STOCHASTIC DOMINANCE APPROACH TO PORTFOLIO OPTIMIZATION

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STOCHASTIC DOMINANCE APPROACH TO PORTFOLIO
OPTIMIZATION

• Nesrin Alptekin
• Anadolu University, TURKEY

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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

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Markowitzs Mean-Variance Analysis

• Maximize Return subject to Given Variance

Subject to
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Markowitzs Mean-Variance Analysis

• Minimize Variance (risk) subject to Given Return

Subject to
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Criticisms of Mean-Variance Analysis

• Mean-variance rules are not consistent with
  axioms of rational choice.
• Probability distribution of returns is normal.
• Decision makers utility function is quadratic.
  Beyond some wealth level the decision makers
  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.

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Criticisms of Mean-Variance Analysis

• 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) gt 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.

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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.

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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.

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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.

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Stochastic Dominance Rule
FIRST ORDER STOCHASTIC DOMINANCE
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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.

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

• 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.

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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.

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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 Yaaris, 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.

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STOCHASTIC DOMINANCE APPROACH TO PORTFOLIO
OPTIMIZATION

• 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

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STOCHASTIC DOMINANCE APPROACH TO PORTFOLIO
OPTIMIZATION

• 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

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STOCHASTIC DOMINANCE APPROACH TO PORTFOLIO
OPTIMIZATION

• 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.

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STOCHASTIC DOMINANCE APPROACH TO PORTFOLIO
OPTIMIZATION

• Quantile Form of Stochastic Dominance Rules
• 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.

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STOCHASTIC DOMINANCE APPROACH TO PORTFOLIO
OPTIMIZATION

• 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

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STOCHASTIC DOMINANCE APPROACH TO PORTFOLIO
OPTIMIZATION

• Quantile Form of Stochastic Dominance Rules
• 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.

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STOCHASTIC DOMINANCE APPROACH TO PORTFOLIO
OPTIMIZATION

• Q , i 1,,M j 1,,N1 matrix of
  consisting of
• the 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.)

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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.

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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.