Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of 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 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 of asset prices as a proxy for risk.^[1]

Economist Harry Markowitz introduced MPT in a 1952 essay,^[2] for which he was later awarded a Nobel Memorial Prize in Economic Sciences; see Markowitz model.

Mathematical model

Risk and expected return

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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-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 ${\displaystyle \sigma _{p}}$ is a function of the correlations 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:

${\displaystyle \operatorname {E} (R_{p})=\sum _{i}w_{i}\operatorname {E} (R_{i})\quad }$

where ${\displaystyle R_{p}}$ is the return on the portfolio, ${\displaystyle R_{i}}$ is the return on asset i and ${\displaystyle w_{i}}$ is the weighting of component asset ${\displaystyle i}$ (that is, the proportion of asset "i" in the portfolio).

• Portfolio return variance:

${\displaystyle \sigma _{p}^{2}=\sum _{i}w_{i}^{2}\sigma _{i}^{2}+\sum _{i}\sum _{j\neq i}w_{i}w_{j}\sigma _{i}\sigma _{j}\rho _{ij}}$,

where ${\displaystyle \sigma _{i}}$ is the (sample) standard deviation of the periodic returns on an asset i, and ${\displaystyle \rho _{ij}}$ is the correlation coefficient between the returns on assets i and j. Alternatively the expression can be written as:

${\displaystyle \sigma _{p}^{2}=\sum _{i}\sum _{j}w_{i}w_{j}\sigma _{i}\sigma _{j}\rho _{ij}}$,

where ${\displaystyle \rho _{ij}=1}$ for ${\displaystyle i=j}$ , or

${\displaystyle \sigma _{p}^{2}=\sum _{i}\sum _{j}w_{i}w_{j}\sigma _{ij}}$,

where ${\displaystyle \sigma _{ij}=\sigma _{i}\sigma _{j}\rho _{ij}}$ is the (sample) covariance of the periodic returns on the two assets, or alternatively denoted as ${\displaystyle \sigma (i,j)}$, ${\displaystyle {\text{cov}}_{ij}}$ or ${\displaystyle {\text{cov}}(i,j)}$.

• Portfolio return volatility (standard deviation):

${\displaystyle \sigma _{p}={\sqrt {\sigma _{p}^{2}}}}$

For a two-asset portfolio:

• Portfolio return: ${\displaystyle \operatorname {E} (R_{p})=w_{A}\operatorname {E} (R_{A})+w_{B}\operatorname {E} (R_{B})=w_{A}\operatorname {E} (R_{A})+(1-w_{A})\operatorname {E} (R_{B}).}$
• Portfolio variance: ${\displaystyle \sigma _{p}^{2}=w_{A}^{2}\sigma _{A}^{2}+w_{B}^{2}\sigma _{B}^{2}+2w_{A}w_{B}\sigma _{A}\sigma _{B}\rho _{AB}}$

For a three-asset portfolio:

• Portfolio return: ${\displaystyle \operatorname {E} (R_{p})=w_{A}\operatorname {E} (R_{A})+w_{B}\operatorname {E} (R_{B})+w_{C}\operatorname {E} (R_{C})}$
• Portfolio variance: ${\displaystyle \sigma _{p}^{2}=w_{A}^{2}\sigma _{A}^{2}+w_{B}^{2}\sigma _{B}^{2}+w_{C}^{2}\sigma _{C}^{2}+2w_{A}w_{B}\sigma _{A}\sigma _{B}\rho _{AB}+2w_{A}w_{C}\sigma _{A}\sigma _{C}\rho _{AC}+2w_{B}w_{C}\sigma _{B}\sigma _{C}\rho _{BC}}$

Diversification

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An investor can reduce portfolio risk (especially ${\displaystyle \sigma _{p}}$) simply by holding combinations of instruments that are not perfectly positively correlated (correlation coefficient ${\displaystyle -1\leq \rho _{ij}<1}$). 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. 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

Efficient Frontier. The hyperbola is sometimes referred to as the 'Markowitz Bullet', and is the efficient frontier if no risk-free asset is available. With a risk-free asset, the straight line is the efficient frontier.

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.^[3] 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 parabolic,^[4] and the upper part of the parabolic 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 parabolic boundary is the capital allocation line (CAL).

Matrices are preferred for calculations of the efficient frontier.

In matrix form, for a given "risk tolerance" ${\displaystyle q\in 0,\infty )}$, the efficient frontier is found by minimizing the following expression:

${\displaystyle w^{T}\Sigma w-q\times R^{T}w}$

where

• ${\displaystyle w}$ is a vector of portfolio weights and ${\displaystyle \sum _{i}w_{i}=1.}$ (The weights can be negative);
• ${\displaystyle \Sigma }$ is the covariance matrix for the returns on the assets in the portfolio;
• ${\displaystyle q\geq 0}$ is a "risk tolerance" factor, where 0 results in the portfolio with minimal risk and ${\displaystyle \infty }$ results in the portfolio infinitely far out on the frontier with both expected return and risk unbounded; and
• ${\displaystyle R}$ is a vector of expected returns.
• ${\displaystyle w^{T}\Sigma w}$ is the variance of portfolio return.
• ${\displaystyle R^{T}w}$ 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 developed a specific procedure for solving the above problem, called the critical line algorithm,^[5] 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,^[6] in JavaScript^[7] and in a few other languages.

Also, many software packages, including MATLAB, Microsoft Excel, Mathematica and R, provide generic optimization 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 ${\displaystyle R^{T}w.}$ This version of the problem requires that we minimize

${\displaystyle w^{T}\Sigma w}$

subject to

${\displaystyle R^{T}w=\mu }$

for parameter ${\displaystyle \mu }$. This problem is easily solved using a Lagrange multiplier which leads to the following linear system of equations:

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