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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