Microscopic Simulation of the Stock Market: the Effect of Microscopic Diversity

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Microscopic Simulation of the Stock Market: the Effect of Microscopic Diversity

Moshe Levy, Haim Levy, Sorin Solomon

To cite this version:

Moshe Levy, Haim Levy, Sorin Solomon. Microscopic Simulation of the Stock Market: the Ef- fect of Microscopic Diversity. Journal de Physique I, EDP Sciences, 1995, 5 (8), pp.1087-1107.

�10.1051/jp1:1995183�. �jpa-00247116�

Classification Physics Abstracts

07.75+n 02.50-r 01.90+g

Microscopic Simulation of trie Stock Market: trie Elfect of Micro-

scopic Diversity

Moshe

Levy,

Haim

Levy(*)

and Sorin Solomon

llacah Institute of Physics, Hebrew University, Jerusalem, Israel

(Received

10 Marcll 1995, received in final form 3 April 1995, accepted 10 April

1995)

Abstract.

Although

trie representative individual framework has been shown _to be

generally

illegitimate and _erroneous, it continues to be widely used for lack of

a more suitable approach.

In this paper, we present an alternative methodology for _economic study trie microscopic

simulation

(MS)

approach. We employ the MS methodology to study _a basic model of the

stock market formulated at trie microscopic level trie level of the individual investor. We

contrast macroscopic market behavior _arising from heterogeneous _as opposed to homogeneous

(representative)

investors. We study trie macroscopic effects of different forms of trie microscopic heterogeneity: heterogeneity of preference, expectation, strategy, investor-specific _noise, and their combinations. We show thon representative investor models lead to unrealistic market phenomena such _as periodic booms and crashes. These penodic booms and crashes persist

even when _we relax trie assumption of homogeneous preference. Orly when heterogeneous expectorions _are introduced does the market behavior become realistic.

Introduction

Very

few financial

analysts

would

deny

that agents

operating

in trie market _are diverse in their

preferences

and in their frameworks of

forming expectations. Yen,

in order to obtain

analytical results,

_many macroeconomic models _{assume a}

"representative" agent.

^This

approach

is analo- gous ^to trie _mean field

approximation

in statistical

physics.

The

"representative"

_agent models often

yield

results that _{are very} diflerent from observed market behavior. For instance, the

representative

individual framework leads to the unrealistic scenario of _no trade in the market.

The

implication

of such _a result negates ^the existence of _a

meaningful

stock market

[1-4].

Another _erroneous result

emanating

^from the

representative

agent

assumption

is reflected in the

Sharpe-Lintner capital

asset

pricing

model

(CAPM).

In this

model,

which derives

equilib-

rium

prices

of

risky

_assets, ail agents _are assumed _to be risk-averse.

Heterogeneous preference

is

allowed,

yen, ^ail agents _are assumed to agree on trie

expected

returns and _on trie variance- covariance matrix

ii-e-, they

have

homogeneous expectations).

Based _on these

assumptions

the CAPM is derived where "beta" _is trie risk _measure of the individual asset.

However,

once

(*) Hebrew University aud University of Flonda.

@ Les Editions de Physique 1995

trie agents are diverse with respect to their

expectations

_or the information _at their

disposai,

or when

they

_incur dilferent transaction costs, ^{the whole model}

collapses,

beta becomes _mean-

ingless,

and _a much _more

complex

_measure of risk is called for.

Thus,

when the

assumption

of

representative

investors

(with

respect to

expectation)

is

relaxed,

the derivation of the risk-

retum

relationship

leads to a result that is very dilferent _to

Sharpe-Lintner's CAPM(~).

Kirman,

in his review of the

problematic

nature of the

representative

individual concept, concludes:

"In

particular,

I will argue that

heterogeneity

of agents may, in

fact, help

to _save the standard model.. it is clear that the

"representative"

agent ^deserves a decent

burial,

_{as an}

approach

to

economic

analysis

that is non

only primitive

but

fundarnentally

erroneous" (9].

In this

study,

_we suggest

microscopic

simulation _{as an} alternative to the

representative

individual frarnework.

Microscopic

simulation

(MS)

is _a

methodology

that is used to

study

the macroscopic ^{behavior of} a system

composed

of _many basic

interacting

elements

acting according

to a known microscopic rule. In

microscopic simulation,

the

microscopic

elements and their actions _are simulated

by

_computer

according

to the

microscopic

rule.

The simulation _con be

thought

of _{as an}

acting

ouf of

reality by

the computer. For _a concrete

exarnple

consider _an ideal _gas in _a box. The

microscopic

elements _are the _gas atoms, and the

microscopic

rule states that when _an atom collides with another atom, _or with the

watt,

it will bounce off like _a billiard

hall; otherwise,

it will

simply

continue to move at the _same

velocity.

One _cari write _a computer program that will accept as

input

the initial

positions

and velocities of ail the atoms, and will simulate the

development

of the system

through

time. In the _case of the ideal _gas, the

macroscopic properties

of the system cari also be derived

analytically.

However,

if the

microscopic algorithm

is somewhat _more

complex, analytic

methods fait. In contrast, this additional

complexity

does non constitute _a

problem

in the simulation method.

Any

_system with any "wild" microscopic

algorithm

_cari be simulated. MS is _a

powerful

tool because it is non restricted

by

the constraints of

analytical

treatment.

MS is

widely

used in

physics

_[ID,

iii.

More

recently,

MS has been

implemented

in

biology (see,

for

example, (12-16])

and in

sociology Ii?i.

MS has _proven ta be _{a very} fruitful

approach

in these

fields, producing

new

understanding

and

insight

into the behavior of

complex

systems.

We believe that MS will be _as fruitful _in finance and in economics _as it has been in

physics, biology

and

sociology.

The purpose of this _paper is _to

apply

MS to the

study

of the stock _mar-

ket. We construct a microeconomic model of the stock market and _we

analyze

the

resulting

macroeconomic market behavior via MS. In

particular

_we

analyze

and contrast the _macroeco-

nomic behavior

produced by

two microeconomic models: _one with

homogeneous

agents

(the

representative

agent

approach),

^{and the} other with diverse

agents.

^{In both models} we assume

investors to be

expected utility

maximizers.

MS allows _us to

analyze

the elfects of agent

heterogeneity

_on the macroeconomic results

and the _errors obtained

by assuming

the unrealistic

assumption

of the

representative

investor.

We then go on to

analyze

vanous sources of

heterogeneity,

and show that _some

homogeneity assumptions

are non crucial to the macroeconomic results but others

completely change

the results relative to the

non-homogeneous

case.

Agents

con be

heterogenous

in several respects. The main

heterogeneity

factors

analyzed

in this paper are:

1)

^Preference

(risk

aversion

measure).

2) Agent-specific

noise

ii-e-,

^the agent is _an

expected utility

maximizer but _may deviate from the

optimal

investment strategy due to random

noise).

(~) ^For more details _on trie effect of agent' heterogeneity _on equilibrium results, _see references [5-8]

3) Expectation

Even if ail investors have the _same

preference (utility function), they

_may have

heterogeneous expectations regarding

the future distribution of the random variable

(retum

in the stock

market).

For

exarnple,

ail

agents

estimate the future distribution of retums

by looking

ai the historical _rates of retum, but

they

are

heterogeneous regarding

their memory sparts; some look bock ai trie last rive

years'

retums whereas others _may look ai the last ten

years'

retums.

4) Strategy Agents

_may ^have dilferent investment

strategies.

Some _may

adjust

their

expectations

and their investment

policies frequently,

whereas

others,

with

higher

trans- action costs, _may ^have a

longer

_rua

approach.

The microeconomic model that _we present ^{here does} non

yield

to

analytical

treatment _even

in the most

simple

_case where agents are

homogeneous.

MS allows _{us trot}

only

to

study

the

resulting macroscopic

behavior in the

homogeneous

and

heterogeneous

_cases, but also to

study

the elfects of each of trie above factors of

heterogeneity separately,

_as well _as in combination.

Like

Kirman,

_{we come to} trie conclusion that the introduction of

heterogeneity

_{is necessary}

m order _to

produce

results that conform to the stock market's observed behavior.

Relying

on

representative

agents ^leads to the absurd result of _{a no} trade market. Trie introduction of

agent-specific

noise does trot have _a

mitigating

^elfect:

unacceptable results,

such _as

Sharp

booms and crashes in the stock

market,

_are

obtained;

_moreover, these booms and crashes

are

periodic

and therefore

predictable,

which

grossly

contradicts the concept of _an eilicient market.

Only

when

heterogeneous expectation

is introduced do _we obtain _more realistic stock

price

^behavior.

The _structure of the paper is as follows: Section

provides

the model. Section 2

provides

the data and the results, and Section 3 concludes the paper.

1. Trie Model

The

microscopic

"element" of _our model is trie individual investor. Individual investors interact via the

buying

and

selling

of stocks and bonds.

The model

presented

here is the _most basic model attainable in which ail the crucial elements of the stock market _are included. We have

consciously

made certain

simplifying assumptions

and omitted _some of the features of the real market. This _was done in order to focus _on the main elements of the market and trot because of _any limitations of the MS method. As

mentioned

before,

_any model _cari be

microscopically

simulated.

We

begin by describing

the basic model where investors _are

homogeneous

in ail respects.

Later _we will describe how diflerent forms of

diversity

_are introduced.

1.1. THE HOMOGENEOUS INVESTOR MODEL. Dur stock market consists of _two investment

options:

_a stock

(or

index of

stocks)

and _a bond. The bond is assumed _to be _a riskless asset, and the stock is _a

risky

asset. The stock _{serves as a proxy} for the market

portfolio, je-g-,

the Standard & Poors

index).

The extension from _one

nsky

asset to many

risky

assets is

straightforward. However,

_one stock

(the index)

is sulficient for _our present

analysis

because

we restrict ourselves to

global

market

phenomena

and do non deal with distributions _across several

risky

assets. The investors _are allowed _to revise their

portfolio

at

given

lime

points,

1-e-, we discuss _a discrete lime model.

Trie bond is assumed to be _a riskless investment

yielding

_a constant retum at trie end of each lime

period.

The bond is exogenous and investors _can

buy

from it _as much _as

they

wish at _a

given

rate. We denote this riskless rate of retum

by

_r. Thus, _an investment of W dollars at lime t will

yield W(1+ r)

at lime t +1.

The return _on trie stock is

composed

of _two elements:

ii) Capital gain (loss):

The

price

of trie stock is determined

collectively by

ail investors

by

the law of

supply

and demand. If _an investor holds _a stock any rise

(fait)

in the

price

of the stock contributes _{to an} increase

(decrease)

in the investors' wealth.

(ii)

Dividends: The _{company earns}

income1 (a

random

variable)

and distributes dividends.

We _assume that the firm _{pays a} dividend of

Dt

_Per share at lime t. We will elaborate _on

Dt in the next section.

Thus,

the overall return _on stock in

period

t, ^denoted

by Ht,

is

given by:

Ht ₌

~

j~~

^{~ ~~}

^Il)

t-i

where Pt is the stock _puce at lime t.

In order to decide _on the

optimal

diversification between the

risky

and the riskless asset, one

should consider the ex-ante returns.

However,

since in

practice

these returns _are

generally

non

available,

we assume that the ex-post distribution of retums is

employed

_{as an} estimate of the ex-ante distribution.

In _our model investors

keep

track of the last k retums _on the stock which _we call the stock's

History.

We _assume that investors have _a bounded recall in that

they

believe that each of the last k

history

elements at time t

Hj,j=t,t-i,...,t-k+i

has _an

equal probability of1/k

to _reoccur in the next time

period ii +1)(~).

The bounded recall framework has been

employed

in other

garne

theory analyses (18-20].

Investors derive "well

being"

_or

"utility"

from their wealth. Each investor is characterized

by

_a

utility function, U(W), reflecting his/her personal preference.

The

utility

is _a monoton-

ically increasing

^{function of}

wealth,

denoted

by W,

and is also

usually

considered _concave:

U'(W)

> 0,

U"(W)

_< 0. In the

homogenic

model _{we assume} the _same

utility

function for ail investors, ^and we take this function _to be

logw.

Later on, when _we introduce

heterogeneity,

we allow diverse

preferences.

In _a situation with

uncertainty

^trie

objective

of each investor is to maximize the

expected

value of

his/her utility.

Investors divide their _money between the

two investment

options

_in the

optimal

_way which maximizes their

expected utility.

We will elaborate _on this

point

^below.

1.1,1. Trie

Dynamics.

To illustrate the

dynamics

of _our model consider the _state of the market at _some

arbitrary

lime t. We denote the price of the stock at this time

by

Pt. The stock's

history

at this lime _{is a set} of the last k retums _on the

stock, Hj,j=t,t-i,...,t-k+1

We denote the wealth of the ith investor at time _t

by

Wt

Ii),

and the number of shares held

by

this

investor

by Nt(1).

Now,

let _{us see} what

happens

at the _next trade

point,

lime t +1.

Income Gain

First, note that the investor accumulates wealth in the interval between lime t and time t +1.

He/she

_receives

Ni(1)Dt

in dividends and

(Wi(1) Ni(1)Pi)r

_m interest.

(Wt(1) Ni(1)Pi

is the _money held in bonds _as Wi

Ii)

is the total wealth and Ni

(1)Pi

is the wealth held in

stocks).

Thus,

^before the trade at time t +1, the wealth of investor1 is:

(~) We also consider models where the equal probability expectation _is replaced with the expectation of exponentially decaying reoccurrence probability.

wiji)

+

Niji)Di

₊

jwiji) Niji)pi)r j~) During

the interval between time _t and time _t +1 there is _no trade, ^therefore the share _puce does not

change

and there is _no

capital gain

or loss.

However,

at the next

trade,

at the lime t + 1,

capital gain

_or loss _{can occur, as}

explained

below.

The Demand Function for Stocks

We denve the aggregate ^{demand function for various}

hypothetical

prices

Ph,

and then find

P[

=

Pt+i,

the

equihbrium price

at lime t + 1.

Suppose

that at the trade at lime t +1, the

price

of the stock is set at _a

hypothetical price

Ph. How many shares will investor want to hold _at this

price? First,

let _us observe that

immediately

alter the trade the wealth of investor1 will

change by

the _amount

Nt(1)(Ph Pt)

due to

capital gain

or loss. Note that there is

capital gain

_or loss

only

_on the

Nt(1)

shares held

belote

the

trade,

and not on shares

bought

_or sold at the time t +1 trade.

Thus,

if the

hypothetical

_puce is

Ph,

the

hypothetical

wealth of investor1 after the t +1 trade, Wh

Ii),

will be:

Whli)

=

Wzli)

+

Nli)Di

₊

lwili) Nili)Pi)T

+

Nili)lPh Pi),

₁₃₎

where the first three terms are from

equation (2).

The investor has to decide at time t +1 how to invest this wealth.

He/she

will attempt to maximize

his/her expected utility

at the next

period,

lime _{t +} 2. As

explained before,

the _ex- post distribution of _returns is

employed

_{as an} estimate for the ez-ante distribution. If investor

invests at time t +1 _a

proportion Xii)

of

his/her

wealth in the

stock, his/her expected utility

at titre t + 2 will be _given

by:

t-~+i

Eujxji))

=

i/k fl _{iog iii} xj))w~ji)ji

+

r)

₊

xji)w~j)ji

₊

a~ )j, j4)

j=t

where the first term in the square brackets _is the bond's contribution _to

his/her

wealth and tbe second term is the stock's contribution.

The investor will choose the investment

proportion, Xii),

that maximizes

his/her expected utility(~)

As the optimum investment

proportion

is

generally dependent

_on

Ph, (through Wh),

[et _us denote this

proportion by

Xh

Ii).

The amount of wealth that investor will froid in stocks at trie

hypothetical

price

Ph

is

given by Xh(1)Wh(1). Therefore,

the number of shares that investor will want to hold at the

hypothetical price Ph

will be:

Xhii)Whli)

~hi~>~h)

15)

"

Ph

This constitutes the

personal

demand _curve of investor 1.

Summing

the

personal

demand functions of ail investors, we obtain the

following

collective demand function:

(~)As _no borrowing _or shortselling _is allowed, _we have 0 < X < 1. However, _we introduce _a constraint asserting thon 0 < _a < X, < b < where b is _very close to 1, le-g-, 0.99) and _{a is very} close to _zero.

Thus, the assuption _is thon, _even if the pure mathematical solution advocates 100% investment in stocks, to guarantee some money for emergency needs, investors will Dot invest _more thon b in stock,

and they will keep _{some money m}

(riskless)

bonds. If _{we were} to introduce borrowing, _we would still

have _a upper bound _on X, set by the bank. In this _{case we} would have b _{> 1.}

Nh(Ph)

_"

~ _{Nh(1, Ph). (6)}

z

Equilibrium

As the number of shares in the

market,

denoted

by N,

is assumed to be

fixed,

the collective demand function determines the

equilibrium

price

P[. P[

is

given by

the intersection point of the aggregate demand function and the

supply function,

which is _a vertical

fine(~). Thus,

the

equflibrium price

of the stock at lime t +1, ^denoted

by Pi+i,

will be

P[.

History Update

The _new stock price,

Pi+i

and dividend

Di+i, give

_{us a new} retum _on the stock,

Hi+1

~

~t+1

~t ₊

Dt+1

t+Î p

t

We

update

the stock's

history by including

finis _{most recent} retum, and

eliminating

the oldest

retum

Ht-k+i

from the

history.

This

completes

one lime

cycle. By repeating

this

cycle,

_we simulate the evolution of the stock market

through

time.

l.1.2. Noise. The model described _so far is deterministic. The decision

making

_{process is}

conducted

by maximizing

the

expected utility

it is _a bounded

rational, predictable

decision

making (18-20].

In _more reahstic

situations,

investors _are influenced

by

_many factors other than rational

utility

maximization (21]. The net eflect of _a

large

number of uncorrelated random

influences is _a

normally

distributed random influence _or "noise".

Hence,

_we take into account ail the unknown factors

influencing

decision

making by adding

a normal random variable to the

optimal

investment

proportion(~).

To be _more

specific,

_we

replace Xii)

with X*

ii)

where

X*li)

=

Xli)

₊

Eli)

₁₇₎

and

e(1)

is drawn at random from _a normal distribution with standard deviation

a(6).

We should

emphasize

that

Xii)

is the _same for ail investors, but X*

ii)

is trot because

e(1)

is drawn

separately

for each investor.

Thus,

the noise, which is

investor-specific,

is the first factor introduced to induce investor

heterogeneity.

1.2. HETEROGENEOUS INVESTOR MODELS. We _contrast the

homogeneous

investor model

to models in which the investors may be diverse.

First,

_we

study

the _case where investors difler in their expectations. We consider investors with diflerent _{memory sparts} of past retums

leading

to diflerent

expectations regarding

^future retums. Next _we consider

diversity

with respect to

preference.

We take investors with the

following utility

function:

(~) ^{The model} cari easily be modified to incorporate _new stock issues. Note that aise it is assumed that accumulated dividends and interest is reinvested. This _is net necessary and the model _con be easily modified to allow consumption _or additioual saving which is invested in trie market.

(~) Investor-specific noise con be introduced into the system in a vanety ^of other _ways. One possibility

is to _assume asymmetric information _in the market,

Ii,e.,

a diflerent interpretations of the stock's history by diflerent

individuals).

(~)If X*(1)

_{< a or}

X*(1)

> b, we disregard this choice of e(1), and _we draw another e(1) ^instead.

Thus

X*(1)

is normally distributed around

X(1)

with standard deviation _a, but ii is truncated ai _a and ai b.

wl-o,

~~~~~'~ ~

l _Oz ~~~

where the i~~ investor is characterized

by

the risk aversion _{measure a~.} This

function,

_as well _as the

log function,

_are

myopic utility

functions which would dictate _a fixed investment proportion in stocks for _a

given

distribution of rates of _retum. This is non the _case in _our model _as the

distribution

(or History) changes

_every

trading round,

hence the investment

proportion

also

changes.

We also consider trie _case of diflerences in the

strategies employed by

investors. The investors considered _so for

update

their

expectations

and investments at every trade point. ^We now

introduce investors who go for the

long

_run and

disregard

trends. We take the extreme _case

of the investors who

keep

their investment

proportion

in the stock constant,

regardless

of the

history.

In the _next section _we describe the results of _our

simulations,

and the eflects of these diflerent types ^of

heterogeneity

_on market

dynamics.

2. Data and Results

In the simulations described in this paper, we choose the lime

period

between each trade to be

one year.

Accordingly,

_we chose the _rest of the parameters

realistically.

We take the annual interest rate to be 4%. The initial

History,

consists of _a discrete distribution of returns with

a rnean of 4.15% and _a standard deviation of 0.3%. With these pararneters, trie investment

proportion

in the

risky

asset in trie first round is about

50%,

thus trie bond and stock _{are more}

or less

compatible

in the initial stage.

Trie number of investors is 100 and the number of

outstanding

shares _is

10,000.

The initial wealth of each investor _is

Sl,000.

The initial share

price

is

$4.00(~).

The initial divide~ld is taken _to be $0.20

(dividend yield 5%).

We increase the dividend

by

5%

annually(~).

This is close to the

long

rua average dividend

growth

rate of the STOP.

We should stress that _our results _are

general

and that there _{was no}

fine-tuning

of the _pa-

rameters. The initial conditions do non affect the main features of the

dynamics.

Let _us first examine the

dynamics

of the

homogeneous

investor market.

Figure

1

plots

the

price

of the stock _{as a} function of time

(in years)

in _a market of

homogeneous

investors with _{memory spart} k

= 15 and

logarithmic utility

functions. Two _{cases are} shown: _one with

perfectly

rational investors

ii-e-,

no

noise,

_{a =}

0)

and the other with _a moderate

degree

of noise or

irrationality ii-e-,

a =

0.2).

We

begm by discussing

the no-noise

dynamics.

In this _case ail investors _are

completely homogeneous

and will therefore have identical de- mand functions. As _a

result,

the shares will

always

be divided

equally

between ail investors.

Investors

always

end up

holding

the _same number of shares: the total number of shares in the (~) ^{In the initial} stage we ensure thon trie total wealth is greater ^{than the market value of the stock.}

As _we do trot allow borrowmg, _we preclude _a situation where the market value of the stocks

lai

54.00 per

share)

is greater thatn trie total wealth of ail investors which would contradict equibbrium. The

initial price _is determined arbitranly _as if there _were

no history at ail to determine the _puce,

as in the

case of _an initial public oflenng. However, _{we con} reach the _same results by assuming that the initial puce is determined by _some history, and then induce _some initial shock in the system. Trie shock _con take the form of

a change _in the wealth, _a change _in trie interest rate, etc. After trie shock, _{a new} history _is created and the _{process is} exactly

as given in this _paper.

(~) We aise studied the effects of fluctuating dividends. We do _trot report these results _in this paper for trie sake of brevity.

MEMORYSPAN 15

~~i

io~

a=oo

10~ à=0 2

S

g g Î

8

10~

~

10'

'~o

0 50 100 150 200

TIME (YEARS)

Fig. 1. Stock price _{as a} function of time with and without investor-specific _noise

(100

investors with logarithmic utility furetions and memory span 15).

market divided

by

the number of investors.

Thus,

shares do _not

change hands,

and _we have _a

no trade market.

We _see that the stock

price

increases

sharply

at first

and, thereafter,

it increases at a

steady exponential

rate

(a straight

fine on the

semi-logarithmic scale).

Let _us first focus _on the

sharp

initial increase. The rate of retum on the stock from the first trade is

relatively high.

This creates _a distribution of returns that is "better" than the initial

History,

where "better" _means that investors _are

willing

to increase their investment proportion m

equity. Changes

in the

investment proportions,

especially

_near the _maximum investment proportion

allowed,

induce dramatic

changes

in the stock

price,

_as demonstrated in

Appendix A(9).

After this

Sharp

initial increase, the stock's

History

of returns becomes very attractive, and the investment

proportions

in the

nsky

asset are fixed _at the maximum. As shown in

Appendix

B, under the condition of

homogeneous

investors and fixed investment

proportions,

the

long

run

capital appreciation

becomes:

Î~

=

11

₁₉₎

where _{g is} the constant

growth

rate of the dividend. This

explains

the constant

slope

of the

exponential

climb. In _{our case g}

=

5%,

_{r =} 4% and the

slope

becomes 1.05. The 5% annual

growth

rate of the stock _puce that _we obtain _is realistic. The

compound

annual

growth

rate of the STOP due to

capital appreciation during

the _years 1925-1992 _was 5.4$io

(source:

Ibbotson

(9) If the rate of retum _on the stock from the first trade _is low, the process is opposite and _we observe

on a Sharp initial decrese _m stock _puce. As will become evident, the initial behavior _is irrelevant to the long _run dynamics (see reference [22])

Associates).

The _{reason we} reach _very

high

stock

prices

is that _{we are}

looking

at a very

long

time horizon

(200 years).

See also

Figure

13.

Now let _us tutu to the _case with _noise.

Again

_{we see a}

sharp

initial increase in the stock

price.

However,

_now the

sharp

_mcrease stops at _a lower

price(~°).

After the

sharp

increase the

price

mcreases at the _same

steady exponential

rate _as in the _zero noise _case.

However,

_as

opposed

to the smooth increase of the _zero noise _{case, now} the

price

fluctuates around the trend. This is _a result of the noise. The noise induces fluctuations _m the investment proportions which in tutu induce these

price

fluctuations. These fluctuations _are also the _cause of the crash that

occurs around time 60. In order to understand this crash _we must first _note that _one of the

consequences of the results of

Appendix

B

is,

that in the _no noise _case the

History

of returns on the stock becomes

homogeneous:

Ht ₌

~

j~~

^~

^~~

- g +

il

+

g) (°

_{= g +}

_(~ (g

>

r) (10)

t-i o o

where Dl _is the first _year dividend. When _a moderate

degree

^of noise _is introduced the

returns will fluctuate around this value, but will remain

relatively homogeneous.

Because of

this

homogeneity,

_a small

change

in _one of the returns _can induce _a dramatic

change

in the investment proportions.

Let _us elaborate. The investor's terminal

wealth,

W, ^is

given by:

W=X(1+É)+(1-X)(1+r)=1+X(H-r)+r _iii)

where

jÎ

_{is the random return}

on the stock. The

expected utility

_is

given by:

EU(W)

= EU

il

+

X(jÎ _r)

₊

rj ⁽¹²⁾

Expand EU(W) by

the

Taylor

series around the _mean

il

+

X(E(ù) _r)

₊

_r)

a EW _to obtain:

EU(W)

= U

il

+

X(E(jÎ) _r)

₊

rj

+

U'(EW)E _(W E(W)]

~,,

+

-(EW)a$

₊

₍₁₃₎

Because the second term

vanishes,

_we have:

EU(W)

= U

il

+ r +

X(E(jÎ) _r)j

₊

~"a$

_+..

₍₁₄₎

We have _seen that Ht _{- g} +

Il

₊

g)

^~°

la

constant

number).

Therefore

jÎ

-

E(jÎ)

and

a$

Po

(not

to mention

higher moments),

will

approach

_zero. Thus, if

E(jÎ)

_>

r the

optimum

solution will be X

= 1

(or

in _{our case} X

=

b), and,

^for

E(jÎ)

_<

r, the optimum solution will be X ₌ 0

(or

X

=

a).

To _see this

"discontinuity"

property, ^consider the

following example:

_r

=

10%,

(~°)When

there _{is no} noise, the Sharp initial increase stops ^when the investment proportions _m ^the stock reach the

maximum ii-e-, X

=

0.99).

The _noise generally widens the distribution of investment

proportions. When the investment proportions _{are near} the _maximum, the _{noise con} only drive them down

(see

footnote

(6)).

Thus, when optimal diversification advocates maximal investment in stocks,

the _noise has the efiect of lowering the average ^investment proportion. This _is the _reason trie sharp

initial _increase stops at _a lower _puce when _{noise is} introduced.

WEALTH DISTRIBUTION

20.0

MEMORY 15

ce ce O

lime

~ O à

time 20

~'~0.0 _{50.0 100.0 150.0}

200.0

OF AVERAGE WEALTH

Fig. 2. Wealth distributions at difiereut times. This data corresponds to trie simulation depicted

in Figure 1

(100

investors with logarithmic utility furetions and _{memory spart} 15. _a

=

0.2).

and the _{retum on} trie stock has _two

possible values,

^{8% and}

12%,

with

equal probability.

In this case, maximization of the

expected logarithmic utility yields

X

= 0. A

slight

increase in

one of the

possible

retums _on the stock from 12% to 12.1% is

enough

to

push

the

proportion

of investment to the other extreme, ^X = 1.

Thus,

_a small

change

in the investment

proportion (due

to the

noise)

_may create _a small

change

in the return _on the stock which will

change

the

history

_so that

E(jÎ)

_{becomes smaller than}

r. This will _{cause a}

sharp

decrease _in X

resulting

in a crash. This is similar to the market

discontinuity

^described in reference (23]. ^{This result}

is robust and does not

depend

_on the assumed

preference.

Fluctuations in the investment

proportions

induce

price fluctuations,

and in tum, ^fluctua- tions in the return _on the stock. One of the fluctuations is

big enough

to _{cause a} discontinuous crash. This

happens

first _at around time 60. Once the

price

_is

down,

the dividends become

significant again

in their contribution to the returns on the stock, and the retums are

high.

After 15

periods,

^{when the}

catastrophic

return

resulting

^{from the crash} is

beyond

^the _memory

spart of the investors, the stock

price jumps

back up. This

cycle

repeats ^itself _very

regularly

with _a

period

of 30 trades

ils highs,

15

lows).

Due to the

investor-specific

nature of the _noise, trot ail investors behave in the _same way in

spite ^{of the}

homogeneity

of

preference,

_{memory spart,} and strategy. Hence, there will be trade in the market and investors will become diverse in respect to their wealth.

How is the wealth distributed _{across investors}

throughout

^{the run? The} _answer to this question is shown in

Figure

2. The horizontal axis

depicts

the wealth _{as a} percentage of the average wealth

(for

_a

given

time

period),

and the vertical axis

depicts

the number of investors with that wealth. As _cari be seen, ^at time 1, the distribution _{is very narrow} almost ail

investors have the _same wealth the average wealth. Over

time,

the distribution widens

the random element in the decision

making

_{process, e,} affects individual investors

dilferently

MEMORY SPAN 5

io~

io~

io~

W

à

_10~

fi

8

~ 10~

~

io'

g

0 50 100 150 200

TIME(YEARS)

Fig. 3. Stock price _{as a} function of time

(100

investors with logarithmic utifity furetions aud memory spart 5, _a =

0.2).

and

they

become richer _{or poorer} than the _average

(see

time

20).

Some _{investors are}

"lucky"

they happen

to

buy

stocks before the booms and self before the crashes whereas the

"unlucky"

investors do trie

opposite.

As would be

expected,

the wealth distribution in this _{case assumes} the form of _a normal distribution with _a standard deviation that increases _over time.

How is the behavior of the market affected

by

the memory spart of the investors? The

dynamics

of _a market of

homogeneous

investors with

logarithmic utility

functions and _memory spart 5 is shown in

Figure

3. The

dynamics

is very similar to that

depicted

in

Figure

1, except that the

cycle

is reduced to 10 time

periods

^because of the shorter _{memory span}

(5 highs,

5

lows).

We have also studied the _case where the investors'

expectation,

that each of the last k _retums

on the stock has _an

equal probility of1/k

to reoccur, is

replaced

with the

expectation

that the

probability

of reoccurrence is

exponentially decaying

with lime, ^1-e-:

P(Ht+i=Hj)oee~@~ _{j=t,t-1,. ,t-k+1. ils)}

where _T is _a time scale constant. Trie results of this simulation with _T

= 4.0 _are

depicted

in

Figure

4. Trie

cycles

_are shorter because of the shorter effective _memory,

however,

the

cyclic

booms and crashes

persist.

What _{can we} learn from

Figures

1, 3 and 4?

First,

with

perfectly

rational

homogeneous

investors, _we obtain _very unrealistic results: the stock

price

increases

continuously

at _a constant rate

(the straight

fine in

Fig. l). Furthermore,

_as ail investors _are assumed _to be identical in ail respects,

they

ail have identical demand functions and _we have _{a no} trade market.

When _we introduce

investor-specific

noise, there will be trade but _we still obtain unrealistic results: booms and crashes with

cyclical regularity

_occur, berce

they

_are

fully,

_or almost

fully,

predictable.

This is in

complete

contradiction _to market

efliciency. Moreover, changing

the

MEMORY SPAN 15

lÙ~'

io~

OE

fl io~

ce

8

lo~

~

io~

io'

io°

o 50 100 lso zoo

TIME (YEARS)

Fig. 4. Stock puce as a function of time with exponentially decaying reoccurrence probability,

r = 4.0 (100 investors with logarithmic utility functions and _{memory span} 15).

memory spart ^to rive

periods (k

₌

5)

does _trot

change anything

except ^for trie

length

of trie

cycle.

Trie introduction of

exponentially decaying

reoccurrence of returns

probability

leads to shorter

cycles

of booms and

crashes,

but does _trot make trie

dynamics

_more realistic.

Thus,

trie

assumption

of _a

representative

investor leads from reasonable microfoundations _to

totally

unreasonable macroeconomic results. Trie introduction of

investor-specific

_noise does not

help (for

^{further details} _on ^trie elfects of _varying

degrees

of _{noise see}

(22]).

^We now tum to

analyze

trie elfects of other _sources of

heterogeneity.

The first _case of

non-homogeneous

investors studied _was that of _a market with two types of investors: investors with _{memory span} 15 and investors with memory span 5. Ail _investors

were assumed _to have

loganthmic utility

functions and _are divided

equally

between these two groups. The results _are shown in

Figure

5. As _can be _seen, what _we have here is

clearly

not a

simple superposition

of

Figure

and

Figure

3. The

dynamics resulting

from the interaction

between these two investor types ^is _very

complex, perhaps

_even chaotic. We still _see booms and crashes, however,

they

are much less

predictable.

Thus,

heterogeneous

_{memory spans}

induce _more realistic results thon those observed in

Figures

1 and 3. It is

interesting

to note that around lime 100, there _{is a} transition from

cycles

of about 10 periods

(corresponding

to a

5-penod

_memory

span)

to

longer cycles

of about 30

periods (corresponding

to a

15-period

memory

spart).

This _can be

partially

understood from

Figure

6. In

Figure

6 _{we see} that

although

^the two investor groups start with the _same wealth

distribution,

after _a number of time

periods they

form _two distinct wealth classes. The

population

with the

larger

_memory

span dominates most of the money m the market. This is

probably

the _reason for the transition to

longer cycles.

We continued to make investors more

heterogeneous

and for each investor in the market, _we chose _a random _{memory span} between 1 and 20

(utility

functions _are still ail

logarithrnic).

The

5D% MEMORY 5, 5D% MEMORY 15

io~

io~

ii'

§

8

io'

ce

o io3

ÉO

io~

io'

g

o 50 loo lso zoo

TIME (YEARS)

Fig. 5. Stock _{puce as a} function of time resulting from two equal investor groups: one of investors with _{memory spart} 5 and the other investors with _{memory spart} 15 (A total of 100 investors ail with loganthmic furetions, _a

=

0.2).

WEALTH DISTRIBUTION

20.0

time

~- memo~ 5

_ memo~ 15

ce ce

ÉO

~

Z lime 80

~

^{memo~ 5}

à

z

~

lime 80

*' memo~ 15

O.O

0 0 50.0 100.0 150.0 200.0

% OF AVERAGE WEALTH

Fig. 6. Wealth distribution of two investor groups

(k

₌ 5, k

=

là)

at difierent times. This data correspond to the simulation presented in Figure 5.