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Can financial strategiesbased on Information

Theory beat the Market?

Joint work in progress

D.G., D.Herlemont, M.Grasselli

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Out l ine

n Investment & Information Theory

n Benefits of rebalancing

n Universal portfoliosn Empirical studies

n Further issues

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Out l ine


uKelly’s betting rule and formula

uShannon Entropy


n Universal portfolios

n Empirical studies

n Further issues

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Part I - Information Theory & Investment

Kelly’s betting problem (1956)

n Given n bets with random returns r i whatfraction πi of wealth W allocate to bet i ?

Global random return: X = ∑ πi r i = π.r n Repeated games: W t+1 = X t+1W t t=0,1,...

W t = X 1 X 2…X t W 0

n

Assume X 1 , X 2 ,…X t … i.i.d.

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Kelly’s rulen Kelly criterion: Maximize E[ln W t ] <=>

Maximize growth rate g(π) = E[ln X(π)]

n Solution: use same optimal π* for all t

Bet in constant proportions!

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Kelly’s formula

Then optimal asymptotic growth rate

G = E (ln r ) - H(r) Shannon Entropy

H(r) = Entropy of successive returns signal

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Shannon Entropy (1948)

n Information source: discrete r.v. X

P(X = xi ) = pi

Information of message xi : I(X xi) = - log( pi)

n Entropy: expected value of r.v. I(X)

H(X) = E[ I(X)] = -∑ pi log( pi)n H(X) = minimum code size of signal source X

= optimal rate of compression

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Role of additional information

n Choose π* using privileged information q

n Gain in optimal growth rate:

∆G = H(p) - H(q) = I(p,q)

I(p,q) = Mutual information

n Ex: p full ignorance, q know for sure

∆G = log n

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Summary on Kelly

Mutual information I(p,q)Gain from privileged

information

Source i.i.d. (ergodic)Constant Rebalanced

Portfolio

Compressed file sizeWealth Ln(W)

Source Entropy H =

Optimal compression rateOptimal growth rate G

Source probabilities p Portfolio weights π

INFORMATION THEORYINFORMATION THEORYINVESTMENTINVESTMENT

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Out l ine



uVolatility pumping

uRole of correlations

uKelly revisited


n Empirical studies

n Further issues

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Part II - Benefits of rebalancing

Rebalancing vs Buy&Holdn Kelly optimal constant proportions =>

constant mix strategies, rebalancing

active management of allocationn Common practice: reset (monthly)

against risk of drift, but

uTransactions costs

uMissed trends (rebalance = contrarian)

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Classical risk diversificationn Rebalancing example: keep πi = 1/ n

n Static framework (buy & hold) :

n identical uncorrelated assets r i N (m,σ)equally weighted portfolio r = 1/n ∑ r i

E(r ) = m Var(r ) = σ / n

risk reduction

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Luenberger volatility pumpingn Dynamic Black Scholes framework:

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Luenberger volatility pumpingn Growth rate, however:

uGain from volatility

with rebalancing

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Volatility pumping

n Example: «value line»

uEqually weighted portfoliouSimulation on DJ stocks vs DJI

uCan work on other indices

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Role of correlationsn Volatility pumping benefit from

uHigh level of assets volatilities

u

but also anticorrelation of assetsn Idea: Long/short strategies on

correlated assets

n Simulation on long DJ value line / short

DJI

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Kelly revisited: risk aversionn Kelly’s criterion as maximum log-utility

n Power utility more risk-sensitive

γ >1 risk aversion coefficient

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Kelly risk-sensitive solutionµ = excess return vector

V = variance-covariance matrix

n quadratic expected growth rate

n Maximum for πγ = 1/ γ V -1µsame as Merton!

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Part III - Benefits of rebalancing

Growth Optimal Portfolios Efficient Frontier

n Expected growth rate vs Variance

Source: B.F.Hunt, UTS

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Out l ine




u Optimal Coding & Wealth Growthu Cover’s Universal Portfolios

u Extensions of Universal Portfolios

n Empirical studies

n Further issues

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Part III - Universal portfolios

Optimal Coding & Wealth Growthn In practice of communications systems: find

source probability p by optimal coding

n

Portfolio problem: find optimal weights π*

without knowledge of returns distribution (µ,V )

n Tom Cover approach: use on line optimal

coding algorithms to find π* from past returns

n

Naive approach: compress returns fileu Compression rate H ~ optimal growth rate G

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Cover Universal Portfolios (1991)n Imagine we can transport in future to T we

could then compute πT * maximizing

ln(W T (π) / W 0)

πT * = BCRP in hindsight BCRP: Best Constant Rebalanced Portfolio

n Cover’s goal: find successiveapproximations πt

# of πT * using only past

returns

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Cover Universal Portfolion Cover’s weights updating rule:

πt+1# = ∫ W t (π) π d π / ∫ W t (π) d π

• Need computing n -dim integrals overdomain ∑ = {π πi = 0, ∑ πi = 1}

• Discretization & interpretation: distribute(ficticiously) initial wealth amongvarious fixed CRP πk and weight them

according to performance up to t

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Properties of Universal Portfolion Competitivity

lim 1/ T (ln W T # - ln W T

*) = 0

lim 1/ T ln W T * = G

W T # / W T * ¡ 2/ √(T+1)

n Universality

uMinimal assumptions on returnsdistribution

uWorks for all «active» markets (diversity)n Side information

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Extensions of Universal Portfoliosn Same idea:

uDistribute wealth among virtual portfolios

uWeighted on line according to performance

n But portfolios no more restricted CRPu «Expert» portfolios estimate dynamic

rebalancing proportions using nonlineartechniques (nonparametric methods,

machine learning, pattern recognition…)n Faster and overperforms Best CRP

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Out l ine




n Empirical studiesuExtension of Gyorfy-Lugosi

u Tests on DJ stocks and DJ Index

u Tests on FX

u Tests on US futures ratesn Further issues

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Part IV - Empirical studies

Extension of Gyorfy-Lugosi (2003)n Similar nonparametric kernel method

n Runs on + 3000 trading days

uCompute Best CRP in-sample uRun algorithm out-of-sample

n We allow for negative weights

n We introduce VaR constraints and

transactions costs (buy & sell)

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term wealth 1138

yr return 51.16%volatility 18.1%

sharpe 2.58kurtosis 2.32mdd 14.1%

Test DJ stocks & DJI 1992-2004

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term wealth 1.44yr return 2.63%

volatility 7%sharpe0 0.37

kurtosis 7.42mdd 22.2%

Test FX 1992 - Today

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term wealth 30.6

yr return 23.4%volatility 13.6%

sharpe0 1.72kurtosis 23.7mdd 23.1%

Futures US Rates 1992 - Today

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Futures US Rates Portfolio weights

Long 2yr10yr

Short 5yr

30yr

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Out l ine




n Empirical studies

n Further issues

uWeak information (Baudoin)

uRelative arbitrage (Fernholz-Karatzas)

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Conclusions

n Kelly’s GOP (also «numéraire») setuseful target

n Active management of allocation canimprove performance

n Development of statistical learningalgorithms can help if better understood

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