Harry Markowitz’s Portfolio Management Optimization

HARRY MARKOWITZ’S PORTFOLIO MANAGEMENT OPTIMIZATION [pic 1]

1. Issue of Risk Management

1. General objectives

Risk Management is a complex issue, because it depends on the adopted perspective by the holder of the portfolio. Several cases appear:

• When we invest operational way in a certain period of time, we can look for to minimize a measure of temporal risk, which the best known is “Value at Risk”. It’s an optimization issue of the portfolio in a limited view. The question to which we’d like to answer is

“How can we modify the portfolio in a way that the potential loss in a certain view of management might be minimal?”

-If we place the operational way in a short term, we can choose to make insensitive the portfolio the hazard variation on which it depends on, and especially of the underlying portfolio. As a matter of fact, it’s an issue of optimization of local portfolio in time. We answer to the following question:

“How can we modify the portfolio in short term in order not to be worried of the random variation of the market?’

-When we consider the portfolio in all its life length, we can look for minimize a measure of global risk. So it’s a optimization issue of the global portfolio in time. We’d like to answer to the question:

“How can we modify the portfolio in a way that the potential losses are minimal?”

It is always utile to answer to different questions, even though we choose to follow one method of operational management. We will see for instance that the Markowitz type models give a global view of the portfolio in a long term 9 1 year for instance) and allow to take into account the anticipations of the behavior of the market asset. The methods of local coverage favor a fine operational management ( i.e a day ) but don’t guarantee the absence of long term drift of equilibrium of positions in the portfolio.

The common element of the above approaches is to conduct to a temporal optimization of portfolio risk, under feasibility constraints, and earnings …

The following part gives the general elements of mathematical formulation of management issue.

1. Mathematical hypothesis frame of the problem

The mathematical formulation of the risk management issue leans on the measure tools of portfolio risk, and optimization methods.

In a general way, the risk management consists in minimizing the measure of the risk of the portfolio under return constraints and admissibility