The Probability Measure

Paolo Sironi

A poker player may believe that a deck of cards is well shuffled. Yet he may not know all the implications of this belief. He is not likely to know, offhand, the probability of beating 3 aces and 2 jacks; or of beating 4 eights and a king if deuces are wild. It is usually not polite or convenient to employ a computing machine to calculate probabilities during the course of a poker game. In portfolio selection, however, the stakes are higher and decisions should be made on the basis of thorough analysis.

Markowitz (1959)

In this chapter we investigate the probability measure: a priori and a posteriori probability, the probability distribution function, goal-based probability and the analysis of ex post and ex ante performance.

INTRODUCTION

Risk-based portfolio management requires estimation of the probability density function of a multitude of risk factors to model the potential dynamics of asset prices and inform investment decision-making within a consistent risk management framework. Classical approaches to portfolio choice have relied upon the measurement of specific moments of these distributions or tail loss estimates in order to formulate the objective function