Lecture 11

Market selection in general equilibrium models II: Example when markets are incomplete.

The GEI model

• General equilibrium and market selection when markets are incomplete

• Example that irrational traders are not necessarily wiped out by the market in these models • Details on blackboard

• Blume/Easley [5] If You're So Smart Why Aren't You Rich? NB: many(!) (annoying) typos in their example

1 Lecture 12+13

Noise traders and market selection. A detailed discussion

References

The early years (here we get historical):

• De , Shleifer, Summers, and Waldmann. “Noise Trader Risk in Financial Markets,” Journal of Political Economy, 98, 703-738, 1990.

• De Long, Shleifer, Summers, and Waldmann. “The Survival of Noise Traders in Financial Markets,” Journal of Business, 64, 1-19, 1991.

• Predecessors of agent-based models

2 MSH

• Fama argued that irrational traders will be wiped out by market forces because they should lose money to the rational traders

• Fisher Black (1986): excess returns comes from noise traders i.e. traders who trade on “information” that is, in fact, “noise”

Model (OLG, investment decision only, 2 types)

• 2 types: rational agents & noise traders

• 2 assets: 1 risk-free (∞ supply), 1 risky (supply 1) both have same constant r (genius at work!)

• Agents maximize CARA utility fct [=>myopic mean-variance maximizers] • Rational = knowing true distribution of return • Noise traders = mis-perceive return (typically optimistic)

• Additional through noise traders = noise trader risk [in the paper’s parlance]

• Noise traders take more risky positions for which they are compensated => Noise traders’ return can be higher!

3 Some more details

• On blackboard

Applications of this result (in the same paper): Volatility and in Asset Prices Closed-end Mutual Funds Puzzle Equity Premium Puzzle Corporate : Modigliani-Miller (the irrelevance theorem)

Lecture 14

Complex adaptive systems: Trader types; Population and strategy choice dynamics; Price dynamics and stylized facts.

4 Main references

• Cars Hommes. “Financial Markets as Nonlinear Adaptive Evolutionary Systems,” Quantitative Finance, 1, 149- 167, 2001.

• Blake LeBaron. “A Builder's Guide to Agent-Based Financial Markets,” Quantitative Finance, 1, 254-261, 2001.

• Both are survey-style papers with lots of references [the first is better IMHO].

• Thomas Lux. “Herd Behaviour, Bubbles and Crashes,” Economic Journal, 105, 881-896, 1995

• A lot of the work on this topic is computational

Philosophy

• View of financial markets as “evolutionary systems between different, competing trading strategies” (Cars Hommes)

• Agents are boundedly rational: favor strategies that have done well in the past

• In particular simple “technical” rules are permitted

• Expectations or beliefs dynamics

• Rational expectations eq. (REE) as benchmark

• Heterogeneous population, difference of opinion, coexistence

5 Main ingredients

• Rational traders: [why this term?] fundamentalists (smart money)

• Noise traders: chartists, technical analysts (trend, pattern,…)

• Belief dynamics (correlates with past performance) and/or trader type dynamics

Adaptive belief systems

• 1 riskless (infinite supply) & 1 risky asset (zero supply)

• Myopic mean-variance maximizers with heterogeneous ‘beliefs’ about expected returns

• Forecasting of tomorrow’s price [point estimate, e.g. trend (, mean-reversion), fundamentalists (fund. value)]

• Performance indicators: realized/risk-adjusted return

• MAIN FOCUS: price dynamics and stylized facts

• Minor role played by wealth dynamics (bankruptcy prevalent but not monitored)

• Borrow a LOT form the noise trader approach discuss earlier

6 Adaptive belief systems: std model

Dynamics between types

• Performance of types induces flow between trader types (discrete-choice model with intensities) • Forecasting rules (for time lag =1)

forecast * h h * h pt+1 − pt+1 = ft+1 = α ( pt − pt ) + β e.g. fundamentalist α = β = 0 trend follower α > 0, β = 0 biased belief α = 0, β ≠ 0

7 Dynamics of adaptive belief systems (Hommes, p.157+) Here forecasting depends on (discounted) past prices

8 Comparison with noise trader approach and criticism

• Covers the standard noise trader model!

• REE as benchmark [Is that reasonable?]

• Restriction of “universe”: too restrictive as lacks e.g. log-optimum investment

9 More comments

• LeBaron’s “builder’s guide” survey

• etc.

• Price impact

• -selling (“unrealistic” wealth dynamics)

Contagion model for financial markets by Lux [only if time permits]

Motivation • Self-organized process of ‘moods of the market’

• Optimism and pessimism as an infectious ‘disease’ [aka herd behavior]

• Rooted in ‘psychologist’s attack’ on EMH

10 Model equations

• 2N speculative traders

• A trader is either optimist (n+) or pessimist (n-) n − n • Average opinion x = + − ∈[−1,+1] 2N • Change of mood probabilities p+− (x) [− → +], p−+ (x) [+ → −] • Change of average opinion dx = (1− x) p (x) + (1+ x) p (x) dt +− −+

Specification of functions

p+− (x) = v exp(ax), p−+ (x) = vexp(−ax) • Resulting differential eq. for average opinion has the following fixed points (steady states) If a ≤ 1, x=0 (stable) If a > 1, x=0 (unstable)

and two stable x- < 0, x+ 0 [ = - x-] > 0

• Graph on next slide

11 Graph

Prices via demand and supply

• Net excess demand of traders (each can trade tN units)

DN = n+tN − n−tN = x(2NtN ) =: xTN • Only zero (i.e. market clears) if x=0!

• Add additional actors(!): Fundamentalists

DF = TF ( p f − p) • “Market-maker” induces price adjustment process (though not market clearing in any point in time) dp = β (D + D ) = β[xT +T ( p − p)] dt N F N F f

12 Final model

• “Combining the contagion and price dynamics it seems reasonable to include a feedback effect from the price change on the disposition of speculators” (Lux, p.887)

p+− (x) = vexp(a1(dp / dt) / v + a2 x),

p−+ (x) = v exp(−a1(dp / dt) / v − a2 x)

• Now we have a 2-dimensional system

Graphs

13 Bear/Bull switching model

• Add ‘basic mood’ of the market a0

p+− (x) = v exp(a0 + a1(dp / dt) / v + a2 x),

p−+ (x) = vexp(−a0 − a1(dp / dt) / v − a2 x) da 0 =τ[(r +τ −1(dp / dt)) / p − r / p ] dt f • plus instantaneous market clearing dp = (T /T ) dx / dt dt N F

14 Lecture 15

An evolutionary market model I: Model.

Model basics

• I /strategies [‘fund’ remark] • K long-lived assets (), each in supply 1 • Payoff is perishable consumption good [Lucas 1978] K A (s) > 0 Ak (s) ≥ 0, ƒk=1 k

• Strategies (simple) in simplex of dimension K-1 • Time is discrete • S states of nature (i.i.d. for simplicity) • Common savings rate  [consumption rate 1- ]

15 Evolutionary Finance - understanding the dynamics of financial markets - K i i wt+1 = ƒ(Ak (st+1 ) + pt+1,k )θt,k k =1

i i I λt,k wt p = ρ λ j w j t+1,k ƒ j=1 t+1,k t+1 pt,k

Similarities and differences to short-lived asset model

• Investment strategy = budget shares • K+1 markets (K assets, 1 consumption good) • Short-term (temporary) equilibrium • Main innovation Resale value => capital gains

• Asset market applications are now within reach • Role of consumption (not needed previously)

16 Solving the model On blackboard

Steps • Account for consumption • Determine total wealth => relative wealth (wealth shares) • Dealing with the equilibrium problem => map with state [random dividend]

• Verifying that everything is correct

Lecture 16

An evolutionary stock market model II: Results.

17 Assumptions Stationarity relative and strategies stationary

Definition λ∗-market is evolutionary stable if • no strategy can gain against incumbent λ∗ • any µ ≠ λ∗ loses against some mutant strategy

I = 2: Inverse matrix tractable! (Note: similar to Evol Game Th.)

Local Stability - linearization and the variational equation -

Growth rate at fixed point (1,0)

» K 1 t+1 ÿ i (1− ρ) Rk (st+1 ) + ρ λk (s ) i t i rt+1 ≈ …ƒ 1 t λk (s )Ÿ rt k =1 λk (s ) ⁄

» K t+1 ÿ (1− ρ ) Rk (st+1 ) + ρ λk (s ) t g(µ,λ) = E ln…ƒ t µk (s )Ÿ k =1 λk (s ) ⁄

Growth rate of )‘s share at investor ‘s prices

18 Question Is there a λ * ( s t ) such that for allµ ≠ λ*

(1) g λ * ( µ ) < 0 and for some t ? (2) gµ (λ) > 0 λ(s )

» K t+1 ÿ (1− ρ ) Rk (st+1 ) + ρ λk (s ) t gλ (µ) = E ln…ƒ t µk (s )Ÿ k =1 λk (s ) ⁄

Result (finitely many states) IID case * t * λk (s ) =λk = ERk

Markov case 1− ρ ∞ [λ* (s)]s = ρ m[π m R] k k ƒ ρ m=1

19 Sketch of Proof: IID case 1/4 (1) Why is * stable? λk = ERk

~ 0 gλ* (µ) = — g(...) dP *))))))S Ν +)))))), » K * ÿ ρ λk + (1− ρ) Rk (s) 0 ƒπ s ln…ƒ * µk (s )Ÿ s∈S k=1 λk ⁄

Sketch of Proof: IID case 2/4

» K ≈ ’ ÿ ∆ Rk (s) ÷ 0 = ƒπ s ln…ƒ ∆ ρ + (1− ρ) ÷ µk (s )Ÿ s∈S k =1 « ERk ◊ ⁄

0 K (Strictly) concave inµ(s ) ∈∆ Maximum at µ ( s 0 ) = λ * , and value = 0 there

20 Sketch of Proof: IID case 3/4 (2) Why is µ ≠ λ * unstable?

* ~ 0 gµ (λ ) = — g(...) dP *))))))+S Ν)))))), » K ρ µ (s0 , s) + (1− ρ) R (s) ÿ ƒπ ln…ƒ k k λ* Ÿ s µ (s0 ) k &)s∈)S ))' k =)1 )))( k &))'))⁄( K » 0 ÿ ≥ 0, and > 0 if (s0 ) ≠ * 0 * µk (s ) —...dP ≥ ρ Eƒλk ln… 0 Ÿ = 0 k=1 µk (s , s)⁄

Sketch of Proof: IID case 4/4

» K 0 ÿ » K * ÿ * µk (s ) λk ρƒπ s ln…ƒ λk 0 Ÿ + (1− ρ)ƒπ s ln…ƒ Rk (s) 0 Ÿ s∈S k=1 µk (s , s) ⁄ s∈S k=1 µk (s ) ⁄ &))))'))))( &))'))( ≥ 0, and > 0 if (s0 ) ≠ * K » 0 ÿ 0 * µk (s ) —...dP ≥ ρ Eƒλk ln… 0 Ÿ = 0 k=1 µk (s , s)⁄

21 Findings

Only a λ *-market is evolutionary stable In this stationary market assets are priced at “fundamental values“ (relative dividends)

Market selection favors value investors

Only λ * strategy is growth-optimal @ own prices

Kelly rule for assets instead of states

Fundamental value without short sales

Global selection results

• Evstigneev/Hens/Schenk-Hoppé Working paper NCCR 09/2005 i.i.d. states, fixed investment strategies * λk = ρ ERk is single survivor

• Role of market portfolio still under research

• Adapted strategies: computational results

22 Lecture 17

Applications I: Simulation studies for the evolutionary stock market model; Asset price dynamics; Stylized facts.

Applications

• Simulation with DJIA dividends

• Simulation with SMI dividends (Switzerland)

• Theoretical results: Hens/Schenk-Hoppé, JEDC (2006): an evolutionary view on Tobin’s (1958) liquidity preference argument

23 The data

• DJIA, 21 stocks, 1981-2001 • Each year = one state of nature (S = 21) • Then i.i.d. draw to get “artificial time series” of infinite length • =99% • A “realistic” set of strategies

24 The “list” of strategies

λ* = ƒπ R (s) prospect k s k λk = ƒα(π s )Rk (s) s s

illu 1 gop cond.VaR * λk = λ (1/ K) λ (λ ) K

µ−σ * µ−σ * λ (λ ) λε (λ )

25 26 McDonalds

27 Lecture 18

Applications II: Genetic programs as traders; Evolutionary of strategies; Price dynamics; Kelly rule.

28 In a nutshell

• Evolutionary finance model I: short-lived • Dynamics on the level of strategies • Mutation as innovation/creation of investment strategies • Implemented via genetic programming (GP)

Q: 1. Which strategies will emerge? 2. How are assets priced in the long-term? 3. Does the Kelly rule or the market portfolio play any role?

29 Genetic Programming 1/3

• Investment strategies are computer programmes

• Input variables supply by modeler

• Output = strategy (budget shares)

Genetic Programming 2/3

Evolution of investment strategies:

Crossover Pick 4 strategies at random, rank according to wealth, use best two to generate two new strategies (Tournament)

Mutation Use some random computer code

30 Genetic Programming 3/3 Crossover

Parameter settings

31 Kelly rule (benchmark/prediction)

Experiment 1: state known

32 Experiment 1 Asset price dynamics

Experiment 1 Population

33 Experiment 2: state and last corresponding prices

Experiment 2 Population dynamics (types)

34 A long-term survivor

How do successful strategies invest?

• Wealth is split in two investment styles • market portfolio (proxy) protects against being deleted (by GP) reduces volatility betting on the Kelly rule (but price dependent) generates growth Kelly proportions are moderated, i.e. market’s mood is taken into account

35 Long-run asset prices λ* Kelly rule

= long-term investment strategy

Lecture 19

Applications II: The value premium puzzle.

36 References

• Hens/Schenk-Hoppé/Woehrmann. “An Evolutionary Explanation of the Value Premium Puzzle.” Working Paper No. 280. National Centre of Competence in Research ``Financial Valuation and Risk Management,'' Switzerland, March 2006 (under revision)

• Litterman. “Why an Equilibrium Approach?” In Modern Investment Management: An Equilibrium Approach, Chapter 1, pp. 3-6. Bob Litterman and the Quantitative Research Group Goldman Sachs Asset Management (eds.), Wiley, 2003.

• Punch line: market equilibrium as “center of gravity” for asset price dynamics in the medium run

“Thus, we view the financial markets as having a center of gravity that is defined by the equilibrium of demand and supply.” Bob Litterman, page 3.

SML N R U T E R

S S E C X E

BETA

37 Empirical test of predictions derived from our evolutionary finance stock market model

Hypothesis 1: market equilibrium

• market valuation = expected relative dividend λ* [cross-section]

Hypothesis 2: convergence

* • If above violated, convergence of prices to λ

Convergence dynamics

• Derived from evolutionary finance model with long-lived asset p Relative market valuation of stock = t,k = E R p k ƒn t,n Convergence to this valuation (from local analysis)

p p » p ÿ t+1,k = t,k + a …E R − t,k Ÿ p p k k p ƒn t+1,n ƒn t,n … ƒn t,n ⁄Ÿ

38 Price reversal to *

REL DIV vs REL MV 2001 R^2 = 0.58

0.25

0.2

0.15 V M

l e R

0.1

0.05

0 0 0.05 0.1 0.15 0.2 0.25 Rel Div

Time Series: Procter & Gamble

Rel Div Rel MV Procter&Gamble 81-01

0.08

0.07

0.06

0.05 1 0 - 1 8

V 0.04 M l e R 0.03

0.02

0.01

0 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 Rel Div 81-01

39 40 R^2 EVOL versus CAPM 1981-2001

R^2 EVOL vs CAPM 1981-2001

1.2

1

0.8

EVOL 0.6 CAPM

0.4

0.2

0 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

41 Lecture 20

Summary and conclusion; Hot topics for future research.

Evolutionary finance • Dynamics of financial markets

• Heterogeneous agents

• Strategies (a new view)

• Selection - wealth dynamics

• Evolutionary asset pricing

• Darwinian ideas

• Stability of equilibrium

• Mutation, belief/expectations dynamics

42 Short presentations by PhD students

• Urs Schweri

• Andreas Tupak

(15 minutes each)

Hot topics

• Fitting evolutionary finance models

• Flows (internal versus external)

• Forecasting: Classification of status quo

• Non-stationarity of dividends (as in reality)

• Computational studies beyond Santa Fe

• Continuous-time evolutionary finance

43 Conclusion

• Evolutionary finance has come a long way to deal with problems that cannot be satisfactorily explained within classical (“modern”) financial theory

• Evolutionary finance has provided some surprising insights

• But real challenges still lie ahead

44