One of the most basic traits of (rational) human beings is that we always tend to look for the maximum possible reward we can attain, whilst maximizing whatever risks involved in all our endeavors. The same is certainly true when it comes to the world of finance and investments.
Portfolio Optimization is the part of investment analysis that deals with the problem of selecting investment holdings so as to get the maximal return for a given risk that an investor is willing to incur; or conversely, it is the problem of finding getting the minimal risk in order to guarantee a certain expected1 return. The main problem that investment analysts face is, how do we measure risk.
Economist/Mathematician Harry Markowitz, one of the pioneers of Modern Portfolio Theory, modelled risk of a given asset2 as the standard deviation/variance, σ; in his work in his 1952 paper, "Portfolio Selection".
In Markowitz's Model, the investor has a selection of n risky assets (usually stocks), each with expected return μi and standard deviation σi; i = 1,2,...,n., and one non-risky asset (usually a bond) with expected return μn+1 and standard deviation σn+1 ≡ 0. (i.e., zero-risk). The Portfolio Selection Problem involves allocating a proportion xi ≥ 0 of an investors wealth to the i-th asset (i = 1,2,...,n+1). Clearly Σxi = 1, ⇒ xi ≤ 1, ∀ i.
Our aim is to simultaneously maximize total return, μ = Σ μixi and minimize the total risk σ = Σ Σ σijxixj. Here σij denotes the covariance between asset i and j, and σii = σi2, i.e. the variance (the square of the standard deviation); subject to constraints {Σxi = 1, 0 ≤ xi ≤ 1}. Other constraints may be added, for instance lower bounds li and/or upper bounds ui to all or some of the xi (other than, but not contradicting the trivial bounds 0 and 1); however the objective and constraints given above define a plausible portfolio optimization problem.
NB: The simultaneous maximization and minimization is attained by setting the objective function to f(x) = ½ σ - t μ [where x = (x1, x2, ..., xn, xn+1)]. The objective may also be modified to include transaction costs. t ∈ [0, +∞) is the risk tolerance factor, which is the inverse of the risk aversion factor, λ.
The analysis above is called Mean-Variance analysis.
This is a quadratic programming (QP) problem which does have a minimum since the covariance matrix is positive definite. Current research amongst quantitative analysts is centered around devising specialized QP algorithm for various special cases to guarantee fast and efficient computation. Professor Michael Best of University of Waterloo deals with such kind of research. I took a Portfolio Optimization course with him. Other research is concerned with calculation of t, which is linked to choices of the utility function.
Markowitz's results are highly celebrated and they form the basis of Modern Portfolio Theory. The results are also connected with such concepts like the Efficient Frontier (which is the "best" portfolio one can hold under Markowitz's Model); as well as other concepts like Sharpe ratio; Carpital Marekt Line (CML) and The Market Portfolio; Capital Asset Pricing Model (CAPM).
However, there are some shortcomings to the Quadratic (Markowitz) Model; one being that a more natural way for a layman investor to define risk is not through variance, but rather through other measures such as Value at Risk, VaR. i.e., roughly speaking, an investor usually wants to know, what is the probability of incurring a loss of a fraction α of my investment , for a given portfolio that I chose (say I chose to put TZS 3m in TBL shares, 2m in TCC shares and 5m in 182-day T-Bill). VaR analysis is well researched and there are numerous papers online; and it is a newer concept and more of an in-thing in Quantitative Analysis and Financial Engineering as opposed to Mean-Variance Analysis.
VaR analysis is used not just in investment theory, but also in other areas of economics as well as operations research, management science and project analysis for engineers. The Mathematical analysis of Portfolio Optimization using VaR is a bit more technical than Mean-Variance Analysis and it involves some sophisticated concepts in Convex Analysis and Nonlinear Programming. A not-too-technical presentation can be found here (pdf) or in Power point here. It is from a talk I gave in the research skills seminar at Waterloo. It is based on Rockefeller and Ursayev (1999). The talk was aimed at an audience full of Mathematicians, however only basic points and just sketches of proofs are given so as to ease understanding.
Footnote
1. Here expected means the Expected value in probability sense, i.e., the mean value
2. Asset means financial asset, e.g. stock, bond, etc.
This article is aimed at helping people not familiar with Quantitative Analysis, and particularly Portfolio Optimization to get a gist of the kind of stuff Quants do. I promise more articles to come. I welcome any questions and comments
-Nalitolela, P. S.
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