Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the

_{ij} of the component assets, for all asset pairs (''i'', ''j''). The volatility gives insight into the risk which is associated with the investment. The higher the volatility, the higher the risk.

here

/ref> and the upper part of the hyperbolic boundary is the ''efficient frontier'' in the absence of a risk-free 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 level of expected return. Equivalently, a portfolio lying on the efficient frontier represents the combination offering the best possible expected return for given risk level. The tangent to the upper part of the hyperbolic boundary is the capital allocation line (CAL).

_{''i''} is called the asset's _{''i''} is the asset's beta coefficient and SCL is the

^{2}''-Criterion. Journal of economic theory, 69(2), 532-539.
and may recommend to invest into ''Y'' on the basis that it has lower variance. Maccheroni et al. described choice theory which is the closest possible to the modern portfolio theory, while satisfying monotonicity axiom. Alternatively, mean-deviation analysis
is a rational choice theory resulting from replacing variance by an appropriate deviation risk measure.

The Most Rewarding Portfolio Construction Techniques: An Unbiased Evaluation

Prof.

Portfolio Optimization

Prof. Stephen P. Boyd,

New Approaches for Portfolio Optimization: Parting with the Bell Curve

— Interview with Prof. Svetlozar Rachev and Prof. Stefan Mittnik {{DEFAULTSORT:Modern Portfolio Theory Portfolio theories Financial risk modeling

expected return
The expected return (or expected gain) on a financial investment is the expected value of its return (of the profit on the investment). It is a measure of the center of the distribution of the random variable that is the return. It is calculated ...

is maximized for a given level of risk. It is a formalization and extension of diversification in investing, the idea that owning different kinds of financial assets is less risky than owning only one type. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return. It uses the variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...

of asset prices as a proxy for risk.
Economist Harry Markowitz
Harry Max Markowitz (born August 24, 1927) is an American economist who received the 1989 John von Neumann Theory Prize and the 1990 Nobel Memorial Prize in Economic Sciences.
Markowitz is a professor of finance at the Rady School of Management ...

introduced MPT in a 1952 essay, for which he was later awarded a Nobel Memorial Prize in Economic Sciences
The Nobel Memorial Prize in Economic Sciences, officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel ( sv, Sveriges riksbanks pris i ekonomisk vetenskap till Alfred Nobels minne), is an economics award administered ...

; see Markowitz model.
Mathematical model

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 not be the same for all investors. Different investors will evaluate the trade-off differently based on individual risk aversion characteristics. The implication is that arational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...

investor will not invest in a portfolio if a second portfolio exists with a more favorable risk-expected return profile—i.e., if for that level of risk an alternative portfolio exists that has better expected returns.
Under the model:
*Portfolio return is the proportion-weighted combination of the constituent assets' returns.
*Portfolio return volatility $\backslash sigma\_p$ is a function of the correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistic ...

s ''ρ''In general: *Expected return: :$\backslash operatorname(R\_p)\; =\; \backslash sum\_i\; w\_i\; \backslash operatorname(R\_i)\; \backslash quad$ :where $R\_p$ is the return on the portfolio, $R\_i$ is the return on asset ''i'' and $w\_i$ is the weighting of component asset $i$ (that is, the proportion of asset "i" in the portfolio). *Portfolio return variance: :$\backslash sigma\_p^2\; =\; \backslash sum\_i\; w\_i^2\; \backslash sigma\_^2\; +\; \backslash sum\_i\; \backslash sum\_\; w\_i\; w\_j\; \backslash sigma\_i\; \backslash sigma\_j\; \backslash rho\_$, :where $\backslash sigma\_$ is the (sample) standard deviation of the periodic returns on an asset ''i'', and $\backslash rho\_$ is the correlation coefficient between the returns on assets ''i'' and ''j''. Alternatively the expression can be written as: :$\backslash sigma\_p^2\; =\; \backslash sum\_i\; \backslash sum\_j\; w\_i\; w\_j\; \backslash sigma\_i\; \backslash sigma\_j\; \backslash rho\_$, :where $\backslash rho\_\; =\; 1$ for $i\; =\; j$ , or :$\backslash sigma\_p^2\; =\; \backslash sum\_i\; \backslash sum\_j\; w\_i\; w\_j\; \backslash sigma\_$, :where $\backslash sigma\_\; =\; \backslash sigma\_i\; \backslash sigma\_j\; \backslash rho\_$ is the (sample) covariance of the periodic returns on the two assets, or alternatively denoted as $\backslash sigma(i,j)$, $\backslash text\_$ or $\backslash text(i,j)$. *Portfolio return volatility (standard deviation): :$\backslash sigma\_p\; =\; \backslash sqrt$ For a two-asset portfolio: *Portfolio return: $\backslash operatorname(R\_p)\; =\; w\_A\; \backslash operatorname(R\_A)\; +\; w\_B\; \backslash operatorname(R\_B)\; =\; w\_A\; \backslash operatorname(R\_A)\; +\; (1\; -\; w\_A)\; \backslash operatorname(R\_B).$ *Portfolio variance: $\backslash sigma\_p^2\; =\; w\_A^2\; \backslash sigma\_A^2\; +\; w\_B^2\; \backslash sigma\_B^2\; +\; 2w\_Aw\_B\; \backslash sigma\_\; \backslash sigma\_\; \backslash rho\_$ For a three-asset portfolio: *Portfolio return: $\backslash operatorname(R\_p)\; =\; w\_A\; \backslash operatorname(R\_A)\; +\; w\_B\; \backslash operatorname(R\_B)\; +\; w\_C\; \backslash operatorname(R\_C)$ *Portfolio variance: $\backslash sigma\_p^2\; =\; w\_A^2\; \backslash sigma\_A^2\; +\; w\_B^2\; \backslash sigma\_B^2\; +\; w\_C^2\; \backslash sigma\_C^2\; +\; 2w\_Aw\_B\; \backslash sigma\_\; \backslash sigma\_\; \backslash rho\_\; +\; 2w\_Aw\_C\; \backslash sigma\_\; \backslash sigma\_\; \backslash rho\_\; +\; 2w\_Bw\_C\; \backslash sigma\_\; \backslash sigma\_\; \backslash rho\_$

Diversification

An investor can reduce portfolio risk (especially $\backslash sigma\_p$) simply by holding combinations of instruments that are not perfectly positively correlated ( correlation coefficient $-1\; \backslash le\; \backslash rho\_<\; 1$). In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Diversification may allow for the same portfolio expected return with reduced risk. The mean-variance framework for constructing optimal investment portfolios was first posited by Markowitz and has since been reinforced and improved by other economists and mathematicians who went on to account for the limitations of the framework. If all the asset pairs have correlations of 0—they are perfectly uncorrelated—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). If all the asset pairs have correlations of 1—they are perfectly positively correlated—then the portfolio return’s standard deviation is the sum of the asset returns’ standard deviations weighted by the fractions held in the portfolio. For given portfolio weights and given standard deviations of asset returns, the case of all correlations being 1 gives the highest possible standard deviation of portfolio return.Efficient frontier with no risk-free asset

The MPT is a mean-variance theory, and it compares the expected (mean) return of a portfolio with the standard deviation of the same portfolio. The image shows expected return on the vertical axis, and the standard deviation on the horizontal axis (volatility). Volatility is described by standard deviation and it serves as a measure of risk. The return - standard deviation space is sometimes called the space of 'expected return vs risk'. Every possible combination of risky assets, can be plotted in this 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 hyperbolic,see bottom of slidehere

/ref> and the upper part of the hyperbolic boundary is the ''efficient frontier'' in the absence of a risk-free 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 level of expected return. Equivalently, a portfolio lying on the efficient frontier represents the combination offering the best possible expected return for given risk level. The tangent to the upper part of the hyperbolic boundary is the capital allocation line (CAL).

Matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...are preferred for calculations of the efficient frontier. In matrix form, for a given "risk tolerance" $q\; \backslash in\; [0,\backslash infty)$, the efficient frontier is found by minimizing the following expression: :$w^T\; \backslash Sigma\; w\; -\; q\backslash times\; R^T\; w$ where * $w$ is a vector of portfolio weights and $\backslash sum\_i\; w\_i\; =\; 1.$ (The weights can be negative); * $\backslash Sigma$ is the covariance matrix for the returns on the assets in the portfolio; * $q\; \backslash ge\; 0$ is a "risk tolerance" factor, where 0 results in the portfolio with minimal risk and $\backslash infty$ results in the portfolio infinitely far out on the frontier with both expected return and risk unbounded; and * $R$ is a vector of expected returns. * $w^T\; \backslash Sigma\; w$ is the variance of portfolio return. * $R^T\; w$ is the expected return on the portfolio. The above optimization finds the point on the frontier at which the inverse of 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''.Harry Markowitz Harry Max Markowitz (born August 24, 1927) is an American economist who received the 1989 John von Neumann Theory Prize and the 1990 Nobel Memorial Prize in Economic Sciences. Markowitz is a professor of finance at the Rady School of Management ...developed a specific procedure for solving the above problem, called the critical line algorithm, that can handle additional linear constraints, upper and lower bounds on assets, and which is proved to work with a semi-positive definite covariance matrix. Examples of implementation of the critical line algorithm exist inVisual Basic for Applications Visual Basic for Applications (VBA) is an implementation of Microsoft's event-driven programming language Visual Basic 6.0 built into most desktop Microsoft Office applications. Although based on pre-.NET Visual Basic, which is no longer supp ..., inJavaScript JavaScript (), often abbreviated as JS, is a programming language that is one of the core technologies of the World Wide Web, alongside HTML and CSS. As of 2022, 98% of websites use JavaScript on the client side for webpage behavior, ofte ...and in a few other languages. Also, many software packages, includingMATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...,Microsoft Excel Microsoft Excel is a spreadsheet developed by Microsoft for Windows, macOS, Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro programming language called Visual Basic for Ap ...,Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimiza ...and R, provide genericoptimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...routines so that using these for solving the above problem is possible, with potential caveats (poor numerical accuracy, requirement of positive definiteness of the covariance matrix...). An alternative approach to specifying the efficient frontier is to do so parametrically on the expected portfolio return $R^T\; w.$ This version of the problem requires that we minimize :$w^T\; \backslash Sigma\; w$ subject to :$R^T\; w\; =\; \backslash mu$ for parameter $\backslash mu$. This problem is easily solved using aLagrange multiplier In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied ex ...which leads to the following linear system of equations: :$\backslash begin2\backslash Sigma\; \&-R\; \&\; -\backslash \backslash \; R^T\; \&0\; \&\; 0\; \backslash \backslash \; ^T\; \&0\; \&0\; \backslash end\; \backslash beginw\backslash \backslash \backslash lambda\_1\backslash \backslash \backslash lambda\_2\backslash end\; =\; \backslash begin0\backslash \backslash \backslash mu\; \backslash \backslash \; 1\backslash end$

Two mutual fund theorem

One key result of the above analysis is the two mutual fund theorem.Merton, Robert. "An analytic derivation of the efficient portfolio frontier," ''Journal of Financial and Quantitative Analysis
The ''Journal of Financial and Quantitative Analysis'' is a peer-reviewed bimonthly academic journal published by the Michael G. Foster School of Business at the University of Washington in cooperation with the W. P. Carey School of Business at ...

'' 7, September 1972, 1851-1872. 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 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. 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. 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 being funded by the borrowing from the other fund).
Risk-free asset and the capital allocation line

The risk-free asset is the (hypothetical) asset that pays arisk-free rate
The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payments over a fixed period of time that is assumed to meet all payment obligations.
Since the risk-free r ...

. In practice, short-term government securities (such as US treasury bills) are used as a risk-free asset, because they pay a fixed rate of interest and have exceptionally low default risk. The risk-free asset has zero 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 asset is introduced, the half-line shown in the figure is the new efficient frontier. It is tangent to the parabola at the pure risky portfolio with the highest Sharpe ratio. Its vertical intercept represents a portfolio with 100% of holdings in the risk-free asset; the tangency with the parabola represents a portfolio with no risk-free holdings and 100% of assets held in the portfolio occurring at the tangency point; points between those points are portfolios containing positive amounts of both the risky tangency portfolio and the risk-free asset; and points on the half-line beyond the tangency point are portfolios involving negative holdings of the risk-free asset 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 capital allocation line
Capital allocation line (CAL) is a graph created by investors to measure the risk of risky and risk-free assets. The graph displays the return to be made by taking on a certain level of risk. Its slope is known as the "reward-to-variability ratio". ...

(CAL), and its formula can be shown to be
:$E(R\_)\; =\; R\_F\; +\; \backslash sigma\_C\; \backslash frac.$
In this formula ''P'' is the sub-portfolio of risky assets at the tangency with the Markowitz bullet, ''F'' is the risk-free asset, and ''C'' is a combination of portfolios ''P'' and ''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 parabola 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 tangency portfolio is known as the one mutual fund theorem, where the mutual fund referred to is the tangency portfolio.
Asset pricing

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 proportions to each other—namely in the proportions given by the tangency portfolio—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.Systematic risk and specific risk

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.Systematic risk
In finance and economics, systematic risk (in economics often called aggregate risk or undiversifiable risk) is vulnerability to events which affect aggregate outcomes such as broad market returns, total economy-wide resource holdings, or aggr ...

(a.k.a. portfolio risk or market risk) refers to the risk common to all securities—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 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 given. (There are several approaches to asset pricing that attempt to price assets by modelling 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 and short positions within one portfolio, creating a "market neutral" portfolio. Market neutral portfolios, therefore, will be uncorrelated with broader market indices.
Capital asset pricing model

The asset return depends on the amount paid for the asset today. 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 that derives the theoretical required 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: :$\backslash operatorname(R\_i)\; =\; R\_f\; +\; \backslash beta\_i\; (\backslash operatorname(R\_m)\; -\; R\_f)$ *β,Beta
Beta (, ; uppercase , lowercase , or cursive ; grc, βῆτα, bē̂ta or ell, βήτα, víta) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Modern Greek, it represents the voiced labi ...

, is the measure of asset sensitivity to a movement in the overall market; Beta is usually found via 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.
*$(\backslash operatorname(R\_m)\; -\; R\_f)$ is the market premium, the expected excess return of the market portfolio's expected return over the risk-free rate.
The derivation is as follows:
(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. *Updated market portfolio's risk = $(w\_m^2\; \backslash sigma\_m\; ^2\; +;\; href="/html/ALL/l/w\_a^2\_\backslash sigma\_a^2\_\_+\_2\_w\_m\_w\_a\_\backslash rho\_\_\backslash sigma\_a\_\backslash sigma\_m.html"\; ;"title="w\_a^2\; \backslash sigma\_a^2\; +\; 2\; w\_m\; w\_a\; \backslash rho\_\; \backslash sigma\_a\; \backslash sigma\_m">w\_a^2\; \backslash sigma\_a^2\; +\; 2\; w\_m\; w\_a\; \backslash rho\_\; \backslash sigma\_a\; \backslash sigma\_m$ ::Hence, risk added to portfolio = $;\; href="/html/ALL/l/w\_a^2\_\backslash sigma\_a^2\_\_+\_2\_w\_m\_w\_a\_\backslash rho\_\_\backslash sigma\_a\_\backslash sigma\_m.html"\; ;"title="w\_a^2\; \backslash sigma\_a^2\; +\; 2\; w\_m\; w\_a\; \backslash rho\_\; \backslash sigma\_a\; \backslash sigma\_m">w\_a^2\; \backslash sigma\_a^2\; +\; 2\; w\_m\; w\_a\; \backslash rho\_\; \backslash sigma\_a\; \backslash sigma\_m$ ::but since the weight of the asset will be relatively low, $w\_a^2\; \backslash approx\; 0$ ::i.e. additional risk = $;\; href="/html/ALL/l/2\_w\_m\_w\_a\_\backslash rho\_\_\backslash sigma\_a\_\backslash sigma\_m.html"\; ;"title="2\; w\_m\; w\_a\; \backslash rho\_\; \backslash sigma\_a\; \backslash sigma\_m">2\; w\_m\; w\_a\; \backslash rho\_\; \backslash sigma\_a\; \backslash sigma\_m$ *Market portfolio's expected return = $(\; w\_m\; \backslash operatorname(R\_m)\; +;\; href="/html/ALL/l/w\_a\_\backslash operatorname(R\_a)\_.html"\; ;"title="w\_a\; \backslash operatorname(R\_a)\; ">w\_a\; \backslash operatorname(R\_a)$ ::Hence additional expected return = $;\; href="/html/ALL/l/w\_a\_\backslash operatorname(R\_a)\_.html"\; ;"title="w\_a\; \backslash operatorname(R\_a)\; ">w\_a\; \backslash operatorname(R\_a)$ (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, $R\_f$; this is rational if $\backslash operatorname(R\_a)\; >\; R\_f$. :Thus: $;\; href="/html/ALL/l/w\_a\_(\_\backslash operatorname(R\_a)\_-\_R\_f\_)\_.html"\; ;"title="w\_a\; (\; \backslash operatorname(R\_a)\; -\; R\_f\; )\; ">w\_a\; (\; \backslash operatorname(R\_a)\; -\; R\_f\; )$ :i.e. : $;\; href="/html/ALL/l/operatorname(R\_a)\_.html"\; ;"title="operatorname(R\_a)\; ">operatorname(R\_a)$ :i.e. : $;\; href="/html/ALL/l/operatorname(R\_a)\_.html"\; ;"title="operatorname(R\_a)\; ">operatorname(R\_a)$ : $;\; href="/html/ALL/l/sigma\_.html"\; ;"title="sigma\_">sigma\_$ is the "beta", $\backslash beta$ return— theThis equation can becovariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...between the asset's return and the market's return divided by the variance of the market return— i.e. the sensitivity of the asset price to movement in the market portfolio's value (see also ).

estimated
Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is der ...

statistically using the following regression equation:
:$\backslash mathrm\; :\; R\_\; -\; R\_\; =\; \backslash alpha\_i\; +\; \backslash beta\_i\backslash ,\; (\; R\_\; -\; R\_\; )\; +\; \backslash epsilon\_\; \backslash frac$
where αalpha
Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', or ell, άλφα, álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , w ...

, βsecurity characteristic line
Security characteristic line (SCL) is a regression line, plotting performance of a particular security or portfolio against that of the market portfolio at every point in time. The SCL is plotted on a graph where the Y-axis is the excess return on ...

.
Once an asset's expected return, $E(R\_i)$, is calculated using CAPM, the future cash flow
A cash flow is a real or virtual movement of money:
*a cash flow in its narrow sense is a payment (in a currency), especially from one central bank account to another; the term 'cash flow' is mostly used to describe payments that are expected ...

s of the asset can be discounted
Discounting is a financial mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee.See "Time Value", "Discount", "Discount Yield", "Compound Interest", "Efficien ...

to their present value
In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has in ...

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 theory, 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.
Criticisms

Despite its theoretical importance, critics of MPT question whether it is an ideal investment tool, because its model of financial markets does not match the real world in many ways. The risk, return, and correlation measures used by MPT are based onexpected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

s, which means that they are statistical statements about the future (the expected value of returns is explicit in the above equations, and implicit in the definitions of variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...

and covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...

). Such measures often cannot capture the true statistical features of the risk and return which often follow highly skewed distributions (e.g. the log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a norma ...

) and can give rise to, besides reduced volatility, also inflated growth of return. In practice, investors must substitute predictions based on historical measurements of asset return and volatility for these values in the equations. Very often such expected values fail to take account of new circumstances that did not 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, but says nothing about why those losses might occur. The risk measurements used are probabilistic
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...

in nature, not structural. This is a major difference as compared to many engineering approaches to risk management.
Mathematical risk measurements are also useful only to the degree that they reflect investors' true concerns—there is no point minimizing a variable that nobody cares about in practice. In particular, variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...

is a symmetric measure that counts abnormally high returns as just as risky as abnormally low returns. The psychological phenomenon of loss aversion is the idea that investors are more concerned about losses than gains, meaning that our intuitive concept of risk is fundamentally asymmetric in nature. There many other risk measures (like coherent risk measure
In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent ris ...

s) might better reflect investors' true preferences.
Modern portfolio theory has also been criticized because it assumes that returns follow a Gaussian distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...

. Already in the 1960s, Benoit Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of p ...

and Eugene Fama showed the inadequacy of this assumption and proposed the use of more general stable distributions
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be st ...

instead. Stefan Mittnik and Svetlozar Rachev presented strategies for deriving optimal portfolios in such settings. More recently, Nassim Nicholas Taleb
Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist whose work concerns problems of randomness ...

has also criticized modern portfolio theory on this ground, writing:
Contrarian investors and value investors typically do not subscribe to Modern Portfolio Theory. One objection is that the MPT relies on the efficient-market hypothesis
The efficient-market hypothesis (EMH) is a hypothesis in financial economics that states that asset prices reflect all available information. A direct implication is that it is impossible to "beat the market" consistently on a risk-adjusted bas ...

and uses fluctuations in share price as a substitute for risk.
A few studies have argued that "naive diversification", splitting capital equally among available investment options, might have advantages over MPT in some situations.
Extensions

Since MPT's introduction in 1952, many attempts have been made to improve the model, especially by using more realistic assumptions.Post-modern portfolio theory Post-Modern Portfolio Theory (PMPT) is an extension of the traditional Modern Portfolio Theory (MPT), an application of mean-variance analysis (MVA). Both theories propose how rational investors can use diversification to optimize their portfolios. ...

extends MPT by adopting non-normally distributed, asymmetric, and fat-tailed measures of risk. This helps with some of these problems, but not others.
Black–Litterman model optimization is an extension of unconstrained Markowitz optimization that incorporates relative and absolute 'views' on inputs of risk and returns from.
Connection with rational choice theory

Modern portfolio theory is inconsistent with main axioms ofrational choice theory
Rational choice theory refers to a set of guidelines that help understand economic and social behaviour. The theory originated in the eighteenth century and can be traced back to political economist and philosopher, Adam Smith. The theory postula ...

, most notably with monotonicity axiom, stating that, if investing into portfolio ''X'' will, with probability one, return more money than investing into portfolio ''Y'', then a rational investor should prefer ''X'' to ''Y''. In contrast, modern portfolio theory is based on a different axiom, called variance aversion,Loffler, A. (1996). Variance Aversion Implies ''μ-σOther applications

In the 1970s, concepts from MPT found their way into the field of regional science. In a series of seminal works, Michael Conroy modeled the labor force in the economy using portfolio-theoretic methods to examine growth and variability in the labor force. This was followed by a long literature on the relationship between economic growth and 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 portfolio, then psychological outcomes at the level of the individual such as mood and self-esteem should be more stable than when the self-concept is undiversified. This prediction has been confirmed in studies involving human 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 and at the same time minimize the overall uncertainty of the ranked list.Project portfolios and other "non-financial" assets

Some experts apply MPT to portfolios of projects and other assets besides financial instruments. When MPT is applied outside of traditional financial portfolios, some distinctions between the different types of portfolios must be considered. # 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, 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 account. # 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 been initiated cannot be abandoned without the loss of thesunk costs
In economics and business decision-making, a sunk cost (also known as retrospective cost) is a cost that has already been incurred and cannot be recovered. Sunk costs are contrasted with '' prospective costs'', which are future costs that may b ...

(i.e., there is little or no recovery/salvage value of a half-complete project).
Neither of these necessarily eliminate the possibility of using MPT and such portfolios. They simply indicate the need to run the optimization with an additional set of mathematically expressed constraints that would not normally apply to financial portfolios.
Furthermore, some of the simplest elements of Modern Portfolio Theory are applicable 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 risk, but portfolios of assets like major projects do not have a well-defined "historical variance". In this case, the MPT investment boundary can be expressed in more general terms like "chance of 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 of investment.
See also

* * Beta (finance) * Bias ratio (finance) * Black–Litterman model * * Intertemporal portfolio choice * Investment theory * Kelly criterion * Marginal conditional stochastic dominance * Markowitz model * Mutual fund separation theorem *Omega ratio The Omega ratio is a risk-return performance measure of an investment asset, portfolio, or strategy. It was devised by Con Keating and William F. Shadwick in 2002 and is defined as the probability weighted ratio of gains versus losses for some thres ...

*Post-modern portfolio theory Post-Modern Portfolio Theory (PMPT) is an extension of the traditional Modern Portfolio Theory (MPT), an application of mean-variance analysis (MVA). Both theories propose how rational investors can use diversification to optimize their portfolios. ...

* Sortino ratio
* Treynor ratio
* Two-moment decision models
* Universal portfolio algorithm
References

Further reading

* * *External links

The Most Rewarding Portfolio Construction Techniques: An Unbiased Evaluation

Prof.

William F. Sharpe
William Forsyth Sharpe (born June 16, 1934) is an American economist. He 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 University
Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is conside ...

Portfolio Optimization

Prof. Stephen P. Boyd,

Stanford University
Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is conside ...

*New Approaches for Portfolio Optimization: Parting with the Bell Curve

— Interview with Prof. Svetlozar Rachev and Prof. Stefan Mittnik {{DEFAULTSORT:Modern Portfolio Theory Portfolio theories Financial risk modeling