DYNAMIC ASSET ALLOCATION

BY STOCHASTIC PROGRAMMING METHODS

A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF
MANAGEMENT SCIENCE AND ENGINEERING
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

Alexis Collomb
December 2004
c Copyright by Alexis Collomb 2005
°
All Rights Reserved

ii
I certify that I have read this dissertation and that, in my opinion, it is fully
adequate in scope and quality as a dissertation for the degree of Doctor of
Philosophy.

Walter Murray Principal Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully
adequate in scope and quality as a dissertation for the degree of Doctor of
Philosophy.

Gerd Infanger

I certify that I have read this dissertation and that, in my opinion, it is fully
adequate in scope and quality as a dissertation for the degree of Doctor of
Philosophy.

James Primbs

I certify that I have read this dissertation and that, in my opinion, it is fully
adequate in scope and quality as a dissertation for the degree of Doctor of
Philosophy.

Hervé Kieffel

Approved for the University Committee on Graduate Studies.

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iv
Abstract

This thesis addresses two research questions: (i) the predictability of equity returns and
(ii) the variations of optimal asset allocations, within a multi-stage stochastic programming
framework, with respect to the statistical modeling of returns.
The predictability and serial correlation of equity returns has been for many years a
controversial subject among academics and professional financial analysts. We propose here
a novel methodology for detecting local bursts of serial correlation. Moreover, by analyzing
historical data and conditional returns, we show evidence of market predictability for equity
indices over short-term horizons of a few days, with both momentum and reversal effects.
Stochastic programming is widely used for many asset and liability management (“ALM”)
applications. Within ALM models, we look at the limited setting of a multi-stage asset allo-
cation problem and restrict ourselves to a multi-stage stochastic programming framework.
As for any solution technique used for solving a problem with uncertain parameters, the
stochastic modeling of the uncertainty may drive the character of the solution. We com-
pare the optimal asset allocations obtained from a geometric Brownian motion (“GBM”)
model and a vector autoregressive (“VAR”) model of asset class returns. In the process,
we show clear evidence of serial correlation for the returns on Treasury bonds and bills and
compare the forecasting performances of the GBM and the VAR models. In the VAR case,
we show that the allocation results vary significantly depending on the initial conditions.
For both the GBM and VAR models, we also show that results may be very sensitive to
the historical samples used for calibrating the models. To address this instability, we use
a third statistical model of asset class returns, a Bayesian vector autoregressive (“BVAR”)
approach. We show it provides better out-of-sample forecasts of returns, especially when
the different statistical models are calibrated using small samples. We conclude that our
multi-stage stochastic programming model (for which it is trivial to include transaction
costs) is particularly appropriate for assisting allocation decisions with limited information
and significant transaction costs, as is the case for funds of funds.

v
Acknowledgements

First of all, I want to thank my Reading Committee. In particular, I am greatly indebted to

Professor Walter Murray who made me feel welcome and advised me throughout the years.
Day after day, Professor Murray provided me with very valuable research guidance and
colorful conversational topics (from optimization to Monty Python, and all kinds of sports
and politics in between). His support made my time here both intellectually productive
and very pleasant... Just as inspirational and enthusiastic was Professor Gerd Infanger. I
owe him a lot with respect to stochastic programming insights and I very much enjoyed his
contagious optimism throughout the process. I also want to thank Professor James Primbs
for his very useful and pointed comments and his great responsiveness. I am also greatly
indebted to Professor Hervé Kieffel who provided me with profusely detailed and disciplined
feedback.
For the great ride my years in the Systems Optimization Laboratory have been, I would
like to thank (in chronological order): Byunggyoo Kim, Kien-Ming Ng, Michael Friedlander,
Maureen Doyle, Che-Lin Su, Uday Shanbag, Samantha Infeld and Kaustuv. I am also very
grateful to Michael Saunders for his incredibly pointed and efficient corrections, out of pure
benevolence, and to Professor Richard Cottle for his dry humor, on the fly. In addition,
I was honored to be hooded by Professor Peter Glynn (representing Professor Murray).
I should also mention I was extremely inspired by Professor Darrell Duffie for his quasi-
encyclopedic knowledge of finance (and other topics), his class on dynamic asset pricing
theory, and his always sharp and warm responses to my emails. Also, I want to thank
Professor Ken Singleton for allowing me to audit his equally excellent class on empirical
asset pricing and sit in his research and reading group. I am not mentioning for the sake
of conciseness many other people who all taught me a lesson or a proof, at some point in
time, but I hope they will forgive me. I wish I had consulted with all above mentioned more
often.
I would like to thank my family (direct and extended) and numerous friends (too many
to name here) for their continued moral support. Three people I would have liked to thank

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directly for the role models they were in getting me to this point (in very different, yet
equally meaningful, ways) are my grandfathers, Pierre-Charles Wirth and Charles Collomb,
and my godfather, Pierre Faurre. This dissertation is dedicated in part to their memories.
Last but not least, this thesis is dedicated to my parents. I am most thankful for their
support and enduring love over the years. I am greatly indebted to my father, for being
such a father, and to my mother, for being such a mother! I know they will read this thesis,
will be very critical, and will encourage me to do much more!
I hope as I finish writing these lines that the fantastic time I had at Stanford will turn
out to have been not an end in itself, but a beginning and a gate to the end of education
which (I was told) is creation. I hope this work qualifies as the first building block, however
humble, of a hard-working career of trying to make useful contributions.

vii
Contents

Abstract v

Acknowledgements vi

1 Introduction 1
1.1 Representation of an ALM model . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 Asset and Liability Cash Flows . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Optimization Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.3 Set of Possible Investments . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.4 Structural Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Our Research Framework and Contributions . . . . . . . . . . . . . . . . . . 11
1.2.1 Research Focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Asset Allocation Methods and Strategies 13

2.1 Synopsis of Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Risk Aversion and Risk Tolerance . . . . . . . . . . . . . . . . . . . 14
2.1.2 Hyperbolic Absolute Risk Aversion Utility Functions . . . . . . . . . 15
2.2 Synopsis of Asset Allocation Strategies . . . . . . . . . . . . . . . . . . . . . 15
2.3 Idiosyncrasies of Stochastic Programming . . . . . . . . . . . . . . . . . . . 18

3 Comparison of Asset Allocation Methods 20

3.1 Single-Period Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.1 Sensitivity Results on Historical Estimates . . . . . . . . . . . . . . 21
3.2 Multi-Period Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 Buy-and-Hold Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Fixed-Mix Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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 3.2.3 Stochastic Programming Strategy . . . . . . . . . . . . . . . . . . . 29
3.3 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Momentum of Equity Returns 32

4.1 Our Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1.1 Issue of Market Predictability . . . . . . . . . . . . . . . . . . . . . . 32
4.1.2 Our Restricted Formulation . . . . . . . . . . . . . . . . . . . . . . . 34
4.1.3 Intuition for Methodology . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.4 Previous Evidence of Momentum . . . . . . . . . . . . . . . . . . . . 37
4.2 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.1 Conditional Probabilities of Transition . . . . . . . . . . . . . . . . . 41
4.2.2 Conditional Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Analysis of Statistical Significance . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.2 Bootstrap Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.3 Verification of the Markov Property . . . . . . . . . . . . . . . . . . 47
4.4 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Impact of Serial Correlation 52

5.1 Review of Traditional Assumptions . . . . . . . . . . . . . . . . . . . . . . . 53
5.1.1 Utility Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1.2 Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1.3 Stochastic Modeling of Underlying Assets . . . . . . . . . . . . . . . 53
5.1.4 Scope and Organization of the Chapter . . . . . . . . . . . . . . . . 54
5.2 Asset Returns Data and Model Estimates . . . . . . . . . . . . . . . . . . . 54
5.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2.2 VAR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2.3 VAR Model Estimation and Choice of Lag . . . . . . . . . . . . . . . 55
5.2.4 Coefficient Estimations . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.5 Bayesian Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3 Model and Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.1 Our Stochastic Programming Framework . . . . . . . . . . . . . . . 69
5.3.2 Scenario Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3.3 Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . 71
5.3.4 DECIS Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3.5 Advantages of using DECIS and Computational Time . . . . . . . . 73

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 5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4.1 Out-of-Sample Forecasting . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4.2 Results Variations with respect to Final Goal . . . . . . . . . . . . . 74
5.4.3 Comparisons between GBM and VAR results . . . . . . . . . . . . . 75
5.4.4 Analysis of Nonconvexity . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Bayesian Stochastic Programming 80

6.1 The Bayesian Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2 Bayesian Vector Autoregression . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2.1 Minnesota Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2.2 Variants of the Minnesota prior . . . . . . . . . . . . . . . . . . . . . 86
6.2.3 Random-Walk Averaging Prior . . . . . . . . . . . . . . . . . . . . . 87
6.2.4 Estimating the Coefficients of a BVAR model . . . . . . . . . . . . . 89
6.3 Comparison of BVAR vs. VAR forecasts . . . . . . . . . . . . . . . . . . . . 91
6.4 Application to Funds of Funds . . . . . . . . . . . . . . . . . . . . . . . . . 92

7 Conclusion and Further Research 98

A A Brief History of Mathematical Finance 100

A.1 A Brief History of Mathematical Finance . . . . . . . . . . . . . . . . . . . 100
A.1.1 Introduction of Random Walk . . . . . . . . . . . . . . . . . . . . . . 100
A.1.2 Concept of Market Efficiency . . . . . . . . . . . . . . . . . . . . . . 101
A.1.3 Empirical Testings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.1.4 Wedge between Academics and Practicioners . . . . . . . . . . . . . 102
A.1.5 Debate between Active and Passive Management . . . . . . . . . . . 102
A.1.6 Debate between Economic Predictability and Random Returns . . . 103
A.1.7 Reconciliation of Different Views . . . . . . . . . . . . . . . . . . . . 104
A.1.8 Difficulties in Testing the Efficient Market Concept . . . . . . . . . . 104
A.1.9 The More Recent Developments . . . . . . . . . . . . . . . . . . . . 105
A.2 Epistemological Considerations in Finance . . . . . . . . . . . . . . . . . . . 108
A.2.1 Two-Body Problems and Multi-Agent Systems . . . . . . . . . . . . 108

Bibliography 110

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List of Tables

3.1 Comparison of Asset Allocation Strategies . . . . . . . . . . . . . . . . . . . 31

4.1 Characteristics of DJIA’s 10 Largest Drawdowns in 20th Century . . . . . . 40

4.2 Dow Jones Expected Returns by Transition . . . . . . . . . . . . . . . . . . 42
4.3 S&P 500 Expected Returns by Transition . . . . . . . . . . . . . . . . . . . 44
4.4 Nasdaq Expected Returns by Transition . . . . . . . . . . . . . . . . . . . . 45
4.5 Verification of the Markov Property for the DJIA . . . . . . . . . . . . . . . 49
4.6 Verification of the Markov Property for the DJIA . . . . . . . . . . . . . . . 50

5.1 Data Summary Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2 Lag Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Analysis of Computation Time (sec.) . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Mean Absolute Deviations for Out-of-Sample Forecasts . . . . . . . . . . . . 74
5.5 Allocation results for GBM model with varying goal. . . . . . . . . . . . . . 75
5.6 Allocation results when VAR initial conditions are centered around sample
averages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.7 Values of Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.8 Allocation results for VAR initial conditions centered around sample’s last
returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.9 Values for Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.10 First Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.11 Second Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.12 Third Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

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List of Figures

1.1 Growth of US Pension Asset Reserves . . . . . . . . . . . . . . . . . . . . . 2

1.2 Growth of 401(k)’s and IRAs . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Rise of Households Liquid Financial Assets . . . . . . . . . . . . . . . . . . 4
1.4 Optimization Framework for an ALM Model . . . . . . . . . . . . . . . . . 6

3.1 Utility Function Indifference Curves . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Variations of Overall Risk and Return (NYSE case) . . . . . . . . . . . . . 23
3.3 Allocation Weights (NYSE case) . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Variations of Overall Risk and Return (NASDAQ case) . . . . . . . . . . . 25
3.5 Allocation Weights (NASDAQ case) . . . . . . . . . . . . . . . . . . . . . . 26
3.6 Framework for Computational Results . . . . . . . . . . . . . . . . . . . . . 30

4.1 Mapping of price level into st series . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Two series bt with same first-order autocorrelations . . . . . . . . . . . . . . 38
4.3 Corresponding index levels with similar first-order returns autocorrelations 39
4.4 Daily Momentum Analysis of DJIA . . . . . . . . . . . . . . . . . . . . . . . 42
4.5 Daily Momentum Analysis of S&P 500 . . . . . . . . . . . . . . . . . . . . . 43
4.6 Daily Momentum Analysis of Nasdaq . . . . . . . . . . . . . . . . . . . . . . 44
4.7 Sensitivity Analysis on Conditional Transition Probabilities . . . . . . . . . 46
4.8 Bootstrap Analysis of Runs and Drawdowns . . . . . . . . . . . . . . . . . . 48

5.1 Variations of lag choices on a “rolling historical sample” (monthly returns) 57

5.2 Variations of lag choices with respect to a “rolling historical sample” (yearly
returns) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3 t-statistics and coefficient values on stock yearly returns . . . . . . . . . . . 60
5.4 t-statistics of lagged coefficients on stock monthly returns (30 month) . . . 61
5.5 Values of lagged coefficients on stock monthly returns . . . . . . . . . . . . 61
5.6 Value of lagged coefficient for cash on stock monthly returns . . . . . . . . . 62

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5.7 t-statistics of lagged coefficients on stock monthly returns (60 month) . . . 62
5.8 t-statistics for lagged coefficients on bond yearly returns . . . . . . . . . . . 63
5.9 Values of lagged coefficients for bond yearly returns . . . . . . . . . . . . . . 63
5.10 t-statistics for lagged coefficients on bond monthly returns (30 month) . . . 64
5.11 Values of lagged coefficients on bond monthly returns . . . . . . . . . . . . 64
5.12 Value of lagged coefficient for cash on bond monthly returns . . . . . . . . . 65
5.13 t-statistics for lagged coefficients on bond monthly returns (60 month) . . . 65
5.14 t-statistics for lagged coefficients on cash yearly returns . . . . . . . . . . . 66
5.15 Values of lagged coefficients for cash yearly returns . . . . . . . . . . . . . . 66
5.16 t-statistics for lagged coefficients on cash monthly returns (30 month) . . . 67
5.17 Values of lagged coefficients on cash monthly returns . . . . . . . . . . . . . 67
5.18 Value of lagged coefficient for cash on cash monthly returns . . . . . . . . . 68
5.19 t-statistics for lagged coefficients on cash returns (60 month) . . . . . . . . 68
5.20 Scenario Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.1 Frequentist Approach: Fixed Historical Window. . . . . . . . . . . . . . . . 81

6.2 Frequentist Approach: Expanding Historical Window. . . . . . . . . . . . . 82
6.3 Frequentist Approach: Rolling Historical Window. . . . . . . . . . . . . . . 82
6.4 Bayesian Approach: Priors Are Constantly Updated . . . . . . . . . . . . . 83
6.5 Comparing BVAR vs. VAR forecasts (12-month historical sample) . . . . . 92
6.6 Comparing BVAR vs. VAR forecasts (24-month historical sample) . . . . . 93
6.7 Comparing BVAR vs. VAR forecasts (36-month historical sample) . . . . . 94
6.8 Comparing BVAR vs. VAR forecasts (48-month historical sample) . . . . . 95
6.9 Comparing BVAR vs. VAR forecasts (60-month historical sample) . . . . . 96

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Chapter 1

Introduction

In recent years, the quantitative management of financial cash flows for either individuals
or organizations has become increasingly important. The last score of years have seen a
significant development of equity markets around the globe (fuelled to a large extent by the
US markets) and the size of the US Pension Asset Reserves has dramatically increased from
about $2 trillion in 1985 to more than $9 trillion in 2000 as Figure 1.1 shows. This rise
in the size of pension assets under management can partly be explained by the significant
development of individual retirement accounts (“IRAs”) and other types of retirement plans
such as 401(k) (this is illustrated by Figure 1.2). Today more than ever, the quantitative
management of assets and liabilities is of primary importance for either individuals or
institutions.
For individuals, the increasing access in many countries (especially the US) to online
brokerage, financial news channels and the like, has significantly contributed to developing
a taste for financial planning. The many sophisticated tools available to financial planners
have become increasingly available to the general public and this increased availability has
contributed in turn to building the general public awareness of the subject. In the US alone,
the proportion of wealth invested in equity products and other types of financial securities
has greatly increased since 1985 (as shown by Figure 1.3). This trend has been supported in
part by the general public perception that these products are the most appropriate means
for funding (or supplementing) retirement.
For organizations, the recent development and use of sophisticated information technolo-
gies (e.g. corporate intranets) has usually translated into improved financial accounting and
reporting. This greater and more accurate volume of financial information available has in
turn emphasized to many corporate or institutional executives the need for better financial

1
2 CHAPTER 1. INTRODUCTION

Figure 1.1: Growth of US Pension Asset Reserves. Source: 2001 Securities Industry Fact
Book.
 3

Figure 1.2: Growth of 401(k)’s and IRAs. Source: 2001 Securities Industry Fact Book.
4 CHAPTER 1. INTRODUCTION

14
$ Trillion

12

Securities Products

10

Bank Deposits
8

1985 2000

Figure 1.3: This figure shows the progression from 1985 to 2000 of the dollars invested by
US households in liquid financial assets. Over that period, the bank deposits grew from
about $2.3 to $3.5 trillion (a compound annual growth rate of about 2.85%) while the
investments in securities products rose much more rapidly from $2.1 to $13.1 trillion (a
compound annual growth rate of almost 13%). Source: 2001 Securities Industry Fact Book.
1.1. REPRESENTATION OF AN ALM MODEL 5

management. In this case, the paradigm seems to be that accounting and financial infor-
mation flowing seamlessly to decision makers has helped them identify inefficiencies and
improve the firm’s overall financial management.
In general, the use of financial planning tools has been somehow demystified. Hence it
is reasonable to assert that, today more than ever, financial planning and the management
of cash flows is of concern to a large public.
In this chapter, we first provide a general review of Asset and Liability Management
(“ALM”) models and identify from first principles some of the general issues that they
involve. Second, we introduce the particular framework of our research and map out our
specific contributions.

1.1 Representation of an ALM model

ALM models usually require drawing from different fields such as statistics, economics,
financial mathematics and optimization. In a schematic and generic way, we can organize
the different elements of ALM models along three separate poles of interest and a common
set of structural considerations.
Building an ALM model almost always requires performing three tasks before starting
the optimization process. The first task consists in a prior elicitation of future asset and
liability cash flows that are to be expected. The second task is to clearly define the opti-
mization objectives and make sure they are appropriate for the investing entity (whether
an individual or an organization). The third task is to choose the set of possible invest-
ment vehicles that we want to include in the optimization process. Common to these three
tasks are structural considerations that need to be accounted for such as whether to have
a continuous- or discrete-time model, how new information is revealed (i.e. the so-called
information filtration) and other general assumptions. Figure 1.4 schematizes these points.

1.1.1 Asset and Liability Cash Flows

It is important to understand here the different uses that we can make of the word “asset”.
One usage of “asset”, as in “asset cash flow”, refers to the underlying asset that is generating
revenues for the firm or the individual of concern. In other words, if we are dealing with
an individual’s financial planning model, the asset cash flow referred to above would be the
income stream this individual expects to receive during his working career. Another usage of
the word “asset”, as in “asset class”, refers to a specific class of investments. In developing
an ALM model, the nature of the asset and liability streams needs to be carefully estimated.
6 CHAPTER 1. INTRODUCTION

Figure 1.4: Optimization Framework for an ALM Model. This graph, which we suggest,
provides an overview of the three important types of assumptions that optimizing an ALM
model requires defining: (i) the future asset and liability cash flows; (ii) the objectives and
(iii) the possible asset classes or investment vehicles.
1.1. REPRESENTATION OF AN ALM MODEL 7

The precise timing of cash flows may be known in advance or not. In many cases, we may
only have an approximate idea of their temporal distribution. Similarly, the magnitude of
these cash flows may be known in advance precisely or in a fuzzy way. In many (if not
most) cases, we may only have a probabilistic assessment of what to expect. All these
early assumptions need to be carefully assessed as they may be extremely influential on the
optimal investment policy recommendation we are seeking to calculate.

1.1.2 Optimization Objectives

The second task at hand is to properly elicit the optimization objectives. The representation
of these objectives will vary a lot depending on whether we are dealing with an individual
investor or an organization.

Individual Decision Making

The modeling of optimization objectives for an individual decision maker is far more com-
plex than it seems at first glance. As put by Bell, Raiffa, and Tversky (Bell, Raiffa, and
Tversky 1988), we can distinguish between three kinds of theories of decision making un-
der uncertainty. Normative theories state how decision makers or economic agents should
behave (if they were fully “rational”). Descriptive theories tell us how agents do behave
in fact (and usually underline “deviations” from rationality). Last but not least, prospec-
tive theories attempt to offer advice as to how decision makers should behave when faced
with incomplete information and/or complex situations that cannot be fully grasped by our
limited cognitive abilities.

Normative School The normative school of thought focuses on what rational decision
making should be and assumes that an individual’s preferences over a set of alternatives
can pass logical tests of consistency derived from a few axioms. In other words the de-
cision maker is said to be “rational”. For instance, this approach assumes completeness
over the different set of alternatives, meaning that it presupposes that the decision maker is
always able to rank two alternatives (or state his indifference between the two). Assuming
they are consistent, the decision maker’s preferences are then translated into a utility func-
tion supposed to reflect the possible satisfaction the decision maker can derive from wealth
(Friedman and Savage 1948, Friedman 1957, Markowitz 1952b). For an individual (who is
usually risk averse), representing the investor’s objectives would then translate into fitting
an increasing and strictly concave utility function by asking the individual to choose be-
tween a series of monetary deals, and this until a satisfactory bounding of the risk aversion
8 CHAPTER 1. INTRODUCTION

parameter(s) of interest has been achieved.

Descriptive and Prospective Schools The normative school has been seriously ques-
tioned by different investigations in behavioral economics, behavioral finance and psychology
that demonstrated the inconsistency and irrationality of many economic agents. This is a
vast topic and we limit ourselves here to providing some references in the different fields
of behavioral finance (Shefrin and Statman 1985), economics (Arrow 1986, Kahneman and
Tversky 1979, Thaler 1980, Thaler 1981, Tversky and Kahneman 1991), decision sciences
(Bell, Raiffa, and Tversky 1988), marketing (Thaler 1985) and psychology (Kahneman and
Tversky 1973, Kahneman and Tversky 1984).

Organizational Decision Making

By “organizational decision making”, we refer to any decision making situation that involves
a complex organization (e.g. a corporation, an academic institution, etc.) In such a case,
eliciting and quantifying clear objectives can be a significant (and sometimes open-ended)
task. For instance in the case of a large corporation, defining the ALM optimal policy
may be a complex iterative process between different stakeholders such as the executive
management, the auditors, some large shareholders, etc. Tradeoffs between short-term and
long-term objectives, between conflicting interests and different opinions and assessments
of financial and operational risks need to be agreed upon. Thus, our end purpose of clearly
eliciting quantified objectives that can later be “plugged” into an ALM model may never
be met. There has been a vast and still growing literature on designing organizational
mechanisms to make sure that different groups in a complex organization have incentives
to cooperate. Designing such incentive-compatible mechanisms requires a game theoretic
mindset and Thaler suggests using “prescriptive” game theoretic approaches (Thaler 1992).
We also refer the interested reader to the review by Kreps on “mechanism design” (Kreps
1990). Also, Collomb provides a thorough treatment on how to formulate “reasonable”
objectives when facing the possibility of divergent criteria between various management
groups (Collomb 1971).

1.1.3 Set of Possible Investments

There are many possible investment vehicles with different risk profiles that can be used
for investing purposes. We distinguish here between “primary” investment vehicles such
as individual stocks or bonds and “secondary” investment vehicles, such as mutual funds.
1.1. REPRESENTATION OF AN ALM MODEL 9

There are even higher-order investment vehicles, such as fund of funds1 (that we could call
“tertiary” investment vehicles according to our classification).
Among the traditional primary vehicles of investment are stocks, bonds (whether issued
by a corporation or a government), currencies, commodities, etc. According to our classifi-
cation, secondary investment vehicles comprise all funds that invest in the former, such as
mutual funds, exchange-traded funds (“ETFs”), index funds, so-called “hedge” funds, etc.
As an example of one of these funds, a mutual fund is simply a financial intermediary that
allows a group of investors to pool their money together with a predetermined investment
objective. The mutual fund will have a fund manager who is responsible for investing the
pooled money into specific securities (usually stocks or bonds) and investing in a mutual
fund will consist in buying shares (or portions) of the mutual fund and becoming a share-
holder of the fund. What is important to emphasize here is the increasingly cascading
nature of the possible investment vehicles that can be used for investing activities, from
individual securities to high-level funds of funds.
From the investor’s perspective, all these investment opportunities have very different
risk characteristics, performance features, accessibility and fees, among others. Of fun-
damental importance is the representation that an investor has of the dynamics of these
investment vehicles. This usually involves significant statistical work for analyzing their
past performance in an effort to model their uncertain future performance (to the extent
that past performance bears upon future performance).
In the bulk of our work we limit our investing universe to certain asset classes and
their representative indices (considered “secondary” as they represent a pool of individual
securities). In particular we use broad-based equity indices, government bonds and short-
term treasury bills (used as a proxy for cash). In the last chapter, we address the issue of
funds of funds in the context of a particular application.

1.1.4 Structural Assumptions

There are general assumptions that need to be made across most financial models in general,
and ALM models in particular. First, an important issue is whether a continuous- or a
discrete-time model will be used and, if discrete, how fine a time mesh should be chosen.
Another important consideration is whether financial agents can be considered as “price
takers” in a Walrasian context or not. Other general considerations of interest include:
the information filtration (i.e. the way new information will be delivered), observability
conditions of asset prices and/or of new data (e.g. signal/noise ratios), term structure
1
A “fund of funds” is simply a fund that invests in funds instead of individual securities.
10 CHAPTER 1. INTRODUCTION

dynamics, inflationary expectations and other macroeconomic factors.

Continuous and Discrete Times

There is a rich financial literature of both continuous- and discrete-time models. For in-
stance, Duffie provides a dense review of both types of models (Duffie 2001) and Merton
provides an overview and synthesis of finance theory from the perspective of continuous-
time analysis (Merton 1990). Usually it can be shown that for many purposes, such as
contingent claim prices and optimal consumption-portfolio policies, the results derived from
the discrete-time models converge to their corresponding continuous-time limits (He 1989).
However, the fact that financial data used by practicioners is only available on a discrete ba-
sis raises serious practical implications for using continuous-time models. For instance Brigo
and Mercurio have shown the clear limit of approximating discrete time with continuous
time for pricing options (Brigo and Mercurio 2000). And choosing which continuous model
is the most appropriate given discretely sampled data is far from being an obvious question
as the work by Aı̈t-Sahalia and Mykland shows (Aı̈t-Sahalia and Mykland 2004). In addi-
tion, often solving continuous models requires strong assumptions that are unwarranted by
empirical evidence if closed-form solutions are to be obtained. If these assumptions are not
made and numerical solutions are required, discrete methods have to be used.

Walrasian Context

In most financial articles, the asset allocation optimization is performed under the assump-
tion that the optimizing agent (whether an individual or an organization) is a “price-taker”
(i.e. the agent’s market orders and control actions are too small to influence the mar-
kets). Hence the investment decisions taken are separate from and have no bearing on the
dynamics of the investment vehicles and the stochastic processes used to represent them.
In the case of an individual investor, this assumption is (in most cases) largely justified.
However, in the case of a large fund or of an institution executing sizable trades, this Wal-
rasian assumption may be erroneous and inappropriate as feedback mechanisms may exist
(it is difficult to execute large trades in a security without influencing the market in that
security).
1.2. OUR RESEARCH FRAMEWORK AND CONTRIBUTIONS 11

1.2 Our Research Framework and Contributions

1.2.1 Research Focus

Our original research intent was, in a broad sense, to investigate the predictability (if any)
of financial markets and analyze the impact that any such predictability of financial returns
would have on ALM models. We soon realized this original goal was far too ambitious for
our purposes. First, the issue of whether, and to what extent, some asset class returns are
predictable or not has been a vastly studied subject over the years. While it remained a
controversial topic in the case of equity returns, finding anything significantly new there
(assuming there was anything to be found) appeared challenging. Second, ALM models can
vary a great deal depending upon the needs and applications they originate from and hence
defining a generic model for our analysis seemed unlikely to be really fruitful.
Our research eventually settled on two different specific questions that stemmed from
this original interest of studying market predictability and its impact on ALM models.
First, we focused on analyzing short-term market momentum and reversals by introducing
a new (and deceivingly simple) methodology which, to the best of our knowledge, had never
been carried out before on equity indices. Second, to assess the predictability and serial
correlation of different asset class returns, we estimated -against the same large sample of
data- different statistical models of returns. In particular, and in contrast to the standard
log-normal model of returns derived from geometric Brownian motion (“GBM”), we used a
vector autoregressive (“VAR”) and a Bayesian vector autoregressive (“BVAR”) framework
for analyzing asset class returns. We then assessed the forecasting performance of these
different statistical models of returns and compared the allocation results these different
statistical models of returns would yield.

1.2.2 Our Contributions

First, we revisited the issue of predictability of equity returns and have provided a new
framework for detecting it. Our empirical work shows strong evidence of serial correla-
tion in daily returns of equity indices. It underscores statistically significant momentum
and reversal effects for equity indices, conditional upon their past cumulative number of
consecutive upward or downward movements.
Second, we have analyzed the impact of different statistical models of asset class returns
within a multi-stage stochastic programming framework. For a given set of data comprised
of stock, bond and cash returns (for both yearly and monthly data), we have assessed the
12 CHAPTER 1. INTRODUCTION

different forecasting performance of various stochastic models, including vector autoregres-

sive (“VAR”) processes and Bayesian vector autoregressive processes (“BVAR”). We have
clearly shown that a VAR framework can be useful for predicting different bond returns
(and the yield curve in general). We have then shown that for the same sample of data and
the same programming horizon, the differences in the optimal asset allocations calculated
can be significantly different depending upon which model of asset class returns is used.
Third, our BVAR multi-stage stochastic programming framework provides an appropri-
ate model for funds of funds investing. In particular our model addresses two significant
issues that funds of funds have to face: (i) significant transaction costs and (ii) scarce or
limited historical performance data that makes it hard to quantify precisely the allocation
process.
Chapter 2

Asset Allocation Methods and

Strategies

In discrete time, there is a variety of techniques used by professionals and academics to

solve dynamic asset allocation models but essentially we can categorize them as belonging
either to a dynamic programming methodology or a stochastic programming methodology
(or both).
Dynamic programming was first introduced by Bellman (Bellman 1957) and provides,
among other things, a solution methodology for solving dynamic problems with a graph
structure. Stochastic dynamic programming is essentially the same idea except that the
dynamic structure of the problem may not be spelled out explicitly (as with an exhaustive
graph structure describing the universe of possible scenarios) but rather be represented by
continuous distributions.
Stochastic programming is defined (Birge and Louveaux 1997) as the branch of mathe-
matical programming (whether linear or nonlinear) where some of the problem data might
be stochastic (as opposed to fixed parameters).
There exists also other types of formulation of ALM models such as the network for-
mulation for investment planning as developed by Mulvey and Vladimirou (Mulvey and
Vladimirou 1989).
Early use of a dynamic programming formulation for solving investment problems can
be found in Mossin (Mossin 1968) and Samuelson (Samuelson 1969).
Similarly, stochastic programming formulations of investment planning can be found in
Dantzig and Infanger (Dantzig and Infanger 1993) and Infanger (Infanger 1993).
Our purpose in this chapter is to review the main existing asset allocation techniques

13
14 CHAPTER 2. ASSET ALLOCATION METHODS AND STRATEGIES

and emphasize the pros and cons of the stochastic programming approach to investment
planning.
In order to fulfill this goal, we first need to review the different utility functions tradi-
tionally used. They are crucial as they can sometimes drive the optimal investment policy.

2.1 Synopsis of Utility Functions

The concept of a utility function has axiomatic foundations that date back to von Neumann
and Morgenstern (von Neumann and Morgenstern 1944) and Savage (Savage 1954). Utility
functions were originally designed as a tool for choosing between alternatives that would
produce different random wealth variables. The two main principles in their construction
are non-satiation (i.e. more wealth should be preferred to less) and risk aversion (at least for
most individuals) that respectively translate into having an increasing and concave mapping
of wealth to the utility it yields.

2.1.1 Risk Aversion and Risk Tolerance

The degree of risk aversion exhibited by a utility function is related to the curvature of
the function. Traditionally, it is formally defined by the Arrow-Pratt absolute risk aversion
(“ARA”) coefficient (cf (Luenberger 1998)) which is:

U 00 (x)
ARA(x) = − . (2.1)
U 0 (x)

The relative risk aversion (“RRA”) coefficient is:

U 00 (x)
RRA(x) = −W . (2.2)
U 0 (x)

Risk tolerance is defined as the reciprocal of risk aversion. Thus, the absolute and
relative risk tolerance coefficients are:

1
ART (x) = , (2.3)
ARA(x)
1
RRT (x) = . (2.4)
RRA(x)
2.2. SYNOPSIS OF ASSET ALLOCATION STRATEGIES 15

2.1.2 Hyperbolic Absolute Risk Aversion Utility Functions

Usually absolute risk aversion decreases as wealth increases. A general class of utility
functions that meets this criterium is the set of hyperbolic absolute risk aversion (“HARA”)
utility functions. A general formulation for such functions is
µ ¶γ
1−γ βC
U (C) = +η , (2.5)
γ 1−γ

with appropriate values for the parameters and C. This family is rich as by varying the
parameters, a utility function with absolute or relative risk aversion increasing, decreasing,
or constant, can be obtained (Merton 1971). Particular cases of utility functions often used
include:

• Linear or risk-neutral function: U (x) = x

• Quadratic: U (x) = x − 12 cx2

• Exponential: U (x) = −e−ax

• Power: U (x) = cxγ

• Logarithmic: U (x) = ln x

It is important to appreciate that in a one-period setting where the utility function is

quadratic or the asset returns are all normal, maximizing the expected value of the utility
function is equivalent to solving the Markowitz mean-variance problem (Luenberger 1998).

2.2 Synopsis of Asset Allocation Strategies

We review here the different standard allocation strategies by increasing degree of complex-
ity.

Buy-and-Hold Strategy This is the simplest one-period strategy where an initial asset
allocation is chosen and held through until the end of the planning horizon. A simple
allocation choice can be derived by assuming normal returns and calculating the mean-
variance portfolio.
16 CHAPTER 2. ASSET ALLOCATION METHODS AND STRATEGIES

Fixed-Mix Strategy This is used for a multi-period setting. The asset allocation weights
are fixed and the assets rebalanced to those initial weights at each decision point. An
attractive feature of this allocation policy is that it is somehow equivalent to a certain form
of “volatility pumping” as at every period assets are sold “high” (i.e. when they have done
better than most of their peers) and bought “low”. The theoretical properties of fixed-mix
strategies are discussed, among others, by Merton and Dempster, Evstigneev and Schenk-
Hoppé (Merton 1990, Dempster, Evstigneev, and Schenk-Hoppé 2003). Infanger makes the
point that fixed-mix strategies perform well even when some of the assumptions required
for their theoretical justification are relaxed (Infanger 2002).
Fixed-mix strategies are important because they are often used to benchmark portfolio
managers. Different studies have been made to assess the contribution of active asset
selection and management and the value it added on top of the benchmarks. Such studies
can be found in the work of Brinson, Hood and Beebower (Brinson, Hood, and Beebower
1986, Brinson, Hood, and Beebower 1991), Hensel, Ezra and Ilkiow (Hensel, Ezra, and
Ilkiow 1991) and Blake, Lehmann and Timmermann (Blake, Lehmann, and Timmermann
1999).
Because fees are related to performance, usually relative to a benchmark or a peer group,
Blake et al. in their study of the U.K. market showed that:

• U.K. pension fund managers have a weak incentive to add value and face many con-
straints if and when they try to do it.

• Fund managers know that relative, rather than absolute, performance determines their
long-term survival in the industry.

• Fund managers earn fees related to the value of assets under management, not to their
relative performance against a benchmark or their peers, with no specific penalty for
underperforming or reward for outperforming.1

All these studies show the importance of fixed-mix strategies since they are often used
as benchmarks in the fund management industry.
But as Samuelson (Samuelson 1969) and Merton (Merton 1990) show, there is also a
theoretical justification for fixed-mix strategies if:

• Asset return distributions are independent and identically distributed.

• The utility function exhibits constant relative risk aversion.

1
This is significantly different from the case of hedge fund managers whose compensation usually includes
a performance fee of about 20% of the upside.
2.2. SYNOPSIS OF ASSET ALLOCATION STRATEGIES 17

• Only investment income is considered.

• No transaction costs are allowed.

If the utility function is logarithmic, it is worth noticing that non-iid asset returns also
result in a constant allocation strategy.

Stochastic Programming The two previous strategies have the drawback that they do
not use new information from return realizations in their determination. The study by
Cariño and Turner (Cariño and Turner 1998) shows that the recourse flexibility provided
by a multiperiod stochastic programming model improves performance results. Because a
multi-stage stochastic program allows the asset mix at any future stage to adjust to the
current wealth, the objective is improved. Ziemba (Ziemba 2003) shows results to that
effect.
However, Fleten, Høyland and Wallace (Fleten, Høyland, and Wallace 2002) show more
mitigated results due to the differences between an in-sample and out-of-sample analysis for
the portfolio model of a Norwegian life insurance company. In their analysis, the authors
show that the stochastic programming approach works significantly better than a fixed-mix
strategy in an in-sample analysis but only slightly better in an out-of-sample analysis. If,
as the analyst proceeds forward through the time series of asset returns, there is a strong
disconnect between the realized returns and the forecasted ones, the stochastic programming
approach may loose its precision and its advantage over a fixed-mix approach and the
gap between the two strategies closes. As Ziemba puts it, “the stochastic programming
model loses its advantage in optimally adapting to the information available in the scenario
tree”. However, we would still expect the stochastic programming approach to yield some
improvement over a pure fixed-mix strategy.

Stochastic Dynamic Programming Stochastic dynamic programming is another so-

lution technique for solving a multi-stage scenario tree. Whereas, multi-stage stochastic
programming requires solving directly (or indirectly through sampling and decomposition
techniques) a large-scale linear (or nonlinear) program, stochastic dynamic programming
makes it possible to solve a multi-stage scenario tree by backward induction (provided the
state space is of manageable size). Early use of dynamic programming for solving investment
problems go back at least to Mossin (Mossin 1968) and Samuelson (Samuelson 1969).
Whether to use dynamic programming rather than stochastic programming is not of
vital interest to us at this stage. For small problems, it is not particularly relevant, as
both methods should give similar results. The more interesting question is how we forecast
18 CHAPTER 2. ASSET ALLOCATION METHODS AND STRATEGIES

future asset returns in both cases. If the forecasts are out of line with the actual returns,
an out-of-sample analysis of the results is likely to show that a multi-stage model (whether
solved by dynamic or stochastic programming) is of little use versus a fixed-mix or even a
buy-and-hold strategy.
However, especially in the presence of transaction costs, a multi-stage stochastic frame-
work offers a natural and flexible formulation, especially in the case where the investment
planner has strong views about particular dynamic scenarios.2

2.3 Idiosyncrasies of Stochastic Programming

The stochastic programming approach provides, in our view, a flexible and more realistic
approach to dynamic asset allocation models. Among the important reasons for this flexi-
bility is the fact that the modeling of either the utility function or the asset class returns do
not require the use of smooth functions or highly regular distributions. In other words, a nu-
merical approach to the multi-stage stochastic program, usually done by pre-sampling from
the distributions, yields satisfactory results. Some of the idiosyncracies of the stochastic
programming approach, as pointed out by Ziemba (Ziemba 2003) are:

• The often artificial elicitation of a smooth utility function is not needed. For individual
planning, all that is needed is to understand future liabilities and how important the
decision maker considers them. For doing so, the traditional mean-variance objective
can be abandoned for a more tailored objective that the decision maker can easily
understand. In our numerical results, we use a utility function that is piecewise linear
and strictly concave. Each linear segment of the utility function has its own slope,
supposed to reflect the relative penalty cost the decision maker associates with ending
the planning horizon within a certain wealth range, long or short of a desired objective.

• Scenarios supposed to reflect particular beliefs about various elements of the models
(e.g. a particular shock to returns in the event of a major geopolitical setback, etc.)
can be easily included or adjusted. This form of scenario-dependent knowledge is also
usually easy to extract from the decision maker. Adjusting certain scenarios may be
vital in some cases. For instance, in standard asset models, the asset returns corre-
lation matrix is often assumed to be constant throughout the model. This may be a
2
It is worth noting that, in the presence of transaction costs, the problem as formulated becomes path-
dependent and the current wealth cannot be used as the only state variable in the dynamic programming
formulation. The state space needs to be enlarged in a significant way to reflect transaction costs, and the
dynamic programming approach looses some of its attractiveness.
2.3. IDIOSYNCRASIES OF STOCHASTIC PROGRAMMING 19

dangerous simplification. If most of the time, asset returns can be rightly assumed
to follow a certain correlation structure, in times of particular stress on the finan-
cial markets (such as what happened in August 1998 with the Russian government
defaulting on its debt), asset returns tend to be much more correlated and for the
worse. Representing this alteration of the correlation structure is relatively easy to
implement in the case of a scenario-based multi-stage stochastic program.

• It is easy to include transaction costs and different regularity requirements such as

cash reserve constraints and turnover limits to a stochastic program.

The first two points listed above are also valid for a dynamic programming methodology.
The first point pertains to the formulation of the utility function, which is independent of the
solution technique. The second point relates to the formulation of the multi-stage scenario-
based tree, which would be the same for dynamic or stochastic programming. However the
third point is idiosyncratic to a stochastic programming formulation.
However, the stochastic programming approach has its limitations. In particular, the
investment problem represented as a multi-stage program usually chooses the timing of
these stages arbitrarily. A stochastic control approach would perform better. The stochastic
programming approach also places two kinds of constraints on the optimal allocation policy.
First it places an upper bound on the number of reallocation points. If there are significant
transaction costs, it is very well possible that the solution of the “free problem” would
only require a few trades and hence this constraint might not be too drastic. However,
the second constraint is the time distribution of these stages. Assuming there are (n-1)
reallocation stages in addition to the original allocation, we have to be careful in choosing
their temporal distribution. If this distribution reflects “natural” points of reallocation (e.g.
the decision maker only looks at his/her portfolio every year at the beginning of the year),
then presetting these reallocation points seems appropriate. However, if there is no a priori
behavioral constraints relevant to the decision maker, we need to check that the distribution
of reallocation times is not too arbitrary.
Chapter 3

Comparison of Asset Allocation

Methods

In this chapter, we perform both a single-period and a multi-period analysis of different as-
set allocation strategies. For one-period models, we focus on a mean-variance efficient type
of approach, as developed by Markowitz (Markowitz 1952a). For a multi-period model,
we compare the different results obtained with buy-and-hold, fixed-mix and stochastic pro-
gramming strategies.

3.1 Single-Period Model

A single-period model, despite its simplicity, is enough to capture a few important empirical
points in asset allocation strategies. In general, the three following assumptions strongly
influence the dynamic asset allocation results:

• The size of the historical window used for estimating the expected returns and co-
variance structure of the asset classes considered if indeed historical estimates are
used.1

• The risk-free and borrowing rates.

• The investor’s utility function.

1
Though we do not address it here, it is important to bear in mind that other methodologies are more
appropriate for estimating volatility such as using implied volatilities from equity option prices.

20
3.1. SINGLE-PERIOD MODEL 21

Indifference Curves
35

30

25
Expected Return (%)

20
Risk Aversion of 3

Risk Aversion of 4
15

10

Risk Aversion of 2
5

0
0 5 10 15 20 25 30 35 40
Standard Deviation (%)

Figure 3.1: Utility Function Indifference Curves.

3.1.1 Sensitivity Results on Historical Estimates

The first analysis we do is on yearly data from 12/31/1946 to 12/31/2002 using as asset
classes stocks and bonds. The equity class is represented in a first analysis by the NYSE
index and in a second analysis by the NASDAQ index. For the government bond asset
class, we use the 30-yr bond yearly returns.

Utility Function and Risk Aversion

For convenience, we use Matlab Financial Toolbox’s integrated utility function:

U = E(r) − 0.5 A σ 2 (3.1)

where U is the utility function, E(r) is the expected return, σ the standard deviation
and A the risk aversion which will be the varying parameter in our analysis.
Figure 3.1 shows the indifference curves for such a utility function.
22 CHAPTER 3. COMPARISON OF ASSET ALLOCATION METHODS

Results with NYSE Index

The risk aversion in the example is set at 7 for the utility function previously defined. The
size of the historical window is 10 to 50 years. The risk free rate is at 2% and the borrowing
rate is at 3%. These risk free and borrowing rate values are reasonable given that the
historical window used ends in 2002. We proceed as if we were calculating at the beginning
of 2003 a one-year forward allocation until the end of 2003 based upon historical data. At
this point in time (i.e. January 2003), the interest rates were very low and our assumed
values are reasonable.
Figure 3.2 shows the variations of the overall risk (defined as standard deviation) and
the overall return to be expected with respect to the size of the historical window used
for estimating both the stock and 30-yr bond expected returns, as well as their covariance
matrix. The window size varies from 10 to 50 years. This means that we have computed
the estimates using the last 10 to 50 years of historical returns, from the ending date of our
sample.
We can observe that it takes about thirty years of historical data for both the expected
return and risk to stabilize. However there is a inherent tension between the number of
years needed to get seemingly stable results and the validity of using returns belonging to
a distant past.
The next graphs represented in Figure 3.3 show the variations of the fraction of wealth
to be allocated to the risky portfolio, along with the recommended fractions of total wealth
allocated respectively to stocks and 30-yr bonds.

Based on NASDAQ Historical Performance

We perform the same analysis but this time with NASDAQ value-weighted index returns
(including dividends) obtained from the CRSP. These NASDAQ returns are calculated from
1973 to 2003.
We can see from Figure 3.4 and Figure 3.5 that the variations are still quite significant.
It is worth noticing that the inclusion of the first two years in our sample (1973 and 1974)
reduces significantly the expected returns and the allocation to the risky portfolio.

3.2 Multi-Period Model

For multi-period models, we show the advantages of using stochastic programming over
more traditional strategies, such as ”buy-and-hold” and ”fixed-mix” strategies.
3.2. MULTI-PERIOD MODEL 23

0.36 0.14

0.35
0.13

0.34
0.12

0.33
Overall Return

0.11
Overall Risk

0.32

0.1
0.31

0.09
0.3

0.08
0.29

0.28 0.07
10 30 50 10 30 50
Historical Window (Yrs) Historical Window (Yrs)

Figure 3.2: Variations of Overall Risk and Return (NYSE case). The NYSE yearly returns
used are value-weighted and include dividends.
24 CHAPTER 3. COMPARISON OF ASSET ALLOCATION METHODS

1 0.7

0.65
0.95

0.6 Stock
0.9

0.55
Risky Fraction

Risky Weights

0.85
Bond
0.5

0.8
0.45

0.75
0.4

0.7
0.35

0.65 0.3
10 30 50 10 30 50
Historical Window (Yrs) Historical Window (Yrs)

Figure 3.3: Allocation Weights (NYSE case).

3.2. MULTI-PERIOD MODEL 25

29 7

28.5

6.5
28
Overall Risk (Standard Deviation %)

27.5
6
Overall Return (%)
27

26.5 5.5

26

5
25.5

25
4.5

24.5

24 4
10 20 30 10 20 30
Historical Window (Yrs) Historical Window (Yrs)

Figure 3.4: Variations of Overall Risk and Return (NASDAQ case). The NASDAQ yearly
returns used are value-weighted and include dividends.
26 CHAPTER 3. COMPARISON OF ASSET ALLOCATION METHODS

0.7 0.5

0.45 Bond
0.65

0.4
0.6
0.35 Stock

0.55
0.3
Risky Fraction

Risky Weights

0.5 0.25

0.2
0.45

0.15
0.4
0.1

0.35
0.05

0.3 0
10 20 30 10 20 30
Historical Window (Yrs) Historical Window (Yrs)

Figure 3.5: Allocation Weights (NASDAQ case).

3.2. MULTI-PERIOD MODEL 27

3.2.1 Buy-and-Hold Strategy

This strategy is the simplest. As its name indicates, there is no rebalancing of the portfolio
weights once the initial allocation decision has been made.
The initial equation is:

n−1
X
xi0 (1 + btc i ) + xcash
0 = W0 , (3.2)
i=1

where W0 is the initial wealth available2 , n − 1 is the number of asset classes other than
money markets3 and btc i are the transaction costs for investing in asset class i.4 Between
t − 1 and t the following equation applies:

i,ω ,...ωt−1
xt−11 Rti,ωt = xi,ω
t
1 ,...ωt
, i = 1, ..., n − 1 and t = 1, ..., T − 1. (3.3)

The random variable Rti,ωt is the random return of asset class i between times t − 1 and t
drawn in scenario ωt . For the last period, we get:

n−1
X i,ω ,...,ωT −1 cash,ω1 ,...,wT −1
1
xT −1 RTi,ωT (1 − stc i ) + xT −1 RTcash,ωT = WTω1 ,...,ωT , (3.4)
i=1

where stc i are the transaction costs for selling out of the asset class i and WTω1 ,...ωT is the
terminal wealth thereby obtained in the scenario T-tuple (ω1 , ..., ωT ). The utility function
is represented by slack variables uωT 1 ,...,ωT for being above the final goal G and vTw1 ,...,ωT for
being short of the goal, so that:

uωT 1 ,...,ωT − vTω1 ,...,ωT = WTω1 ,...,ωT − G. (3.5)

G is the desired goal. The final objective function, Z, is:

P ω1 ,...,ωT
(ω1 ,...,ωT ) ∈ Ω1 ×...×Ω1 [uT − ψ vTω1 ,...,ωT ]
Z= . (3.6)
[|Ω1 | × ... × |ΩT |]

2
For the sake of clarity, “W ”, in W0 , is not to be confused -and has nothing to do- with “ω”, in ωt , which
is a different letter and designates a time t scenario. We keep these possibly confusing notations however to
follow existing literature.
3
We assume there are no transaction costs for transferring money in and out of money markets and treat
money markets as equivalent to ”cash”, which is our nth asset class.
4
In our model, the transaction costs are assumed to be proportional to the amount invested in the asset
class.
28 CHAPTER 3. COMPARISON OF ASSET ALLOCATION METHODS

In equation 3.6, ψ is the slope of the utility function below the goal G. It is a penalty factor
for being short of this desired final goal.

3.2.2 Fixed-Mix Strategy

The fixed-mix strategy forces the additional constraints that at any reallocation time t,
we rebalance the portfolio to fall back (post transactions costs) upon the initial allocation
weights we had chosen at time t = 0. To deal with transaction costs, we have to introduce
additional variables that represent how much of each asset class we have bought, identified
by yi , or sold, identified by zi . We detail the full model, starting with the initial allocation
equation of the initial wealth W0 into the different asset classes:

n−1
X
xi0 (1 + btc i ) + xcash
0 = W0 ,
i=1

and we calculate from it the fixed-mix proportions xiF M :

xi
xiF M = Pn 0 i
, i = 1...n. (3.7)
i=1 x0

Subsequently, at intermediate times, we have the following set of equations:

i,ω ,...,ωt−1
xt−11 Rti,ωt + yti,ω1 ,...,ωt − zti,ω1 ,...,ωt = xti,ω1 ,...,ωt , i = 1...n − 1,
n−1
X n−1
X
i,ω ,...,ωt−1
1
xcash Rti,ωt − yti,ω1 ,...,ωt (1 + btc i ) + zti,ω1 ,...,ωt (1 − stc i ) = xtcash,,ω1 ,...,ωt ,
i=1 i=1

where yti,ω1 ,...,ωt is the amount of asset class i bought in scenario (ω1 , ..., ωt ) and zti is the
amount sold in scenario t-tuple (ω1 , ..., ωt ). In addition, we have the fixed-mix constraints:

xi,ω1 ,...,ωt

Pn
t
i,ω1 ,...,ωt
= xiF M , i = 1...n. (3.8)
i=1 xt
3.3. COMPUTATIONAL RESULTS 29

The terminal equations are the same as previously:

n−1
X i,ω ,...ωT −1 cash,ω1 ,...wT −1
1
xT −1 RTi,ωT (1 − stc i ) + xT −1 RTcash,ωT = WTω1 ,...ωT ,
i=1

uωT 1 ,...,ωT − vTω1 ,...,ωT = WTω1 ,...,ωT − G,

P ω1 ,...,ωT
(ω1 ,...,ωT ) ∈ Ω1 ×...×ΩT [uT − ψ vTω1 ,...,ωT ]
Z= .
[|Ω1 | × ... × |ΩT |]

It is worth noticing that the introduction of the fixed-mix constraint imposes nonlinear
constraints. Hence this nonlinear program (”NLP”) is solved using MINOS (Murtagh and
Saunders 1983). For further references on optimization, we refer the reader to the work by
Gill, Murray and Wright (Gill, Murray, and Wright 1986).

3.2.3 Stochastic Programming Strategy

The stochastic program strategy is formulated in the same way as the fixed-mix strategy
but without the fixed-mix constraints, i.e. equations (3.7) and (3.8). Because the feasible
domain of the stochastic program encompasses the feasible domain of the fixed-mixed strat-
egy, we know immediately that the stochastic programming strategy should yield a higher
(or equal) objective value.

3.3 Computational Results

Figure 3.6 shows our multi-period framework for comparing different asset allocation strate-
gies. We use a terminal utility function at T=3 and two reallocation times (when permitted
by the strategy) at t=1 and t=2. Units of time are in years. The quantity of interest is the
optimal asset allocation at t=0. The penalty factor, ψ, for being short of the goal, G, is set
at 10. The number of scenarios is |Ω1 | = 50, |Ω2 | = 30, and |Ω3 | = 20, for a total number of
scenarios of 30,000. The initial wealth is set at a standard of W0 = 100 and the final goal
at G = 115.
Table 3.1 shows results for an in-sample analysis. The stochastic programming strategy
performs better than the two others, as expected. What is more surprising is the very close
results between the buy-and-hold strategy and the fixed-mix strategy. The solution time
of the latter (about 8,000 seconds) is much larger than the solution time of the former (i.e.
176 seconds). It is not so surprising however given the fact that the fixed-mix strategy
requires solving an NLP. We suspect that these relatively comparable results are due to a
30 CHAPTER 3. COMPARISON OF ASSET ALLOCATION METHODS

Figure 3.6: Framework for Computational Results.

3.3. COMPUTATIONAL RESULTS 31

Allocation Strategy Stock (%) Bond (%) Cash (%) Obj. Sol. Time (sec.)
Buy-and-Hold 55.9% 19.9% 24.2% -3.79 176.1
Fixed-Mix 55.0% 19.3% 25.7% -3.79 7983.3
Stochastic Programming 67.4% 27.7% 4.9% 4.62 243.9

Table 3.1: Comparison of asset allocation strategies. Computations were carried out on a
Intel Pentium 4 running at 1.6 GHz with 512 Mb of RAM.

large part to the small sample size used. Also, in this set of results, all transaction costs are
set at zero. However, adding transaction costs will make the buy-and-hold strategy more
attractive.
Chapter 4

Momentum of Equity Returns

In this chapter, we present a new methodology for analyzing the predictability of equity
returns. Instead of trying to determine the best statistical fit for daily equity returns, we
study directly the serial properties of daily returns in a restricted framework. Specifically
we analyze how series of consecutive rising (or, conversely, diminishing) returns unfold. Our
original work shows evidence that even in recent years, equity markets still harbored both
momentum and mean-reversion effects.
This chapter is organized as follows. We first introduce the methodology used for our
analysis. Second, we present our results for broad-based equity indices and underline the
evidence supporting market momentum and market reversion.
Also, for further references on the subjects of market predictability, market efficiency,
and the historical developments of models of equity returns, we refer the reader to the
review of mathematical finance in the appendices.

4.1 Our Methodology

4.1.1 Issue of Market Predictability

Our interest in analyzing the serial properties of equity returns stemmed from the contro-
versial issue of market predictability and the historical development of various models to
account for the statistical properties of equity returns.
The issue of market predictability has been studied for a long time and by various com-
munities, from speculators to academic members of the mathematical finance community.
It is not hard to understand that if there was any pattern of predictability that could be

32
4.1. OUR METHODOLOGY 33

found in equity markets, this pattern would be quickly exploited and we would expect arbi-
trageurs to wash it away. Historically however, the issue of predictability of equity returns
has been a consistent topic of research. One of the reasons for this continued interest is
the fact that the topic of market predictability (or lack thereof) has been closely tied to
the issue of market efficiency (we refer the reader to our appendix on some historical ele-
ments of mathematical finance for further references on the subject). Many academics have
contended that markets are unpredictable and many books and articles have been written
on this topic. Starting with Bachelier in 1900 (Bachelier 1900), who introduced Brownian
motion (”BM”) to finance, models of equity returns have been laid out that describe the
price of security as being essentially unpredictable in direction (at least once certain adjust-
ments to returns have been made). We mean by direction that it would be impossible for a
forecaster to guess whether the price is going to move up or down in the next time period.
Later on, for various reasons, academics realized geometric Brownian motion (”GBM”) was
a more accurate model and Osborne provided detailed evidence to support this assertion
(Osborne 1959). References on GBM and its usefulness for describing the stock market are
given in Osborne’s paper. In a discrete setting (which is our focus here as we will look at
daily returns derived from daily closing prices), this implies that daily returns should follow
a random walk. This expression of ”random walk” was popularized to a large extent by
Malkiel in his book ”A Random Walk Down Wall Street” (Malkiel 2003). Malkiel defines
it as:
A random walk is one in which future steps or directions cannot be predicted
on the basis of past actions. When the term is applied to the stock market, it
means that short-run changes in stock prices cannot be predicted... On Wall
Street, the term “random walk” is an obscenity. It is an epithet coined by the
academic world and hurled insultingly at the professional soothsayers. Taken
to its logical extreme, it means that a blindfolded monkey throwing darts at a
newspaper’s financial pages could select a portfolio that would do just as well
as one carefully selected by the experts.
It is this issue of market predictability that we intend to analyze here, with a new method-
ology. It is a very important and controversial issue, and as Malkiel puts it:
By the early 2000s, even some academics [have joined] the professionals in ar-
guing that the stock market was at least somewhat predictable after all. Still,
as [one] can see, there’s tremendous battle going on, and it’s fought with deadly
intent because the stakes are tenure for the academics and bonuses for the pro-
fessionals.
34 CHAPTER 4. MOMENTUM OF EQUITY RETURNS

4.1.2 Our Restricted Formulation

We analyze market predictability in a restricted sense, that is by only looking at the “di-
rectional” moves of the market as a whole, or an individual security. What we mean by
“directional move” is whether the market index or security price of concern is going to move
up or down over the next time period.
We focus here on daily returns derived from closing prices. Formally, the closing price
(or level) of the underlying (whether the underlying is taken to be a market index, a sectorial
index or an individual security price) will be defined as Pt and the simple return rt is defined
as:
Pt+1
rt = − 1, 0 ≤ t ≤ T − 1, (4.1)
Pt
where T is the time horizon of our analysis. From this time series rt , we define:

bt = sign(rt ), (4.2)

where 

 1 if x > 0,
sign(x) = 0 if x = 0,


−1 otherwise.
From the series bt , we derive the series st as s0 = b0 and:
(
st + sign(bt+1 ) if st bt+1 > 0
st+1 = f or 0 ≤ t ≤ T − 1.
sign(bt+1 ) otherwise

The construction of the series st will allow us to track the number of consecutive days
the price process Pt has been going up or down. Figure 4.1 shows how the series st is
constructed.
By only looking at the directional moves of the price process Pt and focusing on the
signs of the daily returns rt , we lose the finer details of the distribution of rt . To understand
this, assume rt to be drawn from the probability density function (”PDF”) of a stationary
distribution fr (r) and to be independent of each other. Also assume that our goal was to
predict over T observations the series bt = sign(rt ) with the greatest precision. Mathemat-
ically, if we call ft 1 our forecast of the underlying move over the next period, we can define

1
The notation used for the series of forecasts, ft , is obviously not to be confused with the PDF fr (r).
4.1. OUR METHODOLOGY 35

Figure 4.1: Mapping of the “underlying” price level into the series st that represents the
number of cumulative and consecutive upward or downward movements of the original series
Pt sampled discretely (at the end of each trading day). We have omitted on this graph the
series rt that is easily derived from the series Pt .
36 CHAPTER 4. MOMENTUM OF EQUITY RETURNS

the forecaster’s performance metric, ZT , as:

PT −1
t=0 1{ft =bt }
ZT = ,
T

where 1{ft =bt } = 1, if ft = bt and 0 otherwise. It is easy to show that the best forecasting
strategy at each t, where 0 ≤ t ≤ T − 1, would be:

 −1 w.p. p−1

ft = 0 w.p. p0


1 w.p. p1 ,

where:
Z 0−
p−1 = fr (r)dr (4.3)
−1

p0 = P r{r = 0} (4.4)
Z +∞
p1 = fr (r)dr. (4.5)
0+

So all we would need to know for our optimal forecast would be the cumulative distribution
R 0−
function (”CDF”) value Fr (0− ) = −1 fu (u)du (which is p−1 ), as well as the probability
mass p0 at 0 if there is any (and consequently p1 = 1 − p−1 − p0 ). So two CDFs Fr1 and Fr2
with the same “cutoff” value at 0 (and, if there exists one, the same punctual probability
mass at 0) would give us the same forecasting policy.
Clearly, for many real applications (such as portfolio management), we would want to
estimate in finer details the PDF fr (r). For instance, if we assume over the time horizon T
that the daily returns rt will be independent and identically distributed as rt ∼ N−1 (0, σ 2 ),
where N−1 is just a normal distribution that has been truncated below at −1 and renor-
malized, we would want to estimate the volatility σ. Hence, our analysis is restricted in
the sense that we only focus on the sign of the returns rt in our mapping of the series rt
into bt . By doing so we disregard the information on the returns distribution embedded
in the distribution moments (second-, third- and higher-order).2 However, as our analysis
will later show, the CDF “cutoff” value is all we need to display strong evidence of serial
correlation in the returns of equity indices.

2
We wish in no way to suggest to the reader that two probability distributions with the same moments
ad infinitum have to be equivalent, as this is not the case.
4.1. OUR METHODOLOGY 37

4.1.3 Intuition for Methodology

There were different intuitive categories for undertaking this particular investigation. First,
we believed that by and large markets did not always reflect all available information and
that when markets close, we should expect to have a gradual diffusion of information among
traders and the formation of consensual views from one day to the next. This, we expected,
should be reflected by some market momentum over a couple of days. Second, we thought
that market movements should be conditional to some extent upon their recent history so
that if a market had been going up for a consecutive number of days x, the probability
that it would go up for another day would decrease with x. This so-called mean reversion
should be observable by looking at conditional market transitions. Last, from a statistical
standpoint, focusing on the distribution of returns alone ignores the crucial information that
is embedded in their development as a time series. This can be addressed by preserving the
information contained in the autocorrelation functions of different orders. But even that is
not enough as these summary statistics focus on the sample as a whole and provide average
results for the sample, hence possibly missing ”local patterns of serial correlation” in the
sample. In other words, any sudden burst of correlation in the studied sample would go
undetected if through the rest of the sample there negative correlation to counterbalance
the local burst. This point is illustrated in Figure 4.2 and Figure 4.3.
Figure 4.3 shows two series with approximately the same first-order autocorrelation but
strikingly different paths. Figure 4.2 shows the cumulative number of days with positive
returns, and an artificially created spike in the first-order autocorrelation of the graph’s
time series A.

4.1.4 Previous Evidence of Momentum

We provide here an analysis on momentum in market indices. This will show that successive
price variations are not necessarily independent. As Sornette shows (Sornette 2003), there
is strong evidence of local burst of serial correlation by looking at drawdowns (defined as a
”persistent decrease in the price over consecutive days”). Sornette writes that:

Drawdowns are indicators that we care about: they measure directly the cu-
mulative loss that an investment may suffer. They also quantify the worst-case
scenario of an investor buying at the local high and selling at the next mini-
mum. It is thus worthwhile to ask if there is any structure in the distribution
of drawdowns absent in that of price variations... Their distribution [captures]
the way successive drops can influence each other and construct in this way a
38 CHAPTER 4. MOMENTUM OF EQUITY RETURNS

20

15 Series A
Cumulative Number of Positive Daily Returns

10

−5

Series B

−10

−15

−20
0 50 100 150 200 250 300 350 400 440
Trading Days

Figure 4.2: The two series were generated as follows. Series bA t (series A on the graph)
was generated first with an artificially constructed local “burst” in serial correlation. The
rest of the series was designed with zero first-order autocorrelation so that, all in all, the
sample’s first-order autocorrelation estimate is slightly positive. Series bBt (Series B on the
graph) was generated by a constrained permutation so that the new sample’s first-order
autocorrelation, ρ1bB , is similar to Series A’s, i.e. ρ1bA ' ρ1bB .
t t t
4.1. OUR METHODOLOGY 39

700

600

500
Series A

400
Index Level

300

200
Series B

100

0
0 50 100 150 200 250 300 350 440
Trading Days

Figure 4.3: The index levels represented here correspond to the two series shown in Figure
4.2. The returns for series A were generated from rt as rt ∼ unif orm(0; 0.2) if we wanted
the corresponding bt > 0 or conversely rt ∼ unif orm(−0.2; 0) to have bt < 0. Series B was
generated by a constrained permutation so that ρ1bA ' ρ1bB and ρ1rA ' ρ1rB and in both cases
t t t t
close to 0.
40 CHAPTER 4. MOMENTUM OF EQUITY RETURNS

Rank Starting Time Index Value Duration (days) Loss

1 87.8 2508.2 4 -30.7%
2 14.6 76.7 2 -28.8%
3 29.8 301.2 3 -23.6%
4 33.5 108.7 4 -18.6%
5 32.2 77.1 8 -18.5%
6 29.9 238.2 4 -16.6%
7 29.8 273.5 2 -16.6%
8 32.6 67.5 1 -14.8%
9 31.9 90.1 7 -14.3%
10 32.7 76.5 3 -13.9%

Table 4.1: Characteristics of DJIA’s 10 Largest Drawdowns in 20th Century. The “starting
time”, instead of being in standard date format, is expressed in “decimal years”.

persistent process.

As Sornette points out, this persistence cannot be captured by the distribution of returns
alone (i.e. by only counting the frequency of returns) as by unravelling the returns series
in this way, we forget everything about the relative positions of returns as a function of
time. Similarly, the local persistence of returns cannot be captured by a global analysis of
the two-point correlation functions in the data sample studied as it measures an average
linear dependence over the whole time series, while the dependence may only appear at
special times, for instance for very large runs. Sornette provides in a study with Johansen
significant evidence that ”large stock market price drawdowns are outliers” (Johansen and
Sornette 2001).
From the DJIA historical values, we can easily extract the characteristics of the 10 largest
drawdowns of the DJIA in the twentieth century. Table 4.1 summarizes these values.

4.2 Our Results

Sornette shows in his study, by means of probabilistic arguments, that financial crashes are
”outliers”. However, Sornette focuses on the special cases of major drawdowns and crashes.
We extend this analysis of local burst of persistence in the returns by searching for any kind
of momentum, not just in the case of major market downturns.
The intuition and methodology behind our investigation are different from those of any
study previously mentioned. Typically a financial crash will involve a few consecutive days
of highly negative returns. Thus the persistence that a crash will exhibit will be very
4.2. OUR RESULTS 41

localized (over a few days) and rare (large negative returns). As we mentioned before, the
basis for our idea was to investigate for momentum (and reversion) in both good and bad
times, independently of the amplitudes of returns. Though in this work we will not dwell
on identifying the causes of directional persistence in market movements, it could reflect a
combination of factors such as the gradual formation and diffusion of new beliefs or other
”herding” effects.
The data we have analyzed covers general stock indices, sectorial indices and individual
stocks, though we will only present here results found for equity indices.

4.2.1 Conditional Probabilities of Transition

We look at daily data for three major indices, namely the Dow Jones Industrial Average,
the S&P 500 and the Nasdaq.

Dow Jones Analysis

The data analyzed comprises the daily Dow Jones adjusted closing levels from 01/02/1970
to 12/31/2003. Figure 4.4 shows the daily momentum analysis for the Dow Jones.

S&P 500 Analysis

The data analyzed comprises the daily closing levels for the S&P 500 from 07/02/1962 to
12/31/2003. Figure 4.5 shows the daily momentum analysis for the S&P 500.

Nasdaq Analysis

The data analyzed comprises the daily levels of the Nasdaq from 12/14/1972 to 12/31/2003.
Figure 4.6 shows the daily momentum analysis for the Nasdaq.

4.2.2 Conditional Returns

Table 4.2 shows the returns corresponding to particular transitions for the Dow Jones. Ta-
ble 4.3 shows the returns corresponding to particular transitions for the S&P 500. Table 4.4
shows the returns corresponding to particular transitions for the Nasdaq.
42 CHAPTER 4. MOMENTUM OF EQUITY RETURNS

2500
Number of occurences

2000

1500

1000

500

0
−15 −10 −5 0 5 10 15
State

0.6

0.55
Probability

0.5

0.45

0.4
−5 −4 −3 −2 −1 1 2 3 4 5
State

Figure 4.4: Daily Momentum Analysis of DJIA. The top graph shows the frequency of the
visits of the series st to each state. The bottom graph zooms in on the estimated transitional
probabilities from states −5 to state 5 excluding state 0. This graph shows for instance
that, conditional on st being in state 1 (resp. 2), the probability that the Dow Jones will
close up another day and st+1 end up in state 2 (resp. 3) is almost 55% chance. Similarly
we see that, conditional on st being in state −1 or −2, the probability that the DJIA will
close down another day is higher than 50% as well. These transitional probabilities for
states −2, −1, 1 and 2 suggest a momentum effect over three days for the DJIA index
variations. Conversely, we can see that for states −5 to −3 and states 3 to 5, we have the
opposite effect as the transitional probabilities for continuing the DJIA consecutive moves
in one direction are less than 50%. This underlines market reversion beyond three days of
consecutive moves for the DJIA equity index.
State s E[r | s → −1] E[r | s → +1] Pr[r | s → −1] E[r | s]
1 -0.0062 0.0076 0.4642 0.0012
2 -0.0061 0.0072 0.4617 0.0011
3 -0.0062 0.0069 0.5542 -0.0002
4 -0.0068 0.0060 0.5274 -0.0007
5 -0.0066 0.0054 0.5268 -0.0009

Table 4.2: Dow Jones Expected Returns by Transition

4.2. OUR RESULTS 43

2500
Number of occurences

2000

1500

1000

500

0
−15 −10 −5 0 5 10 15
State

0.58

0.56
Probability

0.54

0.52

0.5

0.48
−4 −3 −2 −1 1 2 3 4
State

Figure 4.5: Daily Momentum Analysis of S&P 500. The top graph shows the frequency
of the visits of the series st to each state. The bottom graph zooms in on the estimated
transitional probabilities from states −4 to state 4 excluding state 0. This graph shows for
instance that, conditional on st being in state 1, the probability that the Dow Jones will
close up another day and st+1 end up in state 2 is almost 58% chance. Similarly we see
that, conditional on st being in state −1 or −2, the probability that the S&P 500 will close
down another day is higher than 50% as well. These transitional probabilities for states
−2, −1, 1, 2 (we’re ignoring 3 and 4 here) confirm, as for the DJIA, a momentum effect
of at least three days for the S&P 500 index variations. Conversely, we can see that for
states −4 and −3, we have the opposite effect as the transitional probabilities for continuing
the S&P 500 consecutive moves downwards are less than 50%. As for the DJIA, we have
the bell-shaped curve centered around 0 that indicates momentum over the first few days
followed by reversion.
44 CHAPTER 4. MOMENTUM OF EQUITY RETURNS

2000
Number of occurences

1500

1000

500

0
−20 −10 −1 1 10 20
State

0.8
Probability

0.6

0.4

0.2
−10 −5 −1 1 5 10
State

Figure 4.6: Daily Momentum Analysis of Nasdaq. The top graph shows the frequency
of the visits of the series st to each state. The bottom graph zooms in on the estimated
transitional probabilities from states −10 to state 10 excluding state 0. This graph shows
that most transitional probabilities are above 50%. However the most significant states are
states −3 to 3 (excluding 0). We see that conditional upon st being in state 1, 2 or 3, the
probabilities for the NASDAQ to close up one additional day is about 60%.

State s E[r | s → −1] E[r | s → +1] Pr[r | s → −1] E[r | s]

1 -0.0064 0.0068 0.4343 0.0010
2 -0.0055 0.0065 0.4448 0.0012
3 -0.0054 0.0063 0.5020 0.0004
4 -0.0058 0.0053 0.5054 -0.0003
5 -0.0053 0.0049 0.4670 0.0001

Table 4.3: S&P 500 Expected Returns by Transition

4.3. ANALYSIS OF STATISTICAL SIGNIFICANCE 45

State s E[r | s → −1] E[r | s → +1] Pr[r | s → −1] E[r | s]

1 -0.0085 0.0080 0.3767 0.0018
2 -0.0073 0.0065 0.3476 0.0017
3 -0.0064 0.0068 0.4122 0.0013
4 -0.0067 0.0063 0.4000 0.0011
5 -0.0050 0.0058 0.4044 0.0014

Table 4.4: Nasdaq Expected Returns by Transition

4.3 Analysis of Statistical Significance

4.3.1 Sensitivity Analysis

Figure 4.7 shows how sensitive transition probabilities are for higher states that are not
often visited. In this graph we take an artificially constructed sample of market movements
(Series A) that is such that the series sA
t derived from it has a perfectly split distribution
at every state. This is constructed as follows: suppose we have a process that is such that
we have 1024 visits to state 1 (using our previous terminology). Half of these visits to state
1 are continued on to state 2, yielding 512 visits to state 2 and so forth until we reach 2
visits to state 10 and 1 visit at state 11. For this last visit to state 11, we assume this
cumulative run up is then reverted (in other words, there would be a t in our sample such
that st = 11 and st+1 = −1). Hence, for this perfectly split construction, we would estimate
conditional transition probabilities of 50% at every state except for the last state, state 11,
where we would estimate P (st+1 = 12 | st = 11) = 0. Now, suppose we perturb this
process in two different ways to perform a sensitivity analysis of the conditional transition
probabilities. First, we add to our original artificial series a run up to state 12 (this is our
Series B). So Series B would have exactly the same number of visits than Series A at every
state, increased by 1 (1025, 513, etc.). Similarly, we construct another series (Series C) by
subtracting from the original series (Series A) the last visit to state 11. In other words, at
every state, the number of visits of Series C would be the same as Series A, minus 1 (1023,
511, etc.). We compute the conditional transition probabilities for Series B and Series C
and, as expected, can see that they are similar to Series A for the “low” states but become
quite erratic for the “high” states.

4.3.2 Bootstrap Analysis

To measure the statistical significance of the runs or drawdowns observed, we perform a

bootstrap analysis, which assumes independence of returns. There is a close connection
46 CHAPTER 4. MOMENTUM OF EQUITY RETURNS

0.7

0.6
Series B
Transition Probability

0.5
Series A
0.4

0.3

0.2 Series C

0.1

0
1 3 5 7 9 11 13
State s

1200

1000
Number of Visits

800

600

400

200

0
1 2 3 4 5 6 7 8 9 10 11 12 13
State s

Figure 4.7: Sensitivity Analysis on Conditional Transition Probabilities. Series A is the

artificially constructed series sA
t with conditional transition probabilities of 50% (except for
the last state 11. The bottom graph shows the number of visits in each state for Series
A (1024 visits for state 1, 512 for 2, etc.) The top graph shows the conditional transition
probabilities obtained for Series A and its derivatives Series B and Series C. Series B and
Series C are obtained by adding or subtracting a visit to all states, respectively. So Series
B conditional probabilities will be calculated using (1025, 513, 257, etc.) as the series
representing the number of visits to each state.
4.4. FURTHER RESEARCH 47

between the bootstrap method and permutation tests of statistical significance and for
further references, we refer the reader to the chapter contained in the work by Efron and
Tibshirani (Efron and Tibshirani 1993). The principle involved is to reshuffle the returns
randomly from the original sample to see if we observe such runs or drawdowns in the
reshuffled process. If we do not, it implies that the assumption of independence cannot
hold for the original series of historical returns, which is precisely what we want to prove.
Figure 4.8 displays the results for 100 replications with the upper and lower limits observed
for the transition probabilities. The results are conclusive except for the DJIA drawdowns.
The S&P 500 clearly shows statistical significance for states 1 and 2 of runs and state 1 of
drawdowns. However, the most significant of all is for the Nasdaq where states 1 and 2 for
both runs and drawdowns are clearly outside of the min-max band.

4.3.3 Verification of the Markov Property

Though we do not need it to contend that equity indices display evidence of serial correla-
tion, it is interesting to study if the process st could be considered a Markov chain. Formally
what we would need to verify to make this claim is that

P (st+1 = s1 | st = s; st0 = s2 ) = P (st+1 = s1 | st = s; st0 = s3 ) (4.6)

0
for all t < t and for all (s1 , s2 , s3 ) in Z 3 .

Given the finite size of our sample, it is hard to verify this formally for all possible
histories of the process. What this property is simply saying is that the transition prob-
abilities from st to st+1 only depend on st and not on the past history of the process
(st−1 , st−2 , st−3 , . . . ). We verify this property for a memory of order 1. That is we verify
that P (st+1 = s1 | st = s; st−1 = s2 ) = P (st+1 = s1 | st = s; st−1 = s3 ) for all t and
(s1 , s2 , s3 ) in Z 3 . Since we disregard transitions from state 0 (due to their low frequencies)
and all other states only have one possible predecessor, we only need to verify this property
for states −1 and 1. Table 4.5 and Table 4.6 show the empirical results for the DJIA index
and we obtain comparable results for the S&P 500 and the Nasdaq indices.

4.4 Further Research

Further research could be done by performing the same analysis but with a different and
translated sampling scheme. For instance, we could look at the daily opening prices (instead
48 CHAPTER 4. MOMENTUM OF EQUITY RETURNS

Figure 4.8: Bootstrap analysis of runs and drawdowns. On each graph, we represent the
maximum and minimum transition probabilities obtained by our bootstrap procedure. That
is, for each time series obtained by the bootstrap, we compute the transition probabilities
as we have for the historical series (for each index). So we also represent on each graph
the transition probabilities estimated from the historical realized returns. The historical
transition probability is considered statistically significant if it lies outside of the “cone”
defined by the bootstrap min-max. The transition probabilities represented here are the
probabilities of “reversals”. For instance for the “Dow Jones/Runs” graph, given that there
has just been an upward market close (i.e. we are in state 1), the probability of having
a downward market close the next day is about 45%. This is outside of the min-max
cone defined by the bootstrap simulation and is considered statistically significant. The
probabilities of “reversals” from states 2 and 3 are not considered statistically significant
as they lie within the bootstrap min-max cone. Note that for the graphs of drawdowns, the
states are represented in absolute values. In other words, state 1 should be read as state
−1, etc.
4.4. FURTHER RESEARCH 49

st−1 st+1 = −2 st+1 = 1 st+1 = 0 Continuation Ratio

0 8 4 0 0.6667
1 483 450 4 0.5155
2 263 251 3 0.5087
3 158 154 2 0.5032
4 72 76 0 0.4865
5 40 37 0 0.5195
6 21 18 0 0.5385
7 3 10 0 0.2308
8 4 3 0 0.5714
9 1 2 0 0.3333
10 2 1 0 0.6667
11 0 1 0 0
12 0 1 0 0
13 0 1 0 0
14 0 0 0 0

Table 4.5: Verification of the Markov Property for the DJIA. Conditional transitions from
state st = −1. The left column shows the predecessor state to state −1 (possible predecessor
states are 0, 1, 2, 3, . . . ) The second column shows the number of transitions from state −1 on
to state −2. The third column shows reversal from state −1 to state 1. The fourth column
shows the number of transitions from state −1 to state 0. The last column computes the
ratios of transitions to state −2 over the total number of transitions from state −1 with the
particular predecessor state listed in the corresponding row.
50 CHAPTER 4. MOMENTUM OF EQUITY RETURNS

st−1 st+1 = 2 st+1 = −1 st+1 = 0 Continuation Ratio

-14 0 0 0 0
-13 0 0 0 0
-12 0 0 0 0
-11 1 1 0 0.5
-10 0 0 0 0
-9 1 0 0 1
-8 3 3 0 0.5
-7 5 7 0 0.4167
-6 17 15 0 0.5313
-5 41 27 0 0.6029
-4 79 61 0 0.5643
-3 140 132 4 0.5072
-2 279 227 7 0.5439
-1 549 454 5 0.5446
0 16 9 0 0.64

Table 4.6: Verification of the Markov Property for the DJIA. Conditional transitions from
state st = 1. The left column shows the predecessor state to state 1 (possible predecessor
states are . . . , −3, −2, −1, 0) The second column shows the number of transitions from
state 1 on to state 2. The third column shows reversal from state 1 to state −1. The fourth
column shows the number of transitions from state 1 to state 0. The last column computes
the ratios of transitions to state 2 over the total number of transitions from state 1 with
the particular predecessor state listed in the corresponding row.
4.4. FURTHER RESEARCH 51

of closing prices) to see if the same results hold.

Chapter 5

Impact of Serial Correlation

The subject of optimal portfolio management is obviously very much intertwined with the
more statistical subject of how to best model returns of financial assets. In this chapter,
we want to study (for the same sample of data) how a VAR modeling of returns affects
the initial allocation, as opposed to other statistical models of returns, such as GBM. As
we have mentioned before, since Markowitz’ ground breaking work on optimal portfolio
allocation and its mean-variance efficient framework (Markowitz 1952a, Markowitz 1956,
Markowitz 1987b, Markowitz 1987a), a vast body of literature has developed that uses
multi-stage models in discrete time or that continuously calculates the optimal portfolio in
continuous time (Campbell and Viceira 2002). Similarly, bearing a key interaction to the
topic of strategic asset allocation, a vast body of literature was written throughout the last
century on the topic of how to best model financial returns. We discussed this issue in the
previous chapter and pointed out that this subject is closely connected to the traditionally
controversial issue of whether financial returns can be deemed predictable or not. Even
though this latter topic of the predictability of returns has been extensively studied in the
financial community, the question of how any form of returns predictability affects portfolio
optimization has not been so extensively studied. The aim of this chapter is to address
this question in a restricted setting. Our restrictions are twofold: first, we assume that
the problem of portfolio allocation can be properly treated within a multistage stochastic
programming framework; second, we assume serial correlation of financial returns can be
properly captured by means of a vector autoregressive process.

52
5.1. REVIEW OF TRADITIONAL ASSUMPTIONS 53

5.1 Review of Traditional Assumptions

Traditionally, the most significant assumptions that need to be made by researchers for
computing dynamic portfolio allocations are: (i) how to model the investor’s goals and the
type of utility function that should be used; (ii) whether to include transaction costs or not,
and (iii) how to best model the asset class returns involved in the problem.

5.1.1 Utility Function

As is assumed in Merton’s early work (Merton 1969, Merton 1990), the utility function
used is often assumed to exhibit either constant relative risk aversion or constant absolute
risk aversion. However, this assumption while preserving the tractability of the allocation
problem and enabling closed-form solutions to be determined is not always justified from a
practical standpoint. In our approach, this assumption is relaxed. We consider a strictly
concave piecewise linear utility function whose kinks correspond to explicit goals formulated
by the decision-maker, at a prefixed horizon.

5.1.2 Transaction Costs

Taking transaction costs into account can be deemed a key reason for choosing a stochas-
tic programming approach over a dynamic programming methodology. Within a dynamic
programming framework, the introduction of transaction costs significantly increases the
dimension of the state space. It is worth noticing that transaction costs have been histori-
cally decreasing. However, in the case of certain applications (e.g. a fund of funds investing
in different hedge funds), they remain quite significant (e.g. some investment funds have
significant fees when funds are withdrawn by investors and may also have upfront fixed fees
when funds are invested). Thus, it is hard to a priori dismiss transaction costs and our
approach seems all the more valuable in this case.

5.1.3 Stochastic Modeling of Underlying Assets

The third type of assumption needed is on choosing the best stochastic representation of
asset class returns. We refer the reader to the appendix on historical developments of
mathematical finance where the topic is reviewed.
54 CHAPTER 5. IMPACT OF SERIAL CORRELATION

Summary Statistic Stock Govt Bonds Cash

Yearly Arithmetic Average 12.34% 6.09% 4.79%
Yearly Standard Deviation 16.47% 9.24% 3.05%
Monthly Arithmetic Average 0.97% 0.48% 0.39%
Monthly Standard Deviation 4.12% 2.05% 0.25%

Table 5.1: Data Summary Statistics.

5.1.4 Scope and Organization of the Chapter

Our goal in this chapter is not to take a position on this controversial discussion of if and to
what extent markets can be deemed efficient and predictable, vast subjects that would go
significantly beyond the scope of this chapter. Rather, we focus on two fundamental issues:
first, can we detect any kind of serial correlation in financial returns by using a vector
autoregressive framework; second, we want to measure the various impacts that different
statistical fittings of the same data can have on optimal asset allocation decisions.
To perform our analysis, we focus mainly on two particular cases: (a) treating financial
returns as if they were independent and identically distributed (“i.i.d.”) and best described
by geometric Brownian motion (“GBM”) and (b) treating returns using a VAR model.
Hence in the first case, we model asset returns as i.i.d. random variables and fit the latest
historical data in a multivariate GBM framework. In the second case, we fit the same data
in a VAR framework. We can them compare the two sets of results and analyze the impact
of serial correlation on the optimal initial asset allocation.
The chapter is organized as follows. First, we describe the data and the estimation
process used for both our GBM and VAR models. Second, we present our stochastic pro-
gramming framework and its implementation details. Third, we report and discuss our
results. Last, we emphasize the conclusions of this investigation.

5.2 Asset Returns Data and Model Estimates

5.2.1 Data

Table 5.2.1 provides a summary of the yearly and monthly data used, spanning from
12/31/1946 to 12/31/2003. This includes value-weighted returns including dividends for
the NYSE stock index, yearly returns for the 10-year government bonds and yearly returns
for the 30-day treasury bills (assumed to be a good proxy for cash). This data was obtained
from the Center for Research in Security Prices database.
5.2. ASSET RETURNS DATA AND MODEL ESTIMATES 55

5.2.2 VAR Model

The vector autoregressive (”VAR”) model posits a set of relationships between past lagged
values of all variables in the model and the current value of each variable in the model.
An introduction to this type of model can be found in the work by Chatfield (Chatfield
1996) and a more complete description in the works by Hamilton or Judge et al. (Hamilton
1994, Judge, Hill, Griffiths, Lutkepohl, and Lee 1988).
A compact way of writing a VAR model is:

yt = c + B1 yt−1 + · · · + Bp yt−p + ²t , (5.1)

where yt , yt−1 , · · · , yt−p are n-dimensional vectors of the time series yt and all its lagged
values, up to order p. ct represents a vector of constants and ²t an n-dimensional vector of
independent disturbances.
So for a VAR model of lag 1, we get in expanded form:
       
y1t c1   y1,t−1 ²1t
    b ... b1n    
 y2t   c2   11    ²2t 
 =   y2,t−1 + .
 ...   ...  +  ... ... ...   ...   ...  (5.2)
       
bn1 ... bnn
ynt cn yn,t−1 ²nt

Using lag operators, another completely equivalent way of writing a VAR model is:
       
y1t c1   y1t ²1t
    A (l) ... A1n (l)    
 y2t   c2   11   y2t   ²2t 
 =  ...   + ,
 ...   ...  +  ... ...  ...   ...  (5.3)
       
An1 (l) ... Ann (l)
ynt cn ynt ²nt
Pp k,
with the same notations as before and introducing Aij (l) such that Aij (l) = k=1 aij l
where l is the lag operator defined by lk (yt ) = yt−k and p is the lag length specified by the
modeler.

5.2.3 VAR Model Estimation and Choice of Lag

It can be showed that fitting a VAR model by a maximum likelihood procedure amounts
to performing a set of OLS1 regressions. For calculation details, we refer the reader to the
work by Hamilton (Hamilton 1994). We conducted all subsequent estimations in MATLAB,
1
We use here the common acronym for Ordinary Least Squares (”OLS”).
56 CHAPTER 5. IMPACT OF SERIAL CORRELATION

using both the standard statistics package and an econometrics package developed by LeSage
(LeSage 1999). For further references on MATLAB and its use for numerical applications
in finance, the reader can also consult the work by Brandimarte (Brandimarte 2002).
An important issue for estimating a VAR model is choosing the time lag. We summa-
rize below the procedure for doing so and detail some potential issues with this approach.
Subsequently, we provide our VAR coefficient estimates as well as their related regression
statistics.

Choosing the Time Lag

For determining the lag length, we follow the commonly used approach outlined by LeSage
(LeSage 1999) that performs statistical tests of models with various lag lengths. As LeSage
puts it:

The longer lag models are viewed as unrestricted models in contrast to the
shorter lag models, and a likelihood ratio statistic is constructed to test for the
significance of imposing the restrictions. If the restrictions are associated with
a statistically significant degradation in model fit, we conclude that the longer
lag length model is more appropriate, rejecting the shorter lag model.

The test statistic used

LR = (T − c)(log |Σr | − log |Σu |) (5.4)

is chi-squared distributed with degrees of freedom equal to the number of restrictions im-
posed. T is the number of observations and c is a correction factor for the degrees of freedom
proposed by Sims, which is the number of variables in each unrestricted equation of the
VAR model (Sims 1980). |Σr | and |Σu | respectively denote the determinants of the error
covariance matrices from the restricted and unrestricted models.
Table 5.2 shows the likelihood ratio statistics for some sample data. This example shows
the likelihood ratios as well as the marginal probability levels. Depending on the confidence
level chosen (whether 95% or 99%), we would respectively choose a model of order 7 (that
is with all lags up to, and including, the lag of order 7) or a model of order 4.
One of the problems with this procedure is that we may choose significantly different lags
depending on the historical sample used. We show our results carrying out this procedure
for both monthly and yearly data using a “rolling historical sample”.
In Figure 5.1, each historical sample is made up of 120 monthly returns. The full sample
of 696 monthly returns is covered as we roll the historical sample.2 The graph shows the
2
We use interchangeably the term “historical window” or “historical sample”.
5.2. ASSET RETURNS DATA AND MODEL ESTIMATES 57

Lags Compared LR Stat. Probability

10-9 15.4443 0.0794
9-8 8.1336 0.5207
8-7 13.0347 0.1610
7−6 20.4897 0.0151
6-5 15.7754 0.0717
5-4 6.7818 0.6598
4−3 30.0381 0.0004
... ... ...

Table 5.2: Lag Analysis. This table shows the sequential procedure of determining the
“statistically optimal” lag order by starting from a maximum lag order (specified by the
modeler, which is 10 in this example) and implementing successive likelihood ratio tests
down to a minimum order (also specified by the modeler, which would be 1 in this case).
By comparing successively two consecutive lag orders, the modeler can stop the procedure
depending on the confidence level chosen. In this case, if the desired confidence level is 95%,
we would decide in favor of a model of order 7 (as the first time the marginal probability
level is less than 1-95%=5% is for 1.51%, when comparing orders 7 vs. 6). If the desired
confidence level is 99%, we would continue the sequential tests until we find a marginal
probability level less than 1% (hence, in this case, we would decide in favor of a model of
order 4).

4
Lag Chosen

1955 1965 1975 1985 1995 2003

Rolling Historical Sample

Figure 5.1: Variations of lag choices with respect to a “rolling historical sample”. Each
sample is composed of 120 consecutive monthly returns. The date indicated corresponds
to the ending date of the sample. The full sample of returns ranges from January 1946 to
December 2003. Hence the full sample is comprised of 696 monthly observations.
58 CHAPTER 5. IMPACT OF SERIAL CORRELATION

4
Chosen Lag

1975 1985 1995 2002

Rolling Historical Sample

Figure 5.2: Variations of lag choices with respect to a “rolling” historical sample”. Each
historical sample is composed of 30 consecutive yearly returns. The date indicated corre-
sponds to the ending date of the sample. The full sample of returns ranges from 1946 to
2002. Hence the full sample is comprised of 57 yearly observations.

variations of the lag orders that would be chosen by our procedure, starting with a maximum
permissible lag of 5 down to a minimum lag of 1, and a confidence level of 99%. The graph
shows that the lag order chosen varies a lot from one sample to another. Hence we settle for
a lag order of 1. This allows us to fit a much smaller number of coefficients and diminishes
the risk of overfitting.

Figure 5.2 provides the same analysis for yearly returns using a historical window size
of 30 years, a maximum permissible lag of 5 (minimum lag being 1) and a confidence level
of 99%. Similarly, the full sample of 57 yearly returns is covered by rolling the historical
window. The variations are as significant as in the monthly case and justify (as previously
with monthly data) our choice of a lag 1 model.

5.2.4 Coefficient Estimations

We provide here the coefficient estimates for both yearly and monthly data, using rolling
historical windows. What our results underline are the strong variations in the estimated
statistical significance of the coefficients depending on the historical sample used.
5.2. ASSET RETURNS DATA AND MODEL ESTIMATES 59

Stock Coefficients

Figure 5.3 shows the t-statistics for the stock coefficients. The graph shows that the co-
efficients used by the latest historical windows do not seem statistically significant (using
2.0 as the cutoff value for the absolute value of the t-statistic). Furthermore this graph
shows a progressive degradation of the statistical significance (if any) of these coefficients.
A possible interpretation for this trend is that throughout the sampled years (i.e. from 1946
to 2002), there has been an increase in market efficiency so that any form of stock returns
predictability observed in the earlier part of our sample has slowly disappeared. Figure 5.3
graphs the lagged coefficients obtained for the stock returns.
We also include coefficient t-statistics for monthly returns for historical sample sizes of
30 months3 (Figure 5.4) and 60 months (Figure 5.7).

Bond Coefficients

Figure 5.8 underlines the strong statistical significance of the cash returns lagged values
for predicting next year’s bond returns, as well as the autocorrelation of bond returns
from one year to the next (though this effect is less clear). This is consistent with the
traditional models of the yield curve where inter-temporal variations of the short-term
bond returns induces serial correlation for longer-term bond returns. In other words, our
estimates below seem to be consistent with the traditionally assumed persistence of the
yield curve. Figure 5.9 graphs the lagged coefficients obtained for the bond yearly returns.
Figure 5.10 and Figure 5.13 provide the t-statistics for monthly returns.

Cash Coefficients

Figure 5.14 shows we find the same statistical significance for the lagged-1 10-yr bond and
cash returns on the current cash returns, as expected for the reason previously cited for
bond returns. However, another surprising finding is the significance of the stock market
lagged value for predicting the current cash returns. Intuitively one can understand that
there would be a certain historical relationship in how the stock market does one year and
the cash returns (assumed here to be equivalent to the returns on the 30-day treasury bills).
Figure 5.15 underlines the fact that the autocorrelation of cash returns is significantly more
important than the other effects.
Figure 5.16 and Figure 5.19 provide the t-statistics for monthly returns.
3
We wanted to have the same number of data points as in the yearly case.
60 CHAPTER 5. IMPACT OF SERIAL CORRELATION

3
Constant
2.5

1.5 Bond

1
Coefficient t−statistic

0.5

−0.5

−1

Stock
−1.5
Cash

−2

−2.5

1975 1985 1995 2002

Historical Sample

1 Bond
Constant

0
Coefficient Values

Stock
−1

−2
Cash

−3

−4

1975 1985 1995 2002

Historical Sample

Figure 5.3: t-statistics and coefficient values for stock yearly returns. Each historical sample
is composed of 30 consecutive yearly returns. The date indicated corresponds to the ending
date of the sample. The full sample of returns ranges from 1946 to 2002. Hence the full
sample is comprised of 57 yearly observations.
5.2. ASSET RETURNS DATA AND MODEL ESTIMATES 61

4
Bond Cash

2
Coefficient t−statistic

−1

−2

−3 Stock
Constant

−4
1950 1960 1970 1980 1990 2000
Historical Sample

Figure 5.4: t-statistics of lagged coefficients on stock monthly returns. Each sample is
composed of 30 consecutive monthly returns. The date indicated corresponds to the ending
date of the sample. The full sample of returns ranges from January 1946 to December 2003.
Hence the full sample is comprised of 696 monthly observations.

Constant

1
Coefficient Values

Stock
−1

−2 Bond

−3

1950 1960 1970 1980 1990 2000

Historical Sample

Figure 5.5: Values of lagged coefficients on stock monthly returns. This shows the values
of the lags on stock and bond returns, as well as the constant. The cash coefficient value is
shown separately. Each sample is composed of 30 consecutive monthly returns. The date
indicated corresponds to the ending date of the sample. The full sample of returns ranges
from January 1946 to December 2003.
62 CHAPTER 5. IMPACT OF SERIAL CORRELATION

120

100

80

60

40
Coefficient Value

Cash
20

−20

−40

−60

−80
1950 1960 1970 1980 1990 2000
Historical Sample

Figure 5.6: Value of lagged coefficient for cash on stock monthly returns. Each sample is
composed of 30 consecutive monthly returns. The date indicated corresponds to the ending
date of the sample. The full sample of returns ranges from January 1946 to December 2003.

4 Constant
Bond
3 Cash

2
Coefficient t−statistic

−1

−2

−3
Stock
−4

1950 1960 1970 1980 1990 2000

Historical Sample

Figure 5.7: t-statistics of lagged coefficients on stock monthly returns. Each sample is
composed of 60 consecutive monthly returns. The date indicated corresponds to the ending
date of the sample. The full sample of returns ranges from January 1946 to December 2003.
Hence the full sample is comprised of 696 monthly observations.
5.2. ASSET RETURNS DATA AND MODEL ESTIMATES 63

3
Cash

2
Coefficient t−statistic

1
Constant
Stock

−1

Bond

−2

1975 1985 1995 2002

Historical Sample

Figure 5.8: t-statistics for lagged coefficients on bond yearly returns. Each historical sample
is composed of 30 consecutive yearly returns. The date indicated corresponds to the ending
date of the sample. The full sample of returns ranges from 1946 to 2002. Hence the full
sample is comprised of 57 yearly observations.

2.5

Cash
1.5
Coefficient Values

0.5
Stock Constant

Bond
−0.5

1975 1985 1995 2002

Historical Sample

Figure 5.9: Values of lagged coefficients for bond yearly returns. Each historical sample is
composed of 30 consecutive yearly returns. The date indicated corresponds to the ending
date of the sample. The full sample of returns ranges from 1946 to 2002. Hence the full
sample is comprised of 57 yearly observations.
64 CHAPTER 5. IMPACT OF SERIAL CORRELATION

3 Bond Cash

1
Coefficient t−statistic

−1

−2

−3
Constant

−4 Stock

−5
1950 1960 1970 1980 1990 2000
Historical Sample

Figure 5.10: t-statistics for lagged coefficients on bond monthly returns. Each sample is
composed of 30 consecutive monthly returns. The date indicated corresponds to the ending
date of the sample. The full sample of returns ranges from January 1946 to December 2003.
Hence the full sample is comprised of 696 monthly observations.

0.8

Constant
0.6
Bond

0.4

0.2
Coefficient Values

−0.2

−0.4

Stock

−0.6

−0.8
1950 1960 1970 1980 1990 2000
Historical Sample

Figure 5.11: Values of lagged coefficients on bond monthly returns. This shows the values
of the lags on stock and bond returns, as well as the constant. The cash coefficient value is
shown separately. Each sample is composed of 30 consecutive monthly returns. The date
indicated corresponds to the ending date of the sample. The full sample of returns ranges
from January 1946 to December 2003.
5.2. ASSET RETURNS DATA AND MODEL ESTIMATES 65

25

20

15
Cash

10
Coefficient Value

−5

−10

−15

−20

1950 1960 1970 1980 1990 2000

Historical Sample

Figure 5.12: Value of lagged coefficient for cash on bond monthly returns. Each sample is
composed of 30 consecutive monthly returns. The date indicated corresponds to the ending
date of the sample. The full sample of returns ranges from January 1946 to December 2003.

3
Bond
Cash
2

1
Coefficient t−statistic

−1

−2

Constant
−3
Stock

−4

1950 1960 1970 1980 1990 2000

Historical Sample

Figure 5.13: t-statistics for lagged coefficients on bond monthly returns. Each sample is
composed of 60 consecutive monthly returns. The date indicated corresponds to the ending
date of the sample. The full sample of returns ranges from January 1946 to December 2003.
Hence the full sample is comprised of 696 monthly observations.
66 CHAPTER 5. IMPACT OF SERIAL CORRELATION

14

12 Cash

10

8
Coefficient t−statistic

4
Stock

0 Constant

−2
Bond

−4
1975 1985 1995 2002
Historical Sample

Figure 5.14: t-statistics for lagged coefficients on cash yearly returns. Each historical sample
is composed of 30 consecutive yearly returns. The date indicated corresponds to the ending
date of the sample. The full sample of returns ranges from 1946 to 2002. Hence the full
sample is comprised of 57 yearly observations.

1.2

Cash
0.8
Coefficient Values

0.6

0.4

0.2 Stock Constant

−0.2 Bond

1975 1985 1995 2002

Historical Sample

Figure 5.15: Values of lagged coefficients for cash yearly returns. Each historical sample is
composed of 30 consecutive yearly returns. The date indicated corresponds to the ending
date of the sample. The full sample of returns ranges from 1946 to 2002. Hence the full
sample is comprised of 57 yearly observations.
5.2. ASSET RETURNS DATA AND MODEL ESTIMATES 67

25

20

Cash Constant
15
Coefficient t−statistic

10

−2
Bond
Stock
−5
1950 1960 1970 1980 1990 2000
Historical Sample

Figure 5.16: t-statistics for lagged coefficients on cash monthly returns. Each sample is
composed of 30 consecutive monthly returns. The date indicated corresponds to the ending
date of the sample. The full sample of returns ranges from January 1946 to December 2003.
Hence the full sample is comprised of 696 monthly observations.

0.03

0.02
Bond

Stock
0.01 Constant
Coefficient Values

−0.01

−0.02

−0.03

1950 1960 1970 1980 1990 2000

Historical Sample

Figure 5.17: Values of lagged coefficients on cash monthly returns. This shows the values
of the lags on stock and bond returns, as well as the constant. The cash coefficient value is
shown separately. Each sample is composed of 30 consecutive monthly returns. The date
indicated corresponds to the ending date of the sample. The full sample of returns ranges
from January 1946 to December 2003.
68 CHAPTER 5. IMPACT OF SERIAL CORRELATION

1.2

0.8

0.6
Coefficient Value

0.4

0.2

0
Cash

−0.2

−0.4
1950 1960 1970 1980 1990 2000
Historical Sample

Figure 5.18: Value of lagged coefficient for cash on cash monthly returns. This shows the
value of the lag on cash. Each sample is composed of 30 consecutive monthly returns. The
date indicated corresponds to the ending date of the sample. The full sample of returns
ranges from January 1946 to December 2003.

30

25

Cash

20
Coefficient t−statistic

15

10

Stock
5
Constant

−2

Bond
−5
1950 1960 1970 1980 1990 2000
Historical Sample

Figure 5.19: t-statistics for lagged coefficients on cash monthly returns. Each sample is
composed of 60 consecutive monthly returns. The date indicated corresponds to the ending
date of the sample. The full sample of returns ranges from January 1946 to December 2003.
Hence the full sample is comprised of 696 monthly observations.
5.3. MODEL AND IMPLEMENTATION 69

Figure 5.20: Scenario Tree

5.2.5 Bayesian Variants

A VAR model of higher order has empirical limitations that come from the large number
of coefficients that usually need to be estimated. Large samples of observations involving
time series that cover many years are usually needed to estimate the VAR model, and these
are not always available. In addition there’s a inherent tension between the large sample
size needed to obtain more accurate estimates and the decreasing relevance of historical
data as it is further distant in the past. Moreover, the independent variables represent
lagged values, which tend to produce high correlations that lead to degraded precision in
the parameter estimates. It is to overcome these problems that variants of VAR models
using Bayesian prior information were introduced (Doan, Litterman, and Sims 1984). We
address these latter models in the next chapter.

5.3 Model and Implementation

5.3.1 Our Stochastic Programming Framework
This formulation involves a sequence of decisions that react to outcomes (i.e. the asset class
returns) that evolve over time.
We use a multi-stage stochastic linear programming framework as defined in (Infanger
1993). In general, multi-stage stochastic linear programs can be extremely large as the
number of scenarios grows exponentially with the number of stages. Thus, we are limited
in our application to a small number of stages. To deal with transaction costs, we have to
70 CHAPTER 5. IMPACT OF SERIAL CORRELATION

introduce additional variables that represent how much of each asset class we have bought
or sold.

The initial equation is:

n−1
X
xi0 (1 + btc i ) + xcash
0 = W0 , (5.5)
i=1

where W0 is the initial wealth available, n − 1 is the number of asset classes other than
money markets4 and btc i are the transaction costs for investing in asset class i.5

Between time t − 1 and t, we have the following equations:

i,ω ,...,ωt−1
xi,ω
t
1 ,...,ωt
= xt−11 Rti,ωt + yti,ω1 ,...,ωt − zti,ω1 ,...,ωt , i = 1...n − 1,
n−1
X n−1
X
i,ω ,...,ωt−1
xtcash,,ω1 ,...,ωt = xcash
1
Rti,ωt − yti,ω1 ,...,ωt (1 + btc i ) + zti,ω1 ,...,ωt (1 − stc i ),
i=1 i=1

where yti,ω1 ,...,ωt is the amount of asset class i bought in scenario (ω1 , ..., ωt ) and zti is the
amount sold in scenario t-tuple (ω1 , ..., ωt ). The random variable Rti,ωt is the random return
of asset class i between times t − 1 and t drawn in scenario ωt .

For the last period, we get:

n−1
X i,ω ,...,ωT −1 cash,ω1 ,...,wT −1
1
xT −1 RTi,ωT (1 − stc i ) + xT −1 RTcash,ωT = WTω1 ,...,ωT , (5.6)
i=1

where stc i are the transaction costs for selling out of the asset class i and WTω1 ,...ωT is the
terminal wealth thereby obtained in the scenario T-tuple (ω1 , ..., ωT ). The utility function
is represented by slack variables uωT 1 ,...,ωT for being above the final goal G and vTw1 ,...,ωT for
being short of the goal, so that:

uωT 1 ,...,ωT − vTω1 ,...,ωT = WTω1 ,...,ωT − G. (5.7)

G is the desired goal.

4
We assume there are no transaction costs for transferring money in and out of money markets and treat
money markets as equivalent to ”cash”, which is our nth asset class.
5
In our model, the transaction costs are assumed to be proportional to the amount invested in the asset
class.
5.3. MODEL AND IMPLEMENTATION 71

The final objective, Z is:

P ω1 ,...,ωT
(ω1 ,...,ωT ) ∈ Ω1 ×...×Ω1 [uT − ψ vTω1 ,...,ωT ]
Z= . (5.8)
[|Ω1 | × ... × |ΩT |]

In equation (5.8), ψ is the slope of the utility function below the goal G. It is a penalty
factor for being short of this desired final objective.

5.3.2 Scenario Generation

There is a significant number of generally accepted methods for generating scenarios, from
bootstrapping historical data to sampling from continuous distributions to obtain a pre-
sampled problem, which is then solved as a substitute for the original problem.
The specification of the vector autoregressive model should be chosen carefully. Al-
though some inter-temporal relationships between the returns might be weakly significant
based on historical data as the previous section points out, that does not imply that these
relationships are also useful for generating scenarios for a financial optimization model with
a long term horizon. To avoid any problems with unstable and spurious predictability of
returns, some authors avoid using lagged variables for explaining the returns of stocks or
other asset classes in the vector autoregressive model and reserve it for obvious categories
where the time series clearly exhibit some memory and serial correlation. For instance
Boender in his Asset-Liability Management (”ALM”) simulation system for Dutch pension
funds (Boender 1997) only uses a first-order autoregressive process for modeling the returns
on deposits and the variations in wage levels, and treats separately the returns on stock,
bond and real estate returns included in the simulation.

5.3.3 Theoretical Considerations

It is now commonly accepted within the stochastic programming/operations research com-

munity that decomposition algorithms are not so useful when used for solving LPs in ex-
tensive form because standard LP optimizers have grown in power. Thus, when dealing
with a stochastic program where the uncertain parameters are assumed to have continous
distributions, a pre-sampling strategy achieved by plain Monte Carlo sampling should work
fine. However, a theoretical issue that needs to be carefully addressed is to what extent a
pre-sampling strategy leads to a solution similar to the one of the original problem. We
know from theory that as the sample size goes to infinity, the solution of the pre-sample
stochastic program epi-converges to the solution of the original problem. For finite sample
72 CHAPTER 5. IMPACT OF SERIAL CORRELATION

sizes, we resort to bounds on the objective to establish how close the objective of the solved
problem is to the objective of the original problem.
There are also other theoretical issues raised by our framework of which we need to be
aware. Firstly, suppose we prefix a certain number of reallocation points. Unless we inves-
tigate numerically the pertinence of our prefixed distribution of the reallocation points, we
may restrict ourselves to sub-optimal allocation policies (with respect to the performance of
other asset allocation policies with the same number of stages but a different temporal dis-
tribution). Secondly, if we avoid prefixing these reallocation points but decide to optimize
under the sole constraint of having a maximum number of reallocation points (whose time
distribution is left a priori unconstrained), we should investigate if we can find a ”rebal-
ancing rule” (formulated for instance as an optimal stopping problem) that would trigger
a reallocation decision. Thirdly, suppose we have no constraint at all on both the number
and timing of reallocation points, we should compare the performances of our restricted
setting’s solution (where we have a prefixed, in number and temporal distribution, of re-
allocation stages) and the performance of the unconstrained solution (performance which
will necessarily be at least as good as ours as it is unconstrained).6

5.3.4 DECIS Implementation

We use GAMS7 and DECIS to solve our problem. DECIS (Infanger 1997) is a system
for solving large-scale stochastic programs that can use Benders decomposition and Monte
Carlo simulation with importance sampling or control variates as variance reduction tech-
niques. Hence DECIS includes a variety of solution strategies and can solve problems with
numerous stochastic parameters. For solving master and subproblems, DECIS interfaces
with MINOS (Murtagh and Saunders 1983) or CPLEX (CPLEX Optimization 1989).
As we are considering long-term horizon, we set up a 4-stage piecewise linear model.
We then use a two-stage decomposition approach to solve the problem and a pre-sampling
strategy. Hence we sample a certain number of times from our VAR process and then
proceed to solve this pre-sampled stochastic program). We do not address here the potential
issue previously noted on the quality of the solution of the pre-sampled problem relative to
the solution of the original problem. Further references on this issue can be found in the
work by Infanger (Infanger 1999).
6
In the general case, these questions of optimal timing of reallocation points are complicated from an
analytical standpoint and we do not address them in this thesis, nor do we report any numerical trial runs
on these issues.
7
The General Algebraic Modeling System (GAMS) is specifically designed for modeling linear, nonlinear
and mixed-integer optimization problems. Further references can be found at http://www.gams.com.
5.3. MODEL AND IMPLEMENTATION 73

Number of Scenarios GAMS+CPLEX DECIS+GAMS+CPLEX Time Reduction

40,000 574.8 234.6 59.2%
60,000 1,046.3 386.8 63.0%
80,000 2,013.0 472.9 76.5%
100,000 3,493.9 706.3 79.8%
120,000 6,555.5 1,659.6 74.7%

Table 5.3: Analysis of Computation Time (sec.)

5.3.5 Advantages of using DECIS and Computational Time

It is important to understand what advantages there are to using DECIS and a decompo-
sition method in solving the large-scale LP that is the result of our pre-sampling strategy.
For a general discussion of the advantages of decomposition techniques, we refer the reader
to the treatment by Ruszcyński and Shapiro (Ruszcyński and Shapiro 2003). There are two
issues at hand here: (i) memory requirements (that is directly related to the scale of the
LP we’re trying to solve) and (ii) computational time.

Of the first issue, i.e. memory usage, it is worth noticing that in a decomposition
approach, the ”basis” is decoupled. Consequently, the memory requirements for any basis
factorization are reduced. The approach is similar to an iterative process for solving linear
equations such as a block Gauss-Siedel, as opposed to a direct method (Strang 1988). For
some problems, the memory requirements to solve the problem directly may exceed the
memory available. By decomposing the larger linear program into smaller subproblems, the
subsequent memory needed in the optimization process is reduced.

On the second issue of computational time, it can be noted that if we focus here on
a 3-year time horizon, there is little need to worry about computational time as long as
the solution time is not more than a few hours. However, exactly the same programming
framework may be applied to daily data for much shorter horizons (e.g. three days instead
of three years). In this case, computational time is important. Also, it is interesting in
its own sake to compare computational times with or without using DECIS. If we assume
that computational time is more or less linear with the number of elementary operations
performed in the optimization process, then computational time can be used as a first-order
proxy for DECIS contribution to reducing the problem complexity. Table 5.3 summarizes
the computational time results.
74 CHAPTER 5. IMPACT OF SERIAL CORRELATION

Model (Sample size in yrs) Stock Bond Cash

VAR (20) 18.56% 8.41% 1.29%
GBM (20) 14.16% 8.74% 3.11%
VAR (30) 15.52% 8.37% 1.25%
GBM (30) 12.93% 8.44% 3.30%
VAR (40) 14.29% 7.53% 0.99%
GBM (40) 13.46% 8.03% 1.92%
VAR (50) 16.36% 6.77% 0.86%
GBM (50) 15.92% 7.69% 1.17%

Table 5.4: Mean absolute deviations for 1-yr ahead forecasted vs. realized returns. The
historical sample size (in years) is indicated in parentheses.

5.4 Results

5.4.1 Out-of-Sample Forecasting

We compare the out-of-sample forecasts we obtain by using VAR or GBM modeling of

returns. Table 5.4 shows forecasts for both GBM and VAR models of returns8 in the case
of yearly data.9
These results show that the VAR forecasts are slightly better than GBM forecasts for the
bond returns and significantly better for cash returns. However, the one-year ahead forecasts
are better with GBM model for stock returns.10 Also, these results show empirically that
there is a historical window size that minimizes the mean absolute deviations between the
one-year ahead forecasts and the realized returns.

5.4.2 Results Variations with respect to Final Goal

Our model’s end result is the optimal recommendation for wealth allocation between asset
classes at the time the model is run. We first show results in the case of a multivariate
GBM fitting of the full sample of yearly returns and the effects of varying the end goal value.
Table 5.5 displays our results for a pre-sampled scenario tree comprised of 70x(60x50) =
210,000 scenarios. This produced a large-scale LP comprised of 3063 rows, 6187 columns
and 21370 non-zeros elements. Table 5.5 shows how the optimal allocation varies with re-
spect to the final goal.
8
The VAR model is initialized with historical sample’s last year returns.
9
The same results can be made available for monthly data.
10
This was to be expected as our VAR fitting is not statistically significant for stock returns and GBM
model is better for this asset class.
5.4. RESULTS 75

Final Goal Stock Bond Cash Objective

105 82.5% 17.5% 33.9
110 70.3% 16.5% 13.2% 22.0
115 66.6% 14.1% 19.3% 7.5
120 71.9% 20.3% 7.8% -11.1
125 79.4% 20.6% -34.1
130 86.4% 13.6% -61.2

Table 5.5: Allocation results for GBM model with varying goal.

As Table 5.5 shows, the objective value decreases as the goal is increased, as was ex-
pected.11 Also we can notice that cash is not used if the final goal is either too low or too
high. An interpretation for this result is to say that if we are significantly above the goal,
we can take more risk without fearing to be penalized, and if we are significantly below the
goal, we need to take on more risk to achieve the goal, which translates in both cases into
avoiding cash and increasing the stock allocation. Another interpretation for these results
is that if we are either significantly above or under the goal, the problem almost reduces to
maximizing an expected value over a linear function (which is equivalent to dealing with a
risk-neutral investor) and hence we would choose an asset mix that maximizes the expected
value without concern for the increased risk. Hence, far from the goal, we should expect
the stock allocation to be increased, as is the case with our results.

5.4.3 Comparisons between GBM and VAR results

Table 5.6 and Table 5.8 show the variations of the results with respect to whether we model
asset returns under a GBM framework or a VAR model with different initial conditions
(“IC”).12 The initial conditions represent the state of the system at the time the opti-
mization is conducted. So, initial conditions are the last returns available from the data
sample at the time of optimization. However, for research purposes and for the sake of
stability, it can be argued that initial conditions should be chosen as the average returns of
the historical sample used in the analysis.
Tables 5.7 and 5.9 detail the values of the initial conditions used for our analysis. In
Table 5.7, the first set of initial conditions (IC1) corresponds to each asset class initial lagged
value set equal to the historical sample’s mean return. The following initial conditions listed
11
These results were obtained for an initial wealth of 100.
12
This set of results was done on monthly data from a historical window composed of the last 120 months
available from the sample.
76 CHAPTER 5. IMPACT OF SERIAL CORRELATION

Model Stock Bond Cash Objective

GBM 24.6% 49.3% 26.1% -6.9
IC1 31.7% 52.4% 15.9% -6.8
IC2 30.1% 17.1% 52.8% -7.6
IC3 28.6% 71.4% -5.9
IC4 35.3% 62.9% 1.8% -6.1
IC5 25.5% 32.2% 42.3% -7.6
IC6 28.7% 51.0% 20.3% -3.7
IC7 24.4% 32.7% 42.8% -9.6

Table 5.6: Allocation results when VAR initial conditions are centered around sample av-
erages.

Initial Conditions Stock Bond Cash

IC1 0.96% 0.54% 0.34%
IC2 5.06% 0.54% 0.34%
IC3 -3.14% 0.54% 0.34%
IC4 0.96% 2.63% 0.34%
IC5 0.96% -1.55% 0.34%
IC6 0.96% 0.54% 0.47%
IC7 0.96% 0.54% 0.21%

Table 5.7: Values of Initial Conditions

Final Goal Stock Bond Cash Objective Value

GBM 24.6% 49.3% 26.1% - 6.9
IC8 23.2% 76.8% -12.1
IC9 26.8% 73.2% -11.9
IC10 19.4% 14.8% 65.8% -12.1

Table 5.8: Allocation results for VAR initial conditions centered around sample’s last re-
turns.

Initial Conditions Stock Bond Cash

IC8 5.21% 1.03% 0.08%
IC8 9.31% 1.03% 0.08%
IC10 1.11% 1.03% 0.08%

Table 5.9: Values for Initial Conditions

5.4. RESULTS 77

(IC2 to IC7) correspond to the same values with the exception that each asset class initial
return is varied (one after another) by +/- one standard deviation.
In Table 5.9, the first set of initial conditions (IC8) corresponds to each asset class
initial lagged value set equal to the returns of the historical sample’s last period. IC9 and
IC10 respectively correspond to increasing the stock returns initial conditions by +/- one
standard deviation.
We can see from these results that there is a much greater difference in optimal allocation
results than in optimal values of the approximated stochastic programs. For instance, Table
5.8 shows that the difference between GBM and VAR IC1 optimal values is much less
than 1.5% whereas the stock allocations vary by more than 22%. This is in line with the
observation by Dupačová (Dupačová 1999) that:
. . . in general, it is much easier to estimate the precision of the obtained optimal
value than of optimal solutions.
We can also observe that, if the sum of the allocations to bond and cash asset classes is
relatively stable, the allocation between the two asset classes is not and varies greatly. For
instance by looking at the VAR results for IC1, IC2 and IC3, the sum of bond and cash
allocations is between 68 to 72%. However, it is almost as if the bond and cash allocations
respectively at 52.4% and 15.9% in the case of IC1 were swapped for IC2, for which they
respectively become 17.1% and 52.8%.
The results also suggest that by using initial conditions for the VAR simulations that are
exactly or close to the historical sample averages, we find less of a discrepancy between the
GBM and VAR cases, with optimal values much closer than when using initial conditions
equal to the sample’s last year returns. In all cases, the distribution between bond and cash
is highly sensitive to the initial conditions used for simulating the VAR process.

5.4.4 Analysis of Nonconvexity

An important observation in our analysis is the fact that the allocation results are not
convex with respect to the initial conditions.
For instance, IC1 represents initial conditions equal to average values of the historical
sample returns. We could expect the optimal allocation produced by this set of initial values
to be somewhat between the optimal allocations produced by the sets of initial conditions
IC2 and IC3, as IC2 and IC3 are only different from IC1 in the value of the initial stock lag
(set at +/- one standard deviation from the mean). But we notice that it is not the case
for the stock allocation as IC1 produces a recommendation of 31.7%, which is outside the
[28.6%-30.1%] range given by IC3-IC2.
78 CHAPTER 5. IMPACT OF SERIAL CORRELATION

Stock Bond
Returns in Good Economy 1.20% 1.13%
Returns in Bad Economy 1.06% 1.07%
Optimal Allocation 28.6 % 71.4 %

Table 5.10: First Problem

Stock Bond
Returns in Good Economy 1.10% 1.12%
Returns in Bad Economy 0.96% 1.06%
Optimal Allocation 0% 100%

Table 5.11: Second Problem

This somewhat counterintuitive result can be better understood if we consider, without

loss of generality, a simpler example for a one period model with two asset classes.

Simple Example Let’s consider the following one-period problem. We have two asset
classes (e.g. stock and bond) and a two-piece linear concave utility function as previously
described with a goal of 115, a slope of 1 above the goal and a slope of 10 below the goal.
The objective is to compute the optimal allocation starting with an initial wealth of 100
that maximizes the expected value.
We look at three different situations corresponding to three different sets of values for the
asset class returns. The first problem described by Table 5.4.4 yields an optimal objective
value of -41.13. The second problem described by Table 5.4.4 has lower returns (with the
same returns spread in a good state of the economy vs. in a bad state of the economy)
and yields an optimal objective value of -60. The third problem (Table 5.4.4) has returns
that are a convex combination of the two preceding ones (P2 + 0.75 ( P3 - P2)) and has
an optimal objective value of -48.95.
In such a simple setting we could expect to get an optimal allocation for P3 that is
somewhat between those of P1 and P2. The returns of P3 are constructed this way and

Stock Bond
Returns in Good Economy 1.18% 1.128%
Returns in Bad Economy 1.04% 1.068%
Optimal Allocation 47.4% 52.6%

Table 5.12: Third Problem

5.5. CONCLUSION 79

in fact the objective values seem to verify this convexity property (i.e. P3’s optimal value
-48.95 is between P1’s optimal value of -41.13 and P2’s optimal value of -60). However, we
clearly see in this example that P3’s optimal stock allocation of 47.37% is clearly outside
the range of P2-P1, which is 0%-28.57%. Notice that the variations of the returns of both
stock and bond asset classes in this simple setting can be easily obtained from a VAR Lag
1 framework.

5.5 Conclusion
We used a multi-stage stochastic programming framework for analyzing the effect of model-
ing serial correlation by means of a vector autoregressive process on the computed optimal
asset allocations. We have shown that for certain asset classes considered, i.e. the bond
and cash asset classes in our setting, a VAR model improves the out-of-sample forecasting
of returns and is more appropriate than a GBM fitting (the out-of-sample forecasting being
roughly equivalent for the GBM and VAR models of stock returns). Hence using a VAR
model for the bond and cash returns and a GBM model for stock returns seems the most
appropriate combination for improving the accuracy of forecasts.
Our results show that the results of the VAR model can be very sensitive to the initial
conditions used. This effect is particularly significant on the allocation itself (especially
between bond and cash), rather than the objective value. As our convexity analysis shows,
there is an inherent instability of the allocation results when modeling returns via a vector
autoregressive process because of its dependence on initial conditions. As the allocation
results are not necessarily convex with respect to the initial conditions, it makes it more
difficult for us to find simple confidence intervals for our results by using convexity argu-
ments.
Chapter 6

Bayesian Stochastic Programming

It is important to bear in mind that the value of using stochastic programming for asset
allocation purposes is very much tied to our ability to forecast future returns and the
consistency of our future scenarios with what we know of economic realities. This is true of
any stochastic program in general: finding the appropriate means for modeling uncertainty
is often critical and drives the results. The results are useful only insofar as the parameters
of the stochastic program have been correctly estimated.
When using stochastic programming for financial optimization applications, an interest-
ing question is to what extent our modeling of the stochastic returns is going to change as we
evolve within our programming horizon. We can take different points of view on this issue.
A very simplistic one would be to assume that returns are stationary and that we would
not want update our beliefs about asset returns as we progress into the planning horizon.
That is, we can take the point of view that all our estimates should be based upon the
historical window of returns we had available at the origin (if this is how we have estimated
the processes describing the returns). This is shown in Figure 6.1. Another approach is to
say that, as we progress into the planning horizon, we will update the parameters based
upon the additional realized returns we observe. So for instance in this approach, at time
t1 , we will update the stochastic processes based upon an expanded historical window that
includes the original historical window and the additional returns observed in the first pe-
riod. This is shown in Figure 6.2. Another modeling strategy is to use a ”rolling historical
window”. With this methodology, the parameters of the stochastic process are reevaluated
using a new historical window that has been “rolled forward”. This is shown in Figure 6.3.
This approach is often favored by financial modelers who view old information as irrelevant.
Often, the latest historical sample will be cut to reflect a belief of “local” or “recent stabil-
ity” and a distribution of financial returns will be estimated based upon this assumption of

80
 81

Figure 6.1: Frequentist Approach: Fixed Historical Window.

“local stationarity”.
The three previous methods reflect a “frequentist” approach of the world whereby we
assume that, over a certain period of time, returns can be treated as if they were drawn
from a distribution with ”true” parameters and we only choose different historical periods
to try to estimate these parameters.
Another radically different approach, called Bayesian, questions this assumption. The
Bayesian approach considers the underlying parameters used to model the stochastic pro-
cesses as random themselves. The modeler has certain prior beliefs on the random de-
scription of these parameters and updates these beliefs, to obtain posterior beliefs, as new
information is made available. This is shown in 6.4.
Hence, as our stochastic program unfolds over the planning horizon, we would want
to reflect this behavior and have a consistent representation of our future actions. The
differences between these approaches may be minimal if the original historical window is of
significant size relative to the planning horizon. For instance, suppose we are dealing with
a model of monthly returns for which we have twenty years of historical data and assume
we are only considering a planning horizon three months ahead (with reallocations one- and
two-month ahead). We would expect that, whatever method used, the updates made after
the first month, at t1 , and after the second month, at t2 , will be very minor. However,
82 CHAPTER 6. BAYESIAN STOCHASTIC PROGRAMMING

Figure 6.2: Frequentist Approach: Expanding Historical Window.

Figure 6.3: Frequentist Approach: Rolling Historical Window.

6.1. THE BAYESIAN APPROACH 83

Figure 6.4: Bayesian Approach: Priors are constantly updated at every stage by taking into
account the realized returns just observed in the last stage.

in the situation where we plan for a long-term horizon, using limited historical data, the
results produced by these methods may differ significantly.
This chapter investigates whether using a Bayesian VAR or a simple VAR approach
matters for our multistage program previously introduced. It also shows the particular
usefulness of the Bayesian approach in a specific application, computing the allocation for
a fund of funds. The chapter summarizes the important concepts of Bayesian analysis and
the different priors traditionally used. It then compares the allocation results to those of
the previous chapter. Last, it presents a financial application, i.e. computing an optimal
fund allocation for a fund of funds.

6.1 The Bayesian Approach

In contrast to classical statistics that assume the existence of true parameters θ, Bayesian
statistics regard θ itself as a random variable.
For instance, we may assume that a given set of observations y = (y1 , y2 , ..., yT )0 are
drawn from a Gaussian distribution with parameters θ = (µ, σ)0 . We would then compute
our estimator θ̂ based upon the maximum likelihood principle. Royall provides a complete
84 CHAPTER 6. BAYESIAN STOCHASTIC PROGRAMMING

treatment of the likelihood paradigm (Royall 1997). The estimator θ̂ would be found as
maximizing the following expression:

T
Y · ¸
1 −(yt − µ)2
f (y; θ) = √ exp . (6.1)
2πσ 2 2σ 2
i=1

In contrast to this classical approach, the Bayesian view as described by Hamilton is (Hamil-
ton 1994):

...θ itself is regarded as a random variable. All inference about θ takes the form
of statements of probability... The view is that the analyst will always have
some uncertainty about θ, and the goal of statistical analysis is to describe the
uncertainty in terms of a probability distribution.

The sample likelihood 6.1 is viewed as the density of y conditional on the value of the
random variable θ, denoted f (y|θ). The prior density that represents our a priori beliefs on
θ is given by f (θ). And the joint density of y and θ is given by:

f (y, θ) = f (y|θ) f (θ). (6.2)

Once the data y has been observed, we update our prior beliefs on θ by computing θ’s
posterior density:
f (y; θ)
f (θ|y) = . (6.3)
f (y)
R∞
Since f (y) = −∞ f (y, θ) dθ and by way of equation 6.2, we get:

f (y|θ) f (θ)
f (θ|y) = R ∞ , (6.4)
−∞ f (y, θ) dθ

which is Bayes’ law.

6.2 Bayesian Vector Autoregression

VAR models of higher order have limitations that are due to (i) the large number of coeffi-
cients that need to be estimated as well as (ii) high correlation in the variables that leads
to degraded precision in the parameter estimates. On (i), for n variables with p lags, the
total number of coefficients that need to be estimated is n × (n × p + 1). So for 5 variables
with 2 lags, we need to estimate 55 coefficients. On (ii), we need to bear in mind that,
according to the model, the independent variables yi,t are regressed against their lagged
6.2. BAYESIAN VECTOR AUTOREGRESSION 85

values yi,t−1 , yi,t−2 , ..., yi,t−p , and that this structure tends to produce high correlations that
lead to degraded precision in the parameter estimates. Hence large samples of observations
involving time series variables that cover many years are needed to estimate the VAR model,
and these are not always available. Hence, even with a lag p moderately large, the coeffi-
cient values, estimated by unrestricted OLS1 methods, are often not very well determined
in a finite set of data. In particular, Litterman shows that this problem is accentuated
when dealing with economic series that exhibit trends or persistent local levels (Litterman
1980, Litterman 1986). To address these issues, Litterman, Doan and Sims suggest an alter-
native method for estimating the coefficients in these cases that rests on the use of Bayesian
prior information (Doan, Litterman, and Sims 1984).
As Robertson and Tallman put it (Robertson and Tallman 1999):

The idea is to treat the coefficients as random quantities around given mean val-
ues, with the tightness of the distributions about these prior means determined
via a set of hyperparameters. The OLS coefficient estimator is then modified to
incorporate the inexact prior information contained in these distributions. The
main technical issues involve specifying the form of the prior distributions and
determining the form of the estimators.

6.2.1 Minnesota Prior

Models of prior information distinguish between coefficients that should a priori be signif-
icant and coefficients that should not (and a priori should be zero). Each coefficient has
a prior mean and variance. The Minnesota prior is one such model where the coefficients
associated with the first-lagged dependent variables in each equation of the VAR are given
a prior mean of 1. All other coefficients are assigned a prior mean of 0. So in equation i,
the Minnesota prior takes the form:

2
βii1 ∼ N (1, σii1 ) f or k = 1, (6.5)

2
βijk ∼ N (0, σijk ) f or i 6= j or k > 1. (6.6)

To deal with the fact that a VAR model contains a large number of parameters, Doan,
Litterman and Sims suggested using a few hyperparameters to generate a formula that
would specify the standard deviation of the prior imposed on variable j in equation i at lag

1
OLS: Ordinary Least Squares.
86 CHAPTER 6. BAYESIAN STOCHASTIC PROGRAMMING

k (Doan, Litterman, and Sims 1984), namely:

σ̂uj
σijk = θ ω(i, j) k −φ (6.7)
σ̂ui

There are three hyperparameters mentioned in equation (6.7) and a scaling factor. θ, the
first hyperparameter, is the “overall tightness”, reflecting the standard deviation of the
prior on the first lag of the dependent variable. The second hyperparameter, the weight
function ω(i, j), specifies the tightness of the prior for the variable j in equation i relative to
the tightness of the own lags of variable i in equation i. Notice that this matrix of weights
is assumed independent of the lag. The third hyperparameter, the term k −φ is called a lag
σ̂uj
decay function with 0 ≤ φ ≤ 1 reflecting the decay rate. Finally, the ratio σ̂ui is a scaling
factor that adjusts for varying magnitudes of the variables across equations i and j: σ̂ui
is the estimated standard error from a univariate autoregression involving variable i and
similarly for σ̂uj with respect to equation j.
A typical weighting matrix would be:
 
1 0.5 · · · 0.5
 
 0.5 1 0.5 
W=
 .. .. . .
 (6.8)
 . . .. 
0.5 0.5 · · · 1

In each equation, this weighting matrix imposes the prior mean of zero for coefficients on
other variables more tightly than it imposes the prior mean of 1 for the first lag of each
dependent variable.

6.2.2 Variants of the Minnesota prior

Variants of the Minnesota prior usually try to alter the fact that a standard Minnesota
prior will treat all variables in the VAR model in the same way, except the lagged de-
pendent variables. A possible and natural modification is to alter the weight matrix and
hence the prior variances. For instance, LeSage and Pan have constructed a weight matrix
based on spatial contiguity to emphasize variables from neighboring states in a multi-state
agricultural output forecasting model (LeSage and Pan 1995). Also LeSage and Magura
employed inter-industry input-output weights to place more emphasis on related industries
in a multi-industry employment forecasting model (LeSage and Magura 1991).
The general concept at work here is that by leaving the prior means as they are in the
Minnesota formulation, but by adjusting the standard deviations of the prior, we can place
6.2. BAYESIAN VECTOR AUTOREGRESSION 87

more or less emphasis on the sample data versus the prior itself. By increasing the prior
variances, the sample data will have a greater weight in determining the posterior means.
Conversely, by tightening the prior variances, it will be harder for the sample data to shift
the posterior means away from the prior means.

6.2.3 Random-Walk Averaging Prior

Another more recent approach to altering the equal treatment character of the Minnesota
prior is a ”random-walk averaging prior” suggested by LeSage and Krivelyova (LeSage
1999). As noted, the Minnesota prior treats all variables in the VAR model (except the
first lag of the dependent variable) in an identical fashion. The prior proposed by LeSage
and Krivelyova (LeSage 1999) involves both prior means and variances motivated by the
distinction between important and unimportant variables in each equation of the VAR
model. It is a generalization of the Minnesota prior.
In this setting, a weight matrix W0 is set up that is supposed to reflect the important
or unimportant variables in each equation. The weight matrix contains values of unity in
positions associated with important variables in each equation of the VAR model and values
of 0 for unimportant variables. For example, in the example of matrix W0 , the important
variables in the third equation are variables 2 and 4. Notice that in this example, only
variable 4 is considered to have an important autoregressive influence.:
 
0 1 0 1
 
 1 0 1 1 

W0 =   (6.9)
0 1 0 1 
 
1 1 0 1

The matrix W0 is then normalized to produce the matrix C0 :

 
0 0.5 0 0.5
 
 0.33 0 0.33 0.33 
W0 = 
 0

 (6.10)
 0.5 0 0.5 
0.33 0.33 0 0.33

This yields the following equation i in the VAR model:

n
X
yit = αi + cij yj,t−1 + uit (6.11)
j=1
88 CHAPTER 6. BAYESIAN STOCHASTIC PROGRAMMING

and, in the case of our example, the VAR system of equations:

       
y1t α1 0.5y2t + 0.5y4t u1t
       
 y2t   α2   0.33y1,t−1 + 0.33y3,t−1 + 0.33y4,t−1   u2t 
 =   + .
 y   α + 0.5y2t + 0.5y4t   u  (6.12)
 3t   3     3t 
y4t α4 0.33y1,t−1 + 0.33y2,t−1 + 0.33y4,t−1 u4t

In other words, this suggests having a prior mean for the coefficients on variables as-
1
sociated with the first lags of important variables equal to ci , where ci is the number of
important variables in each equation i of the model. So, in our example of equation (6.12),
the prior means associated with the lagged important variables y2,t−1 and y4,t−1 in the first
VAR equation are 0.5. The prior means for the unimportant variables y1t and y3t in the
same equation are 0.
This prior formulation allows us to downweight the lagged dependent variable using a
zero prior mean to discount the autoregressive influence of past values of this variable and
thus is far less restricted than the Minnesota prior. The Minnesota prior can be seen as the
particular case of a simple random-walk yit = αi + yi,t−1 + uit , where the intercept term
reflects the drift and is estimated using a diffuse prior. The random-walk averaging prior is
centered on a random-walk model that averages over important variables in each equation
of the model and allows for drift as well. As in the case of the Minnesota prior, the drift
parameters αi are estimated using a diffuse prior. Also consistent with the Minnesota prior,
this generalization uses zero as a prior mean for coefficients on all lags other than first lags.
It is also important to note that all time series used in this model need to be scaled or
transformed to have similar magnitudes. However, this is not an issue when time series
data can be expressed as percentage changes, as is the case of most financial applications
that focus on returns.
The prior variances in the random-walk averaging prior can vary but respect the same
guiding principles as the Minnesota prior. The prior variances differ according to whether
the coefficients considered are associated with important or unimportant variables. LeSage
(LeSage 1999) states the following guidelines for the prior variances:

• Parameters associated with unimportant variables should be assigned a smaller prior

variance, so the zero prior means are imposed more ’tightly’ or with more certainty.

• First lags of important variables are given a small prior variance, so the prior means
force averaging over the first lags of important variables.
6.2. BAYESIAN VECTOR AUTOREGRESSION 89

• Parameters associated with unimportant variables at lags greater than one will be
given a prior variance that becomes smaller as the lag length increases to reflect the
belief that influence decays with time.

• Parameters associated with lags other than first lags of important variables will have
a larger prior variance, so the prior means of zero are imposed ’loosely’. This is
motivated by the fact that we do not really have a great deal of confidence in the zero
prior mean specification for longer lags of important variables. We think they should
exert some influence, making the prior mean of zero somewhat inappropriate. As for
unimportant variables, lag decay is still imposed on longer lags of important variables
by decreasing prior variance with increasing lag length.

As for the Minnesota prior, LeSage reiterates the two main reasons why prior means for
important variables at lags greater than one are set at zero (LeSage 1999):

First, it is difficult to specify a reasonable alternative prior mean for these vari-
ables that would have universal applicability in a large number of VAR model
applications... The second motivation for relying on inappropriate zero prior
means for longer lags of the important variables is that overparametrization and
collinearity problems that plague the VAR model are best overcome by relying
on a parsimonious representation. Zero prior means for the majority of the large
number of coefficients in the VAR model are consistent with this goal of parsi-
mony and have been demonstrated to produce improved forecast accuracy in a
wide variety of applications.

6.2.4 Estimating the Coefficients of a BVAR model

Consider the regression model:

yt = x0t β + ut , (6.13)

where ut ∼ i.i.d. N (0, σ 2 ), and xt is a (k × 1) vector of exogenous explanatory variables

and β is a (k × 1) vector of coefficients. Assuming we have T observations in our sample
of observations, let:
   
y1 x01
   
 y2   x02 
y(T ×1) =
 .. 
 X(T × k) =
 .. .

 .   . 
yT x0T
90 CHAPTER 6. BAYESIAN STOCHASTIC PROGRAMMING

If we treat β as random but σ 2 as known, we get the likelihood:

T
Y · ¸
1 −(yt − x0t β)2
f (y|β, X; σ 2 ) = √ exp
2πσ 2 2σ 2
i=1
· ¸ (6.14)
1 −(y − Xβ)0 (y − Xβ)
= exp .
(2πσ 2 )T /2 2σ 2

If we assume that prior information is represented by β ∼ N (m, σ 2 M), where M is a

matrix that scales our prior beliefs on how β is dispersed around our prior mean m, we can
write2 : · ¸
2 1 −1/2 −(β − m)0 M−1 (β − m)
f (β; σ ) = |M| exp . (6.15)
(2πσ 2 )k/2 2σ 2
The Minnesota prior (or other variants previously described) can be embedded in this
framework of assuming β ∼ N (m, σ 2 M), provided an adjustment is made on the constant
coefficient (which is traditionally assumed to have a diffuse prior) and a column of 1 is
added to the X matrix of regressors. In each VAR equation, a prior mean for the constant
coefficient can be estimated by performing a univariate regression first. In addition, a
“loose” prior variance can be attributed to the constant coefficient via the matrix M, by
having artificially large values on both the first row and column (so that there is also
little prior covariance restrictions between the constant coefficient and the other regression
coefficients)3 .

As knowledge about the exogenous variables X is presumed to have no effect on the

prior distribution, so that equation 6.15 also describes f (β|X; σ 2 ). Using Bayes’ law:

f (β|y, X; σ 2 ) f (y|X; σ 2 ) = f (y|β, X; σ 2 ) f (β|X; σ 2 ),

2
We also assume M is invertible.
3
We still need to make sure the matrix M stays invertible.
6.3. COMPARISON OF BVAR VS. VAR FORECASTS 91

and using equations 6.14 and 6.15, this yields4 :

· ¸
1 ¯ −1 ¯ ∗ 0 −1 0 ∗
f (β|y, X; σ 2 ) = ¯M + X0 X¯1/2 exp −(β − m ) (M + X X)(β − m )
(2πσ 2 )k/2 2σ 2

(6.16)
· ¸
1 ¯ ¯ −(y − Xm)0 (XMX0 )−1 (y − Xm)
f (y|X; σ 2 ) = ¯IT + XMX0 ¯−1/2 exp
(2πσ 2 )T /2 2σ 2
(6.17)
m∗ = (M−1 + X0 X)−1 (M−1 m + X0 y). (6.18)

This means that the conditional distribution of β given the observed data y is:

f (β|y, X; σ 2 ) ∼ N (m∗ , σ 2 (M−1 + X0 X)−1 )

and the marginal distribution of y given x is:

f (y|X; σ 2 ) ∼ N (Xm, σ 2 (IT + XMX0 )).

6.3 Comparison of BVAR vs. VAR forecasts

We perform the same analysis as in the previous chapter on monthly returns for the 10-year
Treasury notes and for the 30-day Treasury bills. We vary the historical sample size from
12 months to 60 months and graph the improvement of the BVAR model forecast over the
VAR forecast (in percent). These results reflect the average forecasting improvements of
BVAR over VAR calculated over the full sample of returns. For instance, using a 12-month
historical sample size, we averaged the forecasting results over 6555 forecasts realized with
a rolling historical sample. For each forecast, we compared the absolute deviation between
the out-of-sample realized returns and the forecasted returns for both VAR and BVAR
models. We then compared the percentage improvement of the BVAR forecast over the
VAR forecast. If this improvement is negative, it simply means that the BVAR forecast did
not perform as well as the VAR forecast. The BVAR forecast uses a standard Minnesota
prior as noted before. Figures 6.5 and 6.6 show that overall the BVAR model seems to
perform better over the VAR forecasts for small historical samples of 12 and 24 months.
4
Detailed proof is in the work by Hamilton (Hamilton 1994).
5
The full sample size is comprised of 696 monthly returns, the forecasting horizon is up to 30-month
ahead and the historical sample size used is 12-month. Thus, by rolling forward our historical sample, we
perform 696 − 12 − 30 + 1 = 655 out-of-sample forecasts.
92 CHAPTER 6. BAYESIAN STOCHASTIC PROGRAMMING

1.2

1
Improvement of BVAR over VAR (%)

0.8

0.6 30−day T. bill

0.4

0.2

10−yr T. note

−0.1

1 5 10 15 20 25 30
Forecasted Periods

Figure 6.5: Comparing BVAR vs. VAR forecasts (12-month historical sample). Values of
the Minnesota prior are set with a symmetric weight matrix containing off-diagonal values
equal to 0.5, an overall tightness of 0.1 and an overall decay of 1. As the historical sample
of 12-month is rolled forward within our full sample of 696 monthly returns, these results
are averaged over 655 out-of-sample forecasts using a forecasting period of up to 30-month
ahead.

Figures 6.7, 6.8 and 6.9 show that this improvement of BVAR over VAR disappears as the
sample sizes are increased to 36, 48 and 60 months.

6.4 Application to Funds of Funds

A particular application of concern here is the problem of computing asset allocations for
funds of funds. A fund of funds is a pool of money from investors in various financial
markets that invests in “hedge funds”. The term “hedge fund” is misleading. At first, such
funds were designed to provide real hedging services for banks or large financial institutions.
Today, though the term ”hedge” remains (and some hedge funds’ core business is to develop
hedging strategies for their clients), the term primarily designates investment funds with
6.4. APPLICATION TO FUNDS OF FUNDS 93

0.9

0.8

0.7
Improvement of BVAR over VAR (%)

0.6

0.5

0.4

10−yr T. note
0.3

0.2
30−day T. bill
0.1

0
−0.05

1 5 10 15 20 25 30
Forecasted Periods

Figure 6.6: Comparing BVAR vs. VAR forecasts (24-month historical sample). Values of
the Minnesota prior are set with a symmetric weight matrix containing off-diagonal values
equal to 0.5, an overall tightness of 0.1 and an overall decay of 1. As the historical sample
of 24-month is rolled forward within our full sample of 696 monthly returns, these results
are averaged over 643 out-of-sample forecasts using a forecasting period of up to 30-month
ahead.
94 CHAPTER 6. BAYESIAN STOCHASTIC PROGRAMMING

0.16

0.14

0.12
Improvement of BVAR over VAR (%)

0.1
10−yr T. note
0.08

0.06

0.04

0.02

30−day T. bill
−0.02
−0.03

1 5 10 15 20 25 30
Forecasted Periods

Figure 6.7: Comparing BVAR vs. VAR forecasts (36-month historical sample). Values of
the Minnesota prior are set with a symmetric weight matrix containing off-diagonal values
equal to 0.5, an overall tightness of 0.1 and an overall decay of 1. As the historical sample
of 36-month is rolled forward within our full sample of 696 monthly returns, these results
are averaged over 631 out-of-sample forecasts using a forecasting period of up to 30-month
ahead.
6.4. APPLICATION TO FUNDS OF FUNDS 95

0.08

0.06
Improvement of BVAR over VAR (%)

10−yr T. note
0.04

0.02

30−day T. bill

−0.02

−0.03

1 5 10 15 20 25 30
Forecasted Periods

Figure 6.8: Comparing BVAR vs. VAR forecasts (48-month historical sample). Values of
the Minnesota prior are set with a symmetric weight matrix containing off-diagonal values
equal to 0.5, an overall tightness of 0.1 and an overall decay of 1. As the historical sample
of 48-month is rolled forward within our full sample of 696 monthly returns, these results
are averaged over 619 out-of-sample forecasts using a forecasting period of up to 30-month
ahead.
96 CHAPTER 6. BAYESIAN STOCHASTIC PROGRAMMING

0.07

0.06

0.05
Improvement of BVAR over VAR (%)

0.04 10−yr T. note

0.03

0.02

0.01

−0.01

30−day T. bill
−0.02
−0.025

1 5 10 15 20 25 30
Forecasted Periods

Figure 6.9: Comparing BVAR vs. VAR forecasts (60-month historical sample). Values of
the Minnesota prior are set with a symmetric weight matrix containing off-diagonal values
equal to 0.5, an overall tightness of 0.1 and an overall decay of 1. As the historical sample
of 60-month is rolled forward within our full sample of 696 monthly returns, these results
are averaged over 607 out-of-sample forecasts using a forecasting period of up to 30-month
ahead.
6.4. APPLICATION TO FUNDS OF FUNDS 97

a few accredited investors (whether institutions or wealthy private investors) that seek to
maximize “absolute” returns.6 The hedge fund industry has grown explosively since 1980.
If by the late 1980s, the number of funds had increased to about 100, there were more than
1,200 hedge funds in 1997 with more than $200 billion7 of assets under management. By
May 2003, there were more than 6,000 hedge funds worldwide with more than $600 billion
of assets under management (Ziemba 2003).
For an outsider, assessing the performance of hedge funds is difficult due to looser regu-
latory requirements and minimal disclosure policies. The manager of a fund of funds, who
is wondering how to allocate resources to each hedge fund, is facing a difficult allocation
problem. Forecasting the performance of each hedge fund is a difficult endeavor. The track
record of each fund is often short (typically a few years of quarterly results). Reconstructing
a hedge fund’s performance from its holdings is not easy as often hedge funds take positions
that are small enough to escape filing requirements with the Securities and Exchange Com-
mission. So, all in all, the allocation of the fund of funds may have to be made on a great
deal of qualitative judgment rather than on a thorough quantitative assessment. Moreover,
early work on hedge fund performances indicates there is little persistence in hedge fund
performance (Ziemba 2003).
Our work suggests using our multistage stochastic program with a Bayesian VAR mod-
eling of returns for this allocation decision. We argue for the following: firstly, investing in
hedge funds may generate significant transaction costs that can be easily included in our
model. Secondly, because of the shortage of significant performance time series, and the
need for the manager of a fund of funds to assess different intangibles, using a BVAR model
with adjusted prior assumptions seems the most appropriate tool available for forecasting
future performance. A lot of funds should have correlated returns: they have the same type
of trading strategy, they invest in the same markets, etc. So we can capture some of this
knowledge of common influences in the prior’s covariance matrix. Lastly, if we are trying
to optimize over an horizon of a few years (assume for instance three years), it may be very
important that we preserve some Bayesian consistency in our modeling of the fund of funds
reallocation behavior. Hence, for all the reasons previously noted, we would expect a BVAR
model of hedge fund returns to be an appropriate forecasting tool.

6
The term “absolute” returns is used to emphasize the fact that returns are maximized independently of
any “benchmark” returns, as it traditionally the case for many mutual funds.
7
Any reference to $ (”dollar”) is by default US $.
Chapter 7

Conclusion and Further Research

This thesis has addressed two research questions and provided a framework for an important
and current financial problem, the problem of allocating funds between funds.
The first research question we have addressed is the issue of predictability and serial
correlation of equity returns. This question has been for many years and continues to be
a controversial subject among academics and professional financial analysts. Our contribu-
tions is to have proposed a novel methodology for detecting local bursts of serial correlation
and, by analyzing historical data and conditional returns, to have shown new evidence of
market predictability for equity indices over short-term horizons of a few days. Our results
suggest both momentum and reversal effects for the variations of these broad-based equity
indices. Further work could be done by translating our sampling scheme and performing,
for instance, the same analysis with daily opening prices.
Our second area of research was analyzing the variations of optimal asset allocations,
within a multi-stage stochastic programming framework, with respect to the statistical mod-
eling of returns. Restricting ourselves to a multi-stage stochastic programming framework,
we have compared the optimal asset allocations obtained by switching from a geometric
Brownian motion (“GBM”) model to a vector autoregressive (“VAR”) model of asset class
returns. In the process, we have showed clear evidence of serial correlation for Treasury
bonds’ and bills’ returns. Moreover, in the VAR case, we have showed that the allocation
results vary significantly depending on the initial conditions. For both the GBM and the
VAR models, we have showed that the allocation results are very sensitive to the historical
samples used for calibrating the models. We have also introduced a third statistical model of
asset class returns, a Bayesian vector autoregressive (“BVAR”) approach, and have shown
empirically that this model provides better and more stable out-of-sample forecasts of re-
turns, especially when the different statistical models are calibrated using small samples of

98
 99

historical data.
Our previous results suggests that the multi-stage stochastic programming framework
developed here, when coupled with a BVAR modeling of asset class returns, is particu-
larly appropriate for assisting allocation decisions with limited information and significant
transaction costs, as is the case for funds of funds. Further work could be done involving
out-of-sample comparisons of the performances of the different allocations derived from the
different statistical models of asset class returns.
Appendix A

A Brief History of Mathematical

Finance

This appendix provides a review of some historical developments of mathematical finance

in general and the different models of financial returns. A second section briefly addresses
some epistemological considerations.

A.1 A Brief History of Mathematical Finance

A.1.1 Introduction of Random Walk
Bachelier, in his doctoral thesis, was the first researcher to attempt to qualify the stochastic
nature of stock market movements. One of the remarkable feats of Bachelier’s work is
that he is the first researcher to describe stock market returns as following a Gaussian
distribution. As Bachelier states1 :

The determination of these fluctuations (of stock market prices2 ) depends on an

infinite number of factors; it is, therefore to aspire to mathematical predictions
of it. Contradictory opinions concerning these changes diverge so much that at
the same instant buyers believe in a price increase and sellers in a price decrease.

Thus, Bachelier is the first to consider an infinite number of factors that summed up together
(and without dwelling here on the technical conditions required) will produce a Gaussian
distribution (as could have been derived from the Central Limit Theorem).
1
This is from p. 17 of Cootner’s translation listed in the bibliography along with Bachelier’s original
work (Bachelier 1900).
2
This is our addition.

100
A.1. A BRIEF HISTORY OF MATHEMATICAL FINANCE 101

Another important point in Bachelier’s thesis is the fact that he considers that there is
an infinity of factors (of similar magnitude) that determine stock price movements. Implicit
in this statement is that it is completely useless to attempt to predict those movements.
In short and loosely speaking, stock price movements are the results of an infinite sum of
small steps (random walk) and hence follow a Gaussian distribution and are unpredictable.
In other words, Bachelier is the first to apply the concept of random walk in a financial
context and to introduce Brownian motion to finance.

A.1.2 Concept of Market Efficiency

It is very important, we believe, to have a clear idea of this concept, as it is
often used in a very murky way by both practicioners and academics alike. The
idea of ”market efficiency” is that the market prices fully reflect the available information
on the underlying security and the economic reality of the entity they represent. As Walter
states (Walter 2003):

If the relevant information necessary for a transaction is correctly transmitted in

the price, and considered as an exchange mechanism, the market is then called
”efficient”.

With this definition, we see a first aspect of market efficiency: financial markets and the
prices they provide need to correctly reflect the prices of the underlying economic realities.
But there is also a second aspect important to market efficiency (aspect which is not always
alluded to in the murky definition of the concept), i.e. the role of financial markets as
exchanges for allocating risks. In other words, markets are deemed ”efficient” if different
investors with different risk profiles can find the risk-returns tradeoffs most appropriate for
them. This second aspect of efficiency deals with the allocational efficiency of the markets.

A.1.3 Empirical Testings

Subsequent to Bachelier’s early work, there was no direct test of his Gaussian model of
stock prices until after the Crash of 1929. This empirical work amounted essentially to
statistical testing of financial data along two dimensions: (i) verifying that returns, if they
really are unpredictable, are indeed time independent and that there is no serial correlation;
(ii) characterizing the amplitudes of variations of prices and returns (if Bachelier’s work was
assumed to be true, this amounted to fitting a Gaussian distribution on existing data and
estimating the mean and volatility3 parameters).
3
As by tradition, ”volatility” is synonymous with ”standard deviation”.
102 APPENDIX A. A BRIEF HISTORY OF MATHEMATICAL FINANCE

Different tests of serial independence were carried out that concurred with Bachelier’s as-
sertion that markets are random. We mention here the studies by Cowles , Working, Cowles
and Jones, and Kendall (Cowles 1933, Working 1934, Cowles and Jones 1937, Kendall 1953).
The work by Osborne showed that geometric Brownian motion was more appropriate for
the limiting distribution than Brownian motion (Osborne 1959). Building upon known
properties of particle movements in statistical mechanics, Osborne inferred a Gaussian den-
sity in the first differences in the logarithms of quoted prices. The statistical investigations
continued and as Walter states:

One after another, following Roberts’ suggestions, Larson, Working, Houthakker,

Alexander, Cootner, Moore, Granger and Morgenstern bring strong support to
the random walk thesis, and confirmed that successive price changes can be
considered comparable in first approximation to a Brownian motion (Larson
1960, Working 1960, Houthaker 1961, Alexander 1961, Cootner 1962, Moore
1962, Granger and Morgenstern 1964).

A.1.4 Wedge between Academics and Practicioners

The statistical validation of random walk models led to an open conflict between aca-
demics who saw there the proof that markets were unpredictable and practicioners, such as
chartists, whose very jobs seemed seriously threatened by the academic claims. As Alexan-
der writes (Alexander 1961):

There is a remarkable contradiction between the concepts of behavior of specu-

lative market prices held by professional stock market analysts on the one hand
and by academic statisticians and economists on the other. The professional
analysts operate in the belief that there exist certain trend generating facts,
knowable today, that will guide a speculator to profit if only he can read them
correctly.

Partisans of the existence of trends believed that predictability was possible either because
of ”fundamental” reasons (by looking at underlying business basics and their forecasts) or
because of ”technical” reasons (they would chart the history of prices and believed the price
trajectories followed certain patterns supposed to reflect general rules of the market).

A.1.5 Debate between Active and Passive Management

The Gaussian modelling of stock price movements had a major impact on the asset man-
agement industry. It raised the question of whether active management was really possible
A.1. A BRIEF HISTORY OF MATHEMATICAL FINANCE 103

or whether it was some kind of charlatanism. Again, in a brief sweep at the issues, the
reasoning was the following: (i) if markets are efficient, they reflect instantaneously a lot of
exogenous and unpredictable information, hence their price movements are unpredictable;
(ii) if prices are unpredictable, at least in direction, there is little reason to believe an active
manager is really adding value doing stock picking.
There was a flurry of studies on the performance of portfolio managers to see if they were
performing better than the market. This was a natural endeavor if one believed markets
were unpredictable. We provide here a few references such as (in chronological order) the
studies by Treynor, Sharpe and Jensen (Treynor 1965, Sharpe 1966, Jensen 1968, Jensen
1969), all written between 1965 and 1969.
What these studies underlined was that, net of management fees, active managers were
on average doing no better than the market overall. It suddenly seemed that active man-
agement was more a matter of luck than real talent. Some argued that investors were just
as well off investing in a general market index. This led several banks (e.g. Wells Fargo) to
pioneer the development and marketing of index funds to their clients as early as 19714 .

A.1.6 Debate between Economic Predictability and Random Returns

A major argument used by so-called ”fundamentalists” is that the economic reality of a

business is usually changing at a slow pace. It should therefore be reasonable to expect
some predictability (whether medium-term or long-term) in the associated financial returns.
For instance, suppose one values a stock using a discounted cash flow (”DCF”) analysis of
its future dividends. It is often the case that a firm’s dividend policy is predictable a few
months or years in advance, usually displaying some serial correlation. Similarly, usually
the operational accounting metrics used display some serial correlation. Hence it is not a
priori unreasonable that there could be serial correlation in the company valuation itself
if we believe there should be a certain correspondence between a company’s operational
results and its stock price (all the more if we make the simplifying assumption that the
company has no debt).
This debate is again to be understood within the concept of market efficiency and the
translation of economic realities (e.g. the state of a company’s business) into financial
information and prices. As Fama states (Fama 1965):

... In essence, there is as yet no general model of price formation in the stock
market which explains price levels and distributions of price changes in terms of
4
Bernstein provides an account of this development (Bernstein 1992).
104 APPENDIX A. A BRIEF HISTORY OF MATHEMATICAL FINANCE

behavior of more basic economic variables. Developing and testing such a model
would contribute greatly toward establishing sound theoretical foundations in
this area.

Almost forty years later, it seems the question remains open though different elements of
answer have been proposed.

A.1.7 Reconciliation of Different Views

Fama was one of the first ones to reconcile the seemingly mutually exclusive views of having
both efficient and predictable markets. One way for doing so is by considering market
imperfections (e.g. transaction costs). By and large, Fama’s view is that markets can be
considered efficient and that there may be serial dependence in stock price returns but that
this dependence cannot be exploited for excess gains over a standard buy-and-hold model
of the market index. Fama asserts (Fama 1965):

... the independence assumption is an adequate description of reality as long as

the actual degree of dependence in the series of price changes is not sufficient to
allow the past history of the series to be used to predict the future in a way which
makes expected profits greater than they would be under a naive buy-and-hold
model.

A.1.8 Difficulties in Testing the Efficient Market Concept

Taxonomy of Information Sets This classification is due to Roberts (Roberts 1967)

who distinguishes among:

• Weak-Form Efficiency: The information set includes only the history of prices or
returns themselves.

• Semistrong-Form Efficiency: The information set includes all information known

to all market participants (publicly available information).

• Strong-Form Efficiency: The information set includes all information known to

any market participant (private information).

Testing a Joint Hypothesis Testing those concepts and different levels of efficiency is
very difficult in essence. Let us assume first that we are in a financial system that provides
A.1. A BRIEF HISTORY OF MATHEMATICAL FINANCE 105

equal public information to all participants5 . Hence according to the classification above,
we could test for the semistrong-form of efficiency. This situation corresponds intuitively
to the orginal setting of the Bachelier-Osborne model6 where the information shocks are
homogeneous in nature or, loosely speaking, of the same order of magnitude7 . This rep-
resentation of the new information arrival translates into a Gaussian distribution for price
changes. Testing for ”efficiency” in this context boils down to testing for the Gaussian
distribution of price changes. If this statistical test is rejected, it is then the whole afore-
mentioned logic which is rejected but not necessarily the ”efficiency” itself of the markets at
hand. It could very well be, as Mandelbrot will argue (Mandelbrot 1962, Mandelbrot 1971),
that the Bachelier-Osborne assumption of new information shocks of equal importance is
wrong. Mandelbrot develops the case that shocks should be described by a ”wild random-
ness” (which translates into Paretian-stable, fat-tailed distribution) rather than by a ”mild
randomness” (as was assumed until then in the Gaussian-efficiency model). From a scientific
standpoint, we can only test the statistical inferences of these models. If a particular statis-
tical distribution is rejected (whether Gaussian or fat-tailed), strictly speaking we can only
reject our joint hypothesis. But we cannot decide between rejecting the ”market efficiency”
concept (which would translate into considering that markets do not fully and accurately
reflect new information) or rejecting our underlying assumption on the arrivals and nature
of new information shocks (which would translate into saying that new information does
not emerge the way we thought it did).

A.1.9 The More Recent Developments

Lévy’s stable distributions. Mandelbrot is unequivocal in his rejection of a Gaussian

description of price changes and proposes using Lévy’s stable distributions instead of the
standard Gaussian distribution (Mandelbrot 1962). Mandelbrot further refined this idea
convinced that financial markets were subject to fractures and not just smooth transitions
(Mandelbrot 1997). Mandelbrot introduces the concept of ”wild randomness” (or Paretian
randomness) to ”mild randomness” (or Gaussian randomness). As noted, Mandelbrot had
proposed Paretian (or Lévy’s) distributions to model stock prices as early as 1962 but the
5
This is an unrealistic assumption. Even, if strictly speaking, this may be the case as information is
freely available, behavioral finance models show that investors have limited attention spans and capacities
to process information. Getting the information is not just the issue, heeding it, being able to process it and
acting upon it is just as important.
6
To the best of our knowledge, this joint-name attribution was first used by Fama (Fama 1965) and
generally refers to the Gaussian model of stock price movements as originally introduced by Bachelier
(Bachelier 1900) and further developed by Osborne (Osborne 1959).
7
A condition that is required for the application of the Central Limit Theorem.
106 APPENDIX A. A BRIEF HISTORY OF MATHEMATICAL FINANCE

use of these distributions did not spread quickly within the financial community as they did
not provide closed-form solutions to traditional financial problems (e.g. the determination
of a mean-variance portfolio). However, it has become increasingly clear (all the more since
the 1987 crash8 ) that financial returns are too fat-tailed to be lognormal. Quoting a recent
study by Green and Figlewski on the subject (Green and Figlewski 1999), further referenced
by Sornette (Sornette 2003):

There are more realizations in the extreme tails (and the extreme values them-
selves are more extreme) than a lognormal distribution allows for. In other
words, the standard valuation models are based on assumptions about the re-
turns process that are not empirically supported for actual financial markets.

Development of Behavioral Finance. In the last score of years, significant studies

have been done of the Stock Market that emphasize the significant role behavioral finance
considerations can play. For instance, the so-called ”herding” phenomenon has been doc-
umented that reflects the conservative attitude of most fund managers and the fact that
they are often very averse to taking contrarian positions to their peers’. In other words,
for pension managers concerned with preserving their jobs, there is more risk in being right
by themselves than in being wrong with everybody else. Or in other words, if a pension
manager is wrong with many of his peers, he/she will be less likely to be criticized. This
creates a ”herding” phenomenon among some money managers that can sometimes deeply
influence the market in that suddenly a significant number of managers start selling or buy-
ing the same type of equity (a particular industrial sector, a certain size of companies, etc.).
Further references can be found in the works by Hong et al. or Bikhchandani and Sharma
(Hong, Kudik, and Solomon 2000, Bikhchandani and Sharma 2001). Generally speaking, a
wide range of social dynamics can also account for stock price movements as documented
by Shiller (Shiller 1984). All these endogenous phenomena underline the fact that stock
price movements are not just the product of exogenous shocks but also of endogenous fluc-
tuations of opinions, group pressures and other factors. Financial markets have never been
historically immune to fads and fashions9 and lately, the ”Internet boom” of the late 1990s
(and subsequent ”Internet bust” of the early 2000s) shows how excessive the markets can
be due to the emergence of new technologies and its corollary of massive diffusion of new
and unchecked opinions.
8
An interesting article to read on the subject is Business Week’s article ”The Efficient Market was a
Good Idea - and then Came the Crash,” dated February 22nd , 1998, pp. 38-39.
9
Further reference is needed on the story of Law.
A.1. A BRIEF HISTORY OF MATHEMATICAL FINANCE 107

Implications on ALM Models In our view, there are two venues of improvement for
dynamic asset allocation models: (α) improving the statistical description of financial re-
turns and (β) doing a better job of incorporating macroeconomic considerations to try to
distinguish market fads from fundamental trends. On the first issue, the financial returns
used in dynamic asset allocation models need to reflect the acknowledgement that financial
returns may be better described by fat-tailed distributions and the emergence of fractal
descriptions of financial markets (that have the remarkable property that they are immune
to scale changes). In this case, the focus is likely to be on the accuracy of our numerical
solutions. On the second issue, we need to take into account behavioral finance considera-
tions, all the more if they allow us to detect the formation of irrational trends (e.g. market
bubbles) to benefit in the short term and take directional positions in the markets (however
controversial this issue is).

Outstanding Issues Both the inclusion of fat-tailed distributions or the use of behavioral
finance considerations have their own drawbacks. The main weakness for dealing with
fat-tailed distributions is that the extreme events that usually drive the calibrations of
the tails of the distributions (certainly on the down side) are hard to predict and can
vary greatly in nature. Sometimes the market swings wildly because of the apparition
of great uncertainty that no expert is really able to assess. This is the case, we believe,
in situations involving large geopolitical events (e.g. the terrorist attack of 9/11/01 on
the World Trade Center). The far-reaching consequences of such an event are very hard
to estimate immediately. It may be the case that in such extreme situation multi-agent
behavioral finance models are the appropriate way to model the markets’ response. It is
likely that, in such a situation of uncertainty, there are ”anchoring” and ”imitation” effects
between different groups of investors that become prevalent, at least in the short run. In
any event, estimating the frequency of these major events from historical data may not
make any sense. The philosophical question of whether history repeats itself remains open.
Some major market corrections were the product of complex chain reactions and systemic
breakdowns (e.g. the market correction of August and September 1998 that followed the
demise of LTCM). These systemic breakdowns are better addressed now than they were
in the past. However, nothing guarantees that we have anticipated all possible systemic
market breakdowns, which are unlikely to happen twice in the same form. The second
issue involves the complexity of behavioral finance models and their game-theoretic nature.
Some models break down financial agents into different groups and argue that large swings
in markets can be the product of those different groups. One relatively clear instance of
108 APPENDIX A. A BRIEF HISTORY OF MATHEMATICAL FINANCE

this phenomenon was the Internet boom-and-bust of the late 1990s and early 2000s. In this
instance, we believe (and will further address at a later stage) that there was an asymmetric
equilibrium between different groups of investors (modelling and proving this is the case is
not an easy task).

A.2 Epistemological Considerations in Finance

Epistemology is usually defined as the branch of science that studies the nature of knowledge,
its presuppositions and foundations, and its extent and validity. There is a fundamental
epistemological issue with financial theory arising from the fact that financial market dy-
namics are the consequences of the actions of many agents, each with their own views and
beliefs.
It is worth mentioning here a comparison between physics and finance. As is pointed
out by Roehner (Roehner 2002) when he describes the relatively new field of econophysics
(defined as the ”investigation of economic problems by physicists”), there is been little
emphasis put on empirical work in economics or finance compared to other disciplines such
as physics. As Roehner points out, ”several Nobel prizes have been awarded for work which
remains completely theoretical ... one would in vain search for any statistical test in the
works of [Samuelson, Debreu or Allais]10 . This striking contrast emphasizes the fact that
observation and experimental evidence have a completely different status in physics and
economics.”

A.2.1 Two-Body Problems and Multi-Agent Systems

As Roehner further points out (Roehner 2002), ”most of the problems that physics and
biology were able to solve in the nineteenth and early twentieth century were of the two-
body type.” Whether it is the Sun-Mars interaction studied by Kepler (in the 17th century),
the two heat reservoir problem studied by Carnot in thermodynamics (in the 19th century),
the proton-electron model by Bohr or the Sun-Mercury refined trajectories calculated by
Schwarzschild (in the 20th century), these were all works that assumed a limited level of
complexity (the interactions between two bodies).
As Roehner further suggests, much of the early theories of economics or finance were
not tested by empirical evidence because ”no real economic systems matched the two-body
assumptions even in an approximate way.”
10
Our editing.
A.2. EPISTEMOLOGICAL CONSIDERATIONS IN FINANCE 109

Roehner further states that the level of complexity in economics or finance is of the order
of n-non identical body problems. The financial markets’ complexity can be compared to
that of a meteorological system with n-non identical bodies of air and water
masses interacting.
In our view, finance is an even more complex system than physical systems with n
interacting bodies as financial markets can be best represented as multi-agent systems (not
multi-body systems), with each agent having a mind of his/her own. Hence the behavior of
each component of the overall system can be highly complex and unpredictable. It cannot
be assumed to follow a couple of simple laws (such as short and long range interactions, etc.)
Each agent’s behavior is best described by a game theoretic framework where beliefs about
the market in general and about other agents’ behavior need to be taken into account.
This idea was popularized by Soros (Soros 1987) who notes that ”the participants’ think-
ing introduces an element of uncertainty into the subject matter... Situations which have
thinking participants may be impervious to the methods of natural science but they are sus-
ceptible to the methods of alchemy. The thinking of participants, exactly because it is not
governed by reality, is easily influenced by theories. In the field of natural phenomena, scien-
tific method is effective only when its theories are valid; but in social, political and economic
matters, theories can be effective without being valid.”
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