The Evolution of Portfolio Rules and the Capital Asset Pricing Model

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The Evolution of Portfolio Rules and the Capital
Asset Pricing Model

• Emanuela Sciubba

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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
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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 .

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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 .

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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

6

• 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 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 .

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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

• Let denote the product s-field on O
  
• denote the sub-s-field s(?t) 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
  .
• asti the fraction of trader is wealth at the
  beginning of t ,
• that he invests in asset s .
  
  
  

(1)
(2)
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• and (1)
  

(3)
(4)
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• 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)
14
(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)
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• Define a formal notion of dominance

16

• 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

(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 .

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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
(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)
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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

(25)
(26)
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

(28)
(29)

• There is only unit available of each asset

(30)
(31)
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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)
  )

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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
(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.

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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 .

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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

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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.

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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.

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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

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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 n1,N.
• For simplicity we also assume that

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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.

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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.

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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.)
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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.

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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 traders
  portfolio rules.

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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,
  

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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.

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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.

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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.

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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.

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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 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.

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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.

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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,

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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.

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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.

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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.

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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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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.