Chapter 7 Appendix
Stochastic Dominance
1
What can be added to the happiness of a man
who is in health, out of debt, and has a
clear conscience?
- Mark Twain
2
Outline
 Introduction
 Efficiency
revisited
 First-degree stochastic dominance
 Second-degree stochastic dominance
 Stochastic dominance and utility
3
Introduction

Stochastic dominance is an alternative technique
employed in the portfolio construction process
• Stochastic “denotes the process of selecting from
among a group of theoretically possible alternatives
those elements or factors whose combination will most
closely approximate a desired result”
– Stochastic models are not always exact
– Stochastic models are useful shorthand representations of
complicated processes
4
Efficiency Revisited
 Portfolios
are efficient is they are not
dominated by other portfolios
 Portfolios
are inefficient if at least one other
portfolio dominates them
 Rational
investors prefer efficient
investments
5
First-Degree
Stochastic Dominance
 Cumulative
distribution A will be preferred
over cumulative distribution B if every
value of distribution A lies below or on
distribution B, provided the distributions are
not identical
• The distribution lines do not cross
6
Second-Degree
Stochastic Dominance
 Alternative A is
preferred to Alternative B if
the cumulative probability of B minus the
cumulative probability of A is always nonnegative
• SSD can be a significant aid in reducing the
security universe to a workable number of
efficient alternatives
7
Stochastic Dominance
and Utility
 Introduction
 Stochastic
dominance and mean return
 Higher orders of stochastic dominance
 Practical problems with stochastic
dominance
8
Introduction
 Regardless
of how much risk a person can
tolerate, the FSD criterion is appropriate
• Both the conservative investor and the gambler
will prefer a first-degree stochastic dominant
investment over an FSD inefficient alternative
 Investors
who are risk averse can use SSD
to weed out inefficient alternatives
9
Stochastic Dominance and
Mean Return

Alternative A is FSD efficient over Alternative B
if the expected return of A is no less than the
expected return of B

If alternatives are ranked by both geometric mean
and level of stochastic dominance, no FSDefficient portfolio can have a higher geometric
mean return than an SSD-efficient portfolio
10
Higher Orders of
Stochastic Dominance
 For
third-degree stochastic dominance:
• The investor is risk averse
• The investor’s degree of risk aversion declines
as wealth increases
11
Practical Problems With
Stochastic Dominance
 FSD
frequently fails to reduce the security
universe very much
• SSD is a much more powerful screening tool
than FSD
 It
is difficult to calculate higher than thirddegree stochastic dominance
12