Portfolio Selection in Mean-Minimum Return Level-Expected Bounded First Passage Time Framework

Tsotne Kutalia

doi:10.4236/jmf.2019.93012

Abstract

This paper explores the selection of optimal portfolio by replacing the standard Mean-Variance model by Mean-Minimum Return Level (MRL) framework and adding one important dimension—expectation of bounded First Passage Time (FPT) towards the MRL. To measure how much a given portfolio is exposed to risk, the new model can capture both, the amount of the largest possible loss at a certain confidence level and time to such an event occurring. The novelty of this paper is the introduction of bounded first passage time towards MRL and taking its expectation into consideration as an additional factor in portfolio selection decision making. Assuming that the asset price dynamics follow multi-dimensional Geometric Brownian Motion with drift, we obtain a portfolio wealth process for multiple assets and we evaluate the lowest possible value to which it can drop by a high confidence level. Then we extend our examination of the optimal portfolio selection by ultimately obtaining the efficient surface of risky portfolios. As a result, the paper shows that the third dimension can make a significant difference while choosing the asset weights compared to classical models ignoring the portfolio return paths as long as they achieve a desired combination of risk and return.

Highlights

Portfolio selection theories have gone through various improvements since the introduction of its most prominent theory by Harry Markowitz in 1952
This paper explores the selection of optimal portfolio by replacing the standard Mean-Variance model by Mean-Minimum Return Level (MRL) framework and adding one important dimension—expectation of bounded First Passage Time (FPT) towards the MRL
Once having MRLs and portfolio expected returns computed for different sets of asset weights, we extend the framework by introducing expected first passage time bounded by investment horizon as a third dimension used for decision making

Summary

Introduction

Portfolio selection theories have gone through various improvements since the introduction of its most prominent theory by Harry Markowitz in 1952. One may allocate funds into assets in proportions, which while being optimal in Mean-Variance sense, can cause hitting the MRL level faster by having overlooked one important factor—expected time of the portfolio return process towards the minimum level This may be a source of severe problems for investors who are exposed to margin calls or need to raise funds in a short period of time if such an event is realized. Once having MRLs and portfolio expected returns computed for different sets of asset weights, we extend the framework by introducing expected first passage time bounded by investment horizon as a third dimension used for decision making. In a highly volatile environment, portfolio of assets selected by the Mean-Variance framework will hit the lowest possible return level earlier than the portfolio selected by Mean-MRL-FTP framework and at the same time, the latter includes the risk measured by variance as it is reflected in computation of MRL. It combines all the three dimensions and obtains the efficient surface of risky portfolios

Portfolio Wealth Process

Minimum Return Level

Mean-MRL-FPT Framework

Conclusions