04/18/23 1

Consumption, Population, and the Cross-Section of Stock Returns

Tzuling, LinRichard MacMinnLarry Y. Tzeng

04/18/23 2

Background and Motivation

Identifying the empirical linkage between the macroeconomic risks and the financial markets has always been an important economic issue.

Consumption risk is an important factor.

04/18/23 3

Background and Motivation

The real consumption per capita figures of the emerging markets are on average lower than those of the developed countries, but the larger numbers or the larger growth rates of population of these countries appear to imply explosive potential in their financial markets.

The baby boom has been claimed as a contributing factor in explaining the high stock returns and the large increase in stock prices in the 1990s for the US.

We propose a new factor—Population Risk

04/18/23 4

The Main Focus of the paper Population risk is an important factor!1. Hansen and Singleton (1982, 1983), Mankiw

and Shapiro (1986), and Breeden, Gibbons, and Litzenberger (1989) almost fail to find support for the CCAPM.

2. There are a lot of papers regarding the non-traditional one.

3. We conjecture the reason for not supporting the CCAPM is ignoring the dynamic of aggregate population. We derive the CPCAPM and prove that the CPCAPM improves CCAPM in alternative ways.

04/18/23 5

The Main Focus of the paper

Demographics affects the stock returns for the US!

1. There is a substantial literature on the impact of demographics on stock returns.

2. We conjecture that the speeds and stabilities of the population growth including these age groups can give different shocks to the stock market.

3. We extend our original CPCAPM by incorporating different age groups and show that this extension improves the CPCAPM.

04/18/23 6

The CCAPM Literature Improve the performance of traditional

CCAPM by using multiperiod growth rates1. Parker and Julliard (2005) find that the

covariance between an asset’s return during a quarter and cumulative consumption growth over the several following quarters, which they refer to as ultimate consumption risk, explains the cross-section of stock returns very well.

04/18/23 7

The CCAPM Literature

2. Jagannathan and Wang (2007) find that when consumption betas of stocks are computed using year-over-year consumption growth based on the fourth quarter, the CCAPM performs as well as the Fama and French (1993) three-factor model. Since, for the CCAPM to hold at any given time point, investors must make their consumption and investment decisions simultaneously at that time point, they suspect it is more likely to happen during the fourth quarter.

04/18/23 8

The CCAPM Literature Improve the performance of traditional

CCAPM by using different consumption goods 1. Ait-Sahalis, Parker, and Yogo (2004) use

novel data on the consumption of luxury goods and find that the consumption of luxuries covaries significantly more with stock returns than does aggregate consumption.

2. Yogo (2006) finds that durable consumption in conjunction with nondurable consumption can explain the cross-section of stock returns.

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04/18/23 9

The Demographics Literature

DellaVigna and Pollet (2007) forecast future cohort sizes at long horizons by using current cohort sizes in combination with mortality and fertility tables and analyze the consumption patterns of cohort ages in order to forecast the shifts in demand for various consumption goods. They find that an increase in forecasted future

consumption demand growth induced by changes in age structure predicts an increase in stock returns.

9

04/18/23 10

The Demographics Literature

Bakshi and Chen (1994) empirically find that a rise in average age is found to predict a rise in equity risk premium using U. S. data in the post-1945 period. The finding supports the life-cycle

investment hypothesis that an investor allocates more wealth in housing at an early stage and then switches to financial assets at a later stage.

10

04/18/23 11

The Demographics Literature Poterba (2001) investigates the

association between the projected asset demands based on the future age structure of U. S. population and the returns on stocks and bonds.

Ang and Maddaloni (2005) examine the link between equity risk premiums and demographic changes for many countries.

04/18/23 12

The Demographics Literature Mehra et. al.(2002) observe a high equity

premium in the presence of borrowing constraints.

Goyal (2004) empirically finds that, for the U. S. economy, the outflows from the stock market are positively correlated with the changes in the fraction of old people and negatively correlated with those of the middle-aged.

Athanasoulis (2006) numerically illustrates that there is a positive correlation between the proportion of the population that is young and the equity premium.

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04/18/23 13

The Model—CPCAPM

1. denotes the excess return on an asset i from time t to t+1; denotes aggregate consumption;

2. Following Parker and Julliard (2005), we use the Euler equation for the risk-free rate,

between time t+1 and t+1+s

,01

1,

t

ttit Cu

CuRE

fsttR 1,1

1 1 1, 1 1f

t t t t s t su C E R u C

1, tiR

tC

04/18/23 14

The Model—CPCAPM

3. Substituting from equation in 2 into 1 yields

4. Defining and reorganizing bullet 3, the expected returns are given by

(*)

.01

1,11,

t

stfstttit Cu

CuRRE

stf

sttst mRm 11,11

.,cov

1

1,11, s

t

tist

ti mE

RmRE

04/18/23 15

The Model—CPCAPM

5. The population at time t is ; the consumption per capita at time t is . The aggregate consumption can be represented as . Hence, given the power utility function ( )

tNt

ttt NC

.111,11

s

tt

ststfstt

st N

NRm

. tt CCu .1

1

t

t

CCu

04/18/23 16

The Model—CPCAPM

6. Approximating around and using Taylor series to get

(**)

where and

stm 1 tst NN 1 tst 1

t

ststN N

Ng

11,

t

ststg

11,

, 1 , 1

1 1, 1

2 2

, 1

1 1 1

1 11 1

2 2

s ss N t s t

s ft t t s

s s s s sN t

g g

m R

g g

04/18/23 17

The Model—CPCAPM7. Substituting (**) into (*), yields the CPCAPM

:the population beta for (s+1)-period population growth

:the market price for population risk

, 1s s s s

i t N NE R

s

tN

tistNs

N g

Rg

1,

1,1,

var

,cov

1,1,

1,

2

1,1,

,cov

,cov

2

12

var

tistN

tistNs

s

stNss

N Rg

Rg

A

g

04/18/23 18

The Model—CPCAPM :the consumption beta for (s+1)-period consumption growth

:the market price for consumption risk

where

1,1,

1,

2

1,1,

,cov

,cov

2

12

var

tist

tists

s

stss

Rg

Rg

A

g

2

1,

2

1,1,1, 12

11

2

1111 s

tss

tNss

tstNs ggggEA

s

t

tists

g

Rg

1,

1,1,

var

,cov

04/18/23 19

The Model—CPCAPM with Age Groups

1. We denote the number of population for age group as , where

: childhood, : young-aged, : middle-aged, : old-aged.

Since

a omycaN a

t ,,,, c ym o

ot

mt

yt

ctt NNNNN

.11111

1,11

s

tot

mt

yt

ct

sto

stm

sty

stc

stfstt

st NNNN

NNNNRm

04/18/23 20

The Model—CPCAPM with Age Groups

2. Approximating around

and using Taylor series yields

(***)

stm 1 omycaNN a

ta

st ,,,,1

tst 1

1 1, 1 , 1, 1, , ,

22 2

, 1, 1, , ,

1 1 1

1 11 1

2 2

a

a

as f s stt t t s s s tN t

a c y m o t

as s s s s st

tN ta c y m o t

Nm R g g

N

Ng g

N

04/18/23 21

The Model—CPCAPM with Age Groups

3. Putting (***) into (*), yields the CPCAPM with age groups.

:the population beta for age groups

, 1 c c y y

m m o o

s s s si t N N N N

s s s s s s

N N N N

E R

s

tN

tis

tNs

Na

a

a

g

Rg

1,

1,1,

var

,cov

04/18/23 22

The Model—CPCAPM with Age Groups

: the market price for population risk of age group

: the market price for consumption riskwhere

1,1,

1,

2

1,

22

1,

,cov

,cov

2

11~

var

tis

tN

tis

tN

t

ats

t

at

st

at

s

tNss

N Rg

Rg

N

N

N

N

N

N

A

g

a

aa

a

1,1,

1,

2

1,1,

,cov

,cov

2

12~

var~

tist

tists

s

stss

Rg

Rg

A

g

2

1,

2

1,,,,

2

1,1,,,,

12

11

2

1111

~ st

ss

tNomyca t

atss

ts

tNomyca t

at

s ggN

Ngg

N

NEA aa

04/18/23 23

The Data

Consumption data: use real consumption expenditure on two sets of consumption goods (nondurables plus services, and durables and nondurables plus services) for the period 1950 to 2005 from NIPA tables available from the Bureau of Economic Analysis.

Table I

04/18/23 24

The Data

Population data:1. the quarterly number of the

aggregate population from NIPA Table 2.1

2. the annual numbers of population of every age from the Human Mortality Database for the period 1950 to 2005.

Table II

04/18/23 25

The Data

Asset return data: use the returns on 25 book-to-market and size-sorted portfolios, values for Fama and French (1993) three factor (market, SMB, HML) for the period 1951 to 2005 available from Kenneth French’s web site.

Table III

04/18/23 26

Empirical Findings

CPCAPM v.s traditional CCAPM v.s Fama-French model

Population risk v.s Q4 v.s ultimate risk Two consumption goods (nondurables

and service v.s durables, nondurables and services)

Demographics

04/18/23 27

Empirical Findings

Comparison of the CCAPM, the CPCAPM, and Fama and French Three-Factor Model

Three Effects: Population Risk, Q4, and Ultimate Risk

See Tables IV, V, Fig. 1

See Tables VI, VII

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Empirical Findings

The Robustness Evidence

Population Risks of Age Groups

See Tables VIII, IX, X

See Tables XI, XII

04/18/23 29

Empirical Findings

Consumption Betas and Population Betas

Further Comparison of the CCAPM, the CPCAPM, and the CPCAPM with Age Groups

See Table XV, Fig. 2

See Tables XIII, XIV

04/18/23 30

Conclusion

The CPCAPM performs much better than the CCAPM.

1. The stock market actually responds to the dynamic of the aggregate population.

2. An increase in the aggregate population risk will decrease the expected excess return.

04/18/23 31

Conclusion The CPCAPM with age groups improves the

CPCAPM.1. The stock market also responds to the

dynamics of the population of age groups.2. When measuring risks at the best-fitted

horizon, an increase in the population risk of the childhood and the young-aged will decrease the expected excess return whereas an increase in the population risk of the middle-aged and the old-aged will increase the expected excess return.

04/18/23 32

Thank you for your attention!!