PORTFOLIO SELECTION:EXPERIMENTAL COMPARISON OF UNIVERSAL AND

NON-UNIVERSAL ALGORITHMS

Lorenzo Coviello and Petros Mol

June 2, 2011

Universal Information Processing, Spring 2011

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Motivation

• Investing money in the stock market

• How to build a successful portfolio?

• Compare various strategies

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• Universal portfolio selection: provides guarantees on wealth growth rate

• Real market: invest in the most profitable way

• Compare performance of portfolio selection criteria on real data from the stock market

Introduction

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Rest of the talk • Introduction

• Portfolio selection: the model

• Methodology

• Two approaches• Reversal to the mean• Trend is your friend

• Simulations - Comparison

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The model – price relatives• Portfolio: m stocks

• Trading period: T trading days

• Xij: price relative of stock j at day i

• Xi often assumed i.i.d. (strong assumption)

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The model - wealth• Portfolio at day i

• The wealth gain in one day

• The overall wealth gain in T days

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The model - strategy• How to distribute the wealth among the stocks?

• Decision problem: choose a portfolio each day

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Rest of the talk • Introduction

• Portfolio selection: the model

• Methodology

• Two approaches• Reversal to the mean• Trend is your friend

• Comparison

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Methodology

• Data Collected from Yahoo! finance

• Adjusted close price used

• Period: 1996- 2010• 3778 trading days

• No priors on the stocks, no fundamentals

• No transaction costs

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Portfolio: List of Stocks

• Tech (11) : AMD, Apple, AT&T, Cisco, Dell, HP, IBM, Intel, Microsoft, Nokia, Oracle

• Finance (7): American Express, Bank of America, Barclay’s, Citigroup, JP Morgan, Morgan

Stanley, Wells Fargo

•Other (12) : Boeing , BP, Coca-Cola Company, Exxon, Ford, General Electric, J&J, McDonalds,

Pfizer, P&G, Wall Mart, Walt Disney

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Rest of the talk • Introduction

• Portfolio selection: the model

• Methodology

• Two approaches• Reversal to the mean• Trend is your friend

• Comparison

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Two main approaches

• Reversal to mean• Assume stock growth rates stable in the long run, and• Occasional larger returns followed by smaller rates• CRP, Semi-CRP, ANTICOR

• Trend is your friend• Portfolio based on recent stock performance• Histogram portfolio selection, kernel portfolio selection

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Buy and hold• Build portfolio once, let the wealth grow

• Uniform buy and hold (U-BAH)

• Performance guarantees for U-BAH

• Best BAH in hindsight: invest on the best stock

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Simulation

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Rest of the talk • Introduction

• Portfolio selection: the model

• Methodology

• Two approaches• Reversal to the mean• Trend is your friend

• Comparison

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Reverse to mean approach

Assumptions• Stock growth rates stable in the long run• Occasional larger returns followed by smaller rates, and vice

versa

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Constant rebalancing portfolio

• Rebalance portfolio every day according to pmf b

• Uniform CRP:

• Exponential gain if “reversal to the mean” market• Stock 1: constant value• Stock 2: doubles on odd days, halves on even days• Uniform CRP• Wealth grows of 1/8 every 2 days

• Best CRP in hindsight difficult to compute

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Semi-constant rebalanced portfolio

• Reference: Kalai (1998), Helmbold (1998), Kozat (2009)

• Portfolio rebalanced every arbitrary period

• Rebalancing period can be fixed

• Real market: reduced commissions

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Semi-constant rebalanced portfolio• Consider rebalancing every d days

• Uniform target distribution

• The wealth before rebalancing for the kth time

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Semi-CRP with deviation control

• Ref. Kozat (2009)

• Idea: avoid useless rebalancing

• Rebalance only if large distance between target portfolio b and current wealth distribution w

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Simulation (with fixed interval)

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Simulation (with distance threshold)

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ANTICOR algorithm

• Reference: Borodin, El-Yaniv, Gogan (2004)

• Aggressive “reversal to the mean”

• Transfer money from stock i to stock j if• Growth of stock i > growth of stock j over last window• Stock i in second last window and stock j in last window

positively correlated

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ANTICOR algorithm• Define

• Averages of columns of LXk

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ANTICOR algorithm• Cross correlation• stock i over the second last window• stock j over the last window

• Normalization

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ANTICOR algorithm• Transfer money from stock i to stock j if

• In an amount proportional to

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Simulation (with variable window length)

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Simulation (smaller window length)

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Simulation (zoom in)

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Simulation (zoom in)

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Simulation (zoom in)

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Rest of the talk • Introduction

• Portfolio selection: the model

• Methodology

• Two approaches• Reversal to the mean• Trend is your friend

• Comparison

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The trend is your friend

• Portfolio based on stock performance

• Prefer performing (trendy) stocks

• Use the market history to determine the current portfolio

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Histogram portfolio selection

• Ref: Gyorfi and Schafer (2003)

• Rectangular window of width w days

• Distribute the wealth uniformly among k best stocks

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Simulation (variant window)

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Simulation (variable #active stocks)

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Kernel portfolio selection• Higher weight to the recent past

• Window size of w days

• Window shape• Linear• Exponential

• Example: score of stock j at day i+1

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Kernel portfolio selection

• Each day the scores determine the portfolio

• Examples• Follow the best stock• Uniform distribution between k best stock• Proportional to score for best k stocks

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Simulation

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Summary of Cases

Reversal to the mean Trend is your friend- Constant Rebalancing (CRP)- Semi-CRP- ANTICOR

- Buy and Hold- Histogram- Kernel

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Comparing the winners (w/o Anticor)

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Conclusion

Put all your money in Anticor!

But choose the right window!!!

THANKSLorenzo Coviello and Petros Mol