Modern portfolio theory

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.

[1] Often, the historical variance and covariance of returns is used as a proxy for the forward-looking versions of these quantities,[2] but other, more sophisticated methods are available.

In 1940, Bruno de Finetti published[4] the mean-variance analysis method, in the context of proportional reinsurance, under a stronger assumption.

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 frontier in its entirety is parametric on q. Harry Markowitz developed a specific procedure for solving the above problem, called the critical line algorithm,[9] 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,[10] in JavaScript[11] 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...).

This problem is easily solved using a Lagrange multiplier which leads to the following linear system of equations: One key result of the above analysis is the two mutual fund theorem.

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.

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. 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 hyperbola does at every possible risk level.

The price paid must ensure that the market portfolio's risk / return characteristics improve when the asset is added to it.

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.

[15] In practice, investors must substitute predictions based on historical measurements of asset return and volatility for these values in the equations.

Options theory and MPT have at least one important conceptual difference from the probabilistic risk assessment done by nuclear power [plants].

And, unlike the PRA, if there is no history of a particular system-level event like a liquidity crisis, there is no way to compute the odds of it.

If nuclear engineers ran risk management this way, they would never be able to compute the odds of a meltdown at a particular plant until several similar events occurred in the same reactor design.

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.

Already in the 1960s, Benoit Mandelbrot and Eugene Fama showed the inadequacy of this assumption and proposed the use of more general stable distributions instead.

[19][20][21] More recently, Nassim Nicholas Taleb has also criticized modern portfolio theory on this ground, writing:After the stock market crash (in 1987), they rewarded two theoreticians, Harry Markowitz and William Sharpe, who built beautifully Platonic models on a Gaussian base, contributing to what is called Modern Portfolio Theory.

Simply, if you remove their Gaussian assumptions and treat prices as scalable, you are left with hot air.

The Nobel Committee could have tested the Sharpe and Markowitz models—they work like quack remedies sold on the Internet—but nobody in Stockholm seems to have thought about it.

[22] One objection is that the MPT relies on the efficient-market hypothesis and uses fluctuations in share price as a substitute for risk.

Sir John Templeton believed in diversification as a concept, but also felt the theoretical foundations of MPT were questionable, and concluded (as described by a biographer): "the notion that building portfolios on the basis of unreliable and irrelevant statistical inputs, such as historical volatility, was doomed to failure.

"[23] A few studies have argued that "naive diversification", splitting capital equally among available investment options, might have advantages over MPT in some situations.

Post-modern portfolio theory extends MPT by adopting non-normally distributed, asymmetric, and fat-tailed measures of risk.

Alternatively, mean-deviation analysis[29] is a rational choice theory resulting from replacing variance by an appropriate deviation risk measure.

[31] Recently, modern portfolio theory has been applied to modelling the uncertainty and correlation between documents in information retrieval.

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.

When risk is put in terms of uncertainty about forecasts and possible losses then the concept is transferable to various types of investment.

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 ellipsoid is the contour of constant variance. The plane is the space of possible portfolios. The other plane is the contour of constant expected return. The ellipsoid intersects the plane to give an ellipse of portfolios of constant variance. On this ellipse, the point of maximal (or minimal) expected return is the point where it is tangent to the contour of constant expected return. All these portfolios fall on one line.
Illustration of the effect of changing the risk-free asset return rate. As the risk-free return rate approaches the return rate of the global minimum-variance portfolio, the tangency portfolio escapes to infinity. Animated at source [2] .