Lorenzo Coviello and Petros Mol PORTFOLIO SELECTION: experimental comparison of Universal and non-universal Algorithms Lorenzo Coviello and Petros Mol Universal Information Processing, Spring 2011 June 2, 2011

Motivation Investing money in the stock market How to build a successful portfolio? Compare various strategies

Introduction 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

Rest of the talk Introduction Portfolio selection: the model Methodology Two approaches Reversal to the mean Trend is your friend Simulations - Comparison

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)

The model - wealth Portfolio at day i The wealth gain in one day The overall wealth gain in T days

The model - strategy How to distribute the wealth among the stocks? Decision problem: choose a portfolio each day

Rest of the talk Introduction Portfolio selection: the model Methodology Two approaches Reversal to the mean Trend is your friend Comparison

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

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

Rest of the talk Introduction Portfolio selection: the model Methodology Two approaches Reversal to the mean Trend is your friend Comparison

Two main approaches Reversal to mean Trend is your friend 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

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

Simulation

Rest of the talk Introduction Portfolio selection: the model Methodology Two approaches Reversal to the mean Trend is your friend Comparison

Reverse to mean approach Assumptions Stock growth rates stable in the long run Occasional larger returns followed by smaller rates, and vice versa

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

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

Semi-constant rebalanced portfolio Consider rebalancing every d days Uniform target distribution The wealth before rebalancing for the kth time

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

Simulation (with fixed interval)

Simulation (with distance threshold)

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

ANTICOR algorithm Define Averages of columns of LXk

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

ANTICOR algorithm Transfer money from stock i to stock j if In an amount proportional to

Simulation (with variable window length)

Simulation (smaller window length)

Simulation (zoom in)

Simulation (zoom in)

Simulation (zoom in)

Rest of the talk Introduction Portfolio selection: the model Methodology Two approaches Reversal to the mean Trend is your friend Comparison

The trend is your friend Portfolio based on stock performance Prefer performing (trendy) stocks Use the market history to determine the current portfolio

Histogram portfolio selection Ref: Gyorfi and Schafer (2003) Rectangular window of width w days Distribute the wealth uniformly among k best stocks

Simulation (variant window)

Simulation (variable #active stocks)

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

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

Simulation

Summary of Cases Reversal to the mean Trend is your friend Constant Rebalancing (CRP) Semi-CRP ANTICOR Buy and Hold Histogram Kernel

Comparing the winners (w/o Anticor)

Put all your money in Anticor! Conclusion Put all your money in Anticor! But choose the right window!!!

Lorenzo Coviello and Petros Mol THANKS Lorenzo Coviello and Petros Mol