Universal Portfolio Selection: Application of Information Theory in Finance
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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

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Universal Portfolio Selection: Application of Information Theory in Finance- SC500 Project Presentation -

Gudrun Olga Stefansdottir

May 5 2007

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Outline Theory in Finance

  • Introduction

  • Information Theory in Gambling

  • Information Theory and the Stock Market

  • Concepts and Terminology

  • Cover’s Universal Portfolio (CUP)

  • Simulations

  • Conclusion

  • References

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Introduction Theory in Finance

  • 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

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Information theory in Gambling Theory in Finance

  • 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.

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We can represent the Theory in Financestock 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

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Concepts and Terminology Theory in Finance

  • 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!

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Portfolio selection Theory in Finance

  • 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

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Portfolio selection – cont. Theory in Finance

  • 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.

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Constant Rebalanced Portfolio Theory in Finance

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

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Cover’s Universal Portfolio (CUP) Theory in Finance

“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

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Universal portfolios Theory in Finance

  • 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:

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CUP – cont. Theory in Finance

  • 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???

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Simulations Theory in Finance

  • 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.

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Increased volatility, x8 Theory in Finance

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“A good gambler is also a good data compressor” Theory in Finance

  • 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

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Incorporating Side Information Theory in Finance

  • 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.

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Side Information Theory in Finance

  • 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.

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Conclusion Theory in Finance

  • 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.

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References Theory in Finance

  • [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!