STOCHASTIC DOMINANCE APPROACH TO PORTFOLIO OPTIMIZATION

STOCHASTIC DOMINANCE APPROACH TO PORTFOLIO OPTIMIZATION

Nesrin AlptekinAnadolu University, TURKEY

OUTLINE

Mean-Variance Analysis

Criticisms of Mean-Variance Analysis

Stochastic Dominance Rule First Order Stochastic Dominance Rule Second Order Stochastic Dominance Rule

Advantages of Stochastic Dominance Rules

Stochastic Dominance Approach to Portfolio Optimization

Quantile Form of Stochastic Dominance Rules

Linear Programming Problem of Portfolio Optimization With SSD Further Remarks

Markowitz’s Mean-Variance Analysis

Maximize Return subject to Given Variance

N

iii wrMaximize

1

N

ii

T

w

wQwk

1

2*

,1

..2

Subject to

.0iw

Markowitz’s Mean-Variance Analysis Minimize Variance (risk) subject to Given

Return

wQwk

Minimize T ...2

N

ii

N

iii

w

rwr

1

1*

,1

Subject to

.0iw


Mean-variance rules are not consistent with axioms of rational choice.

Probability distribution of returns is normal.

Decision maker’s utility function is quadratic. Beyond some wealth level the decision maker’s marginal utility becomes negative.

When considering the risk, variance which is the risk measure of mean-variance rule, is not always appropiate risk measure, because of left sided fat tails in return distributions.


According to this rule, the random variable X will be preferred over the random variable Y, if and

and there is at least one strict equality. However, with empirical data E(X) > E(Y) and

inequalities are common. In such cases, the mean-variance rule will be unable to distinguish between the random variables X and Y.

E(X) E(Y)22YX

22YX

Stochastic Dominance Rule

Stochastic dominance approach allows the decision maker to judge a preference or random variable as more risky than another for an entire class of utility functions.

Stochastic dominance is based on an axiomatic model of risk-averse preferences in utility theory.

Stochastic Dominance Rule The decision maker has a preference ordering over all

possible outcomes, represented by utility function of von-Neumann and Morgenstern.

Two axioms of utility function are emphasized: the Monotonicity axiom which means more is better than less and the concavity axiom which means risk aversion.

Stochastic dominance rule theory provides general rules which have common properties of utility functions.

Suppose that X and Y are two random variables with distribution functions Fx and Gy, respectively.

Stochastic Dominance RuleFirst order stochastic dominance Random variable X first order stochastically dominates

(FSD) the random variable Y if and only if Fx Gy.

No matter what level of probability is considered, G always has a greater probability mass in the lower tail than does F.

The random variable X first order stochastically dominates the random variable Y if for every monotone (increasing) function u: R R, then

is obtained. This is already shows that FSD can be viewed as a “stochastically larger” relationship.

[ ( )] [ ( )]E u X E u Y


return

Cumulativeprobability

G

F

FIRST ORDER STOCHASTIC DOMINANCE

Stochastic Dominance RuleSecond order stochastic dominance

The random variable X second order stochastically dominates the random variable Y if and only if

for all k.

X is preferred to Y by all risk-averse decision makers if the cumulative differences of returns over all states of nature favor Fx. The random variable X second order stochastically dominates the random variable Y if for u: R R all monotone (increasing) and concave functions u: R R, that is; utility functions increasing at a decreasing rate with wealth:

, then is obtained.

( ) ( )k k

F t dt G t dt

0, 0u u [ ( )] [ ( )]E u X E u Y


+

-

+

G

F

return

F,G

SSD-not FSD

Geometrically, up to every point k, the area under F is smaller than the corresponding areas under G.


Criteria have been developed for third degree stochastic dominance (TSD) by Whitmore (1970), and for mixtures of risky and riskless assets by Levy and Kroll (1976). However, the SSD criterion is considered the most important in portfolio selection.

Stochastic dominance approach is useful both for normative analysis, where the objective is to support practical decision making process, as well as positive analysis, where the objective is to analyze the decision rules used by decision makers.

ADVANTAGES OF STOCHASTIC DOMINANCE

APPROACH TO MEAN-VARIANCE ANALYSIS Stochastic dominance approach uses entire probability

distribution rather than two moments, so it can be considered less restrictive than the mean-variance approach.

In stochastic dominance approach, there are no assumptions made concerning the form of the return distributions. If it is fully specified one of the most frequently used continuous distribution like normal distribution, the stochastic dominance approach tends to reduce to a simpler form (e.g., mean-variance rule) so that full-scale comparisons of empirical distributions are not needed. Also, not much information on decision makers’ preferences is needed to rank alternatives.

ADVANTAGES OF STOCHASTIC DOMINANCE APPROACH TO MEAN-VARIANCE ANALYSIS

From a bayesian perspective, when the true distributions of returns are unknown, the use of an empirical distribution function is justified by the von-Neumann and Morgenstern axioms.

Stochastic dominance approach is consistent with a wide range of economic theories of choice under uncertainty, like expected utility theory, non-expected utility theory of Yaari’s, dual theory of risk, cumulative prospect theory and regret theory. However, mean variance analysis is consistent with the expected utility theory under relatively restrictive assumptions about investor preferences and/or the statistical distribution of the investments returns.


In the stochastic dominance approach to portfolio optimization, it is considered stochastic dominance relations between random returns.

Portfolio X dominates portfolio Y under the FSD(first order stochastic dominance rule) if,

Relation to utility functions: X FSD Y

F(R(x)) G(R(Y))

E(u(X)) E(u(Y)) nondecreasing u( )


Second order stochastic dominance rules are consistent with risk-averse decisions in decision theory.

For X and Y portfolios, risk-averse consistency:

X SSD Y E(u(X)) E(u(Y)) nondecreasing u( )


• Up to now, first and second order stochastic dominance rules are stated in terms of cumulative distributions denoted by F and G.

• They can be also restated in terms of distribution quantiles.

• These restatements allow to decision maker to diversify between risky asset and riskless assets.

• They are also more easily extended to the analysis of stochastic dominance among specific distributions of rates of return because such extensions are quite difficult in the cumulative distribution form.



The Pth quantile of a distribution is defined as the smallest possible value Q(P) for hold:

For X random variable, the accumulated value of probability P up to a specific x value is denoted by xP. Thus xP value is equal to Q(P), it is also Pth quantile.

Pr(X Q(P)) P (0 P 1)


Quantile Form of Stochastic Dominance Rules For a strictly increasing cumulative distribution denoted

by F, the quantile is defined as the inverse function:

Theorem 1: Let F and G be cumulative distributions of the return on two investments. Then F FSD G if and only if:

for all

1PQ(P) x F (P).

)()( PQPQ GF 10 P



Theorem 2: Let F and G be two distributions under consideration with quantiles and , respectively. Then F SSD G, if and only if

for all

Finally, this theorem holds for continuous and discrete distributions alike.

)(PQF )(PQG

0)()(0

dttQtQP

GF 10 P


Q = , i = 1,…,M; j= 1,…,N+1 matrix of consisting ofthe stratified sample of combinations of returns of a group of N candidate assets

: weights of asset j, j = 1,…,N ( )

Using of the quantile form of the SSD criterion, define:

: reference return (market index, existing portfolio,etc.)

)( ijq

jw 0jw

Mkqzk

iijkj ,...,1,

1

1, NkY

LINEAR PROGRAMMING PROBLEM OF PORTFOLIO OPTIMIZATION WITH SSD

Maximize rP =

Subject to

The objective function maximizes the expected return of the portfolio.

The set of M constraints requires the computed portfolio to dominate the reference return by SSD.

N

jjj wr

1

MkYwz Nk

N

jjkj ,...,11,

1

N

jjw

1

,1 Njw j ,...,1,0

Further Remarks

This work in progress. The next step is to find solving this problem in practice.

For this LP problem of portfolio optimization with SSD, we need optimality and duality conditions.

Finally, its computational results must be compared with M-V analysis consequences.

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STOCHASTIC DOMINANCE APPROACH TO PORTFOLIO OPTIMIZATION