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The Evolution of Portfolio Rules and the Capital

Asset Pricing Model

- Emanuela Sciubba

0. Abstract 1. Introduction 2. The Model 2.1

The Dynamics of Wealth Shares 2.2 Types of

Traders 3. Dynamics with Traders who Believe in

CAPM 3.1 Trivial Cases 3.1.1 No

Aggregate 3.1.2 Constant Absolute Risk

Aversion 3.2 Existence of Equilibrium

3.3 The main Result 3.4 Extensions 4.

Genuine Mean-Variance Behavior 5. Concluding

Remarks

Abstract

- The aim test the performance of the standard

- version of CAPM in an evolution framework .
- Prove traders who either believein CAPM
- and use it as a rule of thumb ,or are endowed

with - genuine mean-variance preferences ,under some

very - weak condition ,vanish in the long run .
- A sufficient condition to drive CAPM or mean

variance - traders wealth shares to zero is that an

investor endowed with - a logarithmic utility function enters the

market .

1. Introduction

- 1.1 Motivation
- Imagine a heterogeneous population of

long-lived agents - who invest according to different portfolio

rules and ask - what is the asymptotic market share of those

who happen - to behave as prescribed by CAPM .
- The result proves
- 1.CAPM is not robust in an evolution sense
- 2.it triggers once again the debate on the

normative appeal - and descriptive appeal of logarithmic

utility approach as - opposed to mean-variance approach in

finance .

- The debate originates from the dissatisfaction

with the mean- - variance approach which fails to single out a

unique optimal - portfolio .
- Kelly criterion That a rational long run

investor should - maximise the expected growth rate of his

wealth share and - should behave as if he were endowed with a

logarithmic - utility function .
- The evolutionary framework adapted in this paper

suggests - that maximising a logarithmic utility function

might not make - you happy ,but will definitely keep you alive

- 1.2 Related Literature
- Debate on bounded rationality in economics and

find - motivation in the simple idea that individuals

may be - irrational and yet markets quite rational
- Becker (1962) and numerous studies
- Evolutionary model of an industry
- Luo (1995)
- Noise trading
- Shefrin and Statman (1994)
- De long et al. (1990,1991)
- Biais and Shadur(1994)

- Blume and Easley (1992,1993)
- in the long run ,traders who are endowed with

a logarithmic - utility function will survive ,as well as

successful imitators . - Cannot directly apply Blume and Easley results
- Two major reasons
- 1 .Blume and Easleys result on logarithmic

tradersdominance - do not necessarily imply that CAPM traders

would vanish . - 2 .both CAPM and mean-variance trading rules

do not satisfy - a crucial boundedness assumption which

Blume and Easley - impose .

2. The Model

- Time is discrete t
- There are S states of the world s
- States follow an i.i.d process with distribution

- Let denote the product s-field on O

- denote the sub-s-field s(?t) of

.

- wst total wealth in the economy at time t if

state s occurs . - the price of asset s at date t .
- denotes his demand of asset s at time t

. - asti the fraction of trader is wealth at the

beginning of t , - that he invests in asset s .

(1)

(2)

- and (1)

(3)

(4)

- In equilibrium ,prices must be such that markets

clear , - i.e. total demand equals total supply

(5)

(6)

(4)

- Market prices are related to wealth shares .

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2.1 The Dynamics of Wealth Shares

- Trader is wealth share

(8)

- Market saving rate

(9)

(10)

(11)

- Using our price normalisation

(12)

- Trader is wealth share will increase if he

scores a payoff - which is high than the average population

payoff . - The fittest behaviour is that which maximises the

expected - growth rate of wealth share accumulation .
- is a weighted average across

traders of , - where weight are given by wealth shares at

the beginning of - period t .

(15)

- Define a formal notion of dominance

- Blume and Easley justify the word dominates

as follows - When saving rates are identical a trader who

dominates - actually determines the price asymptotically .
- His wealth share need not converge to one

because - there may be other traders who asymptotically

have - the same portfolio rule ,but prices adjust
- so that his conditional expected gains

converge to zero - Assumption 1 For all t and all i ,
- and

- Assumption 2 There exists a real number
- such that ,for all i

for all s .

(12)

the indicator function that is equal to 1 if

state s occurs at date t and equal 0 to

otherwise .

- The expected values of

conditional on the information - available at time t-1

(13)

(14)

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- Intuitions
- 1 .the dominating traders are those who are

better than the others - in maxinising the expected growth rate of

their wealth shares . - 2 .condition (c) implies that conditions (b) and

a fortiori(a) fail . - condition (c) puts a restriction on the rate

at which - diverge .
- 3 .if all traders have the same rate ,the

dominating trader - determines market prices asymptotically and

his wealth share - need not converge to 1 because there might be

other suriving - traders .

- Proof Under simplifying assumption all traders

have - identical savings rates .

2.2 Types of Traders

- Three different types of traders Type CAPM,Type

L,Type MV

- First type Agents who believe in CAPM(Type

CAPM)

Second type Agents who are endowed with a

logarithmic utility function(Type L) and

who maximise the growth rate of their wealth

share and invest according to a simple

portfolio rule

(22)

- More generally ,a rational trader i will choose
- so as to maximise
- subject to the constraint that investment

expenditure at each date is less than or equal to

the amount of wealth saved in the pervious period

. - If is logarithmic ,it follows that
- and that

(23)

(1)

Third type Agents who display a genuine

mean-variance behavior (Type MV) and are

endowed with a quadratic utility function

where

(24)

- Substituting (24)into (23) and solving for

using the first - order conditions ,we obtain
- where

(25)

(26)

is the wealth share of

mean-variance traders at date t

- According to (1),(4),(8)and (25)

(27)

- If for some s ,then both

and - so that theorem 1 in section 2.1 does not apply

.

(19)

3. Dynamics with Traders who Believe in CAPM

- Assumption
- Only two types of traders in the economy
- 1.believe in CAPM
- 2. Logarithmic utility function(MEL

traders) - is the quantity (share) of each

asset s - that trader i demands at time t .

- is the share of aggregate wealth

which belong to type L - is the share of aggregate wealth

which belong to - type CAPM at the beginning of period t .
- The degree of risk aversion is homogeneous in the

population - of traders who believe in CAPM ,so that
- and

3.1 Trivial Cases3.1.1 No Aggregate Risk

- Remark 1 With no aggregate risk ,in a

population of traders - who believe in CAPM and traders with

logarithmic utility - function ,the behavior of traders who believe

in CAPM and - traders with a logarithmic utility function

coincide . - Formally ,if then

- Intuition Because market and risk-free

portfolio coincide , - traders who believe in CAPM invest only

according to the - market portfolio ,so that their behaviour is

purely imitative . - As a result ,when a logarithmic utility

maximiser enters the - economy ,everyone invests according to his

portfolio rule .

3.1.2 Constant Absolute Risk Aversion

- All investors are risk averse and that the

degree of risk aversion - does not change with wealth i.e.constant

absolute risk aversion . - Remark 2 under the CARA assumption ,in a

population of - traders who believe in CAPM and traders with

logarithmic utility - function .if the

behaviour of traders who believe - in CAPM and traders with a logarithmic utility

function - coincides .i.e.

3.2 Existence of Equilibrium

- Two types of traders 1. believe in CAPM

and - 2. endowed with a logarithmic utility

function - Traders demands are

(28)

(29)

- There is only unit available of each asset

(30)

(31)

Definition 3 Market clearing equilibrium at

date t for for this economy is an array of

portfolios and assets prices

such that ,

- Proposition 2 Provided that

,at each date - there exists a unique market clearing

equilibrium .

(31)

- A corollary of equation 31 if all traders

behave according to - CAPM rule that there is no market clearing

equilibrium . - Intuition in such an economy (CAPM) every

trader would - like to invest his whole wealth in the

risk-free portfolio . - However ,as long as there is aggregate

uncertainty ,for an - equilibrium to exist some traders must bear the

risk . - A unique equilibrium exists in an economy

populated only by - traders who are endowed with a logarithmic

utility function . - Equilibrium prices are equal to probabilities
- (Substituting into(31)

)

- Characterise the limiting behavior of prices as
- equilibrium prices move towards a vertex of of

the price - simplex .Only the market of asset 1(the asset

with the lowest - payout) clears with a strictly positive price

. - Proposition 3 When

while - In compact notation

(pf)In the limit ,non-negativity of prices

requires while market clearing requires The

unique limiting value for

that satisfies both is

(32)

Implies

- Consequence of proposition 3 that portfolio

weights of traders - who believe in CAPM are not bounded away from

zero on those - sample paths where So theorem 1

does not apply .In particular,we can not use it

to show that log traders dominate, since we would

need to assume their dominance( ) in

order to apply the theorem.

Corollary 4 according to (28)(29)(31)(32)

- Notice that

,so that there is - market clearing
- Both types of traders invests

- only CAPM traders invest in asset 1 .

3.3 The Main Result (1)

- In this section we prove our results under a

simplifying assumption - Assumption 3
- We present our first two main results as separate

propositions which accords with Blume and Easley

(1992) - -Proposition 5Under assumption 1 and 3, in a

population of traders who believe in CAPM and

traders who are endowed with a logarithmic

utility function, the latter dominate almost

surely. Formally - (pf steps)

converge almost surely to

The Main Result (2)

- -Proposition 6Under assumption 1 and 3, in

a population of traders who believe in CAPM and

traders who are endowed with a logarithmic

utility function, the latter dominate almost

surely,so that, - (Note)MEL dominate
- Because it is possible that

and yet - Extinction of traders who believe in CAPM is the

last main result, and one could not directly

anticipate that through Blume and Easleys

theorem 1.In fact, We have examined two trivial

cases as examples that traders who believe in

CAPM survive because they behave as MEL. To prove

this result,we need to make a further assumption

on traders behavior towards risk.

The Main Result (3)

- Assumption 4The portion of wealth that traders

who believe in CAPM decide to invest in the risk

free portfolio, ,is a monotonic function of

their level of wealth, - -Proposition 7Under assumption 1, 3 and 4 and in

presence of aggregate uncertainty, in a

population of traders who believe in CAPM and

traders who are endowed with a logarithmic

utility function, the former vanish almost

surely. - (Intuitive Proof)Dominance of MEL requires that

in the long run all surviving traders invest

according to the Kelly criterion.We prove that

the CAPM rule does not succeed in fully imitating

the behavior of MEL traders.We find that the

market portfolio weights converge to

probabilities,but risk-free portfolio do not if

there is aggregate uncertainty.And under

assumption 4, there is no sample path for such

that CAPM traders asymptotically invest only

according to the market portfolio.

3.4 Extensions

- In this section, our aim is to check the

robustness of our main results in three more

general settings - A Multipopulation Model
- Heterogeneous Risk Attitudes
- Traders with Different Savings Rates

A Multipopulation Model (1)

- Consider a population of traders who believe in

CAPM, and suppose a MEL trader enters the market

with N other types of traders with portfolio

rules and n1,N. - For simplicity we also assume that

A Multipopulation Model (2)

- Assumption 5 allows us to apply corollary 4.1 in

Blume and Easley (1992). - Assumption 6 is without loss of generality even

if

all the results in this section would

still apply by proposition 5, 6 and 7. - It is possible to show that, provided that

, then a market clearing equilibrium exists at

each date.In particular,as ,equilibrium

prices for some s and therefore

for some s, so that, despite ass.5,

theorem 1 is not applicable.

A Multipopulation Model (3)

- Proposition 8Under assumptions 1,3 and 5,given a

population of traders who believe in CAPM,

suppose that a trader with log utility function

and N other traders with portfolio rules

and n1,N, enter the market.Traders endowed

with a log utility function will dominate almost

surely and determine asset prices asymptotically.

- (Pf Steps)We first show that log utility

maximizers outperform each of the N new types of

traders.We then prove that LOG traders dominate

by similar arguments to those used for

proposition 5.

A Multipopulation Model (4)

- Let be the

limiting values of - respectively,

as t?8. - Proposition 9Under assumptions 1,3,4,5,and 6,

given a population of traders who believe in

CAPM, suppose that a trader with log utility

function and N other traders enter the

market.Unless the evolution of the system is such

that,

(36)

Traders who believe in CAPM vanish.(

a.s.)

A Multipopulation Model (5)

- Condition (36)can also be express as follows
- What (36) requires is that the N new rules

should complement CAPM behavior so that we could

think of them as of a single trader whose

portfolio rules are asymptotically equal to

probabilities.As a result, even no traders

asymptotically behaves as a log utility

maximizer, all traders survive. - This condition is severe,so we claim that

extinction of CAPM believer is generic.Survival

of CAPM traders is not robust to small change to

the set of the new N types of traders introduced

in the market.

Heterogeneous Risk Attitudes (1)

- In this section, we show that our results are

robust when allowing for heterogeneity in the

degree of risk aversion among CAPM traders. - In fact, we can deal with heterogeneity thinking

of a population of traders endowed with different

degrees of risk aversion as of a single average

trader whose portfolio rules are given by an

appropriate weighted average of each traders

portfolio rules.

Heterogeneous Risk Attitudes (2)

- Consider a population of CAPM traders, indexed by

trader js portfolio rules at t

will be -

, and assumption 4 holds for each j.

- Denote by and the wealth shares of MEL

traders and of CAPM trader j, respectively. - Proposition 10Under assumption 1,3 and 4, log

utility maximizers dominate and drive to

extinction a population of heterogeneous traders

who believe in CAPM.Formally,

Heterogeneous Risk Attitudes (3)

- (pf steps)
- We first show that log utility maximizers

dominate in a world of aggregate uncertainty. - Again, an immediate corollary of this result is

that price converge to probabilities.

Finally, assuming that

is a monotonic function of wealth is a

sufficient condition for all CAPM traders to

vanish.

Traders with Different Saving Rates (1)

- If saving rates are different across traders, by

theorem 1, trader i dominates on those sample

paths where - So, the market selects for most patient

investors, i.e., those whose savings rate is

larger w.r.t. the average . - Obviously, if , the MEL

traders will dominate and drive CAPM traders to

extinction.

Traders with Different Saving Rates (2)

- Proposition 11Under assumptions 14, in a

population of traders who believe in CAPM and of

log utility maximizers, the latter dominate,

provided that their savings rate is at least as

large as the average savings rate, and drive to

extinction the population of traders who believe

in CAPM.Formally,if - then,
- However,by assuming that ,we

ignore the fact that MEL traders have a

comparative advantage, so we will prove their

dominance under a weaker assumption.

Traders with Different Saving Rates (3)

- Proposition 12Under assumptions 1 and 4, in a

population of traders who believe in CAPM and

traders with a log utility function, the latter

dominate and drive CAPM traders to extinction if - a.s.
- This condition is weaker than

Namely -

, while the - converse is not true. It is not the weakest

one could impose however, it shows that

in Blume and Easley (1992) can be

relaxed.

4. Genuine Mean-Variance Behavior

- Traders who believe in CAPM do not display a

genuine mean-variance behavior they know what

the two-fund separation theorem prescribes,

believe it works in reality and only partially

optimize between the risk-free and market

portfolios. - In this section, we show that, in an evolutionary

framework, traders with mean-variance preferences

will not do any better than traders who believe

in CAPM. - 4.1 Existence of Equilibrium
- 4.2 The Evolution of Wealth Shares

4.1 Existence of Equilibrium (1)

- Suppose that there are two types of rational

traders in the markettraders who are endowed

with a quadratic utility function(and display a

genuine mean-variance behavior)and traders who

are endowed with a log utility function. - From an analytical point of view, the equilibrium

existence problem in this setting is equivalent

to the general equilibrium problem in a pure

exchange economy.

Existence of Equilibrium (2)

- Definition 13 At each date t?0, an equilibrium

for this economy is an array of portfolio

compositions

and a price vector

s.t. - and markets clear
- This is clearly not a pure exchange economy

traders are not endowed with assets shares but

with exogenous wealth. However, we can consider

as if it

was an endowment vector in assets shares for

trader i and we can study equilibrium existence

as if we were facing a pure exchange general

equilibrium model.

Existence of Equilibrium (3)

- Proposition 14When there are two types of

traders- traders who are endowed with a log

utility function (traders of type L)and traders

who display a genuine mean-variance

behavior(traders of type MV)-there always exists

an equilibrium. - Proposition 15Equilibrium prices have a strictly

positive lower bound.Formally,

4.2 The Evolution of Wealth Shares (1)

- Recall (27) that a rational trader endowed with a

quadratic utility function chooses a portfolio - Proposition 15 allows us to claim that are

bounded away from 0.Therefore theorem 1 apply. - Proposition 16Under assumption 1 and assuming

that in a

population of log utility maximizers and of

traders who display a genuine mean variance

behavior, the former dominate and determine asset

prices asymptotically. Formally, -

a.s.

The Evolution of Wealth Shares (2)

- Proposition 17Under assumption 1 and assuming

that , a population of

traders who display mean-variance behavior will

be driven to extinction by traders who behave as

log utility maximizers.Formally, - (pf steps)We first show that, in presence of

aggregate uncertainty, will not

converge to probabilities.We then prove that

dominance of MEL traders and price convergence to

probabilities implies that the wealth share of

mean-variance traders must converge to 0 a.s.

The Evolution of Wealth Shares (3)

- In an economy where some traders display a

genuine mean-variance behavior and others believe

in CAPM, both types will be driven to extinction,

should a log utility maximizer enter the

market.Formally, - The proof is straightforward since the results we

proved in the multipopulation framework apply.

5. Concluding Remarks (1)

- In the evolutionary setting for a financial

market developed in Blume and Easley (1992), we

consider three types of traders traders who

believe in CAPM, traders who display a genuine

mean-variance behavior, and MEL traders. - Our main result are obtained in a simple setting

where traders have constant and identical saving

rates.We prove that MEL traders dominate.

Furthermore, in presence of aggregate

uncertainty, traders believing in CAPM are driven

to extinction.

5. Concluding Remarks (2)

- We then show the robustness of these results

removing some of the initial simplifying

assumption. Firstly, we allow for more than two

types of traders in the market.Secondly,we allow

for heterogeneous degree of risk aversion among

CAPM traders.Finally, we allow for different

saving rates across traders. - We also deal with an economy populated by genuine

mean-variance traders.We show that if a log

utility maximizer enters the market, he

dominates, determines market prices

asymptotically and drives to extinction the

population of mean-variance traders.