1. 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

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

3. 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

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

5. 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)

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

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

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

9. Methodology Data Collected from Yahoo! finance
Adjusted close price used
Period:
3778 trading days
No priors on the stocks,
no fundamentals
No transaction costs

10. 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

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

12. 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

13. 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

14. Simulation

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

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

17. 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

18. 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

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

20. 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

21. Simulation (with fixed interval)

22. Simulation (with distance threshold)

23. 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

24. ANTICOR algorithm
Define
Averages of columns of LXk

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

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

27. Simulation (with variable window length)

28. Simulation (smaller window length)

29. Simulation (zoom in)

30. Simulation (zoom in)

31. Simulation (zoom in)

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

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

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

35. Simulation (variant window)

36. Simulation (variable #active stocks)

37. 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

38. 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

39. Simulation

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

41. Comparing the winners (w/o Anticor)

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

43. Lorenzo Coviello and Petros Mol
THANKS
Lorenzo Coviello and Petros Mol