AntiCor Revisited

8/8/2019 AntiCor Revisited

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Portfolio Management :

An empirical study of the Anticor

algorithm

Danny Castonguay

Master of Engineering

Electrical and Computer Engineering

McGill University

Montreal,Quebec

2007-05-01

A thesis submitted to McGill University in partial fulfillment of the requirements ofthe degree of Masters of Engineering (M.Eng.) in Electrical and Computer

Engineering

cDanny Castonguay, 2007


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ACKNOWLEDGMENTS

I thank Shie Mannor for his advice. I thank Hasan Mirza and Chantale Cardinal-

Watkins for reviewing my text. I thank my parents for their support.

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ABSTRACT

The Anticor algorithm for portfolio selection, developed by Borodin, El-Yaniv,

and Gogan, is empirically studied. In their original presentation of this algorithm,

Borodin et al. provided results on historical markets, demonstrating that the Anticor

algorithm not only beats the market, but can also beat the best stock. Our study

of the Anticor algorithm extends these results in several ways. First, we examine how

the Anticor algorithm performs on more recent market data. Second, we run Anticor

on several simulated markets, as part of an attempt to explain its performance.

Finally, we examine how the Anticor algorithms performance is affected when some

of the underlying assumptions, such as zero transaction costs, are removed.

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ABREGE

Lalgorithme Anticor pour la selection de portefeuilles, developpe par Borodin,

El-Yaniv et Gogan, est empiriquement etudie. Dans la presentation originale de cet

algorithme, Borodin et al. donnent des resultats bases sur des marches financiers

historiques qui demontrent que lalgorithme Anticor non seulement bat le marche

mais peut aussi surperformer le meilleur titre. Notre etude de lalgorithme Anticor

ajoute a ces resultats de plusieurs faons. Premierement, nous examinons comment

lalgorithme Anticor performe sur les marches financiers recents. Deuxiemement,

nous appliquons lalgorithme Anticor a des marches simules afin de tenter dexpliquer

ce qui determine une bonne performance. Finalement, nous examinons comment

la performance de lalgorithme Anticor est affectee lorsque certaines hypothses, telle

que de ne pas avoir de frais de transactions, sont enleves.

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TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

ABREGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Agents and Monetary Resources . . . . . . . . . . . . . . . . . . . 11.2 Financial Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Efficient Market Hypothesis . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Weak-Form Efficient Market Hypothesis . . . . . . . . . . . 41.3.2 Semi-Strong-Form Efficient Market Hypothesis . . . . . . . 51.3.3 Strong-Form Efficient Market Hypothesis . . . . . . . . . . 51.3.4 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . 6

2 Portfolio Selection Problem . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Simplifying Assumptions . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Simplified view of market operation . . . . . . . . . . . . . 82.2.2 Infinitely small agent . . . . . . . . . . . . . . . . . . . . . 102.2.3 Frictionless transactions . . . . . . . . . . . . . . . . . . . . 102.2.4 Tax-free profits . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Portfolio Selection Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Passive Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.1 Choosing a good b for BAH online . . . . . . . . . . . . . . 153.2 Active Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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3.2.1 Constant rebalancing . . . . . . . . . . . . . . . . . . . . . 153.2.2 The Universal Portfolio Algorithm . . . . . . . . . . . . . . 16

4 The Anticor Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1 Notation preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 174.2 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Compounded Algorithms . . . . . . . . . . . . . . . . . . . . . . . 224.4 Compounding the Anticor Algorithm . . . . . . . . . . . . . . . . 234.5 Anticor Explorer . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Transaction Cost Considerations . . . . . . . . . . . . . . . . . . . . . . 25

5.1 Brokerage Scheme Examples . . . . . . . . . . . . . . . . . . . . . 255.2 Proportional Commission Model . . . . . . . . . . . . . . . . . . . 26

5.2.1 Modifications to the Proportional Commission Model . . . 266 Markets Used For Simulation . . . . . . . . . . . . . . . . . . . . . . . . 28

6.1 Old Historical Market Data . . . . . . . . . . . . . . . . . . . . . 286.2 Recent Historical Market Data . . . . . . . . . . . . . . . . . . . . 286.3 Simulated Market Data . . . . . . . . . . . . . . . . . . . . . . . . 29

6.3.1 Modified Random Walk . . . . . . . . . . . . . . . . . . . . 296.3.2 Modified Autoregressive Model . . . . . . . . . . . . . . . . 32

7 Empirical Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.1.1 Total Return . . . . . . . . . . . . . . . . . . . . . . . . . . 357.1.2 Cumulative Return . . . . . . . . . . . . . . . . . . . . . . 367.1.3 In-Hindsight Geometric Mean Return . . . . . . . . . . . . 377.1.4 Sharpe Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.2 Market Detailed View . . . . . . . . . . . . . . . . . . . . . . . . . 397.2.1 Old Historical Market Data . . . . . . . . . . . . . . . . . . 397.2.2 Recent Historical Market Data . . . . . . . . . . . . . . . . 42

7.3 Simulated Market Data . . . . . . . . . . . . . . . . . . . . . . . . 547.3.1 Modified Random Walk . . . . . . . . . . . . . . . . . . . . 547.3.2 Modified Autoregressive Model . . . . . . . . . . . . . . . . 54

8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

8.1 Future extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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A Dependence Matrix Generation Algorithm . . . . . . . . . . . . . . . . . 64

B Indices Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B.1 Old Historical Market Data . . . . . . . . . . . . . . . . . . . . . 65B.2 Recent Historical Market Data . . . . . . . . . . . . . . . . . . . . 65

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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LIST OF TABLESTable page

21 Example of an order book . . . . . . . . . . . . . . . . . . . . . . . . . 9

22 Effective order book under the simplified view of market operation . . 9

23 The before and after tax return of two strategies . . . . . . . . . . . . 12

61 Daily Sampled Statistics of Old Historical Markets . . . . . . . . . . . 28

62 Daily Sampled Statistics of Recent Historical Markets . . . . . . . . . 30

63 Daily Sampled Statistics of Simulated Markets . . . . . . . . . . . . . 31

71 Sharpe ratio comparisons . . . . . . . . . . . . . . . . . . . . . . . . . 38

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LIST OF FIGURESFigure page

41 Anticor Explorer Graphical Interface . . . . . . . . . . . . . . . . . . 23

61 Noisy Feedback Model . . . . . . . . . . . . . . . . . . . . . . . . . . 33

71 Results for datML1.txt . . . . . . . . . . . . . . . . . . . . . . . . . . 40

72 Results for djia.txt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

73 Results for nyse.txt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

74 Results for sp500.txt . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

75 Results for tse.txt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

76 Results for dja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

77 Results for dji . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

78 Results for dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

79 Results for iix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

710 Results for ndx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

711 Results for nwx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

712 Results for nyi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

713 Results for nyy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

714 Results for oex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

715 Results for soxx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

716 Results for xau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

717 Results for xmi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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718 Results for mrw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

719 Results for mrw3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

720 Results for mam0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

721 Results for mam1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

722 Results for mam2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

723 Results for mam3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

724 Results for mam4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

725 Results for mam5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

726 Results for mam6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

727 Results for mam7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

728 Results for mam8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

729 Results for mam9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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CHAPTER 1Background Information

We start with some background information on the portfolio selection problem

in a semi formal context. We explain how groups of agents come together to exchange

monetary resources thus forming financial markets. We then provide a brief overview

of the efficient market hypothesis. Moving into the core subject matter, we present

formally the portfolio selection problem. We carefully list simplifying assumptions

made to render the portfolio selection problem more manageable. We then present

a series of portfolio selection algorithms, namely UBAH (the uniform buy-and-hold

strategy), BAH* (the optimal in hindsight buy-and-hold strategy), UCBAL (the

uniform constant rebalancing strategy) and UCBAL* (the optimal constant rebal-

ancing strategy). We then present the markets used for simulation: old historical

markets, recent historical markets, simulated markets. Subsequently, we measure

and compare the performance and the risk of all the algorithms presented.

1.1 Agents and Monetary Resources

Finance studies the ways agents allocate monetary resources over time. An

agent could be a person or an organization, while a monetary resource is anything

to which a numerical dollar value can be assigned.

An agent holding some monetary resources can decide either to consume them,

or to invest them. When an agent consumes a monetary resource, it modifies the

resource in such a way that the agent becomes relatively happier, and (usually)

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the value of the resource decreases. To invest a monetary resource is, simply, to not

consume it. Buying a house and living in it is both an investment and a consumption.

As the price of the house changes over time, a profit (or loss) may be realized and

that is the investment part. But by not leasing out the house, some revenue is not

earned and that is the consumption part.

The foremost example of a monetary resource is money. To own money, in a

particular currency, without spending it, is an investment in that currency. The

value of an amount of money changes over time as exchange rates vary. However,

spending money does not, according to the definition above, constitute consuming

the money; instead, it just involves exchanging it for another monetary resource.

In economics, the agents increase in happiness is measured by a numerical utility

function. Agents are usually assumed to always act in a way that maximizes their

utility. This characteristic is called rationality. It has been suggested, however, that

humans are not always rational. In [10], Kahneman and Tversky present a critique

of expected utility theory and, instead, propose an alternative model, called prospect

theory. We mention this because, as we will see in Section 1.3, the Efficient Market

Hypothesis assumes that agents are rational.

It is hoped that this elementary introduction encourages and motivates the

reader to learn more about investing. Before presenting the Anticor algorithm, which

is the main focus of this thesis, we need to define what financial markets are.

1.2 Financial Markets

The mechanisms which allow agents to exchange monetary resources are called financial markets. There are many types of financial markets, such as stock markets,

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bond markets, commodities markets, futures markets, and foreign exchange markets.

In general, when a group of agents come together and exchange their monetary

resources, a financial market is created.

A commonly used tool for evaluating the characteristics of a group of related

securities is a security market index, which is a statistic reflecting the composite

value of the securities in the group. These securities usually share some common

feature, such as belonging to the same industry, or being traded on the same market

exchange. Indices are defined by news or financial-services firms, and are often used

to benchmark the performance of portfolios. We will use indices in a similar way to

benchmark the Anticor algorithm.

In the real world, the large gains that can be achieved by trading wisely on

the financial markets have attracted a lot of agents over the last few centuries. The

competition over the finite (but usually growing in value over time) amount of mon-

etary resources has captivated the attention of many researchers both in and out of

academia. The fierce competition has lead some to the formulation of the efficient

market hypothesis, which suggests that security prices adjust rapidly and rationally

to new information.

Since many researchers believe in the efficient market hypothesis and since these

researchers (those who firmly believe in the efficient market hypothesis at least) would

probably find little interest is learning about the Anticor algorithm, we will present

briefly the essence of the efficient market hypothesis. Following that, we will present

formally the portfolio selection problem which the Anticor algorithm attempts tosolve.

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1.3 Efficient Market Hypothesis

The question of whether or not financial markets are efficient is controversial.

The efficiency of a given financial market is usually hard to assess, and unquestionably

always open for debate. If the efficient market hypothesis holds true, then agents

should not be able to consistently achieve above-average performance. In other

words, the likes of Warren Buffet are akin to lottery winners: gamblers who got

lucky. The efficient market hypothesis is based on the following assumptions:

The number of agents trading in a financial market is large

These agents are rational

New information comes to the market randomly

New information is instantly reflected in agents buying/selling prices

The efficient market hypothesis states that if a financial market satisfies these as-

sumptions, then the prices on the securities traded in this market reflect all known

information.

There are three ways to define what is meant by information, leading to three

different forms of the efficient market hypothesis: the weak-form, the semi-strong-

form and the strong-form.

1.3.1 Weak-Form Efficient Market Hypothesis

The weak-form efficient market hypothesis defines information as all the histor-

ical market data (expressible as large matrices of real numbers) including sequences

of prices, rates of return, trading volume, odd-lot transactions, block trades, and

exchange specialist transactions.

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Consequently, the weak-form efficient market hypothesis states that technical

analysis will not be able to consistently produce excess returns. As we will see later,

this implies that algorithms such as the Anticor algorithm (which uses only historical

sequences of prices) should not be able to achieve consistent excess returns. On the

other hand, it acknowledges that some forms of so-called fundamental analysis,

which also consider qualitative factors, may still provide excess returns.

1.3.2 Semi-Strong-Form Efficient Market Hypothesis

The semi-strong-form efficient market hypothesis defines information as all the

publicly available data (including both market and non-market data) such as weak-

form information, earnings and dividends announcements, price-to-earnings ratios,

dividend-yield ratios, stock splits, news about the economy and politics, and more.

Consequently, the semi-strong-form efficient market hypothesis states that the

collective beliefs and expectations are rapidly (or instantly) reflected in the assets

prices. This implies that it is not possible to consistently outperform the market

using the information that the market already knows. The semi-strong-form implies

that neither technical nor fundamental analysis will allow agents to consistently

outperform the market. The only way to consistently outperform the market is

through luck or by obtaining and trading on material non-public information.

1.3.3 Strong-Form Efficient Market Hypothesis

The strong-form efficient market hypothesis defines information as all publicly

available data, as in the semi-strong-form, plus all the data that is not publicly

available. Thus, under the strong-form efficient market hypothesis, even insidertrading cannot lead to consistent above average performance.

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1.3.4 Empirical Evidence

In [6] and [7], Fama presents the efficient market theory in terms of a fair game

model: an agent can be confident that current market prices fully reflect all available

information. We present in Chapter 2 the Portfolio Selection Problem and then

present empirical results of the performance of the Anticor algorithm.

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CHAPTER 2Portfolio Selection Problem

2.1 Problem Definition

In the portfolio selection problem, we have a market of m securities which are

traded over T days. At the end of each day, each security j 1, . . . , m has a closing

price vt(j). For convenience, we define the relative price of security j on day t as

xt(j) = vt(j)vt1(j) . An investment of d (dollars) in stock j before day t yields dxt(j)

dollars at the end of day t. The vector [vt(1), . . . , vt(m)] of all prices is called

the price vector on day t, denoted vt; similarly, the vector [xt(1), . . . , xt(m)] of

all relative prices is called the market vector, denoted xt. The matrix [x1, . . . , xT]

containing all market vectors over the T days is called the market sequence, denoted

by X.

At the start of each day t, an agent chooses a portfolio bt = [bt(1), . . . , bt(m)],

satisfying bt(j) 0 for all j and

j bt(j) = 1, where each entry bt(j) is the fraction

of the agents wealth invested in security j. These portfolios produce a total returnTt=1 bt

xt over the T days. The agent might not have access to the full market

sequence X when choosing each bt. (In practice, for example, an agent does not

have access to future market vectors.) Loosely stated, the goal in the portfolio

selection problem is to choose bt to achieve a good return, given the information

available to the agent.

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A portfolio selection algorithm A is a (deterministic or randomized) rule for

specifying the portfolio sequence b1, . . . , bT. We define retX(A) as the total return

ofA for the market sequence X. In practice, the market sequence X is not known

in advance and hence it is a random process. In this context, retX(A) is a random

variable even ifA is not random.

We will explore a few algorithms in Chapter 3 but first, it is important to explain

the assumptions inherent in this problem formulation.

2.2 Simplifying Assumptions

2.2.1 Simplified view of market operation

The problem formulation given above is based on a highly simplified view of

how markets operate. Specifically, it assumes that there is a given current price

for each security, which is set at the start of each trading day, and any agent can

always buy or sell any amount of any security at its current price. Below, we present

a more realistic view of market operation.

Throughout each trading day, agents enter the market at various times, seeking

to buy or sell securities. When an agent decides to buy (respectively, sell) a security,

it places an order specifying the quantity desired (respectively, for sale), and the

maximum (respectively, minimum) price per unit that it is willing to pay (respectively

sell for). This order is recorded in a tabular structure called an order book, which

contains two lists: one containing all the unfulfilled purchase orders, called the bid

list, and one containing the unfulfilled sale orders, called the ask list. The contents

of an order book change as orders are placed, fulfilled, and retracted; an example isgiven in Table 21.

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Table 21: Example of an order book

Bid Ask

Qty. Price Qty. Price

100 50$ 200 51$150 49$ 100 52$

The difference between the highest price in the bid list and the lowest price in the

ask list is called the bid-ask spread. Whenever the bid-ask spread is negative or zero,

a transaction occurs between the agent who placed the highest-price purchase order

and the agent who placed the lowest-price sale order. Whichever order has a higher

quantity is removed from the order book; the other has its quantity appropriately

reduced. Transactions continue to occur in this way until the bid-ask spread is strictly

positive, at which time the market becomes idle until an agent places or retracts an

order, changing the bid-ask spread. Note that a single purchase order can be fulfilled

by multiple sale orders with different prices, and vice versa.

In effect, the simplified view given above assumes that throughout each trading

day, the order book always appears as shown in Table 22. The price p is set at the

beginning of the day and does not change throughout the day.

Table 22: Effective order book under the simplified view of market operation

Bid Ask

Qty. Price Qty. Price p$ p$

The assumption that the bid and ask quantities are both infinite is known as

infinite liquidity, while the assumption that the bid and ask prices are equal is known

as zero spread.

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2.2.2 Infinitely small agent

Under this assumption, the agent is considered infinitely small compared to

the market, so its actions do not affect the future evolution of the market.

2.2.3 Frictionless transactions

Under this assumption, no brokerage fees are incurred when a transaction takes

place. As we shall see later, it is possible to relax this assumption to penalize overly

active strategies. However, brokerage fees are difficult to represent accurately, since

they vary widely with the amounts being traded, the type of securities being traded,

and other considerations such as soft dollars.

2.2.4 Tax-free profits

Under this assumption, there is no tax incurred when a profit is realized. This

assumption is often overlooked, but as we will see, a strategys after-tax profit can

sometimes be significantly less than its before-tax profit.

Although the tax rate is the same for all strategies, different strategies will pay

different amounts of taxes. The reason for this is that taxes are paid whenever

securities are sold for a profit. For example, buying a security for d dollars then

selling it later for 1.5d dollars will result in a before-tax profit of 0.5d dollars and an

after-tax profit of (1 )0.5d dollars, where is the tax rate. Note however that

taxes are calculated on the aggregate of the profit and losses of all the securities in

the portfolio.

In practice, it is preferable to compare the relative performance of two strategies

on an after tax basis. The following example demonstrates an example where the

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However, every time he sells, he pays a 20% tax on any profits made since the previous

purchase and his after-tax profit is

{[1 + (1 0.2) (1.125 1)]10 1} 10, 000$ = 15, 937.42$

Thus, over the the 10-year period, Bobs total before-tax profit is greater than Alices,

but his after-tax profit is lower. Table 23 shows the before- and after-tax return of

the two strategies in greater detail. The conclusion to draw from this example is that

although Bob correctly chooses the best performing securities, he should consider the

effect of taxes before making his investment decisions. Fortunately, he can go back

in time and rectify his investments to account for taxes.

Table 23: The before and after tax return of two strategies

X Before tax After tax

Year S1 S2 retX(A) retX(B) retX(A) retX(B) Cash(A) Cash(B)

1 1.125 1.115 1.125 1.125 1.125 1.1 11,250 11,0002 1.115 1.125 1.115 1.125 1.115 1.1 12,544 12,1003 1.125 1.115 1.125 1.125 1.125 1.1 14,112 13,310

4 1.115 1.125 1.115 1.125 1.115 1.1 15,735 14,6415 1.125 1.115 1.125 1.125 1.125 1.1 17,701 16,1056 1.115 1.125 1.115 1.125 1.115 1.1 19,737 17,7167 1.125 1.115 1.125 1.125 1.125 1.1 22,204 19,4878 1.115 1.125 1.115 1.125 1.115 1.1 24,757 21,4369 1.125 1.115 1.125 1.125 1.125 1.1 27,852 23,579

10 1.115 1.125 1.115 1.125 1.115 1.1 31,055 25,937

ROI 3 .1055 3.1055 3.1055 3.2473 2.6844 2.5937 26,844 25,937-

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We have given here a simplicied view of how taxes affect return on investments.

However in practice, taxes are significantly more complicated than that; hence, we

decided to ignore the effect of taxes.

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CHAPTER 3Portfolio Selection Algorithms

We now present algorithms that attempt to solve the portfolio selection prob-

lem (as defined in Chapter 2) and we classify them as either passive or active

algorithms. Presentation of the Anticor algorithm will follow in Chapter 4.

3.1 Passive Algorithms

Passive algorithms are algorithms that never re-invest any money. The main ex-

ample of such an algorithm is BAHb (buy-and-hold). This algorithm is parametrized

by an initial portfolio b; it invests according to b on the first trading day, then never

re-invests any money henceforth. This results in a portfolio sequence given by

bt+1 =1

xtbt[xt(1)bt(1), . . . , xt(m)bt(m)]

There are two important special cases:

The U-BAH (uniform buy-and-hold) algorithm is BAHb with b = [1m

, . . . , 1m

].

The performance of U-BAH is often used as an indication of the overall perfor-

mance of the market when benchmarking other algorithms. (In practice, how-

ever, stock market indices such as the Dow Jones use non-uniform weights.) If

an algorithm A has retX(A) > retX(U-BAH), then we say that A beats the

market.

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The BAH algorithm is BAHb with b = arg maxb retX(BAHb). This is the op-

timal in hindsight buy-and-hold strategy, and is often used in offline bench-

marks. The portfolio b assigns a weight of 1 to the best stock, and a weight

of 0 to all others.

3.1.1 Choosing a good b for BAH online

In practice one could use a fundamental [9] approach such as that of Warren

Buffet or Peter Lynch, but such approaches are informal and require the evaluation

of intangible factors, such as the quality of management of the company selling the

stock. Alternatively, one could use a behavioral approach. In [15], Shleifer argues

that less than fully rational investors trade against arbitrageurs whose resources are

limited by risk aversion, short horizons, and agency problems. We are focusing our

work on quantitative approaches that require only the market sequence X.

3.2 Active Algorithms

3.2.1 Constant rebalancing

This algorithm maintains a fixed portfolio b throughout the entire trading period

by appropriately re-investing money at the end of each trading day. As with BAH,

there are two important special cases:

U-CBAL, where b = [ 1m

, . . . , 1m

].

CBAL*, where the fixed portfolio is the optimal in hindsight portfolio.

We have that

retX(CBAL) retX(BAH

)

Cover and Gluss [3] present an interesting example involving a hypothetical no

growth market, where U-CBAL yields a return that is exponential in T. Specifically,

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consider the market sequence

XT =2 12 2 12

1 1 1 1

;

we have

retXT(U-BAH) = 1

but

retXT(U-CBAL) =

9

8

T/2which is exponential in T.

In [5], Cover and Thomas prove that, if a random market sequence X = [x1, . . . , xT]

consists of i.i.d. daily market vectors, then for any online algorithm A,

EX {retX(CBAL)} EX {retX(A)}

However, in [11], McKinlay and Lo argue that the daily market vectors xt are not

i.i.d., but instead have memory. It would be preferable to have an online algorithm

which drops the i.i.d. assumption and makes use of the memory between the xts.

3.2.2 The Universal Portfolio Algorithm

Cover and Ordentlich [4] present an algorithm, called the Universal Portfolio

Algorithm, which they prove guarantees a sub-exponential ratio (in n) between its

return and the return of CBAL for any market sequence over n days. This result

is surprising, as it implies that the Universal algorithm can track the potentially

exponential returns of CBAL

; however, real markets rarely provide exponential

returns, so it is not particularly useful in practice.

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CHAPTER 4The Anticor Algorithm

By attempting to systematically follow the constant rebalancing philosophy, the

Anticor algorithm is capable of some extraordinary performance in the absence of

transaction costs, or even with small transaction costs. The Anticor algorithm was

initially formulated by Borodin et al. in [2]. In our view, their presentation can be

simplified and we propose here a new formulation of the algorithm.

4.1 Notation preliminaries

To simplify the presentation of the Anticor algorithm, we introduce some special

notation for indexing matrices and for performing operations on matrices.

For any m n matrix A, we denote by Ak,...,l the sub-matrix consisting of

columns k through l of A; that is, for any k, l with 1 k < l n, we define

Ak,...,l =

a1k a1l

......

amk aml

Next, for an m n matrix A, we denote by Log(A) (note the capital L) the

element-wise logarithm of A:

Log(A) =

log a11 log a1n

... ...

log am1 log amn

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For two m n matrices B and C, we denote by B C their element-wise product

(also known as the Hadamard product), and by BC their element-wise quotient:

BC =

b11c11 b1nc1n

......

bm1cm1 bmncmn

BC =

b11/c11 b1n/c1n...

...

bm1/cm1 bmn/cmn

Finally, we define the row-wise mean operator

Mean(A) =

1n

ni=1 a1i...

1n

ni=1 ami

and the row-wise standard deviation operator

StdDev(A) =

1n

ni=1

a1i

1n

nj=1 a1j

2...

1n

ni=1

ami

1n

nj=1 amj

2

which produce m1 column vectors containing, respectively, the mean and standard

deviation of each row of A.

4.2 The algorithm

The Anticor algorithm evaluates changes in stocks performance by dividing

the sequence of previous trading days into equal-sized periods called windows, each

with a length of w days. w is an adjustable parameter called the window size.

The Anticor algorithm is based on a reversal to the mean approach: wealth istransferred from recently high-performing stocks to anti-correlated low-performing

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stocks. Specifically, whenever the algorithm detects that (i) a stock i outperformed

a stock j during the last window, but (ii) is performance in the last window is anti-

correlated to js performance in the second-to-last window, then it transfers wealth

from i to j. We present the algorithm more formally below.

Using the notation introduced above, we define

L1 = Log(Xtwt2w+1)

L2 = Log(Xttw+1)

which are m w matrices containing the logarithms of the daily market vectorsduring the second-to-last and last windows. We take logarithms because ordering

logarithms of arithmetic means is equivalent to ordering geometric means, though

analytically simpler.

Next, we derive centered versions L1 and L2 of L1 and L2 by subtracting the

mean of each row from that row. Let

1 = Mean(L1)

2 = Mean(L2)

which are m 1 matrices; then

L1 = L1

1 1

L2 = L2 2 2

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Now, we let

Mcov =1

w 1L1L

T2

For each i and j, Mcov(i, j) is the covariance between the log-relative prices of stock

i over the first window and stock j over the second window.

Finally, we let

1 = StdDev(L1)

2 = StdDev(L2)

and let Mcor be given by

Mcor(i, j) =

Mcor(i,j)(i)(j)

(i),(j) = 0

0 otherwise

procedure Anticor(w, t, Xt, b)

if t < 2w then

return b

end if

L1 Log([Xt]t2w+1,...,tw)

L2 Log([Xt]tw+1,...,t)

1 Mean(L1)

2 Mean(L2)

L1 L1 1 1L2 L2

2 2

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Mcov 1

w1L1L

T2

1 StdDev(L1)

2 StdDev(L2)

for all i, j {1, . . . , m} do

if 1(i) = 0 and 2(j) = 0 then

Mcor(i, j) Mcov(i,j)1(i)2(j)

else

Mcor(i, j) 0

end if

end for

for all i, j {1, . . . , m} do

if 2(i) 2(j) and Mcor(i, j) > 0 then

claimij Mcor(i, j) [Mcor(i, i)] [Mcor(j,j)]

else

claimij 0

end if

end for

for all i {1, . . . , m} do

if claimij = 0 for some j then

for all j {1, . . . , m} do

transferij b(i) claimij/

mj=1 claimij

end forelse

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transferij 0

end if

end for

for all i, j {1, . . . , m} do

b(i) b(i) transferij + transferji

end for

end procedure

Note that output of ANTICORw for day t cannot be directly fed back into

ANTICORw+1 as the next days input; we must first compute the effect of the market

vector xt on bt:

bt =1

bt xtbt xt

The resulting vector bt can then be fed into ANTICORw as input for day t + 1.

4.3 Compounded Algorithms

The Anticor algorithm is parametrized by the window size w, which signifi-

cantly affects the algorithms performance. We can thus view the Anticor algorithm

as not a single algorithm, but rather a family of algorithms, indexed by the pa-

rameter w. Since it is not possible to choose w in hindsight when applying the

Anticor algorithm online, the authors of [2] (effectively) suggest viewing the dif-

ferent ANTICORw algorithms as stocks in a market and applying a portfolio

selection algorithm to these stocks. In a simple case, we can apply a uniform

buy-and-hold on all ANTICOR2, . . . , ANTICORW (2 < w W) algorithms. Hence,

the BAHW(ANTICORw) algorithm is used in practice rather than using a single

ANTICORw.

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4.4 Compounding the Anticor Algorithm

If we can apply the BAH algorithm to a set of Anticor algorithms, then it

should also be possible to apply the Anticor algorithm to a set of Anticor algorithms

(effectively treating the various ANTICORw as stocks). Similarly to the authors

of [2], we compound twice and then use a BAH investment strategy resulting in

BAHW(ANTICORw(ANTICORw)).

4.5 Anticor Explorer

In order to explore the algorithm in action, we have implemented a graphical

user interface that enables us to walk through the algorithm as it trades. In Figure

41 we show a screenshot of the graphical interface. Of particular interest are the

first two columns which show the portfolios bt and bt+1. The frequency with which

Figure 41: Anticor Explorer Graphical Interface

we execute the Anticor algorithm can vary anywhere from split seconds to years.

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In this context, we apply it daily and this qualifies the Anticor algorithm as a high

frequency trading strategy. Furthermore, the algorithm has a high turnover ratio

every time it is applied as is clearly demonstrated in 41.

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CHAPTER 5Transaction Cost Considerations

The effect of transaction costs associated with brokerage fees is non negligible.

The following example is used to demonstrate the various ways of computing trans-

action vectors. One way is to compute the change in monetary value of the securities

and the other is to compute the change in number of units (or shares, for the sake

of the example).

5.1 Brokerage Scheme Examples

Let B1 = b1d1 where d1 = 100$, b1 = [0.5, 0.5], and hence B1 = [50$, 50$].

Now, if we wanted to know how many shares of each security were owned, we would

need to know the price of the shares. Let P1 = [50$/s, 100$/s]. Thus, we have

Q1 = B1P1 = [50$, 50$] [50$/s, 100$/s] = [1s, 0.5s], where is the element

wise division as defined in Section 4.1.

Next, we wish to determine what happens one time step later. So suppose that

P2 = [100$/s, 50$/s]. For convenience, we define X1 = P2 P1 = [2, 0.5] which

is a unitless measure of each shares growth over the period. Let us assume that the

algorithm outputs b2 = [0.5, 0.5]. Hence we can compute d2 = B1 X1 = 125$ and

as before B2 = b2d2 = [62.5$, 62.5$].

At this point, two alternatives exist for us. If the broker charges us a per share

fee, then we will need to compute Q2 = B2P2 = [62.5$, 62.5$][100$/s, 50$/s] =

[0.625s, 1.25s]. The transfer in shares is simply TS = |Q1 Q2| = [0.375s, 0.75s].

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If the broker charges us a per monetary value fee, then we first need to compute the

value of B at the end of period 1, right before the transfer occurs, which we denote

by B1. Hence, B1 = B1 X1 = [100$, 25$]

and the transfer is Tm =B1 B2

=[100$, 25$] [62.5$, 62.5$] = [37.5$, 37.5$], where is the element wise multi-

plication as defined in Section 4.1.

5.2 Proportional Commission Model

The problem with these two methods is that one requires knowledge of the

price of the underlying securities to compute the transfer vectors. The authors

of [2] suggest using the proportional commission model which assumes a fraction

(0, 1) that an investor pays at a rate of 2

for each buy and for each sell. The

model specifies that the return of a sequence b1, . . . , bn of portfolios with respect to

a market sequence x1, . . . , xn is

t

btxt

1

j

2

bt(j) bt(j)

where

bt = 1btxt

(bt xt)

5.2.1 Modifications to the Proportional Commission Model

We believe that the model as it is stated is wrong. Perhaps it is a typo but in

any case we think that it really should be either

t

btxt

1

j

2

bt+1(j) bt(j)

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if the transaction costs are to be included in the previous days performance or

tbtxt1

j

2bt(j) bt1(j)

if the transaction costs are to be included in the next days performance. We have

arbitrarily decided that the transaction costs be included in the previous days per-

formance measure.

A very important point to consider here is that brokerage fees are not the same

for all types of securities. If we were to apply the Anticor algorithm to deriva-

tive products (such as options, futures, or swaps) which usually incur much smaller

transaction fees, then making the zero transactions fee assumption would be more

acceptable.

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CHAPTER 6Markets Used For Simulation

The experimental study was performed using three different types of data, de-

scribed in the Sections below.

6.1 Old Historical Market Data

Our first data set consisted of historical data for DJIA, SP500, NYSE and

TSE, obtained from the authors of [2]. Running our implementation of the Anticor

algorithm on this data allowed us to verify their results, and to ensure that our

implementation was correct. Another source is the London Stock exchange data set,

DATML1. Table 61 gives the daily sampled mean, variance, skewness and kurtosis

of the old historical market data.

Table 61: Daily Sampled Statistics of Old Historical Markets

Index Start Date End Date Mean Variance Skewness Kurtosis

datML1.txt N/A N/A 1.0004 0.00044 0.21604 13.9545djia.txt 2001-01-14 2003-01-14 0.9997 0.000662 -0.8938 26.557nyse.txt 1962-07-03 1984-12-31 1.0006 0.000399 1.0445 17.7132sp500.txt 1998-01-02 2003-01-31 1.0005 0.000656 0.13304 8.074tse.txt 1994-01-04 1998-12-31 1.0004 0.00057745 1.5791 71.435

6.2 Recent Historical Market Data

Our second data set consisted of recent trading data, obtained from Yahoo [8], for

several market indices, each containing about 30 stocks. Note that the composition

of these indices is given in Appendix B.2. Each index was treated as one market

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for the purpose of the algorithm. Although we could have assembled markets from

other collections of securities, market indices are more representative of practical

trading situations, and are easier to obtain data for. Also, they are more likely to

approximately represent the simplifying assumptions presented earlier.

Several flaws in this data set make it difficult to use with the algorithms:

Stocks that cease to exist during the considered time period are completely

omitted from the provided data set, even for the time when they did exist.

This is difficult to compensate for, as the data set contains no evidence that

the stocks were ever in the index, and it introduces a bias towards stocks that

survived through the entire time period.

A stock may be added to an index mid-way through the time period. When

this occurs, we assume the stock to have a constant price, equal to its earliest

known price, on every day before it was added.

Each stock may have gaps in its sequence of prices, where the last traded

price is unknown for one or more consecutive days. These gaps are filled in by

interpolating between the nearest known prices.

Table 62 gives the daily sampled mean, variance, skewness and kurtosis of the

recent historical market data.

6.3 Simulated Market Data

6.3.1 Modified Random Walk

Gaussian Noise

In this model, we draw each relative price xs(t) (relative price of security s attime t) from a Gaussian distribution with mean and variance 2. For each security

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Table 62: Daily Sampled Statistics of Recent Historical Markets

Index Start Date End Date Mean Variance Skewness Kurtosisdja 1998-11-16 2007-02-02 1.0006 0.00053639 0.68273 62.7138

dji 1998-11-16 2007-02-02 1.0004 0.00043574 -0.032271 11.3017dot 1998-11-16 2007-02-02 1.0012 0.0015245 0.62122 15.9012iix 1998-11-16 2007-02-02 1.0012 0.0022224 0.84644 15.4131ndx 1998-11-16 2007-02-02 1.0012 0.0012992 1.3052 39.0859nwx 1998-11-16 2007-02-02 1.0008 0.0017904 0.37211 12.332nyi 1998-11-16 2007-02-02 1.0007 0.00045597 8.6386 770.3089nyy 1998-11-16 2007-02-02 1.0006 0.00075616 1.6179 88.5836oex 1998-11-16 2007-02-02 1.0006 0.0005303 -0.13936 22.0578soxx 1998-11-16 2007-02-02 1.0009 0.0014157 0.36659 7.8734xau 1998-11-16 2007-02-02 1.0012 0.0011922 0.5639 10.9352xmi 1998-11-16 2007-02-02 1.0003 0.00036708 -0.080012 11.591

s, this produces a price sequence vs(1), . . . , vs(T) that is similar to a random walk,

except that the noise at each time step is scaled according to the price level. Indeed,

a (scalar-valued) Gaussian random walk process is given by

v(t) = v(t 1) + n2(t)

where each n2

(t) is a Gaussian random variable with mean 0 and variance 2

; thisproduces a relative price sequence

x(t) =v(t)

v(t 1)= 1 +

n2

v(t 1)

Hence the larger v(t 1) is, the smaller the variance of x(t) is. This, we feel, is not

representative of real markets; in other words, we believe it is not the case that a

100$ security has a relative price variance approximately 10 times smaller than a 10$

security. Thus, we instead use the modified model given above, which gives the price

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sequence

v(t) = v(t 1) ( + n2(t))

To determine values for and 2, we computed the first four moments of the

historical market data sets. Thus, we set the yearly expected mean to be conserva-

tively = 1.07 (or 7%) and the daily variance to be 2 = 0.0005. Assuming 252

days of trading a year, this corresponds to a daily mean of 1.000268. Comparing

this with the values in Table 61 and Table 62 we see that 7 percent annual return

is smaller than all real market data.

Table 63: Daily Sampled Statistics of Simulated Marketsmam Mean Variance Skewness Kurtosismrw0 1.0003 0.00049848 0.0087546 2.961mrw1 1.0003 0.00049848 0.0087546 2.961mam0 1.0002 0.00049841 0.056457 2.9856mam1 1.0003 0.00051299 0.062445 3.0189mam2 1.0003 0.00054372 0.061123 3.0098mam3 1.0001 0.0005965 0.054341 3.0443mam4 1.0003 0.00072091 0.046434 3.0126mam5 1.0003 0.00088263 0.012532 3.0171

mam6 1.0003 0.0012225 0.013761 3.0878mam7 1.0004 0.0018012 0.0097862 3.2389mam8 1.0003 0.0033626 -0.002742 3.377mam9 1.0003 0.0051434 0.0093402 4.0548

It should be noted here that under such random markets, one cannot expect to

perform well. Hence if the Anticor algorithm performs poorly on mrw0, this should

not come as a surprise.

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Log-normal Noise

Alternatively, we tried to use log normal distributed noise since markets tend to

be positively skewed. In this version of the model, the log-normal distribution has

the same mean and variance as the previous model. Hence, the distribution of the

log-normal relative price is obtained by taking the exponential of the normal

vt = vt1 (e1m+n2)

where parameters are

2 = log1 + V AR(X)(E(X)

2

)

and

= log(E(X))2

2

However, experimenally, the difference between normal and log-normal distributions

was not noticeable as far as the relative performance of the algorithms is concerned.

6.3.2 Modified Autoregressive Model

In this model, the joint movement of all securities relative prices are generatedby the following Rm-valued random process:

x(t) = 1m1 +Ll=1

Dl (1m1 x(t l)) + n(t)

Here, n(t) is a noise process, which may be either normally or log-normally dis-

tributed. The matrices D1, . . . , DL, each of size mm, are parameters that can be

used to express dependencies between different securities. Specifically, the i, j entry

of Dl expresses how much the price of security j will influence the price of i, l days

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later. This overcomes an important limitation of the modified random walk model

given above, which assumes the securities prices to be independent.

This model can be visualized as the system diagram shown in Figure 61.

x(t)

1 + n(t)

z1

+ +

z1

+

z1 . . .

. . .

D2 D1DL

+ 1

+

Figure 61: Noisy Feedback Model

Note that ifL = 1 and D1 = 0, then this model reduces to the modified random

walk model.

For some choices of the matrices D1, . . . , DL, the relative price sequence x(t)

may grow unboundedly, because of positive feedback. We have not explored the exact

conditions under which this happens, but we have found a method for constructingthe D matrices which, intuitively and experimentally, seem to avoid unbounded

growth in x(t). The algorithm for generating these matrices is shown in Appendix

A; by ensuring that each matrix Dl is lower triangular, it avoids cyclical dependencies

between securities, thus preventing positive feedback.

We should mention that stocks are expected to grow. What we are trying to

ensure is that the stocks grow by a limited amount. More work is needed to

determine the bounds of the growth, but experimentally, as shown in Table 63, the

model behaves as we would expect.

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CHAPTER 7Empirical Comparison

We implemented several software systems to perform this experiment. We will

not discuss the implementation details here, however we have provided online a

readme file which explains how to obtain the source code. The readme file is

available at:

http : //www.cim.mcgill.ca/ dcasto4/anticor/readme tex.txt.

In this chaper, we present empirical results for every market mentioned in Chap-

ter 6. We focus our attention on four graphs (as presented in Section 7.1) which we

use to compare the relative performance of the algorithms.

To avoid overcrowding the graphs, we do not display all of the benchmarks. As

stated before, retX(CBAL) retX(BAH

), since we know that BAH* invests only

in the best stock(s), and this strategy is a special case of CBAL*. We thus decided

to hide BAH*. The performance of UBAH and UCBAL are usually similar, so only

one should suffice. However, the performance of UBAH is unaffected by transaction

costs, so we decided to hide it as well. Hence the algorithms presented are as follows:

UCBAL (as defined in Chapter 3)

CBAL* (as defined in Chapter 3)

BAHW(ANTICORw): abbreviated as ANTI1 (as defined in Section 4.3)

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BAHW(ANTICORw(ANTICOR

w)): abbreviated as ANTI2 (as defined in

Section 4.4)

To account for transaction costs, we used the method described in Section 5.2.1.

The friction coefficient used is equal to one percent and the performances of the

algorithms after transaction costs are incurred are shown as a dotted line (and are

denoted by f-name, where f stands for friction).

It is difficult to provide a completely unbiased view, but we hope that these four

graphs provide the reader with enough information to assess the performance and

the risk (as will be discussed in Section 7.1.4) of the Anticor algorithm. In Section

7.1 we discuss each of the four graphs in turn and present a summary of the salient

points observed in the simulations. For a more thorough investigation, we present

(in Section 7.2) specific observations for every market mentioned in Chapter 6.

7.1 Overview

We will now present each of the four graphs and make general observations

about what each graph enables us to see.

7.1.1 Total Return

The first graph we consider shows the total return versus the window sizes. The

benchmark algorithms are not parametrized by the window sizes and so are displayed

as straight lines. The most relevant curves plot the total return of ANTICORw (and

similarly, ANTICORw(ANTICORw)) algorithms, which are the components of

ANTI1 (and similarly, ANTI2).

It is interesting to note that the choice of w heavily impacts the total return ofthe Anticor algorithm, and that compounding the Anticor algorithm does not always

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lead to better performance. In some but not all cases, the Anticor algorithm beats

CBAL* for both old and recent historical markets. For simulated markets, it is clear

that as the dependence factor increases in magnitude, so does the comparative

performance of the Anticor algorithm. For most of these simulated markets, ANTI1

provides higher returns than ANTI2, which could be attributed to the the simplicity

of the simulated markets: ANTI2 is attempting to exploit complex interdependencies

that do not exist. Note that the maximum lag is 30, so it is not surprising that the

performance of the Anticor algorithm declines greatly for window sizes between 30

and 50.

The total return versus window size enables us to see how ANTICORw performs

with respect to the window size parameter. If ANTICORw does better than the

benchmark algorithms for all window sizes, we know that for the specific period

between the start date and the end date the Anticor algorithm (irrespective of the

window size used) was a better way to invest. However, the total return versus

window size graph tells us little about risk and performance over time. Indeed, it is

heavily biased by the specific choice of start date and end date.

7.1.2 Cumulative Return

The cumulative return graph enables us to look at return over time, removing

the bias associated with choosing a specific end date (but retaining the bias from the

start date). It also allows us to obtain an idea of the risk by looking at the volatility

of the cumulative return.

The cumulative return graphs plot, as a function of the number n of days sincethe start, the return obtained during the first n days. These plots show the growth

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(in absolute terms) of the portfolio given a strategy. This view enables us to avoid

the bias associated with the starting point, as we can see that certain periods account

for much of the growth.

7.1.3 In-Hindsight Geometric Mean Return

Given a particular ending date, the in-hindsight geometric mean return (IGMR)

over T days is the average yearly return that we would have obtained if we had started

investing T days before this end date. (Note that larger values of T correspond to

earlier start dates.) By examining this quantity, we avoid the bias associated with

choosing a start date, but still incur bias from the chosen end date.

It is obvious that, in principle, we would want to always invest in the strategy

with the highest IGMR at every t, to maximize the amount of return obtained at the

end date. Also interesting to note are the cross-overs between strategies; cross-over

signifies that the strategy going up will do better than the strategy going down

for the coming while.

Another interesting point to note is that UCBAL is usually in the 10 percent

region while CBAL* is in the 20 percent region. ANTICORw produces returns near

50 percent on some indices, which is quite exceptional.

7.1.4 Sharpe Ratio

The Sharpe ratio enables us to look at risk-adjusted return, which is a measure

of investment performance that accounts for both the return obtained and the risk

incurred.

The finance literature on balancing risk and return, and the proposed metricsfor doing so, are far too large to survey here (see [1], Chapter 4 for an overview).

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Among the most common methods are the Sharpe ratio [14], and the mean-variance

(MV) criterion, of which Markowitz was the first proponent [12].

Even after taking the transaction costs into account, ANTI1 and ANTI2 have a

higher Sharpe ratio than UCBAL for some indices. Table 71 summarizes how the

Sharpe ratios of ANTI1 and ANTI2 compare to those of CBAL.

Table 71: Comparison of Sharpe ratios of ANTI1/ANTI2 with CBAL. The Bet-ter column contains the data sets where ANTI1 and ANTI2 always had betterSharpe ratios than CBAL; similarly for the Worse column.

Better Mixed WorseOld market data

djia.txt datML1.txtsp500.txt

tse.txtnyse.txt

Recent market datadji soxx djaiix dot ndx

nyy nwxxau nyi

oex

xmi

This distribution is impressive and puts the Anticor algorithm in a good light.

Whether such Sharpe ratios will exist in the future is a difficult question to answer.

On the other hand, it is clear that the results provided by the authors of [2] paint

a more positive picture than the results we obtained. We also see that even with

small commissions, the Sharpe ratio is significantly affected. The performance is

diminished, but we see that the risk stays the same, which serves as a sanity check

of the results.

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7.2 Market Detailed View

7.2.1 Old Historical Market Data

This Section covers the empirical comparisons for old offline markets. We

consider the four markets considered by the authors of [2] and one extra market,

datML1, for the London Stock Exchange.

It should be noted that the dates on the graphs are not meaningful because the

results where obtained offline and we do not have access to the exact dates at which

the data were recorded.

datML1

Figure 71 shows the empirical results obtained for the datML1 historical market

data set. The total return graph shows that the Anticor algorithm performed better

than the UCBAL but worse than CBAL*. We also note that when transaction

costs are taken into consideration the Anticor Algorithm still performs better than

UCBAL.

The Sharpe ratio shows that CBAL* offers more return for a greater risk. On a

return per unit of risk basis CBAL* is also the clear winner. In the IGMR graph it is

interesting to note that for a short while in the middle section the ANTI2 algorithm

actually surpassed CBAL*. Hence, an investor starting to invest during this short

time window could have expected to obtain a greater return from ANTI2 than from

CBAL*.

In the cumulative return graph we observe that during a short period of time

ANTI1 had less cumulative return than even UCBAL but somewhere after the mid-point ANTI1 managed to surpass UCBAL. The clear winner for datML1 is CBAL*

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but since CBAL* requires knowledge of the future, ANTI1 or ANTI2 would have

given excellent returns.

0 50 100 150 200 2500

50

100

150

Time (days)

YearlyR

eturn(%)

IGMR (up to February 02, 2007)

0 100 200 300 400 5000

1

2

3

4

5

Time (days)

CumulativeReturn

Cumulative Daily Return

0 0.01 0.02 0.03 0.040

0.5

1

1.5

2

2.5

3x 10

3

Daily Risk

DailyReturn

Risk

freeRate=0.0

4

Sharpe Ratio

0 20 40 601

1.5

2

2.5

3

3.5

4

4.5

Window Size 1

TotalReturn

Total Return

ucbalfucbalcbal*fcbal*anti1

fanti1anti2fanti2

datML1.txt

Figure 71: Results for datML1.txt

djia

Figure 72 shows results that are consistent with those of the authors of [2]. It

is interesting to note that both ANTI1 and ANTI2 performed extremely well bothin terms of total return and Sharpe ratio. These market data clearly demonstrate

that historically the Anticor algorithm could have generated high returns.

nyse

Figure 73 confirms the extraordinary results obtained on the nyse by authors

of [2]. All the graphs confirm that ANTI1 and ANTI2 offer much higher returns and

risk-adjusted returns than the benchmark algorithms. Interestingly CBAL* offers

a lower Sharpe ratio than UCBAL, however this could be due to the fact that this

data set contains more trading days than the other ones.

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0 100 200 30020

0

20

40

60

80

100

Time (days)

YearlyReturn(%)


0 200 400 6000.5

1

1.5

2

2.5

Time (days)

CumulativeReturn


0 0.005 0.01 0.015 0.02 0.0255

0

5

10

15

20x 10

4

Daily Risk

DailyR

eturn

Risk

freeR

ate=0.0

4

Sharpe Ratio

0 20 40 600.5

1

1.5

2

2.5

Window Size 1

TotalReturn

Total Return

ucbalfucbalcbal*fcbal*

anti1fanti1anti2fanti2

djia.txt

Figure 72: Results for djia.txt

0 1000 2000 30000

50

100

150

200

Time (days)

YearlyReturn(%)


0 2000 4000 60000

0.5

1

1.5

2

2.5 x 108

Time (days)

CumulativeReturn


0 0.005 0.01 0.015 0.02 0.0250

0.5

1

1.5

2

2.5

3

3.5x 10

3

Daily Risk

DailyReturn

Risk

freeRate=0.0

4

Sharpe Ratio

0 20 40 600

2

4

6

8

10x 10

8

Window Size 1

TotalReturn

Total Return

ucbalfucbalcbal*fcbal*anti1fanti1

anti2fanti2

nyse.txt

Figure 73: Results for nyse.txt

sp500

The Anticor algorithm performed well on the sp500 as shown in Figure 74. The

most notable aspect is that in the IGMR graph, the yearly return of all strategies

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goes down as time increases. This suggests that as time passes, over the period

covered by the the sp500 data set, the overall market performed worse.

0 200 400 600 80020

0

20

40

60

Time (days)

YearlyR

eturn(%)


0 500 1000 15000

1

2

3

4

5

6

7

Time (days)

CumulativeReturn


0 0.005 0.01 0.015 0.02 0.025 0.030

0.5

1

1.5x 10

3

Daily Risk

DailyReturn

Risk

freeRate=0.0

4

Sharpe Ratio

0 20 40 600

2

4

6

8

10

12

14

Window Size 1

TotalReturn

Total Return


fanti1anti2fanti2

sp500.txt

Figure 74: Results for sp500.txt

tse

The Toronto Stock Exchange historical market empirical results shown in Figure

75 are also consistent with what the authors of [2] found. Of particular interest isthat even though in terms of absolute return ANTI2 performs better than ANTI1,

the risk-adjusted return of ANTI1 is superior to that of ANTI2. In fact, the Sharpe

ratio of f-ANTI2 is approximately equal to that of CBAL*.

7.2.2 Recent Historical Market Data

dja

Figure 76 shows the four graphs for the dja market index. We should note that

the window size parameter has a significant effect. Indeed, for small window sizes

both ANTI1 and ANTI2 perform worse than CBAL* but perform better for larger

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0 200 400 600 8000

20

40

60

80

100

120

Time (days)

YearlyReturn(%)


0 500 1000 15000

5

10

15

20

25

30

Time (days)

CumulativeReturn


0 0.01 0.02 0.03 0.04 0.050

0.5

1

1.5

2

2.5

3x 10

3

Daily Risk

DailyR

eturn

Risk

freeR

ate=0.0

4

Sharpe Ratio

0 20 40 600

5

10

15

20

25

30

35

Window Size 1

TotalReturn

Total Return



tse.txt

Figure 75: Results for tse.txt

window sizes. Furthurmore, we find interesting that the Sharpe ratio of CBAL* is

the best. Hence for an equal amount of risk (assuming that the investor can borrow

at the risk free rate of four percent), an investor should prefer CBAL* over ANTI1

or ANTI2.

dji

The dji market is one of the new historical markets where the Anticor algorithm

performs best. As shown in Figure 77 both ANTI1 and ANTI2 perform better than

UCBAL and CBAL*. In addition, the Sharpe ratio of the Anticor algorithms are

better than CBAL* even after transaction costs. However, as can be observed from

the IGMR graph, most of the return is accumulated prior to 2002. In fact, if we look

at the cumulative return graph it appears that the period 2002 to 2005 resulted in a

negative return.

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Nov 1998 Dec 200210

20

30

40

50

60

70

80

Time (days)

YearlyReturn(%)


Nov 1998 Dec 2002 Feb 20070

5

10

15

20

25

30

35

Time (days)

CumulativeReturn


0 0.01 0.02 0.03 0.04 0.050

0.5

1

1.5

2x 10

3

Daily Risk

DailyR

eturn

Risk

freeR

ate=0.0

4

Sharpe Ratio

0 20 40 600

10

20

30

40

Window Size 1

TotalReturn

Total Return



dja

Figure 76: Results for dja

Nov 1998 Dec 20025

10

15

20

25

30

35

40

Time (days)

YearlyReturn(%)


Nov 1998 Dec 2002 Feb 20070

2

4

6

8

10

12

14

Time (days)

CumulativeReturn


0 0.005 0.01 0.015 0.020

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

3

Daily Risk

DailyReturn

Risk

freeRate=0.0

4

Sharpe Ratio

0 20 40 600

5

10

15

20

25

Window Size 1

TotalReturn

Total Return


anti2fanti2

dji

Figure 77: Results for dji

dot

Similarly to dji, the Anticor algorithms performs well on the dot market. How-

ever, it should be noted that when the transaction costs are taken into consideration

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the ANTI1 performs worse than CBAL*. In the cumulative return graph shown

in Figure 78, we note that most of the gains of ANTI2 were accomplished after

2002. In that respect, the performance of the Anticor algorithm on the dot market

is negatively correlated with the performance on the dji market.

Nov 1998 Dec 20020

20

40

60

80

100

Time (days)

YearlyReturn(%)


Nov 1998 Dec 2002 Feb 20070

10

20

30

40

50

60

70

Time (days)

CumulativeReturn


0 0.01 0.02 0.03 0.040

0.5

1

1.5

2

2.5x 10

3

Daily Risk

DailyReturn

Risk

freeRate=0.0

4

Sharpe Ratio

0 20 40 600

50

100

150

200

250

Window Size 1

TotalReturn

Total Return


anti2fanti2

dot

Figure 78: Results for dot

iix

The performance of the Anticor algorithm for the iix market suggests that the

extremely high returns on the old historical nyse are not an isolated case. Indeed,

the Anticor algorithms would have obtained over 60 percent of yearly return had an

investor used either ANTI1 or ANTI2 since 1998 as shown in Figure 79. In addition,

this would have been accomplished at a small level of risk. Indeed, the daily risk of

ANTI1 is equal to the daily risk of CBAL*.

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Nov 1998 Dec 20020

20

40

60

80

100

120

140

Time (days)

YearlyReturn(%)


Nov 1998 Dec 2002 Feb 20070

50

100

150

200

250

Time (days)

CumulativeReturn


0 0.01 0.02 0.03 0.04 0.050

0.5

1

1.5

2

2.5

3x 10

3

Daily Risk

DailyR

eturn

Risk

freeR

ate=0.0

4

Sharpe Ratio

0 20 40 600

200

400

600

800

1000

1200

Window Size 1

TotalReturn

Total Return



iix

Figure 79: Results for iix

ndx

In Figure 710, we observe the four graphs for the ndx market. As in many

other instances, both ANTI1 and ANTI2 perform better than UCBAL even when

transaction costs of one percent are taken into consideration. However, unlike for

other markets, the total performance of CBAL* is between that of the two Anticor

algorithms. Most interestingly, a closer examination of the risk-adjusted performance

suggests that in this case the preferred strategy would be to adopt CBAL* (if it were

possible to view in the future). As shown in the IGMR graph, at all times it is clearly

preferable to invest in either ANTI1 or ANTI2 rather than UCBAL.

nwx

Similarly to the ndx market, the nwx market differentiates the performance of

ANTI1 versus ANTI2 by a large factor. However, as shown in Figure 711, the

Sharpe ratios of both Anticor algorithms outperform CBAL*. It is interesting to

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Nov 1998 Dec 20020

20

40

60

80

100

120

140

Time (days)

YearlyReturn(%)


Nov 1998 Dec 2002 Feb 20070

100

200

300

400

500

Time (days)

CumulativeReturn


0 0.01 0.02 0.03 0.040

0.5

1

1.5

2

2.5

3x 10

3

Daily Risk

DailyR

eturn

Risk

freeR

ate=0.0

4

Sharpe Ratio

0 20 40 600

100

200

300

400

500

600

700

Window Size 1

TotalReturn

Total Return



ndx

Figure 710: Results for ndx

note that most of the gain realized by ANTI2 occured post 2002, a period during

which ANTI1 tracked approximately the performance of CBAL*.

Nov 1998 Dec 200220

0

20

40

60

80

Time (days)

YearlyReturn(%)


Nov 1998 Dec 2002 Feb 20070

10

20

30

40

50

60

Time (days)

CumulativeReturn


0 0.01 0.02 0.03 0.040

0.5

1

1.5

2x 10

3

Daily Risk

Daily

Return

Risk

freeRate=0.0

4

Sharpe Ratio

0 20 40 600

20

40

60

80

Window Size 1

Tota

lReturn

Total Return



nwx

Figure 711: Results for nwx

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nyi

Figure 712 shows impressive total returns earned by ANTI1 and ANTI2 be-

tween November 1998 and December 2002. Here we need to emphasize that the total

return graph is not in units of percent; it is the multiple times the initial investment.

The IGMR graph shows that over that time period, UCBAL had yearly returns

slightly less than 20 percent, which is also very good. It is astonishing to see that,

even after accounting for transaction fees, both ANTI1 and ANTI2 could have yielded

returns in excess of 60 percent, every year.

In terms of the Sharpe ratio, ANTI1 has the clear lead with similar returns

to ANTI2 but lower risk. Hence ANTI1 would have been the preferred strategy in

terms of risk-adjusted return.

Nov 1998 Dec 20020

20

40

60

80

100

120

Time (days)

YearlyReturn(%)


Nov 1998 Dec 2002 Feb 20070

20

40

60

80

100

120

Time (days)

CumulativeReturn


0 0.01 0.02 0.03 0.04

0

0.5

1

1.5

2

2.5x 10

3

Daily Risk

DailyReturn

Risk

freeRate=0.0

4

Sharpe Ratio

0 20 40 60

0

50

100

150

200

250

Window Size 1

TotalReturn

Total Return


fanti1anti2fanti2

nyi

Figure 712: Results for nyi

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nyy

Figure 713 shows impressive returns for the nyy. As surprising as the results

on nyi were, the results for nyy are superior. Both of these markets start with the

letters ny (for New York Stock Exchange), however they have little overlap. On

the one hand, nyi contains international stocks from all industries, while on the other

hand, nyy contains technology, media and telecommunications stocks.

One interesting observation is that we can clearly observe a sharp decline in the

IGMR around 2000-2001, the time when the so called dot-com bubble burst.

Nov 1998 Dec 20020

20

40

60

80

100

120

140

Time (days)

Yea

rlyReturn(%)


Nov 1998 Dec 2002 Feb 20070

100

200

300

400

500

Time (days)

Cum

ulativeReturn


0 0.01 0.02 0.03 0.040

0.5

1

1.5

2

2.5

3x 103

Daily Risk

DailyReturn

Risk

freeRate=0.0

4

Sharpe Ratio

0 20 40 600

100

200

300

400

500

Window Size 1

TotalReturn

Total Return


fanti1anti2fanti2

nyy

Figure 713: Results for nyy

oex

The total return graph for the oex market (S&P 100 Index - American) is shown

in Figure 714. This graph shows that the performance of ANTI1 and ANTI2 can

be strongly affected by the window size used. Indeed, we observe a trend where the

higher the window size the higher the total return. One interesting observation is

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that even though ANTI1 and ANTI2 have higher Sharpe ratios than CBAL*, when

transactions costs are included the three strategies appear to have equal Sharpe

ratios. In the cumulative return graph we observe that much of the growth occured

in early 2002.

Nov 1998 Dec 20020

20

40

60

80

Time (days)

YearlyReturn(%)


Nov 1998 Dec 2002 Feb 20070

10

20

30

40

Time (days)

CumulativeReturn


0 0.01 0.02 0.03 0.040

0.5

1

1.5

2x 10

3

Daily Risk

DailyReturn

Risk

freeRate=0.0

4

Sharpe Ratio

0 20 40 600

5

10

15

20

25

30

35

Window Size 1

TotalReturn

Total Return

ucbalfucbalcbal*

fcbal*anti1fanti1anti2fanti2

oex

Figure 714: Results for oex

soxxFigure 715 shows the four graphs for the soxx (Philadelphia Stock Exchange

Semiconductor Sector) market. The ANTI1 and ANTI2 curves for the total return

versus window size graph start high and gradually decrease as the window size in-

creases. It is interesting to note in the cumulative return graph that a sharp increase

in wealth occured between 1998 and 2000 followed by a strong decline. In early 2002,

it seems that all four strategies resulted in an equal total return. However, during the

period between December 2002 and February 2007, ANTI2 (and to a lesser extent

ANTI1) has shown stalwart performance.

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Nov 1998 Dec 200210

0

10

20

30

40

50

Time (days)

YearlyReturn(%)


Nov 1998 Dec 2002 Feb 20070

5

10

15

Time (days)

CumulativeReturn


0 0.01 0.02 0.03 0.040

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

3

Daily Risk

DailyR

eturn

Risk

freeR

ate=0.0

4

Sharpe Ratio

0 20 40 600

10

20

30

40

50

Window Size 1

TotalReturn

Total Return



soxx

Figure 715: Results for soxx

xau

Of all the recent historical markets, xau is perhaps the most curious one. Figure

716 shows the exceptionally high total return for ANTI1 and ANTI2. If an investor

had decided to invest in the xau market in November 1998 using the ANTI2 strategy,

he would have obtained over 100 percent return every year until December 2002. On

the other hand, we can observe that starting to invest in this market using ANTI2 at

a later time resulted in marginally smaller yearly return every following year. Yet,

even for the investor joining in 2002 would have earned over 20 percent per year, an

excellent return when compared to the market.

xmi

Figure 717 shows the results obtained for the xmi market. In the total return

graph we observe that ANTI1 with transaction costs results in a worse total return

for window size of 2 than UCBAL but in a better performance than both UCBAL

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Nov 1998 Dec 200220

40

60

80

100

120

140

Time (days)

YearlyReturn(%)


Nov 1998 Dec 2002 Feb 20070

200

400

600

800

Time (days)

CumulativeReturn


0 0.01 0.02 0.03 0.040

0.5

1

1.5

2

2.5

3

3.5x 10

3

Daily Risk

DailyR

eturn

Risk

freeR

ate=0.0

4

Sharpe Ratio

0 20 40 600

500

1000

1500

2000

Window Size 1

TotalReturn

Total Return



xau

Figure 716: Results for xau

and CBAL* for window size 50. The inverse performance relationship exists for

ANTI2.

It is also interesting to note how the IGMR graph is distributed. For the period

before 1999, the ANTI1 and ANTI2 strategies provided returns better than CBAL*.

However, after 2001, ANTI2 performed worse than CBAL* and ANTI1 performed

worse than UCBAL. This market contains major stocks and mirrors the Dow Jones

Industrial Average. That the results are mixed suggests, perhaps, that major stocks

are priced more efficiently and behave more randomly than others.

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Nov 1998 Dec 20025

0

5

10

15

20

25

30

Time (days)

YearlyReturn(%)


Nov 1998 Dec 2002 Feb 20070

1

2

3

4

5

6

Time (days)

CumulativeReturn


0 0.005 0.01 0.015 0.020

2

4

6

8x 10

4

Daily Risk

DailyRetu

rn

Risk

freeRate

=0.0

4

Sharpe Ratio

0 20 40 600

2

4

6

8

10

12

Window Size 1

TotalReturn

Total Return

ucbalfucbalcbal*f

cbal*anti1fanti1

anti2fanti2

xmi

Figure 717: Results for xmi

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7.3 Simulated Market Data

7.3.1 Modified Random Walk

Figures 718 and 719 show the four graphs for the modified random walk

simulated market as described in Section 6.3.1. The relative prices are independent

and identically-distributed random variables and so no correlation exists between

the relative prices. It is not surprising therefore to find that ANTI1 and ANTI2

performed very poorly on these markets. This suggests that in real markets, relative

prices are not independent and identically-distributed.

0 500 1000 150040

20

0

20

40

60

Time (days)

Yea

rlyReturn(%)

IGMR (up to March 15, 2007)

0 1000 2000 30000

10

20

30

40

Time (days)

Cum

ulativeReturn


0 0.005 0.01 0.015 0.02 0.0251

0.5

0

0.5

1

1.5x 103

Daily Risk

DailyReturn

Risk

freeRate=0.0

4

Sharpe Ratio

0 20 40 600

10

20

30

40

Window Size 1

TotalReturn

Total Return

ucbalfucbal

cbal*fcbal*anti1fanti1anti2fanti2

mrw

Figure 718: Results for mrw

7.3.2 Modified Autoregressive Model

Figures starting from 720 up to and including 729 show the simulation results

obtained on the modified autoregressive model as the dependence parameter was

increased from = 010

to = 910

. (The use of the dependence para

AntiCor Revisited

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