Brief Description of the FUND PS program
FUND is a stock Portfolio selection program using Markowitz theory. The Markowitz frontier and the portfolios corresponding to each of its points are calculated.
In terms of functions the program is divided into three parts, which very briefly are as follows:
- In the first part the aim is to find a portfolio that will hold for the immediately following period. For this purpose a strategy is defined and simulated (selection of a portfolio from the frontier based on the strategy and calculation of the returns) over previous periods. Portfolios are assumed to be bought at the beginning of each period and sold at its end (buy and hold).
- In the second part of the program, the Markowitz frontier for a given period is calculated and analysed thoroughly, along with the return of each of its points on dates after the last date taken into account in the calculations, and a wealth of other data
- In the third part of the program, one can enter a portfolio one already holds, and after calculations the program shows portfolios that have either a better return at the same risk, or lower risk at the same return, or some combination of the two. One can also see, at a time later than the calculations, what the various portfolios with better risk and return characteristics yielded, as well as a comparison with the investor’s original portfolio.
The APPENDIX says a few words about the relevant Theory, for which Markowitz received the NOBEL prize.
The chart below shows all the Markowitz frontiers together, for all calculation periods.
The chart below shows the Markowitz frontier (blue line) and the returns that would be achieved, with no cash at all, at various points on it (green line; the return is shown on the right vertical axis)
The chart below shows the percentage share of one stock in the portfolios one would choose at various levels of risk. (there are as many charts as there are stocks on the Athens Stock Exchange)
It thus appears that this stock is placed in portfolios at high risk and, equivalently, at higher expected return. (The blue line is the Markowitz frontier and the green one the percentage share.
In part C, with the help of the Markowitz frontier, an investor’s existing portfolio is restructured so as to become Markowitz-optimal.
Markowitz portfolio theory
Markowitz portfolio theory is a mathematical model of the fluctuations of the prices of the stocks of an exchange, on the basis of which those portfolios are determined that achieve the optimal trade-off between expected (mean) return and risk, that is, portfolios that maximise the expected return for a given level of risk or, equivalently, minimise the risk for a given expected return.
A basic assumption of this model is that the vector of daily fluctuations of the prices of the stocks of an exchange follows a normal (Gaussian) multivariate distribution with constant mean returns and a constant covariance matrix over time.
A portfolio is defined as a vector of percentages invested in each stock, which may or may not be subject to additional constraints depending on the wishes of the user-investor.
The expected return of a portfolio is defined as the mean of the portfolio’s returns, and the risk is defined as its standard deviation.
The portfolios that achieve the optimal risk-return trade-off under the above conditions are called efficient, and the set of efficient portfolios is called the Markowitz frontier.
Determining the Markowitz frontier for a given time sequence of stock price vectors and for a given set of linear constraints on the portfolios reduces to solving a sequence of optimisation problems belonging to the class of convex quadratic programming problems.
These problems are solved using specialised efficient algorithms that work for input data of any kind and size.
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