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.
Economist Harry Markowitz introduced MPT in a 1952 paper, for which he was later awarded a Nobel Memorial Prize in Economic Sciences; see Markowitz model.
In 1940, Bruno de Finetti publishedde Finetti, B. (1940): Il problema dei “Pieni”. Giornale dell’ Istituto Italiano degli Attuari 11, 1–88; translation (Barone, L. (2006)): The problem of full-risk insurances. Chapter I. The risk within a single accounting period. Journal of Investment Management 4(3), 19–43 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.
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 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 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:
where is the return on the portfolio, is the return on asset i and is the weighting of component asset (that is, the proportion of asset "i" in the portfolio, so that ).
Portfolio return variance: , where is the (sample) standard deviation of the periodic returns on an asset i, and is the correlation coefficient between the returns on assets i and j. Alternatively the expression can be written as:
, where for , or
, where is the (sample) covariance of the periodic returns on the two assets, or alternatively denoted as , or .
Portfolio return volatility (standard deviation):
For a two-asset portfolio: Portfolio expected return: Portfolio variance:
For a three-asset portfolio: Portfolio expected return: Portfolio variance:
The algebra can be much simplified by expressing the quantities involved in matrix notation.David Lando and Rolf Poulsen's lecture notes, Chapter 9, "Portfolio theory" Arrange the returns of N risky assets in an vector , where the first element is the return of the first asset, the second element of the second asset, and so on. Arrange their expected returns in a column vector , and their variances and covariances in a covariance matrix . Consider a portfolio of risky assets whose weights in each of the N risky assets is given by the corresponding element of the weight vector . Then:
Portfolio expected return: and Portfolio variance:
For the case where there is investment in a riskfree asset with return , the weights of the weight vector do not sum to 1, and the portfolio expected return becomes . The expression for the portfolio variance is unchanged.
Diversification
An investor can reduce portfolio risk (especially ) simply by holding combinations of instruments that are not perfectly positively correlated (correlation coefficient ). 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.Portfolio Selection, Harry Markowitz - The Journal of Finance, Vol. 7, No. 1. (Mar., 1952), pp. 77-91 The 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 6 here 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" , the efficient frontier is found by minimizing the following expression:
where is a vector of portfolio weights and (The weights can be negative); is the covariance matrix for the returns on the assets in the portfolio; is a "risk tolerance" factor, where 0 results in the portfolio with minimal risk and results in the portfolio infinitely far out on the frontier with both expected return and risk unbounded; and is a vector of expected returns. is the variance of portfolio return. is the expected return on the portfolio. The above optimization finds the point 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, 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, in JavaScript 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 This version of the problem requires that we minimize
subject to
and
for parameter . This problem is easily solved using a Lagrange multiplier which leads to the following linear system of equations:
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
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.