PORTFOLIO SELECTION: experimental comparison of Universal and non-universal Algorithms
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Transcript of PORTFOLIO SELECTION: experimental comparison of Universal and non-universal Algorithms
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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!!!
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THANKSLorenzo Coviello and Petros Mol