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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

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

concepts and terminology
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!
portfolio selection
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
portfolio selection cont
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.
constant rebalanced portfolio
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
cover s universal portfolio cup
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
universal portfolios
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:
cup cont
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???

simulations
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.

a good gambler is also a good data compressor
“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
incorporating side information
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.
side information
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
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
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!