Optimal Portfolios from Ordering Information

1
Optimal Portfolios from Ordering Information

• CQA Annual Meeting
• Neil A. Chriss
• Joint work with Robert Almgren

2
Introduction

• What are sorts and what is ordering information?
• Why are they important for investment management
  and research?
• How do you build optimal portfolios from ordering
  information?
• Do our methods work better than naive methods?

3
Reference

• A full version of this paper is available on the
  web at www.ssrn.com
• search for Almgren or Chriss.
• A smaller, less mathematical version is coming
  soon to www.ssrn.com!

4
Notation

• This talk is about portfolios ... w and v
• The stocks in the portfolios have expected
  returns
• A portfolio w has expected return
• A portfolio of stocks has covariance matrix V.
  The variance of a portfolio is given by wVw.

5
Sorts and ordering information?

• Single complete sort is an ordering of stock
  returns
• Information about the pairwise relations between
  returns

6
There are many different types of ordering
information

• Sector sorts
• Longs and shorts
• Index out/under perform

7
... and more types of ordering information

• Relationships concerning spreads
• Multiple sorts
• Value
• Momentum

8
Constructing Optimal Portfolios (Outline)

• Review of Markowitz mean-variance analysis
• Define portfolio preference relations
• Define efficient portfolios
• Define optimal portfolio

9
Review of Markowitz Mean-Variance Analysis

• Mean-variance theory assumes an expected return
  estimate exists for each stock in the portfolio
• Mean-variance optimization directs us to Find
  the portfolio that maximizes expected return
  subject to a maximum risk constraint.
• This is, conveniently, an optimization problem
  that one can solve using modern computers and
  tools.

10
What can we do with ordering information?

• An optimization function requires an objective
  function (e.g., expected return)
• With ordering information we do not have an
  object function
• Solution Preference Relations!

11
Preference Relations

• Given two portfolios I prefer the one with the
  higher expected return
• This places a preference relation on the space of
  portfolios. For portfolios w and v with expected
  return r we have

12
Preference Relations Related to a Sort

• In what follows we only consider the case of a
  single complete sort.
• Look at the space Q of all expected returns
  consistent with the sort. These all obey

13
Economic Assumption 1
14
Refinement of Preference Relation

• Fundamental portfolios long short in ith and
  i1st stock
• These portfolios have non-negative expected
  returns with respect to the sort.

15
Portfolio Decomposition

• Imagine we have
• where all coefficients are non-negative.
• This portfolio has a non-negative expected
  return. We like this portfolio.

16
We like portfolios that give us more exposure

• Suppose that
• and we have
• then
• And we like this one even more!

17
Preference Relation

• We note there are n-1 fundamental portfolios and
  n stocks. We are missing one dimension.
• Following the above we write

18
Definition of Preference Relation

• Let w and v be portfolios and write
• In other words, we like portfolios that give us
  more exposure to the fundamental portfolios.

19
Decomposition of Portfolios

• We therefore have a decomposition into relevant
  and irrelevant parts

20
Efficient Portfolios

• Given our preference relation, we simply say that
  a portfolio is efficient if it is maximally
  preferable on a budget set.
• Let
• where V is the covariance matrix.

21
Efficient Portfolios

• Main Result 1 A portfolio w is efficient if and
  only if

22
Characterization of Efficient Portfolios

• If a portfolio w is not efficient, then there is
  a portfolio v with the same level of risk such
  that
• Efficient portfolios give the most exposure to
  the fundamental portfolios.

23
Optimal Portfolios

• In Markowitz theory efficient optimal.
• The set of efficient portfolios is not unique!
• The set of efficient portfolios encompasses
  everything you would expect.
• Can we refine the preference relation?

24
The path to optimal portfolios

• Observation Mean-variance optimization depends
  on the direction but not the magnitude of
  expected returns (because of the budget
  constraint)
• Consistent expected returns depend only on their
  directions as well.

25
Modeling assumption 2

• We assume that each expected return direction is
  equally likely to any other expected return
  direction.
• We place a radially symmetric measure on the
  space of consistent expected returns Q.
• This is not ad hoc!

26
The preference relation

• Let µ be the radially symmetric measure. Then for
  w and v we restate the preference relation

27
Now we interpolate

• We state that
• In other words we like w more than v if w has a
  greater expected return more often.

28
Defining and Calculating Optimal Portfolios

• We now have a preference relations.
• We can now define optimal as most preferable on
  a budget set.
• Can we calculate?

29
The Centroid

• It turns out there is a vector c which is the
  center of mass of Q under µ with the amazing
  property that
• This single vector completely characterizes the
  preference relation!
• Works for any ordering information.

30
Portfolio Optimization from Ordering Information

• Portfolio optimization problem is now a linear
  optimization problem
• Find the most preferable portfolio subject to the
  budget constraint.
• We call the result centroid optimal.

31
The Centroid

• For a single complete sort is easily computable
• A 0.4424, B 0.1185, ß0.21.

32
The Centroid In General

• For any ordering information with a consistent
  cone Q there is a centroid.
• Centroid may be calculated via Monte Carlo.
• Once centroid is calculated, requires
  optimization routines.

33
Centroid vs Linear
34
Two Sectors

• Standard Centroid
• Two Sector Centroid

35
Four Sectors

• Standard centroid
• Four sector centroid

36
For Long-Shorts Specified
37
Deciles

• Standard Centroid
• Decile Centroid

38
Empirical Work

• Reversal Studies ... From Ed Thorpe

39
The Tests

• We tested the reversal portfolio on a variety of
  portfolios from 25 to 500 stocks.
• We used four portfolio formation methods
• Linearly weighted portfolios (non-optimized)
• Centroid weighted (non-optimized)
• Linearly optimal (V-1l)
• Centroid optimal (V-1c)

40
Lets Test It!
41
Is this method robust?

• Simulation results when you do not have perfect
  information about the order.
• Generated simulated markets where order of
  expected returns is known and markets have
  different cross-sectional dispersion of
  volatilities.
• Permute order by a fixed permutation distance
  based on a 2-d Gaussian with correlation 1-s2.
• Backtest centroid optimal portfolios.

42
Permutation Distance A measure of information
jumble
43
Results Information Ratio
Linear
Centroid
Efficient Linear
Efficient Centroid

• Vol dispersion 16, permutation distance of 0.5.
  Badly jumbled information on realistic market.

44
How much improvement does this method deliver

• Charts the ratio of centroid optimal to linear
  portfolio returns for various market
• From top to bottom markets have less to more
  volatility dispersion

45
A Final Word

• This methodology works on any ordering
  information.
• The procedure is as follows
• First to define the ordering information.
• Next compute the set Q of consistent expected
  returns.
• Compute the centroid of Q (every Q has one)
• Optimize!

46
For more information

• Paper available on the webwww.ssrn.com
• Axioma Inc. is implementing this in their
  optimization software.