Universal Portfolio Selection: Application
of Information Theory in Finance
- SC500 Project Presentation -
Gudrun Olga Stefansdottir
May 5 2007
Information Theory in Gambling
Information Theory and the Stock Market
Concepts and Terminology
Cover’s Universal Portfolio (CUP)
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 
Discussion meetings with Samuelson (a Nobel Prize
winner-to-be in economics) on information theory and
Growth-rate optimal portfolios
Financial value of side information
Universal portfolios – counterpart to universal data
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:
horse i wins w.p. pi and has payoff oi m
bi =fraction of wealth invested in horse i,i 1
Wealth after n races
Sn S ( X i )
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
x F(x): Joint distribution of the vector of price relatives
Portfolio: Allocation of wealth accross various stocks.
b (b1 , b2 ,..., bm ), bi 0, i and b 1
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
We want to maximize S in some sense and find
the optimal portfolio!
Growth-rate of wealth of a stock market portfolio b:
W (b, F ) log bT xdF (x) E(log bT x)
W * ( F ) max W (b, F )
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
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
log S n W * , hence
* w. p .1
* nW *
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
Individual sequence minimax regret solution
The portfolio used on day i depends on past market
outcomes bi bi ( X1 ,..., Xi1 )
Intuitively: Each day the stock proportions in CUP are
readjusted to track a constantly shifting “center of
gravity” where performance is optimal and investment
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.
ˆ 1 , 1 ,..., 1
m m m
b k 1
CUP – cont.
It has been shown (at various levels of generality )
that there exists a universal portfolio achieving a wealth
S n at time n s.t. ˆ
So what is
for every stock market sequence and for every n, where
S n is the wealth generated by the best constant
rebalanced portfolio in hindsight.
Does not incorporate transaction costs/broker fees
High maintenance: needs to be rebalanced daily
Needs higly volatile stocks
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 
Value line: Equal
proportion invested in each
stock in the portfolio
Wealth relative: Relative
increase in wealth if entire
money invested in that
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
Any sequence in which a gambler makes a large amount of
money is also a sequence that can be compressed by a
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, 
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
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.
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.
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
 T. Cover. Universal Portfolios. Math. Finance, 1(1):1-29, 1991.
 T. Cover and E. Ordentlich. Universal Portfolios with Side Information.
IEEE Transactions on Information Theory, 42(2):348-363, 1996.
 T. Cover. Universal Data Compression and Portfolio Selection. Proc. 37th
IEEE Symp. Foundations of Comp. Science, 534-538, 1996.
 T. Cover and J. Thomas. Elements of Information Theory. 2nd ed., John
Wiley & Sons, Inc., Hoboken, New Jersey. 2006.
 T. Cover. Shannon and Investment. IEEE Information Theory Society
Newsletter. Special Golden Jubilee Issue,1998
 A. Kalai and S. Vempala. Efficient Algorithms for Universal Portfolios.
Journal of Machine Learning Research, 3:423-440, 2002.