1. Universal Portfolio Selection: Application of Information Theory in Finance- SC500 Project Presentation - Gudrun Olga Stefansdottir May 5 2007

2. Outline • Introduction • Information Theory in Gambling • Information Theory and the Stock Market • Concepts and Terminology • Cover’s Universal Portfolio (CUP) • Simulations • Conclusion • References

3. Introduction • How can Information Theory be applied in Finance? • Although Shannon never published in this area he gave a well-attended lecture in the mid 1960s at MIT, about maximizing the growth rate of wealth [5] • Discussion meetings with Samuelson (a Nobel Prize winner-to-be in economics) on information theory and economics • Growth-rate optimal portfolios • Financial value of side information • Universal portfolios – counterpart to universal data compression

4. Information theory in Gambling • Relationship between gambling and information theory was first noted over fifty years ago and has subsequently developed into a theory of investment • Horse Race Problem: • m horses • horse iwins w.p.pi and has payoff oi • bi=fraction of wealth invested in horse i, • Wealth after nraces • where are the race outcomes and S(X)=b(X)o(X) is the factor by which the gambler’s wealth is multiplied when horse X wins.

5. We can represent the stock market as a vector of stocks m: Number of stocks Xi: Price relatives, ratio of price of stock i at the end of the day to the beginning. x  F(x): Joint distribution of the vector of price relatives Portfolio: Allocation of wealth accross various stocks. where bi is the fraction of wealth allocated to stock i. The Stock Market Example: Alice has wealth $100 and her portfolio consists of the following fractions: 50% in IBM, 25% in Disney, 25% in GM

6. Concepts and Terminology • Wealth-Relative: Ratio of wealth at the end of the day to the beginning of the day • We want to maximize S in some sense and find the optimal portfolio!

7. Portfolio selection • Growth-rate of wealth of a stock market portfolio b: • Optimal growth-rate: • A portfolio b* that achieves this maximum is called a log-optimal portfolio (or a growth-rate optimal portfolio). • Lets define the wealth-relative (wealth factor) after n days using the portfolio b* as

8. Portfolio selection – cont. • Let be i.i.d. According to distribution F(x) • It can be shown using strong law of large numbers that , hence • An investment strategy that achieves an exponential growth rate of wealth is called log-optimal. • What investment strategy achieves this? • Constant rebalanced portfolio (CRP):An investment strategy that keeps the same distribution of wealth among a set of stocks from day to day.

9. Constant Rebalanced Portfolio Example: Alice has wealth $100 and she plans to keep it CRP. Day1: Alice makes her initial investment action, she buys 50% in IBM, and 50% in BAC. End of Day1: IBM ↑2%  $51, BAC ↓1%  $49.50. Wealth has increased to $100.50. Needs to sell $0.75 in IBM to buy BAC. Day2: Alice has adjusted her portfolio to the original fractions. • It can be shown that the constant rebalanced portfolio,b*, achieves an exponential growth rate of wealth.  CRP is log-optimal! • But what should the fixed percent allocation be? • The best CRPcan only be computed with knowledge of market performance

10. Cover’s Universal Portfolio (CUP) “A universal online portfolio selection strategy “ • Universal: No distr. assumbtions about sequence of price relatives • Online: Decide our action each day, without knowledge of future • CUP – how does it work? • Establish a set of allowable investment actions • Goal: Achieve the same asymptotic growth rate of wealth as the best action in this set • Uniformal optimization over all possible sequences of price relatives • Individual sequence minimax regret solution • The portfolio used on day i depends on past market outcomes

11. Universal portfolios • Intuitively: Each day the stock proportions in CUP are readjusted to track a constantly shifting “center of gravity” where performance is optimal and investment desirable. • The investor buys very small amounts of every stock in the market and in essence, mimics the buy order of a sea of investors using all possible “constant rebalanced” strategies. • Mathematically:

12. CUP – cont. • It has been shown (at various levels of generality [5]) that there exists a universal portfolio achieving a wealth at time n s.t. for every stock market sequence and for every n, where is the wealth generated by the best constant rebalanced portfolio in hindsight. • Drawbacks: • Does not incorporate transaction costs/broker fees • High maintenance: needs to be rebalanced daily • Needs higly volatile stocks So what is the catch???

13. Simulations • Performed simulations using historical stock market data from http://finance.yahoo.com(01/02/1990-12/29/2006) • Implemented the efficient version of the algorithm [6] Value line: Equal proportion invested in each stock in the portfolio (market average) Wealth relative: Relative increase in wealth if entire money invested in that particular stock.

14. 12/14/1993-12/06/1995 – 500 days

15. Increased volatility, x8

16. “A good gambler is also a good data compressor” • The lower bound on CUP corresponds to the associated minimax regret lower bound for universal data compression Mathematics parallel to the mathematics of data compression • Any sequence in which a gambler makes a large amount of money is also a sequence that can be compressed by a large factor. • High values of wealth lead to high data compression  If the text in question results in wealth then bits can be saved in a naturally associated deterministic data compression scheme. • If the gambling is log optimal, the data compression achieves the Shannon limit H

17. Incorporating Side Information • CUP has also been proposed that uses side information, [2] • Let (X,Y) ~ f(x,y), X: market vector, Y: side information • I(X;Y) is an upper bound on the increase ∆W in growth rate. where is log-optimal strategy corresponding to f(x) and is the log-optimal strategy corresponding to g(x). • Thus, the financial value of side information is bounded by this mutual information term.

18. Side Information • Suppose the gambler has some information relevant to the outcome of the gamble. • What is the incrase in wealth that can result form such information, i.e. the financial value of side information? • Going back to horse race problem: • Increase in growth rate of wealth due to the presence of side information is equal to the mutual information between the side information and the horse race.

19. Conclusion • The developing theory of online portfolio selection has taken advantage of the existing duality between information theory and finance. • Work in statistics and information theory forms the intellectual background for current/future work on universal data compression and investment.

20. References • [1] T. Cover. Universal Portfolios. Math. Finance, 1(1):1-29, 1991. • [2] T. Cover and E. Ordentlich. Universal Portfolios with Side Information. IEEE Transactions on Information Theory, 42(2):348-363, 1996. • [3] T. Cover. Universal Data Compression and Portfolio Selection. Proc. 37th IEEE Symp. Foundations of Comp. Science, 534-538, 1996. • [4] T. Cover and J. Thomas. Elements of Information Theory. 2nd ed., John Wiley & Sons, Inc., Hoboken, New Jersey. 2006. • [5] T. Cover. Shannon and Investment. IEEE Information Theory Society Newsletter. Special Golden Jubilee Issue,1998 • [6] A. Kalai and S. Vempala. Efficient Algorithms for Universal Portfolios. Journal of Machine Learning Research, 3:423-440, 2002. Thank you!