Higher Risk Aversion in Older Agents: Its AssetPricing Implications1

Amadeu DaSilva2

and

Christos Giannikos3

Draft May 2005

Abstract

This paper investigates asset pricing in a three-period overlapping generations (OLG)model economy where each generation lives as young, middle-aged and old. There is oneperishable consumption good in the economy and two types of traded securities in thecapital market: a bond and a share of equity. Implications for asset pricing and securityreturns of an increasing risk aversion are explored by allowing each agent�s coe¢ cient ofrelative risk aversion to vary with his age; the middle-aged consumer has a higher aversionto risk than the young and the old consumers are more risk averse than the middle-agedones. Our model produces high equity premium without requiring very large levels ofconsumer risk aversion; a result more consistent with the U.S. data. We further modifyour model to re�ect the U.S. demographic trend of an increasing share of older age group.This new speci�cation generates an even higher equity premium and a lower risk-freerate of return with an added desirable result of a lower standard deviation for the riskpremium.

1We would like to thank John B. Donaldson for his helpful comments and suggestions. All remainingerrors are ours.

2Columbia University, Department of Economics, ard23@columbia.edu3Columbia University and Baruch College, Graduate Business School, cig3@columbia.edu

1 Introduction

Since Mehra and Prescott�s (1985) seminal work on the equity premium puzzle, researchers

have taken di¤erent approaches in constructing models that are consistent with the large

equity premium as well as with other related stylized facts observed in the data. The au-

thors using reasonable parameters in a representative-consumer exchange economy model

fail to match the high equity premium as well as the return volatilities observed in the data

for the U.S. economy. There is also no explanation why the observed real risk-free rate

(the return on the T-bills) is so low. This is known as the risk-free rate puzzle (Epstein

and Zin (1989)). The above work has spurred a rich era of research in the �eld of asset

pricing, most of which aimed to conciliate the observed stylized facts with reasonably

speci�ed models.

Zhou (1999) introduces information asymmetry and market imperfections into a ratio-

nal expectations equilibrium model. Using an overlapping generations (OLG) model with

informed and uninformed agents, and with a reasonable relative risk aversion coe¢ cient of

5, they conclude that both the equity premium puzzle as well as the risk-free rate puzzle

can be resolved. This suggests that non-standard models can be used to explain these

well chronicled inconsistencies with the data.

Another strand of research focuses on the agent�s preference speci�cations. Among

them are the studies by Abel (1990), Constantinides (1990), Epstein and Zin (1991),

Campbell and Cochrane (1999), and Danthine, Donaldson, Giannikos, and Guirguis

(2004).

Danthine, Donaldson, Giannikos, and Guirguis (2004) explore the case of state depen-

1

dent preferences. In their paper the agent�s relative risk aversion parameter is related to

the output growth rate. By allowing the risk aversion parameter to vary (positively as well

as negatively) with the growth rate of output, they �nd that for standard parametrization

the equity premium is easily matched or exceeded, the risk-free rate is asymptotically too

low, and that the Hansen-Jagannathan bounds are easily satis�ed. However, their model

produces standard deviations of equity and risk free returns that are too large relative to

the data.

Campbell and Cochrane (1999) take a di¤erent approach in dealing with the afore-

mentioned problems. They use a power utility function that incorporates a time-varying

subsistence level (slow-moving habit). Habit is introduced in the model in order to re�ect

the fact that consumers�well-being is more related to recent consumption changes than

to the absolute level of consumption. The e¤ect then is to make the local curvature of

the utility function sensitive to the surplus consumption: if consumption is low relative

to habit then the utility function exhibits high curvature, and vice-versa. This shows

that habit is essentially a form of state dependent preferences. Their model is successful

at producing results consistent with the observed data; both the equity premium puzzle

and the risk-free rate puzzle are explained, and the model even reproduces some of the

remaining stylized facts.

Giannikos and Zhihong (2004) study the case of a pure exchange economy where the

agent�s preferences over a durable and a perishable good are habit forming. That is,

the agent�s past consumption of each of the two goods a¤ects the utility he gets from

his current consumption. They �nd that this two-good economy with habit persistence

2

generates the high level of risk premium observed in the U.S. data. Furthermore, this

economy also produces lower risk-free rate volatility, a result more in line with the U.S.

data.

Another interesting work in the area is the model by Barberis, Huang, and San-

tos (2001), which introduces two concepts borrowed from psychology into the standard

consumption-based approach. The motivation for this comes from observations of risk-

averse agents making choices in the face of risk. The �rst concept, "loss-aversion", re�ects

the experimental evidence that agents are more sensitive to reductions in wealth than to

increases. The second concept, "prior outcomes", has to do with the fact that the utility

from present gains and losses in wealth depends on whether the agent su¤ered losses or

made gains in prior periods. In this setting the agent�s risk-aversion changes over time as

a function of his investment performance. This reproduces the high mean, volatility, and

predictability of stock returns.

As the above brief review of some of the work in the �eld indicates, the favored

approach to solving these documented puzzles is by replacing the constant relative risk

aversion (CRRA) preferences with alternative preference speci�cations, where the agent�s

preference parameter changes. Several studies have shown that the risk aversion parameter

is not constant; it varies as a function of several di¤erent variables. For example, the

work of Bosch-Domenech and Silvestre (1999) investigates through a series of experiments

whether the risk aversion coe¢ cient varies with the level of income at risk. The results

of these experiments suggest that the individuals�decision to buy insurance is positively

correlated with their level of income at risk.

3

Most importantly however for our work, a signi�cant body of recent research supports

the idea that as agents grow older they become more risk averse. One such study, by

Anne-Marie Palsson (1996), examines risk taking by households. She tries to identify

household characteristics that a¤ect their aversion to risk. Using a cross-sectional data of

more than 7000 Swedish households, she �nds that household risk aversion is positively

related with the age of the head of the household.

Another study corroborating this hypothesis is the work of Bakshi and Chen (1994).

They test the life-cycle risk aversion hypothesis that an investor�s relative risk aversion

increases with age. Using post-1945 U.S. data, they �nd that a rise in the average age

predicts a rise in risk premiums. Additionally, Morin and Suarez (1983) studied the

e¤ect of age on the speci�c composition of risky assets in people�s portfolio. Using a

1970 Canadian Survey of Consumer Finance data, they conclude that the risk aversion

coe¢ cient increases with age.

If the individual investor�s aversion to risk changes with age, it becomes very important

to know how this could a¤ect asset pricing, portfolio composition, capital allocation and

ultimately economic growth. Quite simply, if as a consumer ages he becomes more risk

averse, we expect to see his investment behavior change; he will demand increasingly

higher returns to undertake risky investments. In a partial equilibrium setting, the above

speci�cation for the agent�s preferences will imply that the agent will tend to shift his

portfolio wealth out of very risky assets and into relatively safer ones as he ages. Bakshi

and Chen (1994) provide empirical support for this and conclude that individuals do in

fact change their portfolio wealth allocations as they age.

4

The economy-wide e¤ect of these portfolio changes depends on what happens to the

age structure. If overall, as evidence indicates, the population is ageing over time, then

relative asset prices will change as the aggregate demand for riskier assets will tend to

drop. This will result in higher risky asset returns relative to the safer investments. Also,

over time some forms of investments should become more and more attractive in the

market depending on their risk level.

The U.S. population growth rate is quite small. Combined with the increased life ex-

pectancy, this means the share of older population cohorts is increasing. In fact, according

to the U.S. Bureau of Census, currently 12.8 percent of the U.S. population is over the

age of 65, a number that is expected to increase to over 20 percent by the year 2030. The

proportion of households in the U.S. headed by someone 65 or older is also expected to be

about 40% by 2040 compared to 22% in 1996. Thus, this population group is becoming

more and more dominant over time. Just as important, the older age cohort also happens

to be a very active market participant. According to Sheshinski and Tanzi (1989), in 1985

people 65 or older received about 53 percent of all interest, dividend, and estate incomes

in the U.S. as well as one-third of all capital gains. This market participation has been

steadily increasing over the last years. For example, the percentage of total interest and

dividend income received by those 65 and older has risen from 44.1 percent in 1970 to

50.9 percent in 1982.

Thus, the share of older population has been increasing at the same time that their

in�uence on the market is increasing. Given the fact that these older market participants

are also more risk averse, then we should see implications for asset prices and portfolio

5

composition. In particular, we would expect that over time the aggregate demand for

each asset is going to be adjusted to re�ect the risk aversion shifts that come from the

population age group shifts. Goyal (2004) con�rms this by showing that wealth is moved

out of the stock market as the fraction of the population over 65 increases. Malkiel (1996)

takes a similar position and suggests that the proportion of wealth invested in stocks

should decline despite their higher return than the T-bills. Furthermore, Yoo (1994) �nds

that the real U.S. T-bill return is negatively correlated with the size of the age group with

the highest wealth increment.

Therefore the presence in the market of older, more risk averse agents carrying more

and more wealth should work to increase the return of the risky assets relative to the safer

ones, resulting in larger premiums. It would seem reasonable then that by introducing

into the existing pricing models the higher risk aversion of the older age cohorts along

with their increasing signi�cance, we might be able to get an asset pricing behavior closer

to the observed facts. Would it be possible for a model economy where the older agents

are more risk averse to produce the kind of behavior we see in the U.S. data? Will such an

economy generate a large enough equity premium? A low enough real risk free rate? An

important restriction is that these results do not require large unreasonable risk aversion

parameters. This is the goal of the exercise we undertake in this paper.

We build on the overlapping-generations (OLG) model framework of Constantinides,

Donaldson, and Mehra (2002) with three age cohorts. But, in our speci�cation, each

age group has di¤erent levels of aversion to risk. That is, each period�s representative

consumer�s risk aversion will increase as he ages. The middle-aged consumer has a higher

6

aversion to risk than the young and the old consumers are more risk averse than the

middle-aged. In section II we introduce the model. In section III we present the results

and compare these results to the benchmark case where all agents have the same level of

risk aversion. In section IV we introduce a variant of the model where the total population

share of the older age cohort is no longer the same as the one of the other age groups. In

this section we discuss the implications for asset pricing of an increasing, more dominant,

and more risk averse age group. Again, we compare our results with those obtained when

each age cohort is equally weighted as in section III. Section V concludes.

2 The Model

We consider a three-period OLG model, where each generation lives as young, middle-

aged and old. Each consumer generation is modeled by a representative agent so that

we can focus on across-generation di¤erences while ignoring possible heterogeneity within

any particular generation.

There is one consumption good in each period and it perishes at the end of that period.

All the prices, wages, consumption, dividends, and coupons payments are quoted in terms

of this single consumption good.

There is a �nancial market where two types of securities are traded; a bond and a share

of equity, both in�nitely lived. The bond here works as a proxy for long-term government

debt. The (consol) bond is default-free and pays a �xed coupon of b > 0 units of the

consumption good in every period in perpetuity1. Its supply is �xed at B units. The

1Additionally, we compute the returns and the �rst moment conditions as well as the premia of theone-period (twenty-year) discount bond. The one-period bond is in zero net supply. The price of thisbond is a shadow price, determined by the marginal rate of substitution of the middle-aged consumer.

7

aggregate coupon payment is B in every period and represents a portion of the economy�s

capital income. qbt is the ex coupon price of bond in period t. One perfectly divisible

equity share is traded. The equity is the claim to the net dividend stream fdtg, the sum

total of all the private capital income (stocks, corporate bonds, and real estate). Similarly,

the ex dividend price of equity in period t is qet and is the claim to the dividend stream

in perpetuity, beginning with period t+ 1. The total supply of equity is �xed at one.

The consumer born in period t gets a low deterministic w0 > 0 wage income in period

t, stochastic wage income w1t+1 when middle-aged in period t + 1, and zero wage income

in period t + 2 when old. The wage income process of the middle-aged consumer is

exogenous in order to avoid modeling the labor-leisure trade-o¤. Also, for the sake of

model simplicity, consumers do not trade claims on their future wage income.

A consumer born in period t starts with zero endowment of the bond and equity. He

purchases zbt;0 bonds and zet;0 shares of equity in period t (when young). The consumer then

adjusts his position to zbt;1 and zet;1when middle aged. The old consumers sell their bond

and stock holdings and consume the proceeds. This means that zbt;2 = 0, and zet;2 = 0.

Market clearing in period t then requires that the demand for bonds and equity by the

young and the middle-aged consumers equal their �xed supply. Therefore,

zbt;0 + zbt�1;1 = B, (1)

and similarly,

zet;0 + zet�1;1 = 1. (2)

Let ct;j denote the consumptions in period t + j (j = 0; 1; 2) of a consumer born in

8

period t. The budget constraint of the consumer born in period t is:

ct;0 + zbt;0 � qbt + zet;0 � qet � w0 (3)

when young,

ct;1 + zbt;1 � qbt+1 + zet;1 � qet+1 � w1t+1 + zbt;0 � (qbt+1 + b) + zet;0 � (qet+1 + dt+1) (4)

when middle-aged, and

ct;2 � zbt;1 � (qbt+2 + b) + zet;1 � (qet+2 + dt+2) (5)

when old. In the �rst period the young agent receives a relatively low deterministic

endowment income. The middle-aged agent is employed and receives wages so his total

income is made up of wages and proceeds from investing in the securities market. This

means that his income is subject to some uncertainty. The old agent does not work, and

consumes his entire wealth.

Furthermore, we require that ct;0 � 0, ct;1 � 0, and ct;2 � 0, thus ruling out negative

consumption and personal bankruptcy.

There is an increasing sequence fIt : t = 0; 1; :::g of information sets available to

consumers for their decision making each period t. It contains information about all the

past wage income and dividends up to and including period t. Furthermore, It contains

the consumption, bond investment, and stock investment histories of all the consumers

up to and including period t� 1. Consumption and investment decisions made in period

t depend only on information available in period t. A consumption and investment policy

of the consumer born in period t is the collection of the It-measurable (ct;0; zbt;0; zet;0), the

It+1-measurable (ct;1; zbt;1; zet;1), and the It+2-measurable ct;2.

9

The consumer born in period t has utility

E

2Xi=0

�iu (ct;i) jIt

!(6)

where It is the set of all the information available in period t, � is the subjective discount

factor with 0 < � < 1.

The period utility function is

u(ct;i; �i) =c1��it;i � 11� �i

; (7)

where �i > 0 is the risk aversion parameter. This paper departs from Constantinides,

Donaldson, and Mehra (2002) in that �i is assumed to vary with the consumer�s age: the

older consumers exhibit higher risk aversion than the younger ones. Recall that the period

utility function in Constantinides, Donaldson, and Mehra (2002) is u(c) = c1���11�� , since

the level of risk aversion is held constant throughout the agent�s lifetime. But, as we noted

in the introduction, several studies support the idea that the risk-aversion parameter is

not constant but rather it varies with the agent�s age.

We specify the joint process of the aggregate income and the wages of the middle-aged,

(yt; w1t ), where the aggregate income yt is

yt = w0 + w1t + b+ dt. (8)

The joint process of the aggregate income and the wage income of the middle-aged is

modeled as a time-stationary probability distribution. In the calibration, yt and w1t assume

two values each: y1 and y2 and w11 and w12. These form four states denoted st = j; j =

1; :::; 4. The 4x4 transition matrix is denoted by �.

10

Following the work of Constantinides, Donaldson, and Mehra (2002), we study the

borrowing-constrained version of the economy; the young agents earn low wages and would

want to smooth their lifetime consumption by borrowing against future wage income.

They consume part of the loan and invest the rest in equity whose returns are higher than

the loan rate. But, lacking collateral these young consumers �nd it di¢ cult to borrow

against future wage income by shorting the bonds. The young consumers are however

allowed to short equity, but this is immaterial since we calibrate our model economy using

parameters for which the borrowing constraints are not binding and so the young choose

not to short equity. Thus, the young are excluded from the bond market by their limited

(human) collateral, and do not participate in the equity market. By leaving out the young

from the decision making, the state st�1 becomes irrelevant for decisions (and prices) of

period t.

In this borrowing-constrained economy there exists a rational expectations equilibrium

in which the young do not participate in the bond and equity markets. Therefore, lagged

state variables are not present; decisions and prices in period t are measurable with respect

to current state st = j; j = 1; :::; 4, alone.2

Then our �rst order conditions (FOC) with respect to zbt;1 and zet;1 given the consump-

tion constraints and the market clearing conditions are:

u0 (ct;1) � qbt+1 = E(� � u0 (ct;2) � fqbt+2 + bg) (9)

2Equilibirum is de�ned as the set of consumption and investment policies of the consumers born ineach period and the bond and stock prices in all periods such that: (i) each consumer maximizes hisexpected utility taking the price processes as given; (ii) bond and equity markets clear in all periods.

11

and

u0 (ct;1) � qet+1 = E(� � u0 (ct;2) � fqet+2 + dt+2g); (10)

where

ct;1 = w1t+1 + z

bt;0 � (qbt+1 + b) + zet;0 � (qet+1 + dt+1)� zbt;1 � qbt+1 � zet;1 � qet+1 (11)

and

ct;2 = zbt;1 � (qbt+2 + b) + zet;1 � (qet+2 + dt+2): (12)

We can drop the time subscripts.

Since zbt;0 and zet;0 are zero (the young are not in the market), by market clearing

conditions we have zbt;1 = B and zet;1 = 1. Thus, the presence of borrowing constraints

means that equity is solely priced by the middle-aged investors.

Thus, our FOC become:

u0 (c1) � qb(j) = �4Xk=1

(u0 (c2) � fqb(k) + bg) � �jk (13)

and

u0 (c1) � qe(j) = �4Xk=1

(u0 (c2) � fqe(k) + d(k)g) � �jk; (14)

with

c1 = w1(j)�B � qb(j)� qe(j) (15)

and

c2 = B � (qb(j) + b) + qe(j) + d(j) (16)

for each state j of the economy.

12

Since our utility function is u(ct;i; �i) =c1��it;i �11��i , the marginal utilities of the middle-

aged and old consumers are respectively u0 (c1) = c��11 and u0 (c2) = c��22 . Then

qb(j)

(w1(j)�B � qb(j)� qe(j))�1 = �4Xk=1

fqb(k) + bg � �jk(B � (qb(k) + b) + qe(k) + d(k))�2

(17)

and

qe(j)

(w1(j)�B � qb(j)� qe(j))�1 = �4Xk=1

fqe(k) + d(k)g � �jk(B � (qb(k) + b) + qe(k) + d(k))�2

(18)

are the two equations to be estimated.

2.1 Calibration

The subjective discount factor beta is set equal to 0.44 since in the calibration each period

is of length 20 years. This implies an annual discount factor of 0.96, the standard discount

factor used in the literature.

The equilibrium joint distribution of the bond and equity returns is not dependent on

the level of the exogenous macroeconomic variables for a �xed y, w1 correlation structure.

Instead, the distribution depends on the following factors: (i) the average share of income

going to labor, E(w1+w0)=E(y); (ii) the average share of income going to the labor of the

young, w0=E(y); (iii) the average share of income going to interest on government debt,

b=E(y); (iv) the coe¢ cient of variation of the twenty-year wage income of the middle-

aged, �(w1)=E(w1); (v) the coe¢ cient of variation of the twenty-year aggregate income,

�(y)=E(y); (vi) the twenty-year autocorrelation of the labor income, corr(w1t ; w1t�1); (vii)

the twenty-year autocorrelation of the aggregate income, corr(yt; yt�1); (viii) the twenty-

year cross-correlation, corr(yt; w1t ). Therefore, the model is calibrated in accordance with

conditions i-viii.

13

The transition matrix of the joint Markov process on the wage income of the middle-

aged consumers and the aggregate income, is given by:266664(y1; w

11) (y1; w

12) (y2; w

11) (y2; w

12)

(y1; w11) � � � H

(y1; w12) � +� ��� H �

(y2; w11) � H ��� � +�

(y2; w12) H � � �

377775 (19)

where

�+ � + � +H = 1: (20)

There are nine parameters to be estimated: y1=E(y), y2=E(y), w11=E(y), w12=E(y), �, �,

�, H, and �. These are chosen to satisfy the above moment conditions and as well as the

condition (20), making sure that all the matrix entries are positive.

As Constantinides, Donaldson, and Mehra (2002) point out, there are some di¢ culties

in computing the above moment conditions, mostly due to the fact that we are dealing

with twenty year aggregates but only about a century-long data set. As a way to deal with

these estimation problems, they perform sensitivity analysis using the following range of

values.

(i) The average share of income going into labor, E(w1+w0)=E(y), is set in the lower

half of the documented range (0.66,0.70).

(ii) The average share of income going to the labor of the young, w0=E(y), is set in

the range (0.16,0.20). This number needs to be small enough to ensure that the young

would need to borrow and therefore the borrowing constraint is binding in this economy.

(iii) The average share of income going to interest on government debt, b=E(y), is set

at 0.03 which is consistent with the U.S. historical data.

14

(iv) The coe¢ cient of variation of the twenty-year wage income of the middle-aged,

�(w1)=E(w1), is set at 0.25 based on several studies including Cox (1985).

(v) We have very little guidance in choosing the coe¢ cient of variation of the twenty-

year aggregate income, �(y)=E(y). However, we use the values 0.20 and 0.25.

(vi) As we indicated above, we lack su¢ cient time-series data to estimate the twenty-

year autocorrelations and cross correlation of the labor income of the middle-aged and the

aggregate income, corr(w1t ; w1t�1), corr(yt; yt�1), and corr(yt; w

1t ). Therefore, we present

results for a variety of autocorrelation and cross-correlation structures: fcorr(yt; w1t ) =

0:1, corr(w1t ; w1t�1) = corr(yt; yt�1) = 0:1g, fcorr(yt; w1t ) = 0:1, corr(w1t ; w1t�1) = corr(yt; yt�1) =

0:8g,fcorr(yt; w1t ) = 0:8, corr(w1t ; w1t�1) = corr(yt; yt�1) = 0:1g, and fcorr(yt; w1t ) = 0:8,

corr(w1t ; w1t�1) = corr(yt; yt�1) = 0:8g.

The transition matrix of the joint Markov process and as well as the long probabilities

turn out to depend only on what serial and auto correlation values we use in our calibra-

tion. For example for the case fcorr(yt; w1t ) = 0:1, corr(w1t ; w1t�1) = corr(yt; yt�1) = 0:1g,

the calibration exercise results in the transition probability matrix

� =

26640:5298 0:0202 0:0247 0:42530:0302 0:5198 0:4253 0:02470:0247 0:4253 0:5198 0:03020:4253 0:0247 0:0202 0:5298

3775 (21)

and the long run probability

P =

26640:2750:2250:2250:275

3775 . (22)

Table I, from the work of Constantinides, Donaldson, and Mehra (2002), gives the

historical mean and standard deviations of the annualized, twenty-year holding-period

15

return on the S&P 500 total return series; and on the Ibbotson U.S. Government Treasury

Long-Term bond yield. Real returns are CPI adjusted. The annualized mean return

(for both the equity and bond) is de�ned as the sample mean of [log{20-year holding

period return}]/20. The annualized standard deviation of the equity (or bond) return

is de�ned as the sample standard deviation of [log{20-year holding period return}]/ 2p20.

The annualized mean equity premium is de�ned as the di¤erence of the mean return on

equity and the mean return on the bond. The standard deviation of the premium is

de�ned as the sample standard deviation of [{log{20-year nominal equity return}-log{20-

year nominal bond return}]/ 2p20.

From Table I we see that the nominal mean equity return is 9-11 percent with a

standard deviation of around 14 percent; the mean bond nominal return is about 4 percent

with a standard deviation of 7-8 percent; and the mean equity premium is 5-7 percent

with a standard deviation of 14-15 percent.

We are however more interested in the real returns since ours is a real model, and

therefore, we the goal is to match the U.S. real return statistics. The real mean equity

return is 6-7 percent with a standard deviation of 14-16 percent; the mean bond real

return is about 1 percent with a standard deviation of 7 percent; and the mean equity

premium is 5-7 percent with a standard deviation of 14-15 percent as we have seen. Note

that since in the model the equity is de�ned as claim not just to corporate dividends but

also to all the risky capital in the economy, the mean equity premium we want to match

is around 3 percent.

16

3 Results

For all the reported results we set �(y)=E(y) = 0:20, and �(w1)=E(w1) = 0:25. The

one-period discount bond is referred to as the bond. It is in zero net supply. The consol

bond, which is in positive net supply, is referred to as the consol.

Introducing into the model of Constantinides, Donaldson, and Mehra (2002) the fact

that older consumers are more risk averse than the younger ones produces higher security

returns and their standard deviations. The mean premium and its standard deviation

also increase quite substantially. It is apparent from the results that the equity premium

observed in the U.S. data can be easily matched even for small risk aversion parameters.

Tables IIA and IIB shows the result of our calibration exercise for the case of cross-

correlation Corr(y; w1) = 0:1. They include the security returns and their standard

deviations, as well as the market equity premia over the two bonds. We present results

for two autocorrelation cases and several risk aversion pairs RRA middle-aged, RRA old.

The results for (RRA middle-aged=4.0, RRA old=4.0) and (RRA middle-aged=6.0,

RRA old=6.0) replicate the results from Constantinides, Donaldson, and Mehra (2002).

Their model achieves a mean equity premium over the bond of 3% using RRA=4.0 for

both the middle-aged and old agents, as seen in the third column of Table IIA. In our

speci�cation, the old consumers are made more risk averse relative to the middle-aged

consumers3, and this works to increase the mean risk premium: both the mean equity

return and the bond returns increase relative to the base case, but the increase in the

equity is larger, resulting in higher premium.

3The middle-aged consumers are more risk averse than the younger ones, but this is made irrelevantby the fact the the young consumers do not participate in the bond and equity markets.

17

The e¤ect of linking the consumer�s coe¢ cient of risk aversion to his age can be quite

signi�cant; for example to match the 3.4 percent premium when all agents have an RRA

of 6.0 (Table IIA, �fth column), all that is required is that we set RRA middle-aged=2.0

and RRA old=2.25. Thus, even though the average risk aversion level in our economy is

much lower, the e¤ect of having the older age cohort more risk averse is so strong that it

drives up the premium. The middle-aged agents know they will become more risk averse

when they are older and thus dislike consumption variation even more at the time (when

older). This explains why these agents are going to require much higher equity returns in

order to invest in equity given its uncertain payo¤s in the future. This is not surprising

in light of our discussion in the beginning of the paper. It also conforms to the numerous

studies showing that investors tend to shift their portfolio wealth out of stocks as they

age.

However, it appears that an additional factor is at work here. We can see that the

(one-period) bond return also increases as we make the older agents more risk averse The

middle-aged agents, who will be more risk averse in the future, are no longer as willing

to give up consumption today in order to invest in the two securities (bond and equity).

They will now demand higher returns as compensation for giving up some of today�s

consumption. We will discuss this consumption shift some more later on in the paper.

Thus, the overall e¤ect is for all the security returns to increase, with the equity returns

increasing by more than the bond returns. The implication of more risk averse older

agents is that our model generates the security returns as well as risk premia found in the

U.S. data without relying on very high levels of risk aversion, unlike some of the earlier

18

work in this area. But, this speci�cation has the undesirable e¤ect of further increasing

the bond returns (as we just discussed) as well as the security standard deviations relative

to the data.

Looking at Tables II and III, it seems that what is driving the bulk of this result is

the ratio of the two RRA�s, rather than their absolute levels. For example RRA middle-

aged=2.0 and RRA old=2.5 produce higher statistics than RRA middle-aged=6.0 and

RRA old=6.25. This result is similar to that obtained by Danthine, Donaldson, Giannikos,

and Guirguis (2004). In their work, the authors found that the relative outlook e¤ect,

the ratio of the two risk aversions (�2�1), dominates with regards to the mean risk premium

and its standard deviation: the pair �1 = 1:5, �2 = 2:0 (�2�1= 1:333) results in higher

mean risk premium and standard deviation than the pair �1 = 2:0; �2 = 2:5 (�2�1 = 1:11).

Table IV clearly illustrates this point. Even as overall level of risk aversion increases we

see that all the numbers drop as the ratio RRA oldRRA middle�aged drops. Thus, making one group

of the population more risk averse relative to the others a¤ects the results in proportion

to how di¤erent the risk aversion for each group is.

From tables II and III we can also see that the correlation of the labor income of the

middle-aged (w1) and the equity premium is consistently smaller than the correlation of

the labor income of the middle-aged and the dividend. As Constantinides, Donaldson,

and Mehra (2002) explain, this is another reason the young consumers �nd equity so

attractive. Thus, they want to invest in equity for its high return and because equity

has low correlation with their future consumption if most of that consumption comes

from their future wage income, w1. In the model we presented however, the young are

19

borrowing constrained, and so will not participate in the market.

One interesting question is then why the middle-aged consumers will invest in bonds

despite the very high mean premium. Table V (similar to that from Constantinides,

Donaldson, and Mehra (2002)) illustrates what exactly makes bonds attractive. The

consumption of the old is quite variable, leading the middle-aged to invest some of their

wealth in bonds. The young simply consume their endowment, while the consumption of

the middle-aged is relatively smooth.

Table VI shows the consumption pattern of each age group in the case when old

consumers are more risk-averse relative to the young ones. As we discussed above, the

middle-aged investors are now less willing to give up some of their current consumption in

return for higher future consumption. They will demand higher returns in order to invest

some of the current income and thus provide for future consumption. However, these

middle-aged investors now dislike future consumption so much more that these higher

returns are not enough to get them to part with their current consumption. Instead, they

will consume relatively more now (compared to when everyone has the same risk aversion).

The overall e¤ect, as table V and VI show, is that the consumption of the middle-aged

increases while the old agents�consumption across all states decrease. Additionally, the

consumption of the middle-aged becomes much more variable relative to the consumption

of the old. The standard deviation of the middle age consumption increases from 4938 to

7970 while the standard deviation of the consumption of the old remains around 19,000

(not shown in the table). This is true even when we use lower risk-aversion numbers for

both types of consumers: panel B shows that old agent�s consumption is now even lower

20

and the middle-aged�s is more variable relative to the benchmark case (RRA=4.0 and

RRA=4.0).

Once again, it is the ratio RRA oldRRA middle�aged that is driving these results. As the ratio in-

creases the ratio Consumption oldConsumption middle�aged decreases while the ratio

STD Consumption oldSTD Consumption middle�aged

also decreases. The latter is true even though both consumption of the middle-aged and

the consumption of the old become more and more variable as the ratio RRA oldRRA middle�aged

increases. This happens because STD Consumption old increases by less and less as the

risk aversions ratio increases.

4 Additional Results

In this section we want to study impact of changing population age structure. As we saw

in the introduction, the proportion of the older age group has been increasing over time

not only in the U.S. but in most developed nations as well. Table VII shows the U.S.

demographic structure over last few decades as well as the projections for the next 50

years. Clearly the U.S. population is ageing as the median age indicates. For example,

the median age was 30.0 in 1980, 35.7 in 2000, and is expected to be 38.1 by the year 2050

according to the U.S. Census Bureau projections. The table also shows that the share of

the older population is increasing, with the percent of population 40-64 increasing from

26.3 in 1970 to 30.4 in 2000. It is expected to remain around 30 percent for the next three

decades. The discussion up to now would suggest that if the share of the population with

higher risk aversion increases, then we would see a further increase in the risk premium

relative to the benchmark case in Constantinides, Donaldson, and Mehra (2002).

We modify our model to introduce a population growth factor n; the population now

21

grows at an exogenously �xed rate n. Population at t = 0 is normalized to consist of

(1 + n) young people, 1 middle-aged person, and ( 11+n) old people. At any time t in the

future there will be (1 + n)t+1 young, (1 + n)t middle-aged, and (1 + n)t�1 old people.

Therefore, the stationary age distribution of the population measured as fractions of the

total number of individuals is: Young 11+ 1

1+n+ 1(1+n)2

, Middle-aged 1(1+n)+1+ 1

1+n

, and Old

1(1+n)2+(1+n)+1

4.

As n increases the fraction of the young people in the population increases while the

fraction of the middle-aged and old decreases. We are especially interested in seeing the

e¤ects of a decreasing n on steady state asset prices and risk premium. Lower values of

n translate into a population structure with greater fractions of both the middle and old

age cohorts. As we saw above, this is the case observed in the U.S. data over the last

few years. Table VIII shows the fraction of the three age cohorts for a given population

growth rate. When n = 0, the three age groups are equally represented in the total

population. For n = �0:04 we see that the population is ageing, the younger age group

is now a minority making up only 0.3198 share of the total population. The middle-aged

and the old combine now for about 68 percent of all the population (the old age cohort

fraction is 0.3470).

Table IX shows the results for risk aversions pair RRA middle-aged=6.0 and RRA

old=6.25. The case for n = 0 corresponds to our benchmark case, when the three age

4The output, wages, dividends and aggregate interest will similarly grow at the rate n. That is, periodt aggregate output, yt, is given by yt = w0t + ~w1t + bt + dt, where yt = (1 + n)ty, w0t = (1 + n)t+1w0,~w1t = (1 + n)

t ~w1, bt = (1 + n)tb, and dt = (1 + n)td. The supply of the securities must also grow at therate n, so that in period t we have (1 + n)t equity shares and (1 + n)tb bonds. This increased supply ofsecurities is distributed in proportion to each agent�s holdings: the holdings of the young (middle-aged)increase by a fraction (1 + n) when they are middle-aged (old).

22

groups are distributed equally. The last column is the case in Constantinides, Donaldson

and Mehra (2002) when the proportion of each age group is the same and all the agent

have the same risk aversion parameter.

We can see that as n decreases the mean risk premium in the economy increases

while its standard deviation goes down. Comparing the last two columns it is clear that

increasing the share of the older, more risk averse consumers in our economy has the

overall e¤ect of raising the risk premium by about a half percent.

5 Conclusion

In this paper we suggest a new approach to asset pricing, one that takes into consideration

a vast body of literature linking an agent�s level of risk aversion coe¢ cient to his age. This

approach also incorporates the factual evidence from the U.S. data that the population age

structure is changing, with the fraction of the older age cohorts increasing. Our experiment

show that these small changes introduced in the standard models by our speci�cation lead

to very interesting and signi�cant results. By making the consumers�aversion to risk a

positive relation to their age, the equity premium and the consumption patterns of the

agents will change quite signi�cantly. If the old agents make up a third of the total

population, then even a small increase in the risk aversion coe¢ cient of this segment of

the population (while keeping the average risk coe¢ cient for the population at large close

to its initial number) has the e¤ect of increasing the equity premium in the economy.

Other e¤ects of this change are: the security returns increases as their price drop, and

the old consumers� consumption decreases across all states of the economy relative to

consumption of the middle-aged agents (the young agents�consumption remains the same,

23

they still consume just their endowment). We also studied the implications of a changing

population age structure. We introduce a population growth rate factor n that works

to change the relative proportion of each age cohort in our model economy to match

the U.S. population demographic trends of the last few decades. Not surprisingly, by

increasing the share of the older, more risk averse age cohort in the population we obtain

further increases in the equity premium but with a lower standard deviation. Therefore,

the growth rate n produces results more in accordance with those observed in the data.

What makes our results interesting is that they do not rely on large levels of risk aversion

coe¢ cient.

Our work shows that a way to conciliate the stylized data facts with the current asset

pricing models may require modifying these models to re�ect the population trends and

the empirical evidence of changing risk aversion parameters. Clearly, a lot more work is

needed in this area before we are satis�ed, but there is no doubt that this is promising

avenue.

24

Appendix

Table I: U.S. Nominal and Real Returns (in percent)

U.S. Nominal Returns1/1889-12/1999 1/1926-12/1999

Equity Bond Premium Equity Bond Premiummean 9.20 3.86 5.34 10.55 3.97 6.58std 13.88 7.27 14.32 14.47 8.49 15.21

U.S. Real Returns1/1889-12/1999 1/1926-12/1999

Equity Bond Premium Equity Bond Premiummean 6.15 0.82 5.34 6.71 0.14 6.58std 13.95 7.40 14.32 15.79 7.25 15.21

Note: The mean and standard deviations of the annualized, twenty-year holding-period return on

the S&P 500 total return series and on the Ibbotson U.S. Government Treasury Long-Term bond yield.

Real returns are CPI adjusted. The annualized mean return (for both the equity and bond) is de�ned

as the sample mean of [log{20-year holding period return}]/20. The annualized standard deviation of

the equity (or bond) return is de�ned as the sample standard deviation of [log{20-year holding period

return}]/ 2p20. The annualized mean equity premium is de�ned as the di¤erence of the mean return on

equity and the mean return on the bond. The standard deviation of the premium is de�ned as the sample

standard deviation of [{log{20-year nominal equity return}-log{20-year nominal bond return}]/ 2p20.

27

Table IIA: Equity and Bond Returns and First Moments

Corr(y; w1) = 0:1 & Corr(w1; w1) = Corr(y; y) = 0:1RRA Middle-Aged 2.00 4.00 4.00 6.00 6.00RRA Old 2.25 4.00 4.25 6.00 6.25Mean E. Return 0.138 0.079 0.126 0.084 0.118Std. of E. Return 0.277 0.206 0.281 0.230 0.291Mean Bond Return 0.099 0.051 0.085 0.051 0.076Std. of Bond Return 0.209 0.137 0.219 0.154 0.226Mean P/Bond 0.038 0.028 0.042 0.034 0.042Std. P/Bond 0.238 0.206 0.273 0.236 0.304Mean Consol Return 0.087 0.040 0.063 0.037 0.054Std. of Consol Return 0.216 0.172 0.238 0.191 0.252Mean P/Consol 0.050 0.040 0.063 0.047 0.064Std. P/Consol 0.300 0.240 0.303 0.267 0.305Corr(w1; d) -0.424 -0.424 -0.424 -0.424 -0.424Corr(w1; P remium) -0.043 -0.037 -0.046 -0.039 -0.045

Note: The results correspond to �(y)=E(y) = 0:20, and �(w1)=E(w1) = 0:25. Combinedwith the correlation coe¢ cients, the long-run probabilities for this economy are: p1=0.275, p2=0.225,

p3=0.225, and p4=0.275. The one-period discount bond is referred to as the bond. It is in zero net

supply. The consol bond, which is in positive net supply, is referred to as the consol.

28

Table IIB: Equity and Bond Returns and First Moments

Corr(y; w1) = 0:1 & Corr(w1; w1) = Corr(y; y) = 0:8RRA Middle-Aged 2.00 4.00 4.00 6.00 6.00RRA Old 2.25 4.00 4.25 6.00 6.25Mean E. Return 0.144 0.085 0.142 0.093 0.138Std. of E. Return 0.331 0.239 0.370 0.265 0.376Mean Bond Return 0.118 0.069 0.107 0.067 0.098Std. of Bond Return 0.265 0.193 0.294 0.208 0.303Mean P/Bond 0.025 0.017 0.035 0.026 0.040Std. P/Bond 0.107 0.102 0.167 0.137 0.198Mean Consol Return 0.098 0.048 0.075 0.044 0.065Std. of Consol Return 0.243 0.182 0.258 0.190 0.262Mean P/Consol 0.046 0.038 0.067 0.049 0.073Std. P/Consol 0.225 0.189 0.251 0.215 0.264Corr(w1; d) -0.424 -0.424 -0.424 -0.424 -0.424Corr(w1; P remium) -0.030 -0.022 -0.071 -0.021 -0.062

Note: The results correspond to �(y)=E(y) = 0:20, and �(w1)=E(w1) = 0:25. Combinedwith the correlation coe¢ cients, the long-run probabilities for this economy are: p1=0.275, p2=0.225,

p3=0.225, and p4=0.275. The one-period discount bond is referred to as the bond. It is in zero net

supply. The consol bond, which is in positive net supply, is referred to as the consol.

29

Table IIIA: Equity and Bond Returns and First Moments

Corr(y; w1) = 0:8 & Corr(w1; w1) = Corr(y; y) = 0:1RRA Middle-Aged 2.00 4.00 4.00 6.00 6.00RRA Old 2.25 4.00 4.25 6.00 6.25Mean E. Return 0.150 0.077 0.132 0.079 0.119Std. of E. Return 0.168 0.160 0.211 0.185 0.243Mean Bond Return 0.137 0.060 0.114 0.058 0.097Std. of Bond Return 0.158 0.130 0.197 0.152 0.225Mean P/Bond 0.013 0.017 0.019 0.021 0.022Std. P/Bond 0.204 0.226 0.290 0.262 0.345Mean Consol Return 0.135 0.063 0.111 0.062 0.096Std. of Consol Return 0.169 0.145 0.209 0.166 0.234Mean P/Consol 0.015 0.015 0.021 0.018 0.023Std. P/Consol 0.169 0.138 0.178 0.158 0.180Corr(w1; d) 0.364 0.364 0.364 0.364 0.364Corr(w1; P remium) -0.049 -0.022 -0.065 -0.025 -0.060

Note: The results correspond to �(y)=E(y) = 0:20, and �(w1)=E(w1) = 0:25. Combined withthe correlation coe¢ cients, the long-run probabilities for this economy are: p1=0.45, p2=0.05, p3=0.05,

and p4=0.45. The one-period discount bond is referred to as the bond. It is in zero net supply. The

consol bond, which is in positive net supply, is referred to as the consol.

30

Table IIIB: Equity and Bond Returns and First Moments

Corr(y; w1) = 0:8 & Corr(w1; w1) = Corr(y; y) = 0:8RRA Middle-Aged 2.00 4.00 4.00 6.00 6.00RRA Old 2.25 4.00 4.25 6.00 6.25Mean E. Return 0.152 0.080 0.140 0.084 0.127Std. of E. Return 0.160 0.128 0.181 0.150 0.192Mean Bond Return 0.146 0.072 0.128 0.069 0.112Std. of Bond Return 0.129 0.088 0.132 0.106 0.146Mean P/Bond 0.007 0.008 0.011 0.014 0.016Std. P/Bond 0.087 0.083 0.120 0.113 0.140Mean Consol Return 0.142 0.068 0.123 0.067 0.107Std. of Consol Return 0.137 0.089 0.139 0.102 0.146Mean P/Consol 0.010 0.012 0.017 0.017 0.021Std. P/Consol 0.143 0.137 0.177 0.160 0.186Corr(w1; d) 0.364 0.364 0.364 0.364 0.364Corr(w1; P remium) -0.135 -0.067 -0.106 -0.022 -0.036

Note: The results correspond to �(y)=E(y) = 0:20, and �(w1)=E(w1) = 0:25. Combined withthe correlation coe¢ cients, the long-run probabilities for this economy are: p1=0.45, p2=0.05, p3=0.05,

and p4=0.45. The one-period discount bond is referred to as the bond. It is in zero net supply. The

consol bond, which is in positive net supply, is referred to as the consol.

31

Table IV: Relationship between Risk Premium and the Ratio of Risk AversionCoe¢ cients

ratio RRA oldRRA middle�aged 1.500 1.125 1.042

RRA Middle-Aged 2.00 4.00 6.00RRA Old 3.00 4.50 6.25Mean E. Return 0.446 0.195 0.118Std. of E. Return 0.503 0.381 0.291Mean Bond Return 0.329 0.128 0.076Std. of Bond Return 0.349 0.325 0.226Mean P/Bond 0.116 0.067 0.042Std. P/Bond 0.306 0.307 0.304Mean Consol Return 0.329 0.098 0.054Std. of Bond Return 0.349 0.305 0.252Mean P/Consol 0.117 0.097 0.064Std. P/Consol 0.343 0.321 0.305

Note: The results correspond to the case of Corr(y; w1) = 0:1, Corr(w1; w1) = Corr(y; y) =0:1. The one-period discount bond is referred to as the bond. It is in zero net supply. The consol bond,which is in positive net supply, is referred to as the consol.

Table V: Consumption and the Conditional First Moments of the Returns

State 1 State 2 State 3 State 4Probability 0.275 0.225 0.225 0.275 1Young Cons. 19,000 19,000 19,000 19,000 19,000M-aged Cons. 38,768 34,430 26,821 27,979 32,137Old Cons. 60,432 25,168 72,379 31,619 47,262Mean E. Return 0.051 0.049 0.115 0.104 0.079Mean Bond Return 0.033 0.009 0.069 0.089 0.051Mean P/Bond 0.017 0.040 0.045 0.015 0.028Mean Consol Ret. 0.031 -0.010 0.045 0.085 0.040Mean P/Consol 0.020 0.060 0.070 0.018 0.040

Note: Consumption of the young, middle-aged, and old and the conditional �rst moments of the

returns at the four states of the economy. This results correspond to the case Corr(y; w1) = 0:1,Corr(w1; w1) = 0:1, and Corr(y; y) = 0:1. The one-period discount bond is referred to as thebond. It is in zero net supply. The consol bond, which is in positive net supply, is referred to as the

consol. This is the benchmark case of Constantinides, Donaldson, and Mehra (2002). Both the middle-age

and the old agents have a RRA of 4.00.

32

Table VI: Consumption and the Conditional First Moments of the Returns

Panel AState 1 State 2 State 3 State 4 1

Young Consumption 19,000 19,000 19,000 19,000 19,000Middle-aged Cons. 49,759 42,505 30,520 32,188 38,966Old Consumption 49,441 17,093 68,680 27,410 40,433Mean E. Return 0.097 0.073 0.160 0.172 0.126Mean Bond Return 0.073 0.013 0.089 0.151 0.085Mean P/Bond 0.023 0.060 0.071 0.021 0.042Mean Consol Ret. 0.055 -0.009 0.062 0.131 0.063Mean P/Consol 0.042 0.083 0.098 0.040 0.063

Panel BState 1 State 2 State 3 State 4 1

Young Consumption 19,000 19,000 19,000 19,000 19,000Middle-aged Cons 53,091 49,639 31,344 32,341 41,715Old Consumption 46,109 9,959 67,856 27,257 37,684Equity Return 0.131 0.087 0.152 0.173 0.138Mean Bond Return 0.114 0.027 0.083 0.157 0.099Mean P/Bond 0.017 0.060 0.069 0.017 0.038Mean Consol Ret. 0.103 0.015 0.069 0.145 0.087Mean P/Consol 0.028 0.072 0.084 0.028 0.063

Note: Consumption of the young, middle-aged, and old and the conditional �rst moments of the

returns at the four states of the economy. The long run state probabilities are: p1=0.275, p2=0.225,

p3=0.225, and p4=0.275. Panel A is the case where RRA Middle� aged = 4:0 and RRA Old =4:25. Panel B corresponds to the case of RRA Middle� aged = 2:0 and RRA Old = 2:25. Theone-period discount bond is referred to as the bond. It is in zero net supply. The consol bond, which is

in positive net supply, is referred to as the consol.

33

Table VII: Actual and Forecast U.S. Demographic Structure

Year Median Age % of Population 40-64 Population 40-64Population 65+

1970 27.9 26.3 2.71980 30.0 24.7 2.21990 32.8 25.7 2.12000 35.7 30.4 2.42010 35.7 30.4 2.42020 37.6 30.5 1.82030 38.5 28.0 1.42040 38.6 27.9 1.42050 38.1 27.6 1.4

Note: U.S. Census Bureau historical data and projections from CPS Reports P25-1130. Average age

over 20 computed using the midpoint in 5-year age intervals as the average age for all persons in that

interval, and assuming that the average age for persons 85 and older is 90.

Table VIII: Relationship between the Growth Rate and Age Group Share

n -0.08 -0.04 0.00 0.04 0.08share of the young 0.306 0.3198 0.3333 0.3465 0.3593share of the m-aged 0.3326 0.3331 0.3333 0.3332 0.3327share of the old 0.3615 0.3470 0.3333 0.3203 0.3080

Note: The stationary age distribution of the population measured as fractions of the total number of

individuals is: Young 11+ 1

1+n+ 1(1+n)2

, Middle-aged 1(1+n)+1+ 1

1+n

, and Old 1(1+n)2+(1+n)+1

.

34

Table IX: Relationship between the growth rate and the equity premium.

n 0.00 -0.04 -0.08 0.00RRA Middle-Aged 6.00 6.00 6.00 6.00RRA Old 6.25 6.25 6.25 6.00Mean E. Return 0.121 0.120 0.119 0.084Mean Risk-Free Return 0.093 0.091 0.089 0.061Mean Premium 0.028 0.029 0.030 0.024Std. Premium 0.303 0.299 0.296 0.225

Note: The risk-free rate return is the return to one-period risk-free discount bond in zero net supply.

This security pays one unit of consumption in all states of nature. The last column in the table is the

benchmark case in Constantinides, Donaldson, and Mehra (2002) where we have equal population shares

and all the agents have the same aversion to risk.

35

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