1

Lecture 1

Your lecturer; Introduction of students; Comments on assessment;

Lecture plan; Overview of Evolutionary Finance; Historical context; Relevance;

Applicability; etc.

Your lecturer

Klaus Reiner Schenk-Hoppé

• Centenary Chair in Financial Mathematics• Professor at Leeds University Business School and School of

Mathematics, University of Leeds

• Research interests: Finance (evolutionary and quantitative), Economics, Appl. Mathematics

• Ongoing research with Prof. Thorsten Hens (visit April 7-May 19)

• PhD students in Computational Finance, Game-theory and Fin Markets, Emerging Markets and Sovereign Bonds

• Programme director, MSc in Financial Mathematics

• Zurich 1999-2002, Copenhagen 2002-2004, Leeds 2005-

Introduction of students

• Brief statement• Name, background, future plans• Main field of study• Etc.

Comments on assessment

• Written summary of research text/texts

• In English or German

• Maximum length 3,000 words

• Topic to be agreed with lecturer (me)

• Deadline (tentative): July 31 (noon), by email

Lecture plan• Hand out

• Lectures 1-3: Introduction and background• Lectures 4-5: Kelly rule (log-optimum investment)• Lectures 6-8: Evolutionary Finance model I (simple)• Lectures 9: Exercises/Applications• Lectures 10-11: GE / GEI approach (Economics)• Lectures 12-13: Noise trader approach (Finance)• Lectures 14: Complex adaptive systems (Econophysics)• Lectures 15-16: Evolutionary Finance model II (less simple)• Lectures 17-19: Simulation studies, GP, Applications• Lecture 20: Wrapping up

What sparked Evolutionary Finance?Let’s start with a few simple questions:

• Classical finance takes on board the rationality assumptions prevalent in economics, doesn’t it?

• Do humans act perfectly rational? Or do they rather show behaviorthat is heuristic, adaptive and sometimes unpredictable?

• Are traditional models’ predictions in agreement with actual data?

• What notions of “equilibrium” have you been taught? Are they realistic?

• How do practitioners think about financial markets?

• NB There would be no Evolutionary Finance without classical finance!

2

Evolutionary Finance is not easy to define

• Researchers from different disciplines:Finance, Economics, Physics, Computer Science, Mathematics, Biology, …

• Traits of this approach: Dynamics, Heterogeneity, Bounded Rationality, Co-existence/Selection(Darwin seems to be looking over one’s shoulder)

• Overlaps with: Agent-based finance, behavioral finance plus market equilibrium. Many approaches are sold as continuation of std. theory

• Handbook of Financial Markets: Dynamics and Evolution(edited joint w. Th. Hens), published by Elsevier, Handbooks in Finance series (edited by William Ziemba)

Contributors: Hens/Schenk-Hoppé, Hirshleifer, Hommes, Kurz, Evstigneev/Dempster, Blume/Easley, Lux, Doyne Farmer, Chiarella, Föllmer/Horst, Boehm, Bouchaud

Most closely related approach to the study of financial markets

Agent-based models(usually!)

• one risky and one risk-free asset

• normally distributed dividends

• myopic mean-variance maximizers

• dynamics of types (typically 2 investor types only)

Taking on incumbents, e.g. EMH• Strong and weak forms

(informational (all information, in particular, private is reflected in prices)vs. allocational (capital is with most efficient firms, marginal productivity) efficiency)

• Absence of arbitrage

• Prices follow a random walk (the more efficient the market, the more price changes are uncorrelated/independent) i.e. unpredictability [Samuelson 1965]

• Fundamental values as unique valuation

• Fama used evolutionary reasoning along the line that irrational traders who can destabilize a market will be exploited by rational traders (‘smart money’). Thus non- rational traders lose money and eventually disappear. [Irony: this reasoning sounds good but does not hold water as shown by evolutionary models!]

• Grossman and Stiglitz (1980): impossibility of informationally efficient markets if there are costs to determine fundamental value of stocks.

Robert Shiller (AER 1981)

The life-cycle of theories

• Big bang; then flourish• Minor crisis: Stretch • Major crisis: Eventually shift to new paradigm

The Efficient markets hypothesis (EMH) has undergone this process

• Modern portfolio theory (or ‘traditional finance’): EMH, CAPM, expected utility theory

• New risk factors, new utility funct’s and payoffs • Behavioral Finance, Agent-based models, Evolutionary Finance

A Darwinian perspective on Financial Markets

Evolutionary Financedraws analogies to Darwinian,

i.e. evolutionary, biology

���������������� �������������������������������������

���������������������Essay, 1973�

“The Mecca of the economist lies in economic biology rather than in economic dynamics.”

(Alfred Marshall, Principles of Economics, 1890)

3

• Strategy

• Resources

• Selection

• Mutation

• Portfolio/Investment rule

• Market capital

• Gains/Losses

• Innovation

In a nutshell:

“Towards a Darwinian Theory of Financial Markets”

The evolutionary view of financial markets Ingredients to Evolutionary Finance which you will see over and over again

Dynamics of evolution: Competition, mutation, reproduction, natural selection

For financial markets this means:• Investment strategies• Interaction of traders• Wealth dynamics• Asset payoffs• Price formation• Survival (or Utility?)• Evolutionary spite, ‘animal spirit’• Expectation formation• Flow of funds

Alan Grafen (biologist) at the 2002 Evolutionary Finance conference, SWX Zurich

Out of the mind-blowing variety of approaches:The adaptive markets hypothesis

• Andrew Lo: AMH = “Market efficiency from an evolutionary perspective”

• (Lo [25] p.24) “Specifically, the Adaptive Markets Hypothesis can be viewed as a new version of the EMH, derived from evolutionary principles. Prices reflect as much information as dictated by the combination of environmental conditions and the number and nature of “species” in the economy or, to use a more appropriate biological term, the ecology.”

• Ideas akin to those explored in Evolutionary Finance

Historical context

• Thinking about markets in evolutionary terms is by no means new

• Thorstein Veblen “Why is Economics not an evolutionary science” 1898

• EMH: belief that irrational behavior “dies out”(Fama)

• Industrial economics: profit maximization as result of selection (Friedman)

• Agent-based models inspired by findings in behavioral finance/economics

RelevanceA (good) theory needs to

• capture/explain current (i.e. documented) phenomena

• make predictions on not-yet-analyzed cases/situations

• turn out to have correctly anticipated these findings

But note: Theory and measurement go hand in hand(CAPM, Leontief’s input-output analysis)

4

Applicability

• Are the assumptions reasonable?

• Are the “ingredients” observable?

• Can evolutionary models be “fitted”?

• Can they be used to make forecasts?

Advancement of knowledge

• New/modified asset pricing theory (equilibrium, here: in the long-run)

• A new view on short- and medium term fluctuations [viewed as caused by new investment styles or products, e.g. IT bubble, other break-through events such as electricity, related to stylized facts]

Lecture 2

Introduction to Evolutionary Finance research; The two strands:

Market selection hypothesis and evolutionary asset pricing; Main researchers;

Evolutionary vs. behavioral finance.

Introduction to Evolutionary Finance research

Has to make sense of

• Portfolio/Investment rules• Flow of (market) capital• Gains/Losses• Innovation

[Dynamics of evolution: Competition, mutation, reproduction, natural selection]

Needs to accommodate heterogeneous investors/strategies

The two strands:Market selection hypothesis and

evolutionary asset pricing• MSH/EMH: This goes essentially back to the

(controversial) claim that irrational traders vanish in the presence of rational ones. Fama (1970) argues that if markets were not efficient, there are “arbitrage”possibilities that are picked up by rational traders. Thus irrational trader lose money…

• True or not, this remark has generated an amazing amount of literature

• More modest view, efficient markets as an asymptotic outcome of interaction = evolutionary pressure [AMH]

A short pause for thought

‘Investment based on genuine long-term expectation is so difficult as to be scarcely practicable. He who attempts it must surely lead much more laborious days and run greater risks than he who tries to guess better than the crowd how the crowd will behave; and, given equal intelligence, he may make more disastrous mistakes’(Keynes,The General Theory of Employment, Interest and Money, 1936, p. 157).

5

Evolutionary asset pricing• The process of evolution singles out investment styles

which induce particular asset prices (cross-section as well as time series)

• Market dynamics will wipe out deviant behavior

• Equilibrium provides reference point, convergence to eq. if asset prices are ‘exogenously’ pushed away

• Short- and medium term price dynamics (beside long-run outcome)

• Traders’ predictions/forecasts become coordinated (e.g. converge to a REE)

Main researchers

• MSH in GE model context: Larry Blume, David Easley, Alvaro Sandroni, and many others

• Noise traders: Black, DeLong, Shleifer, Summers, Waldman, and many others

• Agent-based models: Hommes, Brock, Lux, LeBaron, Chiarella, and many others

• Evolutionary finance with wealth dynamics: Evstigneev, Hens, Schenk-Hoppé and few others

• Santa Fe Institute: Farmer, Lo, etc.

Other comments/discussion• Thorstein Veblen "Why is Economics Not an Evolutionary Science"

The Quarterly Journal of Economics Volume 12, 1898.

• Representative agent: price taking behavior, concave utility function, time-separablility, exp. utility.

• Evolutionary game theory (But almost nothing done in financial market context. Strategic behavior: Kyle 1985/1989, market makers who make expected profit of zero i.e. always offer and buy at fair price)

• Study of trading behavior in limit-order markets: market reaction/impact function…(e.g. Bertsimas and Lo, J of Fin Markets, 1998)

Lecture 3

Dynamics of financial markets; Market and non-market interaction; The two evolutionary

forces: Mutation and selection.

Dynamics of financial markets 1/4

The term “dynamics”

``The definition of economic dynamics (that much controverted term) which I have in mind here is this. I call Economic Statics those parts of economic theory where we do not trouble about dating; Economic Dynamics those parts where every quantity must be dated.’’

John R. Hicks (Value and Capital, 1939, Chapter IX)

Dynamics of financial markets 2/4

Ragnar Frisch (1936) proposed using the terms static and dynamic to characterize the structural relations of variables that make up the model. In Frisch's definition, a structural relation is called

static, if all variables refer to the same time; otherwise it is dynamic. A model is characterized as

dynamic if it contains at least one dynamic structural relation. (S-H, 2001)

We will see: preceding definitions agree with the GE/GEI approach

6

Dynamics of financial markets 3/4

• Dynamical systems’ point of viewA map that derives tomorrow’s state of the market from today’s state and exogenous factors (such as random events)

• Limited use of equilibrium concepts (though market clearing is a must)

• Often blend (combination) of dynamical systems and temporary/short-run equilibrium concepts

Dynamics of financial markets 4/4

• Agent-based models (computability problems provide a strong push towards dynamical systems models)

• “Expectations” dynamics, updating of forecasting rules

• Complex adaptive systems, application of non-linear systems theory (bifurcations, chaos)

• Population/trader dynamics, “flow of funds”, learning,…

Market and non-market interaction

• Market interaction:through the price system (and whatever it reveals)

[standard approach]

• Non-market interaction:every interaction that is not through prices, i.e. communication, networking, sharing quantitative analysis results, forecast, announcements, spreading rumours,…

[non-standard, hot topic]e.g. Alan Kirman (and co-authors)

We will focus on market-interaction

The two evolutionary forces: Mutation and Selection

MUTATION

• Force that creates and maintains variety

• In financial context: New investment styles, new financial products, alternative valuations, novel type of leverage, behavioral funds, new big players, sharia finance, hedge funds, etc.

Evolution is

not fair

The two evolutionary forces: Mutation and Selection

SELECTION

• Pressure that reduces variety by removing things…[through fight for resources, external perturbation/disturbance, leads to extinction]

• In financial contexts:Selection works through losses, (under-)performance Survivors: Stocks, derivatives, active management, managers of delegated funds [scorpions]

7

Dead since 10,000 years

Alive and kickingsince 350 million years (oldest animal)(horseshoe crab, 60cm, belongs to species Xiphosurida, see previous slide)

Lecture 4

The Kelly rule; Overtaking; Log-optimum investment;

How to becomerich-if you can wait

A “simple” investment problem• Betting at the horse track, win bets only• Odd of a bet = amount to bet : bet’s payoff [no track take = odds sum to one]

(e.g. 1:5 means you get 5� for each 1� bet if this horse wins; otherwise zero)

ExampleSuppose 2 horses with odds 1:2 and 1:2

Bet fraction � on horse 1 and 1-� on horse 2

If horse 1 wins: vt+1 = 2 � vtIf horse 2 wins: vt+1 = 2 (1-�) vt

Repeated investment of previous receipts

After N periods: vN = 2N �W1 (1-�) W2 v0

where W1 is the # of times horse 1 won between time 1 and N (analogously W2)

How to bet IF you want to become rich with certainty when N� � (and as fast as possible)?

Kelly ruleAssume odds are 1:�1 and 1:�2 ; i.i.d. runs, prob. of winning �1 resp. � 2

Growth rate [use blackboard]

( ) ( )( )22

110

]1[][lnln WWN vv λαλα −=

( ) ( )( )22

110

]1[][ln1

ln1 WW

N NN vv λαλα −=

( ) ( )( ) ( ) ( )λπλπλαλα −+ →−+= ∞→ 1lnln1ln2

ln1

2121N

NW

NW

Then

Maximize to find optimal � in [0,1]( ) ( )2211 lnln απαπ ++

121 11and ππλπλ −==−=

Kelly rule: Interpretation• The shares of money placed only depend on the probability of events

• In particular the bets placed are independent of the odds!

• Maximizes growth rate among all constant proportions (rebalancing) strategies

• Asset pricing theory: When are there no excess returns?

• Analogy to financial markets: bets are like call options in a binomial-tree model or like digital call options in general models & also in reality

[Kelly: value of side[private] information / noisy transmission channel / Shannon the father of information theory. Thorp applied these ideas to horse betting and became rich!]

ss πα =

8

More general results• Breiman (1961): Kelly rule the best among all adapted

strategies when payoffs i.i.d. across time

• Algoet and Cover (1988): Maximizing conditional log in every period in time is optimal (in the sense of achieving maximum growth rate) even without any assumption on distribution. [But underlying probability model needs to be known!]

• Cover’s (1996): “Universal portfolios” paper emphasizes the objective properties of log-optimum investment [even for non-stationary time series and without knowledge of underlying distribution]

Constant-proportions strategies have many good properties: Volatility and Growth

Dempster/Evstigneev/Schenk-Hoppé

• The Joy of Volatility, Quantitative Finance (front matter, forthcoming)

• Volatility-induced Growth in Financial Markets, Quantitative Finance (forthcoming)

• Exponential Growth of Fixed-Mix Strategies in Stationary Asset Markets, Finance and Stochastics 7, 263-276, 2003.

A heated debate• Normative content? / Objective criterion?

• Samuelson’s critique on growth optimal [log-optimum] investment [= do not think about it unless your utility function is log]

• Cover’s 2 page piece on Shannon’s indirect contribution to investment theory [positive content of growth-optimum portfolios]

• Luenberger (JEDC): for large wealth many utility functions look like log!

Importance of these considerations

• Evolutionary finance is concerned with selection forces [and so is the market selection hypothesis as well as the AMH]. These forces might need time to unfold their power/impact

• Frequency of trade matters! [High on financial markets]

• “In the long-run we are all dead” (John Maynard Keynes)

• ‘Dead but happy (rather than rich)’ [with apologies to Paul Samuelson]

Lecture 5

Long-run investment performance in complete markets; Arrow securities;

Bayesian updating.

A simple model: Blume/Easley (JET, 1992)

Randomness: state of the worldat time t: (i.i.d., say)

K assets with random payoff(supply = 1, normalization)

)(tk

sA

Arrow securities, i.e. S=K states of the world and one and only one asset pays off in each state (complete market)

Ss ,...,1=t

s

���

≠=

=ks

kssAk ,0

,1)(

9

Investment Strategies

• Portfolio weights / budget shares of investor i

with

Portfolio positions are given by

( ) history ,)(,tti

kti ssλλ =

1)(,0)(1

,, => �=

K

k

tikt

tikt ss λλ

kt

tiktti

kt p

ss

,

,,

)()(

λθ =

Simple model (ctd)

1

1

11111

,

,,1 )()(

+

+++=+++=

=

==+ ==

t

t

ttsktttsk

skt

it

iskti

sktit p

wsAsAw

λθ

• Investor i ’s wealth dynamics (without consumption)

[Can you spot the odds? Interlude on parimutuel betting markets]

• Individual savings rate (e.g. via discounting and utility)

iskt

iit tttsk sAw

111 ,1 )(+++= =+ = θρ

Analysis (thoroughly done on blackboard)

• Look at two investors i and j, both with stationary investment rules

jt

it

js

is

jt

it

ww

ww

t

t

j

i

1

1

1

1

+

+=+

+

λλ

ρρ

j

i

js

is

js

is

t

t

jt

it

ww

ww

t

t

j

i

0

0

1

1

1

1

1

1 ...)(

)(λλ

λλ

ρρ

+

+=+

+

( ) ( )js

is

Tj

T

iT EE

ww

Tji λρλρ lnlnln

1 − →���

���

∞→

( ) ( ) Douglas]-[Cobb lnmax arglnmax args

iss

isEi λπλρ �=

InterpretationDynamics of the model

• Wealth shares• Entropy• Betting your beliefs• Kelly rule• Importance of savings rate

Market selection hypothesis

• Trade off between accuracy of beliefs and consumption • Will the investor with the smallest entropy prevail?

( )sssI πλπλπ ln)(s�−=

“Over”-Interpretation• Rational rules do not necessarily survive

• Irrational rules can dominate

• “Fit rules are not necessarily rational and rational rules are not necessarily fit” (B/E (1992), Abstract)

• NB. Above investment rule can be derived from additively separable utility function with instantaneous utility = logarithm

Bayesian updating

• We have seen: everything depends on how well one knows the objective probabilities

• Issue: Adaptive behavior(Bayesian) learning / search / imitation

• Assume identical saving rates!

10

Main result on Bayesian learners

Theorem 6.1 (B/E, JET 1992)Suppose saving rates are identical and

everyone bets his/her beliefs. Then any Bayesian whose prior has finite support which includes the true distribution will not be overtaken by some other investor

..0inflim saW

W

jj

T

iT

T >�∞→

Proof

• Parts of the proof on blackboard.

Lecture 6

An evolutionary finance model I: Model; Incomplete markets.

Goal / References

• DynamicsStudy the (endogenous) wealth dynamics of heterogeneous investment strategies

• Asset pricingAnalyze in particular the long-run outcome

• SurvivalWhich strategies survive, which dominate?Is there a single survivor?

• Reading list [2,11,13,14] (mainly Evstigneev/Hens/Schenk-Hoppé)

Model basics

! investors/strategies [remark on funds]

" assets, each in supply of 1, short-lived(!)• Payoff is perishable consumption good [Lucas 1978]

• Strategies (simple) in simplex in RK

• Time is discrete• S states of nature (i.i.d. for simplicity)• [Common consumption rate, set to zero here]

,0)( ≥sAk 0)(1

>� =

K

k k sA

ik

tikt s λλ ≡)(,

Equations of the model• Portfolio of investor i at time t: position in asset k

is given by

• Further

• Pricing equation needs some explanation• NB: no assumptions on asset payoff structure made

kt

it

iki

kt pw

,,

λθ =

�=

+ +=K

k

iktk

it t

sAw1

,1 )(1

θ

�=

=I

i

it

ikkt wp

1, λ

11

Arrow securities -> back to B/E 92 with common discount rate

Interesting case: incomplete markets (prices matter!)

Total wealth

Wealth shares

Exercise: Derive dynamics for wealth shares (blackboard!)

]out! way simple [no ???)(

)(

1,

1,

1

1

1

1

==�

�

=

=

+

+

+

+

K

k

jtkk

K

k

itkk

jt

it

t

t

sA

sA

ww

θ

θ

����== ==

++ ++ ===K

kk

I

i

K

k

iktk

I

i

itt tt sAsAwW

11 1,

111 )()(: 11 θ

tit

it Wwr =

Example: an incomplete market

Coexistence of two strategies

���

�

�

���

=70

4422

A

)5,.5(.1 =λ

)75,.25(.2=λ

3/1=sp

Market Shares

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 19 28 37 46 55 64 73 82 91 100

109

118

127

136

145

154

163

172

181

190

199

208

217

226

235

244

253

262

271

280

289

298

time

iid

Another example

Complete vs. incomplete markets

• Discuss on blackboard

Arrow securities market

Example of incomplete market

12

Lecture 7

An evolutionary finance model II: Dynamical systems view; Analysis

Derive model in wealth shares

• on blackboard

Formal definition of investment strategy

Careful derivation of model

Q&A session

Dynamics for wealth shares

��=

=

++ =K

kI

jj

tjk

it

ik

tki

tr

rsRr

11

11 )(λ

λ

{ }1,0: =≥∈=∆ � iiII rrRr

Dynamical systems view

Dynamics of wealth distribution over investment strategies is governed by map

Tomorrow’s state of market is a function of today’s state and exogenous factors.

This dynamics is random: asset payoffs and/or (random) change of strategies.

Mathematical theory of iteration of random maps

Given initial state and adapted strategies

Vertex = strategy owns all wealth

1λ

3λ

2λ

...),,(),()( 210210100 →→→ sssrssrsrSelection

Market dynamics converges to

• one vertex of the simplex => this strategy is selected

• One face of the simplex => all strategies at the vertices of this face disappear(become extinct)

13

Fixed points of market dynamicsSuppose all strategies are different and there are

no redundant assets.

Lemma

(a) Every vertex is a deterministic fixed point (steady state).

(b) There are no steady states in which two or more investors have strictly positive wealth.

Prove of Lemma on steady states

• On blackboard

Heuristic derivation of local (in)stability results

Suppose ( )0,...,0,1≈tr

Then1k

ktp λ≈

�=

+ +≈K

k

it

k

ik

ki

t rsRr t

111 )( 1 λ

λand therefore

Growth rate of ith investor’s wealth shares is (blackboard)

���

���

= �

=

K

k k

ik

ki sREg

11)(ln:)(1 λ

λλλ

Understanding the growth rates

• If you are investor 1, what should you insert for

to achieve

whenever ?

And how to make sure that also ?

KK ∆∈= ),...,( 11

11 λλλ

0)(1 <ig λλ

1λλ ≠i

0)( 1 >λλig

Lecture 8

An evolutionary finance model III: Results; Relation to the Kelly rule

and other insights.

Solving this concave maximization problem

Theorem (evolutionary stable strategy/single survivor)Suppose no redundant assets. [Then incomplete if and only if K<S]

The simple investment strategyis the unique locally stable strategy among all simple ones,i.e.

[Details on blackboard]

*allfor 0)(* λλλλ ≠<g

KkERkk ,...,1,* ==λ

** allfor 0)( and λλλλ ≠>g

14

Other results

No strategy is even locally evolutionary stable:

Local stability of one strategy against the other implies local instability on the ‘opposite end’

Coexistence possible, requires

0)(0)( >�< jiij gg λλ λλ

0)(0)( >> jiij gandg λλ λλ

][ *kk ER=λ

0)~

(g: toclosey arbitraril ~

is there If * >≠ λλλλλ λ

Prices in the long-run

• Kelly rule! Betting your beliefs

• Arrow securities: same results as Kelly

• Asset pricing theory: Fundamental valuation but in relative terms

• Closed market => level not important only relative valuation

• Evolutionary spite transmits via prices

• Prices are endogenous and can (eventually) turn against you!

• Link to parimutuel betting markets (relative wealth: amount bet = total payoffs)

[later: we do calculations for above numerical examples]

kkk pER ==*λ

Comments

• Linearization captures local dynamics of non-linearized system! (MET)

• Log-linearization in macroeconomics tries to mimic this technique. (nonsense)

Global convergence results 1/2

• Q: What could preventfrom taking over everything?

• Market portfolio! (proportional to (1,…,1), risk-free payoff )

Set (Q: How do you do that?)

Then

kk ER=*λ

kMk p=λ

��=

+=

++ ===K

k

Mt

Mttk

K

k k

Mt

Mk

tkM

t rrsRp

rsRr

11

111 )()(

λ

Global convergence results 2/2Main result

(hand-waving version of Amir/Evstigneev/Hens/Schenk-Hoppé, JME 2005)

Under certain assumptions:

is the single survivor among all adapted strategies as long as it is asymptotically distinct from the market portfolio (of the other investors).

In either case, prices in the long run are given by

kk ER=*λ

kk ER=*λ

Game-theoretic interpretation

• Payoff = asymptotic market share (i.e. long-run share)• Two players

<0,=0,>0 possible

>0< 00

),( jig λλ *λλ =i *λλ ≠i

*λλ =j

*λλ ≠j

15

Lecture 9

Applications, Exercises

Exercise

• Calculate the growth rates in the previous numerical examples to check

- local stability (of Kelly rule)

- co-existence of strategies lack of selection (prices turn against you)

- long-term asset prices

Fixed-mix strategiesExample

2 states of the world2 assets with returns

Task:Find growth rates of(a) Buy and hold(b) Fixed-mix (1/2,1/2)(c) Explain result

1/22Asset 2

21/2Asset 1

State 2State 1 Lecture 10

Market selection in general equilibrium models I: Main results for

dynamically complete markets.

References

• Blume and Easley. “If You Are So Smart Why Aren't You Rich? Belief Selection in Complete and Incomplete Markets,” Econometrica, 74, 929-966, 2006.

• Sandroni. “Do Markets Favor Agents Able to Make Accurate Predictions?” Econometrica, 68, 1303-1341, 2000.

[1st reference is readable, 2nd is beyond my abilities]

The GE modelBASICS• Stochastic general equilibrium model• 1 consumption good ! consumers with stochastic endowment streams• Trade!

GOAL• Asymptotic properties of consumption allocation• Market selection hypothesis• Complete markets!• Bayesian learners• [Incomplete markets]

16

Assumptions

• Markets are (dynamically) complete• Expected discounted utility maximizers

• Endowment stream

• Competitive equilibrium (i.e. only price-takers)

( )))(()(0�

∞

==

t tit

iPi cuEcU i σβ

( ) ∞== ,...,0)( tit

i ee σ

Laffont (The Economics of Uncertainty and Information, 1989)

Results

• Survival is determined entirely by discount factors and beliefs, i.e. utility function (or risk-aversion) does NOT matter

• Bayesian learners selected according to the size of their (priors) parameter space

• Incomplete markets: none of these conclusions has to hold. Payoff functions (of assets) matter for survival and the market selection hyp. fails

Complete market result:Some details on the proof

• On blackboard