# Modern portfolio theory Canonical: https://web4.xpay.bi/en/wiki/wp-en-734434/ Language: en Published: 2026-10-26T16:37:27.216Z Updated: 2026-10-26T16:37:27.216Z Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of financial assets such that the expected return 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. ## Key facts ## Overview Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of financial assets such that the expected return 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 principle 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. The variance of return (or its transformation, the standard deviation) is used as a measure of risk, because it is tractable when assets are combined into portfolios. Often, the historical variance and covariance of returns is used as a proxy for the forward-looking versions of these quantities, but other, more sophisticated methods are available. 25-year-old economist Harry Markowitz introduced MPT in a 1952 paper titled "Portfolio Selection", with its ideas further refined by James Tobin and simplified by graduate student William F. Sharpe, the latter under Markowitz's supervision; for these, Markowitz and Sharpe were later awarded a Nobel Memorial Prize in Economic Sciences together in 1990 (see Markowitz model). Although Markowitz's paper had been described by financial historian Peter L. Bernstein as "a landmark in the history of ideas" for its foundational contribution to financial economics, MPT has received varied criticism for not adequately reflecting the reality of financial markets. In 1940, Bruno de Finetti published the mean-variance analysis method, in the context of proportional reinsurance, under a stronger assumption. The paper was obscure and only became known to economists of the English-speaking world in 2006. ## 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 investor will not invest in a portfolio if a second portfolio exists with a more favorable risk vs 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 σ p \{\\displaystyle \\sigma \_\{p\}\} is a function of the correlations ρij 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: E ⁡ ( R p ) = ∑ i w i E ⁡ ( R i ) ## Diversification An investor can reduce portfolio risk (especially σ p \{\\displaystyle \\sigma \_\{p\}\} ) simply by holding combinations of instruments that are not perfectly positively correlated (correlation coefficient − 1 ≤ ρ i j \< 1 \{\\displaystyle -1\\leq \\rho \_\{ij\}\<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. 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, 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 are preferred for calculations of the efficient frontier. In matrix form, for a given "risk tolerance" q ∈ \[ 0 , ∞ ) \{\\displaystyle q\\in \[0,\\infty )\} , the efficient frontier is found by minimizing the following expression: w T Σ w − q R T w \{\\displaystyle w^\{T\}\\Sigma w-qR^\{T\}w\} where w ∈ R N \{\\displaystyle w\\in \\mathbb \{R\} ^\{N\}\} is a vector of portfolio weights and ∑ i = 1 ## Two mutual fund theorem One key result of the above analysis is the two mutual fund 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 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. 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 if held to maturity (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 hyperbola 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 hyperbola 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 (CAL), and its formula can be shown to be E ( R C ) = R F + σ C E ( R P ) − R F σ P . \{\\displaystyle E(R\_\{C\})=R\_\{F\}+\\sigma \_\{C\}\{\\frac \{E(R\_\{P\})-R\_\{F\}\}\{\\sigma \_\{P\}\}\}.\} ## Geometric intuition The efficient frontier can be pictured as a problem in quadratic curves. On the market, we have the assets R 1 , R 2 , … , R n \{\\displaystyle R\_\{1\},R\_\{2\},\\dots ,R\_\{n\}\} . We have some funds, and a portfolio is a way to divide our funds into the assets. Each portfolio can be represented as a vector w 1 , w 2 , … , w n \{\\displaystyle w\_\{1\},w\_\{2\},\\dots ,w\_\{n\}\} ## Markowitz bullet Since we wish to maximize expected return while minimizing the standard deviation of the return, we are to solve a quadratic optimization problem: \{ E \[ w T R \] = μ min σ 2 = V a r \[ w T R \] ∑ i w i = 1 \{\\displaystyle \{\\begin\{cases\}E\[w^\{T\}R\]=\\mu \\\\\\min \\sigma ^\{2\}=Var\[w^\{T\}R\]\\\\\\sum \_\{i\}w\_\{i\}=1\\end\{cases\}\}\} Portfolios are points in the Euclidean space ## Tangency portfolio The tangency portfolio exists if and only if μ R F \< μ M V P \{\\displaystyle \\mu \_\{RF\}\<\\mu \_\{MVP\}\} . In particular, if the risk-free return is greater or equal to μ M V P \{\\displaystyle \\mu \_\{MVP\}\} , then the tangent portfolio does not exist. The capital market line (CML) becomes parallel to the upper asymptote line of the hyperbola. Points on the CML become impossible to achieve, though they can be approached from below. It is usually assumed that the risk-free return is less than the return of the global MVP, in order that the tangency portfolio exists. However, even in this case, as μ R F \{\\displaystyle \\mu \_\{RF\}\} approaches μ M V P \{\\displaystyle \\mu \_\{MVP\}\} from below, the tangency portfolio diverges to a portfolio with infinite return and variance. Since there are only finitely many assets in the market, such a portfolio must be shorting some assets heavily while longing some other assets heavily. In practice, such a tangency portfolio would be impossible to achieve, because one cannot short an asset too much due to short sale constraints, and also because of price impact, that is, longing a large amount of an asset would push up its price, breaking the assumption that the asset prices do not depend on the portfolio. ## Non-invertible covariance matrix If the covariance matrix is not invertible, then there exists some nonzero vector v \{\\displaystyle v\} , such that v T R \{\\displaystyle v^\{T\}R\} is a random variable with zero variance—that is, it is not random at all. Suppose ∑ i v i = 0 \{\\displaystyle \\sum \_\{i\}v\_\{i\}=0\} and v T R = 0 \{\\displaystyle v^\{T\}R=0\} ## Asset pricing The above analysis describes optimal behavior of an individual investor. Asset pricing theory builds on this analysis, allowing MPT to derive the required expected return for a correctly priced asset in this context. Intuitively (in a perfect market with rational investors), if a security was expensive relative to others - i.e. too much risk for the price - demand would fall and its price would drop correspondingly; if cheap, demand and price would increase likewise. This would continue until all such adjustments had ceased - a state of "market equilibrium". In this equilibrium, relative supplies will equal relative demands: given the relationship of price with supply and demand, since the risk-to-reward ratio is "identical" across all securities, proportions of each security in any fully-diversified portfolio would correspondingly be the same as in the overall market. More formally, then, 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. The result for expected return then follows, as below. ## 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 (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. ## Sources - Modern portfolio theory · Wikipedia: https://en.wikipedia.org/?curid=734434 Source: Wikipedia contributors — https://en.wikipedia.org/?curid=734434&oldid=1380421877 Licence: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation notice: Reformatted for XPay; the source may be abridged, restructured or translated. Consult the cited revision for the original.