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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 ,..., Xi1 )
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!

				
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