Portfolio Analysis
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Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the
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 b ...
is maximized for a given level of risk. It is a formalization and extension of
diversification Diversification may refer to: Biology and agriculture * Genetic divergence, emergence of subpopulations that have accumulated independent genetic changes * Agricultural diversification involves the re-allocation of some of a farm's resources to ...
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 numbers ...
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 In finance, the Markowitz model ─ put forward by Harry Markowitz in 1952 ─ is a portfolio optimization model; it assists in the selection of the most efficient portfolio by analyzing various possible portfolios of the given securities. Here ...
.


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 a
rational 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 \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 statistics ...
s ''ρ''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.
In general: *Expected return: : \operatorname(R_p) = \sum_i w_i \operatorname(R_i) \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: : \sigma_p^2 = \sum_i w_i^2 \sigma_^2 + \sum_i \sum_ w_i w_j \sigma_i \sigma_j \rho_ , :where \sigma_ is the (sample) standard deviation of the periodic returns on an asset ''i'', and \rho_ is the
correlation coefficient A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components ...
between the returns on assets ''i'' and ''j''. Alternatively the expression can be written as: : \sigma_p^2 = \sum_i \sum_j w_i w_j \sigma_i \sigma_j \rho_ , :where \rho_ = 1 for i = j , or : \sigma_p^2 = \sum_i \sum_j w_i w_j \sigma_ , :where \sigma_ = \sigma_i \sigma_j \rho_ is the (sample) covariance of the periodic returns on the two assets, or alternatively denoted as \sigma(i,j) , \text_ or \text(i,j) . *Portfolio return volatility (standard deviation): : \sigma_p = \sqrt For a two-asset portfolio: *Portfolio return: \operatorname(R_p) = w_A \operatorname(R_A) + w_B \operatorname(R_B) = w_A \operatorname(R_A) + (1 - w_A) \operatorname(R_B). *Portfolio variance: \sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2w_Aw_B \sigma_ \sigma_ \rho_ For a three-asset portfolio: *Portfolio return: \operatorname(R_p) = w_A \operatorname(R_A) + w_B \operatorname(R_B) + w_C \operatorname(R_C) *Portfolio variance: \sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + w_C^2 \sigma_C^2 + 2w_Aw_B \sigma_ \sigma_ \rho_ + 2w_Aw_C \sigma_ \sigma_ \rho_ + 2w_Bw_C \sigma_ \sigma_ \rho_


Diversification

An investor can reduce portfolio risk (especially \sigma_p) simply by holding combinations of instruments that are not perfectly positively
correlated 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 statistics ...
(
correlation coefficient A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components ...
-1 \le \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 slide
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).
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'' (franchis ...
are preferred for calculations of the efficient frontier. In matrix form, for a given "risk tolerance" q \in [0,\infty), the efficient frontier is found by minimizing the following expression: : w^T \Sigma w - q\times R^T w where * w is a vector of portfolio weights and \sum_i w_i = 1. (The weights can be negative); * \Sigma is the covariance matrix for the returns on the assets in the portfolio; * q \ge 0 is a "risk tolerance" factor, where 0 results in the portfolio with minimal risk and \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 \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 in
Visual 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 supported ...
, in
JavaScript 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 Website, websites use JavaScript on the Client (computing), client side ...
and in a few other languages. Also, many software packages, including
MATLAB 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, implementation ...
,
Microsoft Excel Microsoft Excel is a spreadsheet developed by Microsoft for Microsoft Windows, Windows, macOS, Android (operating system), Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro (comp ...
,
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, optimizat ...
and R, provide generic
optimization 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 \Sigma w subject to :R^T w = \mu for parameter \mu. This problem is easily solved using a
Lagrange 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: :\begin2\Sigma &-R & -\\ R^T &0 & 0 \\ ^T &0 &0 \end \beginw\\\lambda_1\\\lambda_2\end = \begin0\\\mu \\ 1\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 a
risk-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 ra ...
. In practice, short-term government securities (such as US
treasury bills United States Treasury securities, also called Treasuries or Treasurys, are government debt instruments issued by the United States Department of the Treasury to finance government spending as an alternative to taxation. Since 2012, U.S. gov ...
) 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 In finance, the Sharpe ratio (also known as the Sharpe index, the Sharpe measure, and the reward-to-variability ratio) measures the performance of an investment such as a security or portfolio compared to a risk-free asset, after adjusting for its ...
. 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 + \sigma_C \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 aggreg ...
(a.k.a. portfolio risk or market risk) refers to the risk common to all securities—except for
selling short In finance, being short in an asset means investing in such a way that the investor will profit if the value of the asset falls. This is the opposite of a more conventional "long" position, where the investor will profit if the value of the ...
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 CAPM may refer to: * Capital asset pricing model, a fundamental model in finance * Certified Associate in Project Management, an entry-level credential for project managers {{Disambig ...
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: : \operatorname(R_i) = R_f + \beta_i (\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 labiod ...
, 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. * (\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 \sigma_m ^2 + w_a^2 \sigma_a^2 + 2 w_m w_a \rho_ \sigma_a \sigma_m ) ::Hence, risk added to portfolio = w_a^2 \sigma_a^2 + 2 w_m w_a \rho_ \sigma_a \sigma_m ::but since the weight of the asset will be relatively low, w_a^2 \approx 0 ::i.e. additional risk = 2 w_m w_a \rho_ \sigma_a \sigma_m\quad *Market portfolio's expected return = ( w_m \operatorname(R_m) + w_a \operatorname(R_a) ) ::Hence additional expected return = w_a \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 \operatorname(R_a) > R_f . :Thus: w_a ( \operatorname(R_a) - R_f ) / w_m w_a \rho_ \sigma_a \sigma_m = w_a ( \operatorname(R_m) - R_f ) / w_m w_a \sigma_m \sigma_m :i.e. : operatorname(R_a) = R_f + operatorname(R_m) - R_f* \rho_ \sigma_a \sigma_m / \sigma_m \sigma_m :i.e. : operatorname(R_a) = R_f + operatorname(R_m) - R_f* sigma_ / \sigma_ : sigma_ / \sigma_\quad is the "beta", \beta return— the
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 ...
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 ).
This equation can be
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: :\mathrm : R_ - R_ = \alpha_i + \beta_i\, ( R_ - R_ ) + \epsilon_ \frac where α''i'' is called the asset's
alpha 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 , whic ...
, β''i'' is the asset's
beta coefficient In finance, the beta (β or market beta or beta coefficient) is a measure of how an individual asset moves (on average) when the overall stock market increases or decreases. Thus, beta is a useful measure of the contribution of an individual a ...
and SCL is the
security 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", "Efficient ...
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 inte ...
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 on
expected 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 l ...
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 numbers ...
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 normal ...
) 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 (probability theory), 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 ...
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 numbers ...
is a symmetric measure that counts abnormally high returns as just as risky as abnormally low returns. The psychological phenomenon of
loss aversion Loss aversion is the tendency to prefer avoiding losses to acquiring equivalent gains. The principle is prominent in the domain of economics. What distinguishes loss aversion from risk aversion is that the utility of a monetary payoff depends o ...
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. Already in the 1960s, Benoit Mandelbrot and
Eugene Fama Eugene Francis "Gene" Fama (; born February 14, 1939) is an American economist, best known for his empirical work on portfolio theory, asset pricing, and the efficient-market hypothesis. He is currently Robert R. McCormick Distinguished Servic ...
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 sta ...
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 In finance, the Black–Litterman model is a mathematical model for portfolio allocation developed in 1990 at Goldman Sachs by Fischer Black and Robert Litterman, and published in 1992. It seeks to overcome problems that institutional investors ha ...
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 of
rational 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-σ2''-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 In financial mathematics, a deviation risk measure is a function to quantify financial risk (and not necessarily downside risk) in a different method than a general risk measure. Deviation risk measures generalize the concept of standard deviation ...
.


Other applications

In the 1970s, concepts from MPT found their way into the field of
regional science Regional science is a field of the social sciences concerned with analytical approaches to problems that are specifically urban, rural, or regional. Topics in regional science include, but are not limited to location theory or spatial economics, l ...
. 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 the
sunk 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 be ...
(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

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Beta (finance) In finance, the beta (β or market beta or beta coefficient) is a measure of how an individual asset moves (on average) when the overall stock market increases or decreases. Thus, beta is a useful measure of the contribution of an individual as ...
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Bias ratio (finance) The bias ratio is an indicator used in finance to analyze the returns of investment portfolios, and in performing due diligence. The bias ratio is a concrete metric that detects valuation bias or deliberate price manipulation of portfolio assets ...
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Black–Litterman model In finance, the Black–Litterman model is a mathematical model for portfolio allocation developed in 1990 at Goldman Sachs by Fischer Black and Robert Litterman, and published in 1992. It seeks to overcome problems that institutional investors ha ...
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Intertemporal portfolio choice Intertemporal portfolio choice is the process of allocating one's investable wealth to various assets, especially financial assets, repeatedly over time, in such a way as to optimize some criterion. The set of asset proportions at any time defines ...
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Investment theory Investment is the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance, the purpose of investing is ...
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Kelly criterion In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet), is a formula that determines the optimal theoretical size for a bet. It is valid when the expected returns are known. The Kelly bet size is found by maximizing the expec ...
* Marginal conditional stochastic dominance *
Markowitz model In finance, the Markowitz model ─ put forward by Harry Markowitz in 1952 ─ is a portfolio optimization model; it assists in the selection of the most efficient portfolio by analyzing various possible portfolios of the given securities. Here ...
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Mutual fund separation theorem In portfolio theory, a mutual fund separation theorem, mutual fund theorem, or separation theorem is a theorem stating that, under certain conditions, any investor's optimal portfolio can be constructed by holding each of certain mutual funds in ap ...
* Omega ratio *
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. ...
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Sortino ratio The Sortino ratio measures the risk-adjusted return of an investment asset, portfolio, or strategy. It is a modification of the Sharpe ratio but penalizes only those returns falling below a user-specified target or required rate of return, while t ...
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Treynor ratio The Treynor reward to volatility model (sometimes called the reward-to-volatility ratio or Treynor measure), named after Jack L. Treynor, is a measurement of the returns earned in excess of that which could have been earned on an investment that has ...
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Two-moment decision models In decision theory, economics, and finance, a two-moment decision model is a model that describes or prescribes the process of making decisions in a context in which the decision-maker is faced with random variables whose realizations cannot be ...
* Universal portfolio algorithm


References


Further reading

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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. ...
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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 consider ...

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 consider ...
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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