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The Evolution of Portfolio Rules and the Capital Asset Pricing Model

Emanuela Sciubba

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0. Abstract1. Introduction2. The Model 2.1 The Dynamics of Wealth Shares 2.2 Types of Traders3. 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 Extensions4. Genuine Mean-Variance Behavior5. Concluding Remarks

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• The aim : test the performance of the standard

version of CAPM in an evolution framework .

• Prove : traders who either “believe”in 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 .

Abstract

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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 .

1. Introduction

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• 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

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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)

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• 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 Easley’s result on logarithmic traders’dominance

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 .

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2. The Model• Time is discrete : t • There are S states of the world : s• States follow an i.i.d process with distribution

2,1 t ,1

, ,,1 t0

,210

eachforelementtypicalwithSDefine

elementtiverepresentawithSLettt

• Let denote the product σ-field on Ω

denote the sub-σ-field σ(ωt) of .

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• 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 .

•αsti :the fraction of trader i’s wealth at the beginning of t ,

that he invests in asset s .

st

itw 1

. rate savings si'trader :

. occurs s state if t at time income investment si' trader :ist

istr

(2)

(1)

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•

•

• and (1)

. rule investment si' trader theas,

and rule portfolio si' trader as}{

1

1tit

tit

it

(4)

t dateat sasset of pricemarket normalised the:st

(3)

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• In equilibrium ,prices must be such that markets clear , i.e. total demand equals total supply

(6)

(5)

• Market prices are related to wealth shares .

(4)

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2.1 The Dynamics of Wealth Shares• Trader i’s wealth share

• Market saving rate

(9)

(10)

(8)

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•Using our price normalisation :

Trader i’s 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 .

(12)

(11)

(15)

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• Define a formal notion of “dominance”

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• 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 .

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(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 :

(14)

(13)

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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 .

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• Proof :Under simplifying assumption :all traders have

identical savings rates .

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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)

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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 :

• More generally ,a rational trader i will choose

so as to maximise :

subject to the constraint that investment expenditure at each dat

e is less than or equal to the amount of wealth saved in the pervio

us period .

• If is logarithmic ,it follows that

and that

(23)

(22)

(1)

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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:

(26)

(25)

is the wealth share of mean-variance traders at date t

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• 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)

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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

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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 .

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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.

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3.2 Existence of Equilibrium• Two types of traders : 1. believe in CAPM and

2. endowed with a logarithmic utility function• Traders’ demands are :

(31)

(28)

• There is only unit available of each asset :

(29)

(30)

Lst L

t w1

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Definition 3 Market clearing equilibrium at date t for

for this economy is an array of portfolios and assets’ prices

such that ,

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• 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) )11 t

1 t s stw p

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• 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:

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

while market clearing requires

The unique limiting value for that satisfies both is :

Implies

(32)

• 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.

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• Notice that ,so that there is

market clearing

Both types of traders invests ;

only CAPM traders invest in asset 1 .

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

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In this section we prove our results under a simplifying assumption:

Assumption 3: We present our first two main results as separate propositi

ons which accords with Blume and Easley (1992):

-Proposition 5:Under 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

3.3 The Main Result (1)

0inflim tt

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The Main Result (2) -Proposition 6:Under 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 mai

n result, and one could not directly anticipate that through Blume and Easley’s 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.

0t

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The Main Result (3)Assumption 4:The 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 7:Under 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.

t.1

CAPMtw

t

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3.4 ExtensionsIn this section, our aim is to check the robustness of our main results in three more general settings: A Multipopulation ModelHeterogeneous Risk AttitudesTraders with Different Savings Rates

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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 n=1,…N.

For simplicity we also assume that:

S

snst 1

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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 (2)

00 L

0Lt 0st

0CAPMst

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Proposition 8:Under 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 n=1,…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 (3)

S

snst 1

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Let be the limiting values of

respectively, as t→∞. Proposition 9:Under 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, :

A Multipopulation Model (4)

Traders who believe in CAPM vanish.( a.s.)

(36)

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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.

A Multipopulation Model (5)

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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 trader’s portfolio rules.

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Consider a population of CAPM traders, indexed by ;trader j’s 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 10:Under 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 (2)

Jj ,...1

t jt

.,...,10suplim Jjjtt

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(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.

Heterogeneous Risk Attitudes (3)

jt

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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.

st

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Proposition 11:Under assumptions 1&4, 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 ignor

e the fact that MEL traders have a “comparative advantage”, so we will prove their dominance under a weaker assumption.

Traders with Different Saving Rates (2)

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Proposition 12:Under 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 coul

d impose; however, it shows that in Blume and Easley (1992) can be relaxed.

Traders with Different Saving Rates (3)

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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

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4.1 Existence of Equilibrium (1) Suppose that there are two types of rational

traders in the market:traders 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.

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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 (2)

LOGMVi ,

1 St

1021 ...,

Siit

it

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Proposition 14:When 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 15:Equilibrium prices have a strictly positive lower bound.Formally,

Existence of Equilibrium (3)

.0,...1,0 tandSsvv st

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4.2 The Evolution of Wealth Shares (1) Recall (27) that a rational trader endowed with a quad

ratic utility function chooses a portfolio:

Proposition 15 allows us to claim that are bounded away from 0.Therefore theorem 1 apply.

Proposition 16:Under 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.

MVst

tssaMVst

Lst &..

pand ttt 0inflim

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Proposition 17:Under 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 (2)

tssaMVst

Lst &..

..0suplim saMVtt

..saMVst

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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.

The Evolution of Wealth Shares (3)

..0suplim&0suplim saMVtt

MVtt

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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.

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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.

5. Concluding Remarks (2)