The Evolution of Portfolio Rules and
the Capital Asset Pricing Model
Emanuela Sciubba
1
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
2
Abstract
• 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 .
3
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 .
4
• The debate originates from the dissatisfaction with the meanvariance 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
5
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)
6
• 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 .
7
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    0 1, , S  with representative element  1 ,  2, 
Define  t   0t 1,  S with typical element t for each t  1,2 
• Let
denote the product σ-field on Ω
denote the sub-σ-field σ(ωt) of
.
8
• wst :total wealth in the economy at time t if state s occurs .
•
 st :the price of asset s at date t .
•
•αst
:denotes his demand of asset s at time t .
i :the
i
w
fraction of trader i’s wealth at the beginning of t , t 1
that he invests in asset s .
(1)
 rsti : trader i' s investment income at time t if state s occurs .
  sti : trader i' s savings rate .
(2)
9
•
•
{ } as trader i' s portfolio rule and
i
t

t 1
 ,  
i
t
•
i 
t t 1
as the trader i' s investment rule .
and (1)
(3)
 st : the normalised market price of asset s at date t
(4)
10
• 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 .
11
12
2.1 The Dynamics of Wealth Shares
• Trader i’s wealth share
(8)
• Market saving rate
(9)
(10)
13
(11)
•Using our price normalisation :
(12)
 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 .
(15)

is a weighted average across traders of
,
where weight are given by wealth shares at the beginning of
period t .
14
• Define a formal notion of “dominance”
15
• 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 .
16
(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)
17
18
• 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 .
19
• Proof :Under simplifying assumption :all traders have
identical savings rates .
20
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)
21
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 :
(23)
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
and that
is logarithmic ,it follows that
(1)
22
Third type : Agents who display a genuine mean-variance
behavior (Type MV) and are endowed with a quadratic utility
function :
(24)
where
• Substituting (24)into (23) and solving for
order conditions ,we obtain :
using the first
(25)
where:
(26)
is the wealth share of mean-variance traders at date t
23
• 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)
24
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
25
3.1 Trivial Cases
3.1.1 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 .
26
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.
27
3.2 Existence of Equilibrium
• Two types of traders : 1. believe in CAPM
and
2. endowed with a logarithmic
t 1utility function
w
L
L

st
• Traders’ demands are :
(28)
(29)
• There is only unit available of each asset :
(30)
(31)
28
Definition 3
Market clearing equilibrium at date t for
for this economy is an array of portfolios and assets’ prices
such that ,
29
• 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
:
st
s
t

1
 pw
30
(Substituting  t 1  1 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
In compact notation:
while
31
(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.
32
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 .
33
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 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: lim inf t   t  0
(pf steps)
converge almost surely
to
34
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  t  0
 Extinction of traders who believe in CAPM is the last
main 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.
35
The Main Result (3)
Assumption 4:The portion of wealth that traders who believe
in CAPM decide to invest in the risk free portfolio,  t ,is a
monotonic function of their level of wealth, wtCAPM
.
1
-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 riskfree portfolio do not if there is aggregate uncertainty.And
under assumption 4, there is no sample path for  t such that
CAPM traders asymptotically invest only according to the
36
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
37
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
n S
portfolio rules st s 1and n=1,…N.
For simplicity we also assume that:
38
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  0L  0,
then a market clearing equilibrium exists at each
date.In particular,as  tL  0,equilibrium prices  st  0
for some s and therefore  stCAPM  0 for some s, so
that, despite ass.5, theorem 1 is not applicable.
39
A Multipopulation Model (3)
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  stn Ss1and
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.

40
A Multipopulation Model (4)


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,
:
(36)
Traders who believe in CAPM vanish.(
a.s.)
41
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.
42
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.
43
Heterogeneous Risk Attitudes (2)



Consider a population of CAPM traders, indexed
byj  1,... J ;trader j’s portfolio rules at t will be
, and assumption 4
holds for each j.
Denote by  t and  t j 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, lim supt   t j  0j  1,..., J .
44
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  tj is a monotonic function of
wealth is a sufficient condition for all CAPM traders
to vanish.
45
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  st .
Obviously, if
, the MEL traders will
dominate and drive CAPM traders to extinction.
46
Traders with Different Saving Rates (2)


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
ignore the fact that MEL traders have a
“comparative advantage”, so we will prove their
dominance under a weaker assumption.
47
Traders with Different Saving Rates (3)

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
could impose; however, it shows that
in Blume and Easley (1992) can be relaxed.
48
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

49
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.
50
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 t  S 1 s.t.
i  MV , LOG

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
 ti1 ,  ti2 ... 0i  S 1as 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.
51
Existence of Equilibrium (3)


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,
v  0  st  v, s  1,...S andt  0.
52
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  stMVare bounded
away from 0.Therefore theorem 1 apply.
Proposition 16:Under assumption 1 and assuming
that  stL   stMV a.s.s & t 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,
lim inf t   t  0and t  p a.s.

53
The Evolution of Wealth Shares (2)
Proposition 17:Under assumption 1 and assuming
L
MV



that st
st a.s.s & t, a population of traders who
display mean-variance behavior will be driven to
extinction by traders who behave as log utility
maximizers.Formally, lim supt   tMV  0a.s.
(pf steps)We first show that, in presence of aggregate
uncertainty,  stMV a.s.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.

54
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,
lim supt   tMV  0 & lim supt   tMV  0a.s.

The proof is straightforward since the results we
proved in the multipopulation framework apply.
55
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.
56
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.
57