Markowitz Portfolio Theory

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Markowitz Portfolio Theory

Harry Markowitz introduced and later won the Nobel Prize for the Modern Portfolio Theory in his 1952 article and 1959 book. His theory attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets.

The mathematical formulation is the following.

We solve a quadratic programming problem to find the portfolios weights (ω’s) that maximize this objective function, where λ is the risk-adverse coefficient between 0 and ∞.

$$R^T w – \lambda \times w^T \Sigma w$$

One (unity) constraint is:

$$\sum_i w_i = 1$$

In additional, AlgoQuant lets you put any (reasonable) constraints on the ω’s as you like. For example, one common constraint is to disallow short selling. That is,

$$w_i \ge 0$$

References

​http://en.wikipedia.org/wiki/Modern_Portfolio_Theory

Demo

An online demo is available.

Code

http://redmine.numericalmethod.com/projects/public/repository/svn-algoquant/show/core/src/main/java/com/numericalmethod/algoquant/model/portfoliooptimization/markowitz

http://redmine.numericalmethod.com/projects/public/repository/svn-algoquant/entry/core/src/main/java/com/numericalmethod/algoquant/model/portfoliooptimization/markowitz/MarkowitzUtils.java

http://redmine.numericalmethod.com/projects/public/repository/svn-algoquant/entry/core/src/main/java/com/numericalmethod/algoquant/model/portfoliooptimization/markowitz/MarkowitzPortfolio.java