Modern Portfolio Theory: Foundations, Analysis, and New Developments, + Website

Chapter 2

Assessing Risk

This chapter reviews some fundamental ideas about mathematics and statistics that are useful in analyzing portfolios. The chapter also introduces symbols, definitions, and notations to be used throughout the book.

Essentially, this chapter explains the tools with which risk is analyzed. More specifically, the parts of freshman college algebra and finite probability courses that are relevant to investments analysis are reviewed. The rigor of the explicit definitions used in mathematics turns some people off. Allowing yourself to be turned off and dropping out rather than persevering is short-sighted. If you bite the bullet and master this topic, it will not only teach you how to scientifically analyze investment opportunities, it will also raise your level of consciousness in other academic and nonacademic areas. That is, mathematics applied to real-world problems is powerful stuff that can make your life sweeter. Words such as expectation and risk will be given fascinating new definitions.

2.1 Mathematical Expectation

For a fair game paying $1 for heads and $1 for tails on the flip of a coin, the expected value of the outcome from the game is the probability of heads times the $1 loss plus the probability of tails times the $1 gain. Symbolically,

where p(heads) represents the probability that heads occurs, and p(tails) represents the probability that tails ...