The latest draft of the paper can be downloaded at:

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VIIIth Forecasting Financial Markets Conference -
London - May 01 TEAM Seminar - Paris - June
01 SIRIF Conference - Edinburgh - July 01
Classifying Hedge Funds Using Kohonen Maps A
First Attempt
Bertrand MAILLET TEAM - University of Paris
I ESCP-EAP European School of Management AAA (ABN
AMRO Group) bmaillet_at_univ-paris1.fr
Patrick ROUSSET SAMOS - University of Paris I and
CEREQ - Marseille rousset_at_cereq.fr
July 2001 - Preliminary version

• The latest draft of the paper can be downloaded
  at
• http\\panoramix.univ-paris1.fr\TEAM\maillet\confe
  rence.htm

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PLAN OF THE PRESENTATION

• Introduction
• The Preliminary Data
• The Kohonen Algorithm Some Elements
• Visualization of Kohonen Maps
• Characterization of the Hedge Fund Dataset
• Conclusion, Perspectives and Potential
  Applications

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INTRODUCTION

• Aim of the paper
• classify Hedge Funds without any a priori on
  their styles
• define Benchmark of pseudo-styles
• assess the quality of existing typologies
  (Micropal)
• characterise styles using performance measurement
• The Methodology Kohonen classification
• Self Organizing Map to class funds and organize
  them on a grid
• Define Representative Funds
• Representing Micro- and Macro-classes
• General distances and local distances
• Mix of Kohonen Map and Qualitative
  Characterization of funds
• Meta and Micropal two-level typologies
• Sharpes Ratio

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INTRODUCTION (contd)

• Interest of the subject
• Return-based Style Analysis (Sharpe, 1988, 1992)
• Avoid misleading expectations about fund
  behaviors
• Characterize pure styles for fund of funds
• Increase the diversification effect within a fund
  of hedge funds
• Classification Methods in Finance K-Means
  Method, Hierarchical Classification, Multilayer
  Perceptron
• A short selection of recent papers
• Brown and Goetzmann (1997) - Generalized Style
  Classification - Mutual Funds
• Mantegna (1998) - Hierarchical Trees -
  Securities
• Gruber (2001) - Hierarchical clustering - Mutual
  Funds

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INTRODUCTION (contd)

• Grouping individuals using Kohonen maps
• in general fields
• Daily Electrical Power Curves (Cottrell et al,
  1995)
• Skin types (Rousset and Guinot, 2001)...
• in finance
• Interest Rate Curves (Cottrell et al, 1997)
• Mutual Funds (Deboeck, 1997)
• What is different about Kohonen Maps?
• Observations are classified according to their
  similarities (non-linear method), but also
  according to a neighborhood notion
• Various visual representations are available
• Less sensitive to abnormal individuals...

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The Data
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The Data (contd)
8
The Data (contd)
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Kohonen Classification An Introduction
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Kohonen Algorithm Some Elements

• First step
• define a structure (output space)
• string or grid, number of classes
• representation of the grid (squared boxes versus
  octogonal boxes)
• choose a distance between units
• where i and j are coordinates of the box in the
  grid.
• choose a neighborhood function (at step s of the
  algorithm)
• where r(s) is an arbitrary neighborhood function
  such as
• and S is the total number of iterations...

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Kohonen Algorithm Some Elements (contd)

• Second step
• Associate a Code Vector to each unit u,
  whose dimensions is those of the observations
  (Tx1)
• Kohonens algorithm
• i. Initialize the first U Code Vectors (Tx1) -
  random convex combination of observations
• ii. Draw randomly one observation x and find the
  Winning Code Vector in the Grid
• where . is a distance (Euclidean, Mahanobis,
  Khi-squared...)
• iii. up-date the map (computing new
  representative code vectors)
• where is an adaptative
• parameter
• (with )
• Repeat ii., iii., iv. untill sS.

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Kohonen Map of Hedge Funds
Funds are represented in their own class
Representative funds and macro-classes
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Macro-classes

• Regularity of the class organization on the map
  does not give a good idea of the input space
  structure.
• A first technique uses a hierarchical cluster
  (Ward distance) to group representative funds in
  10 macro-classes.
• Other methods consist in representing distances
  between representative funds (local and general
  distances)

Background colors indicate macro-classes
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Distances between Representative Funds
Neighbored Distances
Medium distance
Small distance
Large distance
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Properties of Previous Representation

• This technique gives a visualization of the local
  structure (how similar are close funds in the
  map?).
• But
• It is an imperfect and limited representation of
  the data
• For instance, large distances and an eventual
  folder of the map cannot be visualized

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Distances Between Representative Funds
One-to-one Distances

• The grid is divided in boxes and boxes in units
• (box u, unit u) is assigned to the distance
  between classes u and u
• Darkness corresponds to the distance level (the
  lighter color, the smaller distance)

d(u6, u24)
d(u1, u36)
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Interpretation of One-to-one RepresentativeFund
Distances (contd)
A large central area
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Characterization of K-classes with Fund Styles
Typology

• For instance (artificial example) suppose 33
  of class 13 individuals are Directional Trading
  Funds, 33 are Specialist Credit Funds, 33 are
  Stock Selection Funds. That can be represented
  with a pie

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Characterization of K-classes with a Four-level
Fund Style Typology (MSDW)For a contingency
table for mutual fund styles, see Brown and
Goetzmann (1997), Table 1, page 384.
Multiple Styles
Directional Trading (1) Relative Value
(2) Specialist Credit (3) Stock Selection (4)
Stock Selection or Directional Trading
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Dispatching Funds onto the Map using Fund Style
Typology

• For instance (artificial example) suppose that
  30 of Specialist Credit Funds are located in
  class 13 while 70 of Specialist Credit Funds
  are located in class 25. That can be represented
  with bar charts

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Dispatching Funds onto the Map Interpretation
from a Four-level Fund Style Typology
Contingency of (Fund Style Ç k-class)
nik Contingency of Fund Style ni.
Bar chart size
Location of funds belonging to the same style in
the Map
Directional Trading (1) Relative Value
(2) Specialist Credit (3) Stock Selection (4)
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Dispatching Funds onto the Map Interpretation
from a Four-level Fund Style Typology (contd)

• Directional Trading and Stock Selection Funds
  spread into the whole map
• Relative Value and Specialist Credit Funds are
  mainly placed into the green zone
• Slight tendency for Relative Value Funds to be
  in the north of the green zone and for Specialist
  Credit Funds to be in the south of the green zone

Directional Trading (1) Relative Value
(2) Specialist Credit (3) Stock Selection (4)
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Characterization of K-classes with an
Eighteen-level Fund Style Typology (Micropal)

• Example of a refinement using a less aggregated
  level
• Isolated Stock Selection Funds (in the
  Four-level typology) are (in the Eighteen-level
  typology)
• Grey zone Emerging Market
• Red zone Distressed Securities (Emerging
  Market)
• Cyan zone Convertible Arbitrage
• Yellow zone Distressed Securities

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Characterization of K-classes with a Performance
Measurement

• Choice Sharpes Ratio discretization in Four
  Classes (quartiles)
• Mix Kohonen map classification with qualitative
  variable discrimination

Low Sharpes Ratios (1) Medium-low (2)
Medium-high (3) High (4)
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Characterization of K-classes with a Performance
Measurement (contd)

• Low and Medium-low Sharpes Ratios (magenta and
  blue levels) can mainly be found on the ring zone
  of the map
• Medium-high (yellow level) ones are more often
  in the green zone
• High Sharpes Ratios (grey level) are
  essentially located in the central zone of the
  map (green and magenta zones)

Low Sharpes Ratios (1) Medium-low (2)
Medium-high (3) High (4)
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Characterization of K-classes with a
Performance Measurement (contd)

• A complementary analysis could be proposed using
  directly quantitative variables

Conditional versus Unconditional Sharpes Ratio
density
Conditional versus Unconditional Box-plot of
Sharpes Ratios
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Conclusion

• Self Organizing Maps are useful tools for
  defining clear homogeneous groups of funds with
  little knowledge of the true category of a fund
  and financial strategies involved
• The methodology leads to define and represent
   benchmarks  of hedge fund styles
• We present as set of tools that make easier the
  characterisation and interpretation of the
  dataset (General and Local Distances, Qualitative
  and Quantitative discrimination...)

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Conclusion (contd)

• NEVERTHELESS Problems have to be encompassed
  before drawing any conclusion
• The dataset is small and biased (survivorship and
  backfilling biases, missing values,
  non-synchronicity...)
• Some extra results might be needed concerning
  Kohonen Map methodology (still some general
  results to be obtained, some questions about
  convergence of the map, rotational structure of
  the map...)

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Perspectives

• Applying on a Large Unbiased Database
• Testing convergence properties of the Kohonen
  Map An Empirical studies (see Cottrell and de
  Bodt, 2000) with bootstrapped series, surrogate
  data, missing value analysis and artificial
  measurement errors
• Testing Return-based Style Analysis using
  Representative funds as Benchmarks (see Sharpe,
  1992)
• Testing Stress-test Analysis using Representative
  funds as Benchmarks (see Lhabitant, 2001)

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Perspectives (contd)

• Generalize Qualitative Discrimination of the
  Kohonen Map
• working with re-scaled series (return-to-variabili
  ty rewards)
• and
• using
• Various Risk Definitions (Volatility, DSR,
  VaR...)
• Various Performance Measurements (see Bowden,
  2000, Dacorogna et al, 2001, Chauveau and
  Maillet, 2001...)
• for an extensive list of references see
• http\\panoramix.univ-paris1.fr\TEAM\maillet\refer
  ence.htm

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Perspectives (contd)

• For instance, Chauveau and Maillet, (2001,
  proceedings of EFMA01) propose a new Performance
  Measurement as a Relative Inefficiency Measure
  (based on Data Envelopment Analysis)...

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Perspectives (contd)

• Projections on Planes which are Tangeant to the
  Surface generated by the Kohonen map
• Data set adjustment with a non-linear surface and
  Representation of this surface (Rousset and
  Guinot, 2001)
• The Kohonen map shows local proximity.
• The distance maps gives the surface structure in
  the input space.
• See artificial example hereafter...

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Data set adjustment with a non linear
surfaceKohonen centroids are projected on the
first principal plane
Perspectives (contd)
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Applications

• An Example of Application American Stock Market
  Mutual Funds (277 funds - 06/98 to 06/01 -
  Classification into Five Styles)

Style
Fund
Rank on
Modified Sharpe s Ratio (EA-DSR)
Return since 01/01
Return on Last Month (05/01)
Sharpe s Ratio
Semi-volatility
Volatility
Benchmark MSCI US
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A first draft of the paper could be found
athttp\\panoramix.univ-paris1.fr\TEAM\maillet\c
onference.htm
VIIIth Forecasting Financial Markets Conference -
London - May 01 TEAM Seminar - Paris - June
01 SIRIF Conference - Edinburgh - July
01 Classifying Hedge Funds Using Kohonen Maps A
First Attempt Bertrand MAILLET and Patrick ROUSSET

• Thanks for your attention...See you in a future
  conference for further results