Editing Modern portfolio theory (section)

===Efficient frontier with no risk-free asset===
{{main article|Efficient frontier}}
{{see also|Portfolio optimization}}
[[Image:markowitz frontier.jpg|thumb|269px|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.<ref>Portfolio Selection, Harry Markowitz - The Journal of Finance, Vol. 7, No. 1. (Mar., 1952), pp. 77-91</ref> 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,<ref name="Kempthorne">see bottom of slide 6 [https://ocw.mit.edu/courses/mathematics/18-s096-topics-in-mathematics-with-applications-in-finance-fall-2013/lecture-notes/MIT18_S096F13_lecnote14.pdf 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 [[#Risk-free asset and the capital allocation line|capital allocation line (CAL)]].

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[[Matrix (mathematics)|Matrices]] are preferred for calculations of the efficient frontier.

In matrix form, for a given "risk tolerance" <math>q \in [0,\infty)</math>, the efficient frontier is found by minimizing the following expression:
:<math>  w^T \Sigma w - q R^T w</math>
where
* <math>w\in\mathbb{R}^N</math> is a vector of portfolio weights and <math>\sum_{i=1}^N w_i = 1.</math> (The weights can be negative);
* <math>\Sigma\in\mathbb{R}^{N\times N}</math> is the [[covariance matrix]] for the returns on the assets in the portfolio;
* <math>q \ge 0</math> is a "risk tolerance" factor, where 0 results in the portfolio with minimal risk and <math>\infty</math> results in the portfolio infinitely far out on the frontier with both expected return and risk unbounded; and
* <math>R\in\mathbb{R}^N</math> is a vector of expected returns.
* <math>w^T \Sigma w\in\mathbb{R}</math> is the variance of portfolio return.
* <math>R^T w\in\mathbb{R}</math> 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 method|critical line algorithm]],<ref name="markowitz1956">{{cite journal |author=Markowitz, H.M. |title=The Optimization of a Quadratic Function Subject to Linear Constraints |journal=Naval Research Logistics Quarterly |volume=3 |issue=1–2 |date=March 1956 |pages=111–133 | doi=10.1002/nav.3800030110 }}</ref> 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]],<ref>{{Cite book|title=Mean-Variance Analysis in Portfolio Choice and Capital Markets|last=Markowitz|first=Harry|publisher=Wiley|date=February 2000|isbn=978-1-883-24975-5}}</ref> in [[JavaScript]]<ref>{{Cite web|url=https://github.com/lequant40/portfolio_allocation_js|title=PortfolioAllocation JavaScript library|website=github.com/lequant40|access-date=2018-06-13}}</ref> and in a few other languages.

Also, many software packages, including [[MATLAB]], [[Microsoft Excel]], [[Mathematica]] and [[R (programming language)|R]], provide generic [[Quadratic programming|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 <math>R^T w.</math>  This version of the problem requires that we minimize

:<math>  w^T \Sigma w  </math>

subject to

:<math>R^T w = \mu</math> 

and 

:<math>\sum_{i=1}^{N} w_i = 1</math>

for parameter <math>\mu</math>.  This problem is easily solved using a [[Lagrange multiplier]] which leads to the following linear system of equations:

:<math>\begin{bmatrix}2\Sigma &-R & -{\bf1}\\ R^T &0 & 0 \\ {\bf1}^T &0 &0 \end{bmatrix} \begin{bmatrix}w\\\lambda_1\\\lambda_2\end{bmatrix} = \begin{bmatrix}0\\\mu \\ 1\end{bmatrix}</math>

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