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Universal Portfolio Selection: Application of Information Theory in Finance - SC500 Project Presentation -PowerPoint Presentation

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### Universal Portfolio Selection: Application of Information Theory in Finance- SC500 Project Presentation -

Gudrun Olga Stefansdottir

May 5 2007

Outline Theory in Finance

- Introduction
- Information Theory in Gambling
- Information Theory and the Stock Market
- Concepts and Terminology
- Cover’s Universal Portfolio (CUP)
- Simulations
- Conclusion
- References

Introduction Theory in Finance

- 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 Theory in Finance

- 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.

We can represent the Theory in Financestock 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 MarketExample: Alice has wealth $100 and her portfolio consists of the following fractions: 50% in IBM, 25% in Disney, 25% in GM

Concepts and Terminology Theory in Finance

- 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 Theory in Finance

- 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. Theory in Finance

- 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 Theory in Finance

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) Theory in Finance

“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 Theory in Finance

- 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. Theory in Finance

- 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 Theory in Finance

- 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 Theory in Finance

Increased volatility, x8 Theory in Finance

“A good gambler is also a good data compressor” Theory in Finance

- 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 Theory in Finance

- 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 Theory in Finance

- 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 Theory in Finance

- 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 Theory in Finance

- [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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