Chapter 3 The Constant Expected Return Model

Eric Zivot

Department of Economics

University of Washington

January 6, 2000

This version: January 23, 2001

1 The Constant Expected Return Model of Asset

Returns

1.1 Assumptions

Let Rit denote the continuously compounded return on an asset i at time t. We

make the following assumptions regarding the probability distribution of Rit for i =

1, . . . , N assets over the time horizon t = 1, . . . , T.

1. Normality of returns: Rit ∼ N (µi , σ 2 ) for i = 1, . . . , N and t = 1, . . . , T.

i

2. Constant variances and covariances: cov(Rit , Rjt ) = σ ij for i = 1, . . . , N and

t = 1, . . . , T.

3. No serial correlation across assets over time: cov(Rit , Rjs ) = 0 for t 6= s and

i, j = 1, . . . , N.

Assumption 1 states that in every time period asset returns are normally dis-

tributed and that the mean and the variance of each asset return is constant over

time. In particular, we have for each asset i

E[Rit ] = µi for all values of t

var(Rit ) = σ 2 for all values of t

i

The second assumption states that the contemporaneous covariances between assets

are constant over time. Given assumption 1, assumption 2 implies that the contem-

poraneous correlations between assets are constant over time as well. That is, for all

1

assets

corr(Rit , Rjt ) = ρij for all values of t.

The third assumption stipulates that all of the asset returns are uncorrelated over

time1 . In particular, for a given asset i the returns on the asset are serially uncorre-

lated which implies that

corr(Rit , Ris ) = cov(Rit , Ris ) = 0 for all t 6= s.

Additionally, the returns on all possible pairs of assets i and j are serially uncorrelated

which implies that

corr(Rit , Rjs ) = cov(Rit , Rjs ) = 0 for all i 6= j and t 6= s.

Assumptions 1-3 indicate that all asset returns at a given point in time are jointly

(multivariate) normally distributed and that this joint distribution stays constant

over time. Clearly these are very strong assumptions. However, they allow us to de-

velopment a straightforward probabilistic model for asset returns as well as statistical

tools for estimating the parameters of the model and testing hypotheses about the

parameter values and assumptions.

1.2 Constant Expected Return Model Representation

A convenient mathematical representation or model of asset returns can be given

based on assumptions 1-3. This is the constant expected return (CER) model. For

assets i = 1, . . . , N and time periods t = 1, . . . , T the CER model is represented as

(1)

Rit = µi + µit

µit ∼ i.i.d. N(0, σ 2 )

i

(2)

cov(µit , µjt ) = σ ij

where µi is a constant and we assume that µit is independent of µjs for all time periods

t 6= s. The notation µit ∼ i.i.d. N (0, σ 2 ) stipulates that the random variable µit is

i

serially independent and identically distributed as a normal random variable with

mean zero and variance σ 2 . In particular, note that, E[µit ] = 0, var(µit ) = σ 2 and

i i

cov(µit , µjs ) = 0 for i 6= j and t 6= s.

Using the basic properties of expectation, variance and covariance discussed in

chapter 2, we can derive the following properties of returns. For expected returns we

have

E[Rit ] = E[µi + µit ] = µi + E[µit ] = µi ,

1

Since all assets are assumed to be normally distributed (assumption 1), uncorrelatedness implies

the stronger condition of independence.

2

since µi is constant and E[µit ] = 0. Regarding the variance of returns, we have

var(Rit ) = var(µi + µit ) = var(µit ) = σ 2

i

which uses the fact that the variance of a constant (µi ) is zero. For covariances of

returns, we have

cov(Rit , Rjt ) = cov(µi + µit , µj + µjt ) = cov(µit , µjt ) = σ ij

and

cov(Rit , Rjs ) = cov(µi + µit , µj + µjs ) = cov(µit , µjs ) = 0, t 6= s,

which use the fact that adding constants to two random variables does not a¬ect

the covariance between them. Given that covariances and variances of returns are

constant over time gives the result that correlations between returns over time are

also constant:

cov(Rit , Rjt ) σ ij

corr(Rit , Rjt ) = q = = ρij ,

σiσj

var(Rit )var(Rjt )

cov(Rit , Rjs ) 0

corr(Rit , Rjs ) = q = 0, i 6= j, t 6= s.

=

σiσj

var(Rit )var(Rjs )

Finally, since the random variable µit is independent and identically distributed (i.i.d.)

normal the asset return Rit will also be i.i.d. normal:

Rit ∼ i.i.d. N (µi , σ 2 ).

i

Hence, the CER model (1) for Rit is equivalent to the model implied by assumptions

1-3.

1.3 Interpretation of the CER Model

The CER model has a very simple form and is identical to the measurement error

model in the statistics literature. In words, the model states that each asset return

is equal to a constant µi (the expected return) plus a normally distributed random

variable µit with mean zero and constant variance. The random variable µit can be

interpreted as representing the unexpected news concerning the value of the asset

that arrives between times t ’ 1 and time t. To see this, note that using (1) we can

write µit as

µit = Rit ’ µi

= Rit ’ E[Rit ]

so that µit is de&ned to be the deviation of the random return from its expected value.

If the news is good, then the realized value of µit is positive and the observed return is

3

above its expected value µi . If the news is bad, then µjt is negative and the observed

return is less than expected. The assumption that E[µit ] = 0 means that news, on

average, is neutral; neither good nor bad. The assumption that var(µit ) = σ 2 can be

i

interpreted as saying that volatility of news arrival is constant over time. The random

news variable a¬ecting asset i, eit , is allowed to be contemporaneously correlated with

the random news variable a¬ecting asset j, µjt , to capture the idea that news about

one asset may spill over and a¬ect another asset. For example, let asset i be Microsoft

and asset j be Apple Computer. Then one interpretation of news in this context is

general news about the computer industry and technology. Good news should lead

to positive values of µit and µjt . Hence these variables will be positively correlated.

The CER model with continuously compounded returns has the following nice

property with respect to the interpretation of µit as news. Consider the default case

where Rit is interpreted as the continuously compounded monthly return. Since mul-

tiperiod continuously compounded returns are additive we can interpret, for example,

Rit as the sum of 30 daily continuously compounded returns2 :

29

X

Rd

Rit = it’k

k=0

d

where Rit denotes the continuously compounded daily return on asset i. If we assume

that daily returns are described by the CER model then

Rit = µd + µd ,

d

i it

µd ∼ i.i.d N(0, (σ d )2 ),

it i

cov(µd , µd ) = σ d ,

it jt ij

cov(µd , µd ) = 0, i 6= j, t 6= s

it js

and the monthly return may then be expressed as

29

X

(µd + µd )

Rit = i it’k

k=0

29

X

µd µd

= 30 · +

i it’k

k=0

= µi + µit ,

where

µi = 30 · µd ,

i

29

X

µd .

µit = it’k

k=0

2

For simplicity of exposition, we will ignore the fact that some assets do not trade over the

weekend.

4

Hence, the monthly expected return, µi , is simply 30 times the daily expected re-

turn. The interpretation of µit in the CER model when returns are continuously

compounded is the accumulation of news between months t ’ 1 and t. Notice that

Ã !

29

X

(µd + µd )

var(Rit ) = var i it’k

k=0

29

X

var(µd )

= it’k

k=0

X³ ´

29

d2

= σi

k=0

³ ´

d2

= 30 · σi

and

Ã !

29 29

X X

µd , µd

cov(Rit , Rjt ) = cov it’k jt’k

k=0 k=0

29

X

cov(µd , µd )

= it’k jt’k

k=0

29

X

σd

= ij

k=0

= 30 · σ d ,

ij

so that the monthly variance, σ 2 , is equal to 30 times the daily variance and the

i

monthly covariance, σ ij , is equal to 30 times the daily covariance.

1.4 The CER Model of Asset Returns and the Random Walk

Model of Asset Prices

The CER model of asset returns (1) gives rise to the so-called random walk (RW)

model of the logarithm of asset prices. To see this, recall that the continuously

compounded return, Rit , is de&ned from asset prices via

Ã !

Pit

ln = Rit .

Pit’1

Since the log of the ratio of prices is equal to the di¬erence in the logs of prices we

may rewrite the above as

ln(Pit ) ’ ln(Pit’1 ) = Rit .

Letting pit = ln(Pit ) and using the representation of Rit in the CER model (1), we

may further rewrite the above as

pit ’ pit’1 = µi + µit . (3)

5

The representation in (3) is know as the RW model for the log of asset prices.

In the RW model, µi represents the expected change in the log of asset prices

(continuously compounded return) between months t ’ 1 and t and µit represents the

unexpected change in prices. That is,

E[pit ’ pit’1 ] = E[Rit ] = µi ,

µit = pit ’ pit’1 ’ E[pit ’ pit’1 ].

Further, in the RW model, the unexpected changes in asset prices, µit , are uncorrelated

over time (cov(µit , µis ) = 0 for t 6= s) so that future changes in asset prices cannot be

predicted from past changes in asset prices3 .

The RW model gives the following interpretation for the evolution of asset prices.

Let pi0 denote the initial log price of asset i. The RW model says that the price at

time t = 1 is

pi1 = pi0 + µi + µi1

where µi1 is the value of random news that arrives between times 0 and 1. Notice that

at time t = 0 the expected price at time t = 1 is

E[pi1 ] = pi0 + µi + E[µi1 ] = pi0 + µi

which is the initial price plus the expected return between time 0 and 1. Similarly,

the price at time t = 2 is

pi2 = pi1 + µi + µi2

= pi0 + µi + µi + µi1 + µi2

2

X

= pi0 + 2 · µi + µit

t=1

which is equal to the initial price, pi0 , plus the two period expected return, 2 · µi , plus

P

the accumulated random news over the two periods, 2 µit . By recursive substitu-

t=1

tion, the price at time t = T is

T

X

piT = pi0 + T · µi + µit .

t=1

At time t = 0 the expected price at time t = T is

E[piT ] = pi0 + T · µi

The actual price, piT , deviates from the expected price by the accumulated random

news

T

X

piT ’ E[piT ] = µit .

t=1

Figure xxx illustrates the random walk model of asset prices.

3

The notion that future changes in asset prices cannot be predicted from past changes in asset

prices is often referred to as the weak form of the e¬cient markets hypothesis.

6

Simulated Random Walk

12

E[p(t)]

p(t) - E[p(t)]

p(t)

10

p(t)

8

6

E[p(t)]

pt

4

2

p(t) - E[p(t)]

0

101

1

5

9

13

17

21

25

29

33

37

41

45

49

53

57

61

65

69

73

77

81

85

89

93

97

-2

time, t

The term random walk was originally used to describe the unpredictable move-

ments of a drunken sailor staggering down the street. The sailor starts at an initial

position, p0 , outside the bar. The sailor generally moves in the direction described

by µ but randomly deviates from this direction after each step t by an amount equal

P

to µt . After T steps the sailor ends up at position pT = p0 + µ · T + T µt .

t=1

2 Monte Carlo Simulation of the CER Model

A good way to understand the probabilistic behavior of a model is to use computer

simulation methods to create pseudo data from the model. The process of creating

such pseudo data is often called Monte Carlo simulation4 . To illustrate the use of

Monte Carlo simulation, consider the problem of creating pseudo return data from

the CER model (1) for one asset. In order to simulate pseudo return data, values for

the model parameters µ and σ must be selected. To mimic the monthly return data

on Microsoft, the values µ = 0.05 and σ = 0.10 are used. Also, the number N of

4

Monte Carlo referrs to the fameous city in Monaco where gambling is legal.

7

simulated data points must be determined. Here, N = 100. Hence, the model to be

simulated is

Rt = 0.05 + µt , t = 1, . . . , 100

µt ˜iid N (0, (0.10)2 )

The key to simulating data from the above model is to simulate N = 100 observations

of the random news variable µt ˜iid N(0, (0.10)2 ). Computer algorithms exist which

can easily create such observations. Let {µ1 , . . . , µ100 } denote the 100 simulated values

of µt . The histogram of these values are given in & gure xxx below

Histogram of Simulated Errors

16.00%

14.00%

12.00%

10.00%

Frequency

8.00%

6.00%

4.00%

2.00%

0.00%

-0.241

-0.208

-0.175

-0.142

-0.109

-0.076

-0.043

-0.010

0.023

0.056

0.089

0.122

0.155

0.188

0.221

e(t)

P

The sample average of the simulated errors is 100 100 µt = ’0.004 and the sample

1

t=1

qP

100

1

standard deviation is 99 t=1 (µt ’ (’0.004))2 = 0.109. These values are very close

to the population values E[µt ] = 0 and SD(µt ) = 0.10, respectively.

Once the simulated values of µt have been created, the simulated values of Rt are

constructed as Rt = 0.05 + µt , t = 1, . . . , 100. A time plot of the simulated values of

Rt is given in &gure xxx below

8

Monte Carlo Simulation of CER Model

R(t) = 0.05 + e(t), e(t) ˜ iid N(0, (0.10)^2)

0.400

0.300

0.200

0.100

Return

0.000

-0.100

-0.200

-0.300

100

1

4

7

10

13

16

19

22

25

28

31

34

37

40

43

46

49

52

55

58

61

64

67

70

73

76

79

82

85

88

91

94

97

time

The simulated return data ! uctuates randomly about the expected return value

E[Rt ] = µ = 0.05. The typical size of the ! uctuation is approximately equal to

SE(µt ) = 0.10. Notice that the simulated return data looks remarkably like the

actual return data of Microsoft.

Monte Carlo simulation of a model can be used as a & pass reality check of the

rst

model. If simulated data from the model does not look like the data that the model is

supposed to describe then serious doubt is cast on the model. However, if simulated

data looks reasonably close to the data that the model is suppose to describe then

con&dence is instilled on the model.

3 Estimating the CER Model

3.1 The Random Sampling Environment

The CER model of asset returns gives us a rigorous way of interpreting the time

series behavior of asset returns. At the beginning of every month t, Rit is a random

9

variable representing the return to be realized at the end of the month. The CER

model states that Rit ∼ i.i.d. N (µi , σ 2 ). Our best guess for the return at the end of the

i

month is E[Rit ] = µi , our measure of uncertainty about our best guess is captured by

q

σ i = var(Rit ) and our measure of the direction of linear association between Rit and

Rjt is σ ij = cov(Rit , Rjt ). The CER model assumes that the economic environment

is constant over time so that the normal distribution characterizing monthly returns

is the same every month.

Our life would be very easy if we knew the exact values of µi , σ 2 and σ ij , the

i

parameters of the CER model. In actuality, however, we do not know these values

nancial econometrics is estimating the values of µi , σ 2

with certainty. A key task in & i

and σ ij from a history of observed data.

Suppose we observe monthly returns on N di¬erent assets over the horizon t =

1, . . . , T. Let ri1 , . . . , riT denote the observed history of T monthly returns on asset

i for i = 1, . . . , N. It is assumed that the observed returns are realizations of the

random variables Ri1 , . . . , RiT , where Rit is described by the CER model (1). We

call Ri1 , . . . , RiT a random sample from the CER model (1) and we call ri1 , . . . , riT

the realized values from the random sample. In this case, we can use the observed

returns to estimate the unknown parameters of the CER model

3.2 Estimation Theory

Before we describe the estimation of the CER model, it is useful to summarize some

concepts in estimation theory. Let θ denote some characteristic of the CER model

(1) we are interested in estimating. For example, if we are interested in the expected

return then θ = µi ; if we are interested in the variance of returns then θ = σ 2 . The

i

goal is to estimate θ based on the observed data ri1 , . . . , riT .

nition 1 An estimator of θ is a rule or algorithm for forming an estimate for

De&

θ.

nition 2 An estimate of θ is simply the value of an estimator based on the

De&

observed data.

To establish some notation, let ˆ i1 , . . . , RiT ) denote an estimator of θ treated as

θ(R

a function of the random variables Ri1 , . . . , RiT . Clearly, ˆ i1 , . . . , RiT ) is a random

θ(R

variable. Let ˆ i1 , . . . , riT ) denote an estimate of θ based on the realized values

θ(r

ri1 , . . . , riT . ˆ i1 , . . . , riT ) is simply an number. We will often use ˆ as shorthand

θ(r θ

notation to represent either an estimator of θ or an estimate of θ. The context will

determine how to interpret ˆ θ.

3.2.1 Properties of Estimators

Consider ˆ = ˆ i1 , . . . , RiT ) as a random variable. In general, the pdf of ˆ p(ˆ

θ θ(R θ, θ),

depends on the pdf s of the random variables Ri1 , . . . , RiT . The exact form of p(ˆ may

θ)

10