The Risk Profile Problem for Stock Portfolio Optimization

TL;DR

This paper explores optimal stock portfolio strategies balancing risk and profit, providing exact solutions for two-stock portfolios and approximation methods for larger portfolios, highlighting computational complexities involved.

Contribution

Introduces a fast greedy algorithm for two-stock portfolios and discusses approximation algorithms for larger portfolios, addressing computational challenges.

Findings

Exact solutions for two-stock portfolios using maximum flow algorithms.

NP-hardness results for portfolios with more than two stocks.

Approximation algorithms for high-dimensional portfolio optimization.

Abstract

This work initiates research into the problem of determining an optimal investment strategy for investors with different attitudes towards the trade-offs of risk and profit. The probability distribution of the return values of the stocks that are considered by the investor are assumed to be known, while the joint distribution is unknown. The problem is to find the best investment strategy in order to minimize the probability of losing a certain percentage of the invested capital based on different attitudes of the investors towards future outcomes of the stock market. For portfolios made up of two stocks, this work shows how to exactly and quickly solve the problem of finding an optimal portfolio for aggressive or risk-averse investors, using an algorithm based on a fast greedy solution to a maximum flow problem. However, an investor looking for an average-case guarantee (so is neither aggressive or risk-averse) must deal with a more difficult problem. In particular, it is #P-complete to compute the distribution function associated with the average-case bound. On the positive side, approximate answers can be computed by using random sampling techniques similar to those for high-dimensional volume estimation. When k>2 stocks are considered, it is proved that a simple solution based on the same flow concepts as the 2-stock algorithm would imply that P = NP, so is highly unlikely. This work gives approximation algorithms for this case as well as exact algorithms for some important special cases.