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Universal Portfolio Selection: Application of Information Theory in Finance - SC500 Project Presentation - Gudrun Olga Stefansdottir May 5 2007 Outline Introduction Information Theory in Gambling Information Theory and the Stock Market Concepts and Terminology Cover’s Universal Portfolio (CUP) Simulations Conclusion References 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 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 i wins w.p. pi and has payoff oi m bi =fraction of wealth invested in horse i,i 1 bi 1 Wealth after n races n Sn S ( X i ) i 1 where X 1 , X 2 ,..., X n 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. The Stock Market We can represent the stock market as a vector of stocks X ( X 1 , X 2 ,..., X m ), X i 0, i 1, 2,..., m 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. m b (b1 , b2 ,..., bm ), bi 0, i and b 1 i 1 i where bi is the fraction of wealth allocated to stock i. Example: Alice has wealth $100 and her portfolio consists of the following fractions: 50% in IBM, 25% in Disney, 25% in GM Concepts and Terminology Wealth-Relative: Ratio of wealth at the end of the day to the beginning of the day T S=b X We want to maximize S in some sense and find the optimal portfolio! Portfolio selection Growth-rate of wealth of a stock market portfolio b: W (b, F ) log bT xdF (x) E(log bT x) Optimal growth-rate: W * ( F ) max W (b, F ) b 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 n S n b*T Xi * i 1 Portfolio selection – cont. Let X1 , X 2 , ..., X n be i.i.d. According to distribution F(x) It can be shown using strong law of large numbers that 1 log S n W * , hence * w. p .1 Sn 2 * nW * n 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. 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 CRP can only be computed with knowledge of market performance 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 bi bi ( X1 ,..., Xi1 ) 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: ˆ 1 , 1 ,..., 1 b1 m m m ˆ b k 1 bS (b)db k S (b)db k CUP – cont. It has been shown (at various levels of generality [5]) that there exists a universal portfolio achieving a wealth ˆ S n at time n s.t. ˆ Sn 2 n 1 * Sn So what is for every stock market sequence and for every n, where * S n is the wealth generated by the best constant the catch??? rebalanced portfolio in hindsight. Drawbacks: Does not incorporate transaction costs/broker fees High maintenance: needs to be rebalanced daily Needs higly volatile stocks 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. 12/14/1993-12/06/1995 – 500 days Increased volatility, x8 “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 S n lead to high data compression If the text in question results in wealth S n then log( S n ) 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 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. W W (b f , F ) W (b g , F ) D( f g ) where b f is log-optimal strategy corresponding to b g and f(x) is the log-optimal strategy corresponding to g(x). W I (X; Y ) Thus, the financial value of side information is bounded by this mutual information term. 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. 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. 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!