1. Sharif University of Technology
Electrical Engineering Department
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Information-Theory And Portfolio Selection Strategies
Omid Aminzadeh Gohari
- Bsc.Project Presentation -
4th July 2010

2. Sharif University of Technology
Electrical Engineering Department
Agenda
 Why should we care about the stock market?
 Information Theory and the Stock Market
 Many works
 Chen’s approach
 Cover&Thomas’s approach
 Concepts and Terminology
 A sample result
 Conclusion
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3. Sharif University of Technology
Electrical Engineering Department
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Why should we care about the stock market
 The “stock market” began in May 17th, 1792 when 24 stock brokers and
merchants signed the Buttonwood Agreement.
 Financial Aspects
 50% - A rough estimate of the rise in the share price of Apple in the aftermath of the
iPhone.
 $36.6 trillion - the estimated size of the world stock market at the beginning of October
2008.
 So, let’s start our own business 
 Caveat: Timing is everything!
McDonald's stock $22.5 in 1965 - was worth well over $14000 in
1995.
 In 2003, they'd only be worth $25!!
 Not everyone is a winner! Using information theory to understand the
market

4. Sharif University of Technology
Electrical Engineering Department
Agenda
 Why should we care about the stock market?
 Information Theory and the Stock Market
 Many works
 Chen’s approach
 Cover&Thomas’s approach
 Concepts and Terminology
 A sample result
 Conclusion
4

5. Sharif University of Technology
Electrical Engineering Department
Chen’s approach
M : number of persons in the market,
p: percentage of those who have a product (scarcity),
-log(p): financial value of the product per unit.
Total value of the product:
Mp(−log p)
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6. Sharif University of Technology
Electrical Engineering Department
Chen’s approach: information in a market
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Value of information Number of people who know
it
Low 
High 
High
Low
Free Information 
Transmitter in a
communication
system
 Receiver in a
communication
system
A broker

7. Sharif University of Technology
Electrical Engineering Department
Agenda
 Why should we care about the stock market?
 Information Theory and the Stock Market
 Many works
 Chen’s approach
Cover&Thomas’s approach
 Concepts and Terminology
 A sample result
 Conclusion
7

8. Sharif University of Technology
Electrical Engineering Department
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The Stock Market
 We can represent the stock market as a vector of stocks
1 2 ( , X  X X ,..., Xm ), Xi  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
 Price ratios are i.i.d over time
 Portfolio: Allocation of wealth accross various stocks.
m
b  b b b b   i  b

( , ,..., ), 0, and 1
1 2
m i i
where bi is the fraction of wealth allocated to stock i.
1
i

Example: Ali has wealth $100 and her portfolio consists of
the following fractions: 50% in IBM, 25% in Disney, 25% in
GM

9. Sharif University of Technology
Electrical Engineering Department
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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!
T S = b X

10. Sharif University of Technology
Electrical Engineering Department
T T T T

b X b X b X b X
log( ) log( ) log( ) ... log( )
(log( )) (log( )) (log( )) ... (log( ))
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Portfolio selection
( )( )( )...( )
   
b X b X b X
 Growth-rate of wealth of a stock market portfolio b:
W(b,F)   logbTxdF(x)  E(logbTx)
*
W F W F
( ) 
max ( , )
b
* *
E S nW F
(log( )) 
( ) n
b
1 1 2 2 3 3
1 1 2 2
1 1 2 2
n
n
n
n n
T T T
n n
T T T
n n
S
S
E S  E  E  
E
b X b X b X

11. Sharif University of Technology
Electrical Engineering Department
Portfolio selection – cont.
E S nW F
(log( )) ( )
1
( log( )) ( )
E S W F
 Let be i.i.d. According to distribution F(x)
 It can be shown using strong law of large numbers that
 
*( ) 2
nW F S 
n
1 2 n X ,X , ...,X
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* *
* *
n
n
n


* . .1 * * 1 1
log ( log ) ( ) w p
n n S E S W F
n n

12. Sharif University of Technology
Electrical Engineering Department
Agenda
 Why should we care about the stock market?
 Information Theory and the Stock Market
 Many works
 Chen’s approach
Cover&Thomas’s approach
 Concepts and Terminology
 A sample result
 Conclusion
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13. Sharif University of Technology
Electrical Engineering Department
Side Information
 Suppose the broker has some information relevant to the
outcome of the stock.
 What is the incrase in wealth that can result form such
information, i.e. the financial value of side information?

14. Sharif University of Technology
Electrical Engineering Department
Incorporating Side Information
 Let (X,Y) ~ f(x,y), X: market vector, Y: side information
 I(X;Y) is an upper bound on the increase ΔW in growth rate.
W  I (X;Y)
 Thus, the financial value of side information is bounded by this
mutual information term.

15. Sharif University of Technology
Electrical Engineering Department
Agenda
 Why should we care about the stock market?
 Information Theory and the Stock Market
 Many works
 Chen’s approach
Cover&Thomas’s approach
 Concepts and Terminology
 A sample result
 Conclusion
15

16. Sharif University of Technology
Electrical Engineering Department
Conclusion
 Motivated the Stock Market
 Briefly discussed connections between Information
theory and Stock Market

17. Sharif University of Technology
Electrical Engineering Department
References
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[1] T. Cover and J. Thomas. Elements of Information Theory. 2nd ed., John Wiley &
Sons, Inc., Hoboken, New Jersey. 2006.
[2] J. Chen, “Information Theory and Market Behavior”, SSRN, Dec 2003.
[3] J. Chen, “Information, Entropy And Evolutionary Finance” , SSRN, January
2003.

18. Sharif University of Technology
Electrical Engineering Department
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Thanks For Your Time!