Higher Moments, Utility Functions and Asset Allocation

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Higher Moments, Utility Functions and Asset
Allocation

• J. Williams
• C. Ioannidis

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Preamble

• Dimensionality, in scope and analysis, has been
  with us for thousands of years.
• Currently high dimensional arrays have mostly
  been exploited in the physical and biological
  sciences, for example the PARAFAC model of mixing
  fluorescent liquids.
• Substantial literature argues over the relative
  merits and demerits of the inclusion of higher
  moments in financial decision making.

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Objectives

• How does our choice of Utility function affect
  portfolio selection when considering higher
  moments?
• Can we estimate higher order moments of groups of
  risky assets using higher dimensional arrays.
• How do we express investor preferences with
  regard to higher moments, which has a more
  intuitive relevance than a simple mathematical
  construct, (this is very hard!!).
• How do we integrate all this information into a
  rational mechanism for constructing and choosing
  portfolios of assets.

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The good the bad and the very ugly

• The GOOD Higher dimensional arrays d gt3 and
  their resulting moments SHOULD capture a great
  deal more information than analysis on the first
  2.
• The BAD Arbitrary ceilings on moments, i.e.
  well take the first 3 or 4, may produce decision
  schemes, which would certainly be rationally
  regarded as less than parsimonious.
• The UGLY Computational issues, higher
  dimensional arrays require vast numbers of
  parameters to fully realise their information
  content into measured portfolio moments.

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Some General Literature

• Donoho (2000) addresses these curses and
  blessings with regards to data analysis in a
  series of modern contexts Bioinformatics,
  Finance and Chemistry.
• Andersson and Bro (1999) utilise high dimensional
  arrays to analyse spectral analysis data from
  chemi-flourescence.

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A little bit more literature

• Loistl (1976) argues that the choice of higher
  moments is generally irrelevant and that
  mean-variance information was the optimal choice.
  Secondly argues that the Taylor expansion of a
  utility function has zero remainder.
• Brocket and Garven (2000) illustrate that common
  utility functions may produce incongruent
  selections under certain circumstances. However
  the nearer the efficient frontier a set of
  portfolios lie this, the selection problem is
  reduced.
• Evidence that asset dependency is not
  multivariate normal, Harvey and Siddique (2000),
  Erb et al (1994) and Ang and Bekaert (2001).
• Harvey and Siddique (2000) suggest that skewness
  is priced by the market and must therefore be
  considered in the utility maximisation problem.
• Patton (2002) suggest that investors preferences
  for positive skewness may exhibit itself in terms
  of an asymmetric dependency between assets.
• Various Derivatives literature, Rebonato (2005),
  Pelsser (2000) include non-linearity's in the
  dependence structure of the underlying.

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Previous Specifications of Higher Moment Asset
Allocation Models

• Friend and Westerfield, (1980) (CAPM)
• Athayde and Flores, (1998) (CAPM)
• Kozik and Larson, (2000) (CAPM)
• Hurlimann, (2001) (Copula)
• Jondeau and Rockinger, (2003) (CAPM)

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Non-Linear Dependency
Gaussian Variables no dependency
Gaussian Variables, Gaussian Copula
Gaussian Variables, Clayton Copula
Gaussian Variables, Franks Copula
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Joint Distributions and Copulas
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Moment Preferences
Explicitly the utility function expansion is
derived in terms of wealth
Here is the remainder (note this includes the m1
moment, Loistl (1976) suggests that the remainder
is exact and equal to zero for CRRA and CARA
utility functions.
Scott and Horvarth (1980) set out these
inequalities to denote the direction of moment
preferences
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• Brockett and Garven (1998) illustrate in the
  following example of a maximum expected utility
  problem, that gives a less than parsimonious
  solution, under the Taylor expansion method.

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So how do we deal with these problems?

• We would like to incorporate higher moments in
  the asset allocation system to try and account
  for asymmetries, excess kurtosis etc.
• We need to incorporate some sort of time varying
  conditional co-dependence
• We need to separate the expected utility
  calculations from the efficient set
  specification, to minimise the problems
  illustrated by Brockett and Garven.

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Co-dependency Arrays
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Asset Allocation Weights
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Super-Symmetry and Hyper-Cubes
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Properties of The Hypercube Hamiltonian
Projection
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The hypercube address structure based on the
Khatri-Rao-Bro Product, 5 Moments 5 assets

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Estimating the Efficient Frontier 1
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Estimating the Efficient Frontier 2 Monte Carlo

• Randomly Generate the weights and calculate the
  portfolio moments, then interpolate the frontier.

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Setting Preferences

• Once we have an approximation of the frontier, we
  can utilise our function or preferences to
  evaluate our preferred portfolio.
• We can create a matrix of partial differential
  equations, varying the function of each moment
  against one other holding the remaining moments
  constant.
• Hopefully this ordering method will negate the
  Brockett and Garven Issues

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Estimation A Conditional Co-Moments Model

• Now we know what to do with our co-moment array,
  what we need to do is estimate it.
• Our first method is a simplistic extension of
  Bollerslevs Constant Conditional Correlation
  Model.
• First we estimate the univariate marginal
  distributions, then we estimate the unconditional
  co-dependency array from the innovations (errors)
  of this series.
• This array is normalised (by its diagonal) and
  the univariate moments are then dynamically
  estimated.

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Functional Form
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Univariate Specification
Unconditional Multi-normal Distribution no
spatial distortion
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Auto co-dependency

• The series may exhibit some sort of time varying
  co-dependency in its moments as follows

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Calculate the normalised array
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The Dynamic Extension

• If we consider that the static dependency
  structure is not satisfactory, then we can move
  into the realm of the dynamic estimator,
• However we pay a big price in terms of parameters
  to estimate and interpret.

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Utility Functions
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Some Conclusions

• There is mixed evidence as to the inclusion of
  higher moments with regards to the fundamental
  asset allocation problem.
• If we select portfolios based on the normal
  utility functions we have available to us, then,
  the inclusion of anything above the second moment
  is normally useless.
• If we select certain non-standard utility
  functions then