assets by modeling the stochastic properties of  the moments of assets' returns - these are broadly referred to as conditional asset pricing models.) Systematic risks within one market can be managed through a strategy of using both long an and d short positions within one portfolio, creating a "market neutral" portfolio.

[edit] edit] Capital asset pricing model  Main article:  article: Capital Asset Pricing Model   The asset return depends on the amount paid for the asset toda today. y. The price paid must ensure that the market portfolio's risk / return characteristics improve when the asset is added to it. The CAPM is a model which derives the theoretical required CAPM  re quired expected return (i.e., discount rate) for  an asset in a market, given the risk-free rate available to investors and the risk of the market as a whole. The CAPM is usually expressed:

 

, Beta, is the measure of asset sensitivity to a movement in the overall market; Beta is usually found via  via regression  regression on historical data. Betas exceeding one signify more than average "riskiness" in the sense of the asset's contribution to overall portfolio risk; betas  below one indicate a lower than average risk contribution.

 

is the market premium, the expected excess return of the market  portfolio's expected return over the risk-free rate.





following regression  regression equation: This equation can be statistically  statistically estimated  estimated using the following 

where αi is called the asset's  asset's alpha, alpha, βi is the asset's  asset's  beta beta coefficient  coefficient and SCL is the the  Security Characteristic Line. Line.  Once an asset's expected return, , is calculated using CAPM, the future  future cash flows  flows of the asset can be  be discounted  discounted to their  present  present value  value using this rate to establish the correct price for the asset. A riskier stock will have a higher beta and will be discounted at a higher rate; less sensitive stocks will have lower betas and be discounted at a lower rate. In theo theory, ry, an asset is correctly  priced when its observed price is the same as its value calculated using the CAPM derived discount rate. If the observed price is higher than the valuation, then the asset is overvalued; it is undervalued for a too low price.

 

(1) The incremental impact on risk and expected return when an additional risky asset, a, is added to the market portfolio, m, follows from the formulae for a two-asset portfolio. These results are used to derive the asset-appropriate discount rate.  



Market portfolio's risk = Hence, risk added to portfolio =  but since the weight of the asset will be relatively low, i.e. additional risk =

 



Market portfolio's expected return =

Hence additional expected return = (2) If an asset, a, is correctly priced, the improvement in its risk-to-expected return ratio achieved  by adding it to the market portfolio, m, will at least match the gains of spending that money on an increased stake in the market portfolio. The assumption is that the investor will purchase the asset with funds borrowed at the risk-free rate,

; this is rational if

.

Thus: i.e. : i.e. : is the "beta", return —   — the the  covariance  covariance  between between the asset's return and the market's return divided by the variance of the market return —   — i.e. i.e. the sensitivity of the asset price to movement in the market portfolio's value.

[edit] edit] Criticisms Despite its theoretical importance, critics of MPT question whether it is an ideal investing strategy, because its model of financial markets does not match the real world in many ways. wa ys. Efforts to translate the theoretical foundation into a viable portfolio construction algorithm have  been plagued by technical difficulties stemming from the instability of the original optimization  problem with respect to the available data. Recent research has shown that instabilities of this type disappear when a regularizing constraint or penalty p enalty term is incorporated in the optimization [10]  procedure..    procedure

edit]] Assumptions [edit The framework of MPT makes many man y assumptions about investors and markets. Some are explicit in the equations, such as the use of  Normal  Normal distributions  distributions to model returns. Others are implicit, such of as them the neglect of taxes MPT and transaction fees. None of these assumptions are entirely true, and each compromises to some degree.

 

 

Investors are interested in the optimization problem described above (maximizing the mean for a given variance). In reality, investors have utility functions that may be sensitive to higher moments of the distribution of the returns. For the investors to use the mean-variance optimization, one must suppose that the combination of utility and returns make the optimization of utility problem similar to the mean-variance optimization  problem. A quadratic utility without any assumption about returns is is sufficient. Another  assumption is to use exponential utility and normal distribution, as discussed below. below .

 

Asset returns are ( jointly)  jointly) normally distributed distributed  random  random variables. In fact, it is frequently observed that returns in equity and other markets are not normally distributed. Large swings (3 to 6 standard deviations from the mean) occur o ccur in the market far more [11] frequently than the normal distribution assumption would predict. predict.  While the model can [12][13] also be justified by assuming any return distribution which is  is   jointly jointly elliptical, elliptical,  all the joint elliptical distributions are symmetrical whereas asset returns empirically are not.

 

Correlations between assets are fixed and constant forever. Correlations depend on systemic relationships between the underlying assets, and change when these the se relationships change. Examples include one country declaring decla ring war on another, or a general market crash. During times of financial crisis all assets tend to become beco me positively correlated, because they all move (down) together. In other words, MPT breaks down  precisely when investors are most in need of protection from risk. risk.

 

All investors aim to maximize economic utility (in other words, to make as much money as possible, regardless of any other considerations). This is a key assumption of the  the efficient market hypothesis, hypothesis, upon which MPT relies.

 

the efficient All investors are rational and  and risk-averse. risk-averse. This is another assumption of the  market hypothesis, hypothesis, but we now know from  from  behavioral  behavioral economics  economics that market participants are not  not rational. rational. It does not allow for "herd behavior" or investors who will accept lower  gamblers  clearly pay for risk, and it is possible po ssible that some returns for higher risk.  risk. Casino gamblers stock traders will pay for risk as well.













 

All investors have access to the same information at the same time. In fact, real markets contain  contain information asymmetry, asymmetry, insider trading, trading, and those who are simply better  informed than others. Moreover, estimating the mean (for instance, there is no consistent estimator of the drift of a brownian when subsampling between 0 and T) and the covariance matrix of the returns (when the number numbe r of assets is of the same order of the number of periods) are difficult statistical tasks.

 

Investors have an accurate conception of possible returns, i.e., the probability beliefs of investors match the true distribution of returns. A different possibility is that investors' expectations are biased, causing market prices to be informationally inefficient. This possibility is studied in the field of   behavioral behavioral finance, finance, which uses psychological assumptions to provide alternatives to the CAPM such as the overconfidence-based o verconfidence-based asset  pricing model of Kent Daniel,  Daniel, David Hirshleifer , and Avanidhar Subrahmanyam



(2001).[14]  (2001).

 

 

There are no taxes or transaction costs. Real financial products are subject both to taxes and transaction costs (such as broker fees), and taking these into account will alter  the composition of the optimum portfolio. These assumptions can be relaxed with more [citation needed ] complicated versions of the model.  

 

All investors are price takers, i.e., their actions do not influence prices. In reality, sufficiently large sales or purchases of individual assets can shift market prices for that





demand..) An investor may not even be able to asset and others (via  (via cross elasticity of demand assemble the theoretically optimal portfolio if the market moves too much while they are  buying the required securities.  

Any investor can lend and borrow an unlimited amount at the risk free rate of  interest. In reality, every investor has a credit limit.

 

All securities can be divided into parcels of any size. In reality, fractional shares usually cannot be bought or sold, and some assets have minimum orders sizes.





More complex versions of MPT can take into account a more soph sophisticated isticated model of the world (such as one with non-normal distributions and taxes) but all mathematical models of finance still rely on many unrealistic premises.

[edit edit]] MPT does not really model the market The risk, return, and correlation measures used by b y MPT are based on on  expected values, values, which means that they are mathematical statements about the future (the expected value of returns is variance  and and  covariance covariance)). In explicit in the above equations, and implicit in the definitions of  variance  practice investors must substitute predictions predictions based on historical measurements of asset return and  volatility  and volatility for these values in the equations. Very often such expected values fail to take account of new circumstances which did not n ot exist when the historical data were generated. More fundamentally, investors are stuck with estimating key parameters from past market data  because MPT attempts to model risk in terms of the likelihood of losses, losses, but says nothing about why those losses might occur. The risk measurements used are  are  probabilistic  probabilistic in nature, not structural. This is a major difference as compared to many man y engineering approaches to  to risk  management..  management Options  Options theory and MPT have at least one on e important conceptual difference from the the   probabilistic probabilistic risk assessment  assessment done by nuclear power [plants]. A PRA is what economists would call a  structural model . The components of a system and their relationships are modeled in in  Monte Carlo simulations. simulations. If valve X fails, it causes a loss of back pressure on pump pu mp Y, causing a drop in flow to vessel Z, and so on.  – Scholes  Scholes equation and MPT, there is no attempt to explain an underlying But in the  the Black  structure to price changes. Various outcomes are simply given probabilities. And, unlike the PRA, if there is no history of a particular pa rticular system-level event like aa  liquidity crisis, crisis, there is no way to compute the odds of it. If nuclear engineers ran risk management this way, they th ey would

 

never be able to compute the odds of a meltdown at a particular plant until several similar events occurred in the same reactor design.  — Douglas Douglas W. Hubbard, 'The Failure of Risk Management', p. 67, John Wiley & Sons, 2009. ISBN 978-0-470-38795-5 978-0-470-38795-5   Essentially, the mathematics of MPT view the markets ma rkets as a collection of dice. By examining past market data we can develop hypotheses about how the dice are weighted, but this isn't helpful if  the markets are actually dependent upon a much bigger and more complicated  complicated  chaotic  chaotic system —  the world. For this reason, accurate structural models of real financial markets are unlikely to be forthcoming because they would essentially be structural models of the entire world. Nonetheless there is growing awareness of the concept of  systemic risk  in financial markets, which should lead to more sophisticated market models. Mathematical risk measurements are also useful only to the degree that they reflect investors' true concerns — there there is no point minimizing a variable that nobody cares about in practice. MPT uses the mathematical concept of  variance variance  to quantify risk, and this might be justified under the distributed  returns such as  as normally distributed  distributed returns, but for general assumption of  elliptically distributed return  distributions return distributions  other risk measures (like  (like coherent risk measures) measures) might better reflect investors' true preferences. In particular,  particular, variance  variance is a symmetric measure that counts abnormally high returns as just as risky as abnormally low returns. Some would argue that, in reality, investors are only concerned about losses, and do not care c are about the dispersion or tightness of above-ave above-average rage returns. According to this view, our intuitive concept of o f risk is fundamentally asymmetric in nature. MPT does not account for the personal, environmental, strategic, or  social dimensions of  investment decisions. decisions. It only attempts to maximize risk-adjusted returns, without regard to other  consequences. In a narrow sense, its complete reliance on asset asset   prices  prices makes it vulnerable to all the standard  standard market failures failures  such as those arising from  from information asymmetry asymmetry,, externalities, externalities, and goods.. It also rewards corporate fraud and dishonest accounting. accoun ting. More broadly, a firm  public goods may have strategic or social goals that shape its investment decisions, and an individual investor  might have personal goals. In either case, information other than historical returns is relevant. Taleb  has also criticized modern portfolio theory because Financial economist  economist  Nassim Nassim Nicholas Taleb it assumes a  a Gaussian distribution distribution::   After the stock market crash (in 1987), they rewarded two theoreticians, Harry  Markowitz and William Sharpe, who built beautifully Platonic models on a Gaussian base, contributing to what is called Modern Portfolio Theory. T heory. Simply, if you remove their  Gaussian assumptions and treat prices as scalable, you are left with hot ho t air. The Nobel  Committee could have tested the Sharpe and Markowitz models  –  they work like quack  remedies sold on the Internet   –  –  but nobody in Stockholm seems to have thought about  [15]::  p.279 p.279 [15]   it .

[edit edit]] The MPT does not take its own effect on asset prices into account a ccount

 

Diversification eliminates non-systematic risk, but at the cost of increasing the  the  systematic risk .  Diversification forces the portfolio manager to invest in assets without analyzing their  fundamentals, solely for the benefit of eliminating the portfolio’s non-systematic non -systematic risk (the  (the CAPM  CAPM  assumes investment in all available assets). This artificially increased demand pushes up the  price of assets that, when analyzed individually, would be of little fundamental value. The result is that the whole portfolio becomes more expensive exp ensive and, as a result, the probability of a positive return decreases (i.e. the risk of the portfolio increases). Empirical evidence for this is the price hike that stocks typically experience once they are [citation needed ]   included in major indices like the  the  S&P 500. 500.

[edit] edit] Extensions Since MPT's introduction in 1952, many attempts have been made to improve the model, especially by using more realistic assumptions. theory  extends MPT by adopting non-normally non -normally distributed, asymmetric Post-modern portfolio theory measures of risk. This helps with some of these problems, but b ut not others. model optimization is an extension of unconstrained Markowitz optimization Black-Litterman model  which incorporates relative and absolute `views' on inputs inpu ts of risk and returns.

[edit] edit] Other Applications [edit edit]] Application Applicationss to project portfolios and other "non-financial" assets Some experts apply MPT to portfolios po rtfolios of projects and other assets besides financial [16][17]  When MPT is applied outside of traditional financial portfolios, some instruments.. instruments differences between the different types of portfolios must be considered. 1.  The assets in financial portfolios are, for practical purposes, continuously divisible while  portfolios of projects are "lumpy". For example, while we can compute that the optimal  portfolio position for 3 stocks is, say, say, 44%, 35%, 21%, the optimal position for a project  portfolio may not allow us to simply change the amount spent on a project. Projects might be all or nothing or, at least, have logical units that cannot be separated. A portfolio optimization method would have to take the discrete nature of projects into accoun account. t. 2.  The assets of financial portfolios are liquid; they can be assessed or re-assessed at any  point in time. But opportunities for launching new projects may be limited and may occur  in limited windows of time. Projects that have already alread y been initiated cannot be abandoned aband oned without the loss of the  the sunk costs  costs (i.e., there is little or no recovery/salvage value of a half-complete project).

 

 Neither of these necessarily eliminate the possibility possibility of using MPT and such portfolios. They simply indicate the need to run the optimization with an additional set of mathematicallyexpressed constraints that would not normally apply to financial portfolios. Furthermore, some of the simplest elements of Modern Portfolio Theory are applicable app licable to virtually any kind of portfolio. The concept of capturing the risk tolerance of an investor by documenting how much risk is acceptable for a given return may be applied to a variety of  decision analysis problems. MPT uses historical variance as a measure of o f risk, but portfolios of  assets like major projects don't have a well-defined "historical variance". In this case, the MPT investment boundary can be expressed ex pressed in more general terms like "chance o off an ROI less than cost of capital" or "chance of losing more than half of the investment". When risk is put in terms of uncertainty about forecasts and possible losses then the concept is transferable to various types [16] of investment. investment.  

disciplines [edit] edit] Application to other disciplines In the 1970s, concepts from Modern Portfolio Theory found their way into the field of  regional [citation needed ] modeled the labor force in the science.. In a series of seminal works, Michael Conroy science economy using portfolio-theoretic methods to examine growth and variability in the labor force. [18] This was.followed by a long literature on the relationship between economic growth and   volatility volatility.

More recently, modern portfolio theory has been used to model the self-concept in social  psychology. When the self attributes comprising the self-concept constitute a well-diversified well-diversified  portfolio, then psychological outcomes at the level of the individual such as mood and selfesteem should be more stable than when the self-concept is undiversified. This prediction has [19]  been confirmed in studies involving human subjects. subjects.   Recently, modern portfolio theory has been applied to modelling the uncertainty and correlation  between documents in information retrieval. Given a query, the aim is to maximize the overall relevance of a ranked list of documents do cuments and at the same time minimize the overall uncertainty of  [20] the ranked list. list.  

edit]] Comparison with arbitrage pricing theory [edit The SML and  and CAPM  CAPM are often contrasted with the  the arbitrage pricing theory theory  (APT), which holds a linear function  function of various various  macrothat the  the expected return  return of a financial asset can be modeled as a  economic  factors, where sensitivity to changes in each factor is represented by a factor specific economic coefficient.   beta coefficient. The APT is less restrictive in its assumptions: it allows for a statistical model of asset returns, and assumes that each investor will hold a unique portfolio with its own particular array of betas, as opposed to the identical "market portfolio". Unlike the CAPM, the APT, however, does not itself reveal the identity of its priced factors - the number and an d nature of these factors is likely to change over time and between economies.

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