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International Association for the Study of Insurance Economics études et Dossiers

études et Dossiers No. 369 World Risk and Insurance Economics Congress

25-29 July 2010

Singapore

Working Paper Series ofThe Geneva Association

The Geneva Association - General Secretariat - 53, route de Malagnou - CH-1208 GenevaTel.: +41-22-707 66 00 - Fax: +41-22-736 75 36 - [email protected] - www.genevaassociation.org

International Association for the Study of Insurance Economics Études et Dossiers

Études et Dossiers No. 369

World Risk and Insurance Economics Congress

25-29 July 2010

Singapore

February 2011

Working Paper Series of The Geneva Association

© Association Internationale pour l'Etude de l'Economie de l'Assurance

The Geneva Association - General Secretariat - 53, route de Malagnou - CH-1208 Geneva Tel.: +41-22-707 66 00 - Fax: +41-22-736 75 36 - [email protected] - www.genevaassociation.org

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Population Growth and Asset Pricing

Tzuling Lin Department of Finance

National Taiwan University, Taipei, Taiwan

Richard MacMinn Katie School

Illinois State University

Larry Y. Tzeng Department of Finance

National Taiwan University, Taipei, Taiwan

Current version

Jan. 2010

The Geneva Association________________________Etudes et Dossiers no. 369

Document free to download www.genevaassociation.org

Abstract

We conjecture that the reason for the poor performance of the traditional consumption-based capital asset pricing model could be that it ignores the growth rate of population. In this paper, we extend a consumption-based capital asset pricing model by considering population growth as an additional factor determining returns. The empirical findings show that the population growth affects the stock market and thus population risk plays a substantial role in explaining stock returns.



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I. Introduction

According to the consumption-based capital asset pricing model (CCAPM) developed by Lucas (1978) and Breeden (1979), the only relevant pricing factor is the growth rate of consumption per capita. However, little of the empirical evidence can support the traditional CCAPM, e.g., see Hansen and Singleton (1982, 1983), Mankiw and Shapiro (1986), and Breeden, Gibbons, and Litzenberger (1989). Several papers have improved the performance of the CCAPM by measuring the consumption growth at different points in time. 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 explains the cross-section of stock returns very well; they refer to this as ultimate consumption risk.1 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 investors must make their consumption and investment decisions simultaneously at that time for the CCAPM to hold at any given point, they suspect that it is more likely to happen during the fourth quarter.

Some papers measure risk by using other consumption goods rather than the nondurable consumption goods2 that most papers usually use. Ait-Sahalia, 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 it does with the aggregate consumption. Yogo (2006) finds that durable consumption in conjunction with nondurable consumption can explain the cross-section of stock returns. Savor (2010) proposes that garbage growth is significantly more volatile and more highly correlated with stocks than the growth of NIPA expenditures.

In this paper, we conjecture that the reason for the lack of empirical support for the CCAPM is that the previous work restricts attention to the dynamic of consumption per

1 If consumption is slow to adjust to returns, the ultimate consumption risk may be a better measure of the true risk of a stock. The extant explanations of slow consumption adjustment include measurement errors in consumption, the costs of adjusting consumption, nonseparability of the marginal utility of consumption from factors such as labor supply or housing stocks, which themselves are constrained to adjust slowly, or constraints on information flow or calculation (Parker and Julliard (2005)).

2 The usual concern with using total consumption, that is, the expenditures on durables and nondurables, is that expenditures on durable goods are not part of the flow of consumption to which the theory applies because they represent replacements and additions to a stock, rather than a service flow from the existing stock (Galí (1990), Lettau and Ludvigson (2001), and Parker and Julliard (2005)).



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capita and so ignores the dynamic of aggregate population. If the equity premium is supposed to be decided by the aggregate consumption in the society and the aggregate consumption is equal to the consumption per capita times the size of the population, then the equity premium should be jointly decided by both consumption growth and population growth. We present the argument for population risk as follows: the market requires fewer excess returns on the firms with higher population betas that can provide a better hedge against future consumption since an increase in the aggregate population growth implies an increase in the future consumption.

By building on the previous large literature, we demonstrate, using US data, the importance of population risk by investigating the empirical performance of our model, which considers consumption growth and population growth simultaneously, in alternative ways. In adding the population factor, we empirically show that a substantial part of the variation in historical average returns across different firm types can be explained by differences in their historical exposure to population risk along with consumption risk. Our empirical results are robust when measuring consumption risk with methods used by Parker and Julliard (2005) and by Jagannathan and Wang (2007). Furthermore, considering the population dynamics of age groups can further improve the performance of the CCAPM. These findings reveal that the stock market actually responds to the dynamics of population and so population risk plays a substantial role in explaining the stock returns.

Our empirical results show that the market price for the aggregate population risk is negative and support our argument that the required equity premium is lower when the equity could provide a natural hedge against future consumption. To trace the source of population risk, we find that the market price for aggregate population risk is negative mainly because the market prices for population risk of the childhood and the young-aged groups are negative. The contribution that this paper makes to the literature regarding the demographics and equity premium is that under the support of a theoretical model we empirically prove that the equity premium is closely related to the population risks of different age groups. Finally, it is worth mentioning that population risk can slightly reduce the equity premium puzzle since adding the population factor reduces the risk premium of consumption factor and then the corresponding coefficient of relative risk aversion can be lower.

The remainder of this paper is organized as follows. Section II contains a review of the literature concerning the impact of demographics on the stock market. Section III describes our econometric model containing the population betas along with the



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consumption betas. Section IV provides a description of the data. Section V details the empirical findings. Section VI concludes.

II. Other Related Literature

A substantial empirical literature shows that demographic changes have become crucial for asset pricing due to the life cycle. The difference between our paper and the empirical literature regarding demographics is that the literature investigates the relationship between the stock market and demographic variables whereas we examine the composition of the risk premium of stocks.

Several studies reveal that risky asset holdings have been increasing or have been humped-shaped over the life cycle. Poterba and Samwick (2001) present the empirical evidence on the basic patterns of household asset allocation over the life cycle. Faig and Shum (2002) explain why younger households have larger cash holdings than the middle-aged by considering liquidity. Benzoni, Collin-Dufresne and Goldstein (2007) state that due to the cointegration of labor income and stock dividends, the young agent’s human capital becomes “stock-like” while that of the old agent becomes “bond-like”. These effects create hump-shaped life-cycle portfolio holdings.

The size of age groups affects stock returns in two respects, i.e., consumption and investment, as households consume goods produced by firms and invest in those firms. On the consumption side, if an increase in certain age groups who purchase some consumption goods leads to an increase in the demand for these goods, the stock returns of the firms producing these goods will be driven up due to an increase in operating income. 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. The empirical finding in DellaVigna and Pollet shows that an increase in forecasted future consumption demand growth induced by changes in age structure predicts an increase in stock returns.

On the investment side, if an increase in some age groups who invest more leads to an increase in asset demand, the increase in demand will drive up stock prices. Using U.S. data in the post-1945 period, Bakshi and Chen (1994) empirically find that a rise in average age predicts a rise in risk premium. Poterba (2001) investigates the association between the population structure and the returns on stocks and bonds, but the paper does



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not find any significant relationship between them.3 Ang and Maddaloni (2005) find that demographic variables significantly predict excess returns internationally; they also find that faster growth in the fraction of the retired significantly decreases risk premiums and this predictability is stronger in countries with well-developed social security systems than those with less-developed financial markets.

Some papers incorporate the basic life-cycle theory into the investment demands using an overlapping generations (OLG) framework.4 Goyal (2004) finds empirical support for the life-cycle theory 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. Constantinides, Donaldson, and Mehra (2002) find that the borrowing constraint prevents the young from investing in equity. They observe a high equity premium in the presence of borrowing constraints. Athanasoulis (2006) numerically illustrates that there is a positive correlation between the proportion of the population that is young, constrained from investing in the stock market, and the equity premium.

III. Population Risk and the Cross Section of Asset Returns

Most empirical papers measure consumption risk by using consumption per capita (see, for example, Hansen and Singleton (1982), Mehra and Prescott (1985), Breeden, Gibbons, and Litzenberger (1989), Parker and Julliard (2005), and Jagannathan and Wang (2007)). They need strong assumptions to support their theoretical models. For example, Jagannathan and Wang (2007) assume that all agents are investors of the same type who review their consumption-investment decisions infrequently but at the same predetermined points in time. They also point out that, when the consumption betas are measured using aggregate consumption and all investors belong to the same type and review their decisions simultaneously, the CCAPM holds.

Theoretically, the performance of the CCAPM, where risk is measured by using the consumption per capita, should be the same as that by using the aggregate consumption

3 Goyal (2004) points out that the aspect of changes in the demographic structure is often ignored in the empirical literature, which sometimes finds insignificant relationships between demographics and stock market variables, such as the finding of Poterba (2001).

4 There is a long tradition of using OLG models in the literature. See Auerbach and Kotlikoff (1987), Rios-Rull (1994), Abel (2001, 2003), and Brook (2002, 2004).



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without assuming the investors are the same type.7 However, since the growth of the aggregate consumption is smoother than an individual agent’s consumption growth8, empirically the performance of the CCAPM, where risk is measured by using the aggregate consumption, gets poorer than that using the consumption per capita.

In this paper, we propose that the empirical models should employ aggregate consumption rather than only consumption per capita. The aggregate consumption contains two kinds of information, namely, the aggregate population and the consumption per capita since the aggregate consumption is actually linked to these two variables. Therefore, the factor “population” should be relevant to asset returns in addition to the per capita consumption. We separate the dynamics of the consumption per capita and of the aggregate population. In this way, we can fill the gap between the theoretical model and the empirical procedures without strong assumptions. The aggregate consumption

can be represented as ttt NC θ⋅= , where the aggregate population at time t is tN , and

the consumption per capita at time t is tθ . We also follow Parker and Julliard (2005) to

test the empirical model, which contains ultimate population risk along with ultimate consumption risk. Hence, we yield the empirical model, the population-based CCAPM (P-CCAPM):9

( ), 1 ,s s s si t N NE R θ θλ β λ β+ = + (1)

where sNβ is the population beta for ( )1s + -period population growth; s

θβ is the

consumption beta for ( )1s + -period consumption growth; sNλ is the market price for

( )1s + -period population risk; and sθλ is the market price for ( )1s + -period consumption

risk.

7 See Appendix A for details.

8 Some papers discuss the reasons why the growth of aggregate consumption is smoother than an individual agent’s consumption growth. See, for example, Marshall and Parekh (1999) and Gabaix and Laibson (2001).

9 See Appendix B for details.



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In the P-CCAPM, the risk premium on an asset can be viewed as the sum of a scale multiple of its exposure to consumption risk, i.e., the consumption beta, and that of its exposure to population risk, i.e., the population beta. The intuition behind the P-CCAPM is similar to that behind the CAPM and the CCAPM; it is that investors need additional compensation for taking additional risks. The consumption and population betas are two relevant measures of a stock’s risk. The amounts of the additional compensation are the market prices for consumption risk and population risk.

Although consumption and population betas measure a stock’s volatility relative to the consumption and population, we provide a different explanation for each since the consumption is endogenous and the population is exogenous. An individual agent can endogenously decide his or her consumption plans and passively adjust these consumption plans when equity prices are adjusted due to the impact brought about by the aggregate population. But they cannot control the impact of the aggregate population and the population exogenously changes over time.

First talk about the consumption factor. The market is willing to pay more to buy firms that provide better insurance against consumption risk, i.e., lower volatility with consumption market. These firms can be exposed to less consumption risk in the bad times when people cut down their consumption. Thus, the expected excess returns on these firms with lower consumption betas will be lower and the market price of consumption risk will be positive.

Next, analyze the population factor. The firms whose excess returns are more highly correlated with the aggregate population growth, i.e., higher population betas10, will obtain more profits due to an increase in the future consumption brought about by an increase in the aggregate population growth. Hence, the firms with higher population betas can provide a better hedge against the demand for future consumption. The investors in the market will be content to accept lower expected excess returns on these firms by paying more and the market prices for population risk may turn out to be negative.

The literature indicates that demographic variables affect the risk premiums and also support the basic life-cycle theory. Hence, we assume that the impact of the population dynamics in the stock market varies with age groups. If that is true, we can find stronger

10 An increase in the population beta could be implied by an increase in correlation between the excess return of an asset and the aggregate population growth since

( )( ) ( ) ,( ) ( ) ( , ) ( ) ( )( , ) ( ) ( ) ( )N N N NN N N g Rstd R std g Cov g R std R std gCov g R Var g std R std gβ ρ== =



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support for the P-CCAPM when further considering the population dynamics of age groups. We derive the following model with population risks posed by age groups by incorporating four generations, namely, the childhood, young-aged, middle-aged, and old-aged groups. The aggregate population is the sum of the populations of the four age

groups. We denote the size of the population for age group a as ( )omycaN at ,,,, ∈ ,

where c denotes childhood, y young-aged, m middle-aged, and o old-aged. Since

ot

mt

yt

ctt NNNNN +++= , we obtain the extended model with age groups11:

( ), 1 ,c c y y m m o os s s s s s s s s s

i t N N N N N N N NE R θ θλ β λ β λ β λ β λ β+ = + + + + (2)

where asN

β is the population beta for age group a , ( )omyca ,,,∈ ; asN

λ

is the market

price for population risk of age group a , ( )omyca ,,,∈ ; and sθλ is the market price for

consumption risk under this extended model.

We follow Jagannathan and Wang (2007) to examine the specification in equations (1)and (2) by applying the two-stage regression method of Fama-MacBeth (1973). However, measurement error could arise due to ignorance of the higher moments of the cross-sectional distribution of consumption growth and population growth12. In general, the measurement error in consumption and population also can depend on the return horizon between time t and 1t + , and s as observed by Parker and Julliard (2005). Ideally, we would like to set the return horizon as a calendar year and present our results under a different s . Moreover, the consumption risk resulting from measuring consumption growth based on the fourth quarter can explain the cross section of stock returns very well as noted by Jagannathan and Wang (2007). Hence, in addition to annual-annual growth which is calculated using annual data, we also consider Q4-Q4 growth which is calculated using data for the fourth quarters; that is, investors could review their consumption and investment decisions at the end of every 1s + calendar year.

11 See Appendix B.II for details.

12 Jacobs and Wang (2004) investigate the importance of higher moments of consumption growth for the cross-section of stock returns. They find that the performance of the consumption-based model with the first two moments, consumption growth and consumption dispersion, compares favorably with that of the Fama-French three-factor model.



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IV. Data

A. Consumption Data

We use the annual and quarterly seasonally adjusted13 aggregate nominal consumption expenditures for two sets of consumption goods (nondurables plus services, and durables and nondurables plus services) for the period 1950 to 2005 from the NIPA Table 2.3.5 available from the Bureau of Economic Analysis. We construct the time series of the per capita real consumption figures of the two sets by using population numbers from the NIPA Table 2.1 and the price deflator series from the NIPA Table 2.3.4.

In addition to calculating one-year ( 0=s ) to five-year ( 4=s ) annual-annual consumption growth rates, we also follow Jagannathan and Wang (2007) to measure one-year to five-year consumption growth based on the fourth quarter. Table I provides the summary statistics for the consumption data. The means and standard deviations of the consumption growth rates of durables and nondurables plus services are all larger than those of nondurables plus services. The means and standard deviations of the Q4-Q4 consumption growth rates are all larger than those of the annual-annual ones. The means and standard deviations of the consumption growth rates increase with s.

<Insert Table I>

B. Population Data

We obtain the quarterly numbers of aggregate population from NIPA Table 2.1 and the annual numbers of population for every age from the Human Mortality Database14 for the period 1950 to 2005. We construct time series population numbers for four age groups, including the childhood (age 0 to age 20), the young-aged (age 21 to age 44), the middle-aged (age 45 to age 64), and the old-aged (age 65 and over) groups15. We calculate the one-year ( 0=s ) to five-year ( 4=s ) annual-annual and Q4-Q4 aggregate population growth rates and one-year ( 0=s ) to five-year ( 4=s ) annual-annual ones for the four age groups.

13 We use seasonally adjusted data since we cannot obtain seasonally unadjusted data.

14 Human Mortality Database (available at http://www.mortality.org or http://www.humanmortality.de ), University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany).

15 We follow Goyal (2004) to separate the middle-aged and old-aged by age 45 and Ang and Maddaloni (2005) to separate the childhood and the young-aged by age 20.



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Table II provides the summary statistics for the population data. The summary statistics of the annual-annual and Q4-Q4 aggregate population growth are not too much different. The order of the means of the population growth for the four age groups from high to low is old-aged, middle-aged, young-aged, and childhood. The order of the standard deviations of the four age groups from high to low is young-aged, childhood, middle-aged and old-aged. The order of the max minus the min is almost the same as that of the standard deviations. We can see that the populations of the middle-aged and old-aged groups are growing faster and more stably than the other two groups.

<Insert Table II>

C. Asset Return Data

We use the returns on the 25 book-to-market and size-sorted portfolios, the risk-free rates, and the values for the three factors (market, SMB (small minus big), and HML (high minus low)) of Fama and French (1993) for the period 1951 to 2005 that may be accessed from Kenneth French’s web site. In order to match the one-year to five-year empirical tests, we construct the annual excess return series on the 25 portfolios for five sample periods, 1951-2005, 1951-2004, 1951-2003, 1951-2002, and 1951-2001. Table III reports the average annual excess returns for the equal-weighted portfolios. The excess returns for the five sample periods are not too much different. The variation in the average excess returns across the 25 portfolios is substantial for every sample period.

<Insert Table III>

V. Empirical Findings

The main focuses of the section are twofold. First, we wish to ask whether the population risk along with consumption risk can explain the cross-sectional variation in the expected returns on portfolios of stocks. The second focus is to investigate whether demographics or the population risks associated with different age groups can affect the stock market.

We provide strong support for the importance of population risk in different ways. First, we compare the performance of the P-CCAPM with that of the traditional CCAPM and also take the Fama and French three-factor model as the benchmark for evaluating the performance of the P-CCAPM. Second, since Ait-Sahalia, Parker, and Yogo (2004) and Yogo (2006) prove the importance of other consumption goods as well as nondurables, we measure another consumption risk by using expenditures on durables and nondurables plus services in addition to that on nondurables plus services.



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Third, we refer to the finding in Jagannathan and Wang (2007) as the Q4 effect and that in Parker and Julliard (2005) as the ultimate risk effect. Both of them can improve the performance of the CCAPM surprisingly well. In our paper, we think that population risk can play an important role in explaining the cross-section of stock returns (we refer to this as the population risk effect). Thus, we will further highlight the performance of the population risk effect when involving these two effects and the difference between these three effects. Finally, we compare the performance of the CCAPM, the P-CCAPM, and the P-CCAPM with age groups to further prove the importance of the aggregate population dynamics and the population dynamics for the different age groups.

A. Comparison of the CCAPM, the P-CCAPM, and Fama-French Three-Factor Model

Table IV reports the estimates of the factor risk prices for the CCAPM, the P-CCAPM, and Fama-French three-factor model. The estimation is performed by using a Fama-MacBeth (1973) two-stage regression. When measuring risks based on annual-annual growth, the 2R for the CCAPM is 0.14 or 0.16 and the risk prices for the consumption factor are positive and significant,16 whereas when considering population risk, the 2R for the P-CCAPM is as high as 0.71, the risk prices for consumption are still positive but become insignificant, and those for population are negative and very significant. Even when measuring risks based on Q4-Q4 growth, the 2R for the P-CCAPM is as high as 0.80 or 0.82, which is still higher than that for the CCAPM, i.e., 0.75 or 0.77, and the risk prices for population are all significantly different from zero. These figures provide strong evidence indicating that the P-CCAPM has more explanatory power than the CCAPM. These findings imply that the stock market actually responds to the dynamic of aggregate population. As a result, population risk can be a substantial factor in explaining the cross-section of stock returns.

Moreover, we find that when considering population risk, the market risk premium for bearing consumption risk decreases from 1.05 (1.47) to 0.50 (0.75) based on the annual-annual growth, and from 2.31 (3.56) to 1.48 (2.36) based on the Q4-Q4 growth. The lower risk price corresponds to a lower coefficient of relative risk aversion, and so the equity premium is much less of a puzzle when considering the population factor. We will further discuss the coefficients of relative risk aversion implied in our model in

16 This is consistent with the poor performance of the traditional CCAPM (Hansen and Singleton (1982, 1983), Mehra and Prescott (1985), Mankiw and Shapiro (1986), Breeden, Gibbons, and Litzenberger (1989), Hansen and Jagannathan (1997), and Parker and Julliard (2005)).



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Section G. However, for the P-CCAPM it seems that when the population factor is introduced as a variable, the intercept term will increase, which means that the misspecification of the model is raised. By using annual-annual growth, the intercept terms, 12.03 or 13.82 for the CCAPM and 14.59 or 15.49 for the P-CCAPM, are all very significantly different from zero, whereas when using Q4-Q4 growth, the intercept terms, -1.15 or 1.52 for the CCAPM and 4.59 or 5.96 for the P-CCAPM, are almost not statistically significant at all. For the Fama-French three-factor model, the 2R is 0.79 and the intercept term is 10.15. For the P-CCAPM using annual-annual growth, the performance is slightly worse than for the Fama-French model, but actually surpasses that of their model when using Q4-Q4 growth. When consumption and population factors are simultaneously added as additional explanatory variables in the Fama-French model, the statistical significance of the slope coefficients for the five factors are all depressed although all of the 2R are very high.

<Insert Table IV>

Figure 1 gives the plots of the realized average excess returns of 25 portfolios against their theoretical values drawn from the CCAPM, the P-CCAPM, and Fama-French three-factor model. By comparing the traditional CCAPM with the P-CCAPM (that is, the top two plots using annual-annual growth in Fig. 1), the P-CCAPM has a much better fit in terms of the expected excess returns than the traditional CCAPM. The points are roughly distributed around the 45-degree line for the P-CCAPM whereas the points are displayed along the vertical line for the CCAPM. When adding the Q4 effect (see the middle plots in Fig. 1), the differences between the realized value and the fitted value of the P-CCAPM appear to be lower than those of the CCAPM. The bottom plot displays the U shape for the Fama-French model.

<Insert Figure 1>

Let ( ) iii RE βλλα ˆˆˆˆ 0 ′−−= denote the pricing error of portfolio i, that is, the difference

between the realized expected excess return on portfolio i and the fitted value according to the asset pricing models. Table V reports the pricing errors for 25 portfolios from the CCAPM, the P-CCAPM, and the Fama-French model. Using annual-annual growth, the

maximum value of |ˆ| iα is 4.76% for the P-CCAPM but as much as 8.83% for the

CCAPM; the average value of |ˆ| iα is 1.41% for the P-CCAPM but 2.57% for the

CCAPM. Thus, the pricing error decreases substantially due to involving the population



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dynamic. Using Q4-Q4 growth, the maximum value of |ˆ| iα is 3.65% for the P-CCAPM

and 3.70% for the CCAPM; the average value of |ˆ| iα is 1.21% for the P-CCAPM and

1.42% for the CCAPM. The pricing error still becomes small due to the population risk effect even after involving the Q4 effect. For the Fama-French model, the maximum

value of |ˆ| iα is 3.83% and the average value of |ˆ| iα is 1.21%. The pricing error under

the P-CCAPM with the Q4 effect can be better than that under the Fama-French model.

<Insert Table V>

B. Three Effects: Population Risk, Q4, and Ultimate Risk

Table VI reports ultimate estimates of the factor risk prices for the CCAPM and the P-CCAPM for annual-annual growth and Table VII for Q4-Q4 growth. In Tables VI and VII, in spite of s, the 2R for the P-CCAPM are all higher than those for the CCAPM and the slope coefficients of population risk are statistically significant for both consumption sets17 respectively shown in Panels A and B. This finding provides the robustness for the P-CCAPM.

<Insert Table VI and VII>

From the two tables, we can see the features of the population risk effect, the Q4 effect, and the ultimate risk effect. The population risk effect can raise the 2R and the intercept term. The ultimate risk effect at the best-fitted horizon can raise the 2R and decrease the intercept term. The ultimate risk effect at the two-year horizon ( 1=s ) generally explains a large fraction of the variation in average returns across the 25 Fama-French portfolios with lower intercept terms and higher 2R both for the CCAPM and the P-CCAPM. The Q4 effect can raise the 2R and decrease the intercept term.

As can be seen from these two tables, combining any two of these three effects can yield good model specification. For, example, in Table VI, the P-CCAPM at the two-year horizon ( 1=s ) displays good model specification with the higher 2R , 0.79 or 0.83; in Panel A of Table VII, the P-CCAPM with a one-year ( 0=s ) growth rate displays the best-fitting model with an 2R of 0.80, and intercept term of 4.59; in Panel B of Table

17 Parker and Julliard (2005) also provide similar evidence for the ultimate risk for the two consumption sets.



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VII, the CCAPM or the P-CCAPM with a two-year ( 1=s ) growth rate displays the best model specification with an 2R of 0.85 or 0.86, and an intercept term of 0.52 or 1.28.

As shown in these two tables, the market prices for the consumption risk are all positive and those for the population risk are all negative. We can conclude that the result supports the argument that the consumption hedging effect exists in the US; an increase in the population beta will decrease the expected excess return of an asset.

C. The Robustness Evidence

In this subsection, we examine the ultimate estimates of the P-CCAPM using 25 Fama-French value-weighted portfolios, as shown in Table VIII for annual-annual growth and Table IX for Q4-Q4 growth. The tables provide the evidence supporting the population factor that the 2R for the P-CCAPM are all higher than those for the CCAPM and some of the slope coefficients of population risk are statistically significant. However, we can find that these three effects in the value-weighted portfolios are weaker compared to the equal-weighted portfolios, shown in Tables VI and VII.

<Insert Table VIII and IX>

Besides, we also examine the robustness of our findings using one-year Q4-Q4 growth and some sets of assets, namely, 18 portfolios sorted on size, 18 portfolios sorted on the book-to-market ratio, 19 portfolios sorted on the earning-to-price ratio, 19 portfolios sorted on the cashflow-to-price ratio, and 17 industry portfolios, all taken from Kenneth French’s website. The results are given in Table X.

<Insert Table X>

From the table, the 2R of the P-CCAPM for these sets of assets increase compared to those of the CCAPM, and even some values of 2R are as high as 0.99, although the intercept terms of the P-CCAPM are much larger than those of the CCAPM. The slope coefficients of population risk are also significant for these sets of assets—18 size portfolios, 18 book-to-market portfolios, and 19 cashflow-to-price portfolios. Although the results are not as strong as those for the 25 equal-weighted portfolios, they still provide some evidence that the population factor is an important factor.



16

D. Population Risks of Age Groups

In this subsection we will further divide the aggregate population into four age groups and empirically demonstrate the pricing kernel for the P-CCAPM with age groups. Table XI reports s-year estimates of the factor risk prices for the P-CCAPM with age groups for annual-annual growth and Table XII for Q4-Q4 growth18. In Tables XI and XII, we find that when dividing the aggregate population into the four age groups, the 2R will improve and the intercept terms will decrease compared to the results for the P-CCAPM without age groups, as shown in Tables VI and VII. This finding shows that the population dynamics of age groups affect the stock market in different ways. Consequently, considering these dynamics further can improve the performance of the P-CCAPM. The result also implies that the population dynamics are closely related to the stock returns.

<Insert Table XI and XII>

In looking at some sets of the estimates displaying the best model specification with the higher 2R and the insignificant intercept, it can be seen that for annual-annual growth in Table XI, the estimates for the two-year horizon ( 1=s ) in the consumption set of nondurables and services fit the best model specification with the higher 2R , 0.87, and the insignificant intercept term, 1.29. The slope coefficients of this best model are 2.96, -5.20, -6.27, 1.72, and 2.34, respectively, for the consumption risk and population risks of the four age groups. For Q4-Q4 growth in Table XII, the estimates for the one-year horizon ( 0=s ) in the set of nondurables and services and the horizon of two-year ( 1=s ) in the set of durables, nondurables and services fit the best model specification, respectively, with 2R values of 0.88 and 0.90, and insignificant intercept terms of -0.08 and 0.17. The slope coefficients of the former set are 1.98, -0.96, -0.54, 1.11 and 0.19, respectively, for the consumption risk and population risks of the four age groups; those of the latter are 4.46, -2.54, -4.42, 0.42, and 1.11, respectively. The market prices for population risks of the age groups in these best models are more significantly negative for the childhood and young-aged groups. Thus, these results imply that the negative market prices for the aggregate population risk result from these two age groups; an increase in the population betas of the childhood and young-aged groups will decrease the expected excess returns.

18 Since we do not have quarterly population data for the age groups, we use the annual-annual population growth of age groups and Q4-Q4 consumption growth in the regressions whose results are shown in Table XII.



17

E. Consumption Betas and Population Betas

To further understand the relationships between the cross-section of stock returns and the risks in the CCAPM, the P-CCAPM, and the P-CCAPM with age groups, Table XIII reports the consumption and population betas, and Table XIV reports the population betas for the age groups. The betas are estimated using the first-stage time series regression of excess returns on the one-year ( 0s = ) factors using the Q4-Q4 consumption growth on the set of nondurables and service. In Table XIII, Panel A shows the well-known size and value premiums that the average annual excess returns for the 25 Fama-French portfolios decrease with size for a given book-to-market equity quintile and increase with book-to-market ratio for a given size quintile.

<Insert Table XIII>

Panel B of the table reports the consumption betas. On average, firms that earn a lower return tend to have smaller consumption betas, that is, smaller or value firms are exposed to higher consumption risk compared to bigger or growth firms. Panel C of the table reports the population betas. We find that firms with a lower return on average tend to have larger population betas. That is, the bigger or growth firms are favorably exposed to higher population risk compared to smaller or value firms since the market price is negative. For example, given the big size quintile, the population beta for the firm with the lowest book-to-market ratio is 13.97 and that for the firm with the highest ratio is 7.04. Since the market price for the risk is -0.24, this implies that the market is willing to pay more to buy the stock of the firm whose population beta is 13.97 in which case the expected excess return of the firm will be lower.

Looking at the population betas for the age groups, as shown in Table XIV, we find that firms with a lower return on average tend to have larger population betas in the cases of the childhood and young-aged groups, and have smaller population betas for the middle-aged and old-aged groups. Since we obtain negative market prices for the population risks of the childhood and young-aged groups and positive ones for the population risks of the middle-aged and old-aged groups, bigger or growth firms are favorably exposed to higher population risks of the childhood and young-aged groups and to lower population risks of the middle-aged and old-aged groups compared to smaller or value firms.

<Insert Table XIV>



18

F. Further Comparison of the CCAPM, the P-CCAPM and the P-CCAPM with Age Groups

We further compare the pricing errors generated by the CCAPM, the P-CCAPM, and the P-CCAPM with age groups. Table XV reports the pricing errors of 25 portfolios drawn from the ( )1s + ( 2,1,0=s )-year (annual-annual) estimation of these three models.

For 0=s , the maximum values of |ˆ| iα are 8.83%, 4.76%, and 3.89% and the average

values of |ˆ| iα are 2.57%, 1.41%, and 1.50%, respectively, for these three models. For

1=s , the maximum values of |ˆ| iα are 3.47%, 4.55% and 3.56% and the average values

of |ˆ| iα are 1.40%, 1.21%, and 1.00%, respectively, for these three models. For 2=s ,

the maximum values of |ˆ| iα are 4.34%, 4.08%, and 2.95% and the average values of

|ˆ| iα are 1.37%, 1.18%, and 1.06%, respectively, for these three models. The pricing

errors generally decrease after considering the dynamic of the aggregate population or after further considering the population dynamics of the age groups.

Figure 2 gives plots of the realized average excess returns of 25 portfolios against their theoretical values drawn from these models. Given any s, most of the points from the P-CCAPM approach the 45-degree line more closely than those from the CCAPM, and those from the P-CCAPM with age groups more closely than those from the P-CCAPM without age groups, which is consistent with the findings in Table XV.

<Insert Figure 2 and Table XV>

G. Implied Coefficient of Relative Risk Aversion

In order to obtain the implied coefficients of relative risk aversion, we respectively denote the slope coefficient in the CCAPM as 1λ and those in the P-CCAPM as 2λ , and

3λ in the second-stage cross sectional regressions given by

;,1,

,,1

,01,

ccapmsti

ccapmsci

sccapmstiR ++ ++= εβλλ

, , , ,, 1 0 2 , 3 , , 1 .s our s s our s s our s our

i t i c i p i tR λ λ β λ β ε+ += + + +



19

We denote 1sγ as the implied coefficient of relative risk aversion in the CCAPM and

2sγ and 3sγ as those coefficients in the P-CCAPM in the equations of market prices

given by ( )( )[ ]11

var

1,1

1,11 −−≈

+

+s

ts

stss

gEg

θ

θ

γγ

λ in the CCAPM and ( )( ) ( )[ ]111

var

1,1,2

1,22 −+−−≈

++

+s

ts

tNs

stss

ggEg

θ

θ

γγ

λ and

( )( ) ( )[ ]111

var

1,1,3

1,33 −+−−≈

++

+s

ts

tNs

stNss

ggEg

θγγ

λ in the P-CCAPM. An estimated slope coefficient

corresponds to an implied coefficient of relative risk aversion. Due to the approximation of the slope coefficients and whether the model is best specified, there are two estimates of implied coefficients of relative risk aversion in the P-CCAPM. Table XVI shows the implied coefficients of relative risk aversion for the CCAPM19 and the P-CCAPM. We find that the coefficients of relative risk aversion ( 2γ ) derived from the slope coefficients of consumption risk in the P-CCAPM are much smaller than those ( 1γ ) in the CCAPM,

although the coefficients of relative risk aversion ( 3γ ) from the market prices for

aggregate population risk are very large. However, the differences between 2γ and 3γ

generally become small with s.

<Insert Table XVI>

In a similar manner, we obtain the implied coefficients of relative risk aversion from the slope coefficients in the P-CCAPM with age groups. The slope coefficients can be

denoted as 8,7,6,5,4, =iiλ in the second-stage cross-sectional regressions given by

( ) ( ) ( )( ) ( )

, 2 , 2 , 2 , 2, 1 0 4 , , _ , 1

5, , 6,, , 7, , 8,

,s our s s our s s our s ouri t i c j i p a i t

c yj a m o

R λ λ β λ β ε+ +⎧ ⎫∈⎨ ⎬⎩ ⎭

= + + +∑

19 The implied coefficient of relative risk aversion by using the Q4-Q4 one-year consumption growth of nondurables and services equals about 32 which is quite close to the estimate of 31 in Jagannathan and Wang (2007).



20

where ( )

( )( ) ( )⎥

⎦

⎤⎢⎣

⎡−+−−

≈

++∈

+

∑ 111

var

1,1,,,,

4

1,44

st

stN

omyca t

at

s

stss

ggNNE

g

a θ

θ

γ

γλ

and

( )( )

( )

, 1

, 1, 1, , ,

var

1 1 1

a

a

ast

sj N tts

j as st

sj tN ta c y m o t

N gN

NE g gN θ

γ

λγ

+

++∈

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠≈

⎡ ⎤− − + −⎢ ⎥

⎢ ⎥⎣ ⎦∑

for ( ) ( ) ( ) ( ){ }( , ) 5, , 6, , 7, , 8,j a c y m o∈ . Table

XVII shows the implied coefficients of relative risk aversion for the P-CCAPM with age groups. Based upon the best model specification with the highest 2R and an insignificant intercept term, the coefficients of relative risk aversion corresponding to the slope coefficients for two-year Q4-Q4 consumption growth on the set of durables, nondurables and services are, respectively, 9, 13, 12, 9, and 1120.

We can observe some phenomena from Tables XVI and XVII. By comparing the coefficients of relative risk aversion respectively derived from the CCAPM and the P-CCAPM in Table XVI, adding the population factor can reduce the market price for consumption risk and then decrease the implied coefficient of relative risk aversion from that price. When further considering the population dynamics of age groups, the differences between these coefficients of relative risk aversion derived from these five market prices become small, as shown in Table XVII. Since the estimated risk aversion declines with s, the ultimate risk effect can reduce the magnitude of the equity premium puzzle, but the intercept terms always remain statistically significant. However, the Q4 effect can reduce the intercept terms. Therefore, by combining these three effects, the coefficients of relative risk aversion lie between 9 and 13 for the best-fitting sets of the CCAPM, the P-CCAPM, and the P-CCAPM with age groups.

<Insert Table XVII>

VI. Conclusion

In the paper, we note that previous studies examining the performance of consumption-based models only consider the dynamics of consumption but ignore that of the population. Hence, we propose a model that extends the CCAPM by incorporating the

20 Eisenhauer and Ventura (2003) estimate the relative risk aversion for a broad cross-section of Italian households. Their estimates display a greater variability across socio-demographic groups, ranging from 4.5 to 13.84. Halek and Eisenhauser (2001) use life insurance data to estimate the coefficient of relative risk aversion for each of nearly 2,400 households. They examine the risk aversion across the demographic groups. The mean of relative risk aversion for these 2,400 households is 3.735, and the standard deviation of relative risk aversion is around 24.



21

population factor. The P-CCAPM performs much better than the CCAPM in different ways. We further divide the impact of the aggregate population into those of four generations on the expected excess returns. The empirical results show that considering the population dynamics of age groups can improve the performance of the P-CCAPM since the four age groups affect the stock market in different ways.

While the population risk is able to explain the cross-section of stock returns very well, some questions in our empirical results still require further discussion. First, although our empirical results support the future consumption hedging effect to obtain the negative risk price, each country remains at some stage of population growth, or the different sample periods of a country have different speeds and stabilities of population growth. Hence, investigating this argument by using international data or choosing the different sample periods of a country can give us a more profound understanding of the role that population risk plays in the market.

Second, we lack an economic explanation of the relationship between the value or size premium and population risk. Garleanu amd Kogan (2009) point out that innovation creates the displacement risk, a systematic risk factor due to the lack of inter-generational risk sharing. The more innovative growth firms offer a hedge against the displacement risk and earn lower average returns than less innovative growth firms. The new source of systematic risk can provide an economic explanation of the cross-sectional variation in historical average returns across stocks. This suggests that incorporating inter-generational changes in consumption patterns and demographics could pose a challenge to the consumption-based model.

Finally, we still have difficulty in concretely interpreting the risks that the population dynamic or the population dynamics of age groups are representing. Bansal and Yaron (2004) state that news concerning the growth rate and economic uncertainty alter perceptions regarding the long-run expected growth rates and economic uncertainty. Hansen, Heaton, and Li (2005) point out that characterizing the components of pricing over long horizons can help us understand the implications of macroeconomic growth uncertainty for evaluation. Since the population seems to be a long-run macroeconomic factor, the perspective of long-run risks could be helpful in investigating the relationship between consumption and population volatilities and asset prices.



22

Appendix A: The Asset Pricing Model with Aggregate Consumption

We assume that there are m investors in the economy. We denote tC as the

aggregate consumption at time t and jtC as the consumption of investor j at time t .

We assume that there is a representative agent whose utility function is defined by

( ) ( )1

maxj

t

mj

t j j tC j

u C u Cλ=

= ∑ subject to 1

mj

t tj

C C=

=∑ at time t for some appropriate set

of weights { }lλ in the same way as in Basak and Cuoco (1998), and Basak (2005). For the lifetime expected utility maximization problem faced by the representative agent at time t , the relation will hold

( )( )

1, 1 0.t

t i tt

u CE R

u C

δ ++

⎡ ⎤⎛ ⎞′⋅⎢ ⎥⎜ ⎟ =

⎜ ⎟′⎢ ⎥⎝ ⎠⎣ ⎦ (A1)

Define the stochastic discount factor as ( )( )t

tt Cu

Cum′′⋅

= ++

11

δ . Substituting this into

equation (A1) gives

, 1 1 0.t i t tE R m+ +⎡ ⎤ =⎢ ⎥⎣ ⎦

(A2)

By using the unconditional expectation and the definition of the covariance, we rewrite equation (A2) as

( )[ ]

1 , 1, 1

1

cov ,.t i t

i tt

m RE R

E m+ +

++

−⎡ ⎤ =⎣ ⎦ (A3)

Given the power utility function with the coefficient of relative risk aversion γ , by a first-order condition we can obtain

1

1

1 1 11t t

t t t t

t t t C C

C C C CC C C

γ γ

δ δ γ+

− − −

+ + +

=

⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞−⎪ ⎪≈ −⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

( )( ), 11 1 ,AC tgδ γ +≈ − − (A4)



23

where 1, 1

A tC t

t

CgC

++ = denotes the aggregate consumption growth. Putting equation (A4)

into equation (A3), we obtain

, 1 ,A Ai t C CE R λ β+⎡ ⎤ =⎣ ⎦ (A5)

where ( )( )

, 1 , 1

, 1

cov ,

var

AC t i tA

C AC t

g R

gβ + +

+

= and ( ), 1

, 1

var

1 1

AC tA

C AC t

g

E g

γλ

γ+

+

=⎡ ⎤− −⎣ ⎦

.

The first-order conditions for the lifetime expected utility maximization problem faced by

investor j at time t give ( )( )

1, 1 0.

jj t

t i t jj t

u CE R

u C

δ ++

⎡ ⎤⎛ ⎞′⋅⎢ ⎥⎜ ⎟ =

⎜ ⎟′⎢ ⎥⎝ ⎠⎣ ⎦ Given the power utility

function with the coefficient of relative risk aversion jγ , we can obtain

, 1 ,j ji t C CE R λ β+⎡ ⎤ =⎣ ⎦ (A6)

where ( )( )

, 1 , 1

, 1

cov ,

var

jC t i tj

C jC t

g R

gβ + +

+

= and ( ), 1

, 1

var

1 1

jj C tj

C jj C t

g

E g

γλ

γ+

+

=⎡ ⎤− −⎣ ⎦

with 1, 1

jj t

C t jt

CgC

++ = . Hence,

the CCAPM will hold both for the aggregate consumption and for an individual’s consumption, as shown in equation (A5) and equation (A6).

Appendix B: Linear Consumption and Population Factor Model

In the economy, we assume a representative investor has a time and state separable utility function for lifetime consumption:

( ) ,ut u

u tE u Cδ

∞

=

⎡ ⎤⎢ ⎥⎣ ⎦∑ (B1)

where uC is the aggregate consumption at time u and δ denotes the time discount factor. The representative agent allocates its resources among consumption and different investment opportunities by maximizing the expected lifetime utility, as in equation .



24

When wealth is allocated optimally across assets, the following relationship must be satisfied by all assets:

( )( )

1, 1 0,t

t i tt

u CE R

u Cδ +

+

⎡ ⎤⎛ ⎞′⋅=⎢ ⎥⎜ ⎟⎜ ⎟′⎢ ⎥⎝ ⎠⎣ ⎦

(B2)

where 1, +tiR denotes the excess return on an asset i from time t to 1+t .

Following Parker and Julliard (2005), we use the consumption Euler equation for the

risk-free rate, fsttR +++ 1,1 , between time 1+t and st ++1 ,

( ) ( )1 1 1, 1 1f s

t t t t s t su C E R u Cδ+ + + + + + +⎡ ⎤′ ′= ⎢ ⎥⎣ ⎦

(B3)

to substitute ( )1+′ tCu in equation to yield

( )( )

11

, 1 1, 1 0.s

t sft i t t t s

t

u CE R R

u Cδ +

+ ++ + + +

⎡ ⎤′⋅ =⎢ ⎥′⎣ ⎦

(B4)

Defining stf

sttst mRm ++++++ ⋅= 11,11 and reorganizing the unconditional version of equation ,

the expected excess returns are given by

( )1 , 1

, 11

cov ,.

st i t

i t st

m RE R

E m+ +

++

−⎡ ⎤ =⎣ ⎦ ⎡ ⎤⎣ ⎦

(B5)

B.I. Aggregate Population Factor

The aggregate consumption can be represented as ttt NC θ⋅= , where the aggregate

population at time t is tN , and the consumption per capita at time t is tθ . Given the



25

power utility function with coefficient of relative risk aversion sγ , the pricing kernel

1stm + can be given by

1 1 11 1, 1 .

s

s f s t s t st t t s

t t

Nm RN

γθδθ

−

+ + + + ++ + + +

⎛ ⎞⋅= ⎜ ⎟⋅⎝ ⎠

(B6)

We can approximate stm 1+ in equation (B6) around tst NN =++1 and tst θθ =++1 using a

Taylor series and take the first-order to obtain

1 1 11 1, 1

1 1 11, 1 1

s

s f s t s t st t t s

t t

f s t s t t s tt t s s s

t t

Nm RN

N NRN

γθδθ

θ θδ γ γθ

−

+ + + + ++ + + +

+ + + + ++ + +

⎛ ⎞⋅= ⎜ ⎟⋅⎝ ⎠

⎧ ⎫⎛ ⎞ ⎛ ⎞− −⎪ ⎪≈ − −⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭

( ) ( ){ }11, 1 , 1 , 11 1 1 ,f s s s

t t s s N t s tR g gθδ γ γ++ + + + += − − − − (B7)

where t

ststN N

Ng +++ = 1

1, and t

ststg

θθ

θ++

+ = 11, .

Putting equation (B7) into equation (B5), we can get

( ) ( )( ) ( )

( )( )

( )( ) ( )

( )( )

, 1 , 1 , 1, 1

, 1, 1 , 1

, 1 , 1 , 1

, 1, 1 , 1

var cov ,

var1 1 1

var cov ,.

var1 1 1

s ss N t N t i t

i t ss sN ts N t t

s ss t t i t

ss sts N t t

g g RE R

gE g g

g g R

gE g g

θ

θ θ

θθ

γ

γ

γ

γ

+ + ++

++ +

+ + +

++ +

=⎡ ⎤− − + −⎣ ⎦

+⎡ ⎤− − + −⎣ ⎦



26

It is the P-CCAPM shown in equation (1), where ( )( )s

tN

tis

tNsN g

Rg

1,

1,1,

var,cov

+

++=β ;

( )( )s

t

tis

ts

gRg

1,

1,1,

var,cov

+

++=θ

θθβ ;

( )( ) ( )[ ]111

var

1,1,

1,

−+−−≈

++

+s

ts

tNs

stNss

N ggEg

θγγ

λ ; and

( )( ) ( )[ ]111

var

1,1,

1,

−+−−≈

++

+s

ts

tNs

stss

ggEg

θ

θθ γ

γλ .

B.II. Population Factors of Age Groups

Since ot

mt

yt

ctt NNNNN +++= , we obtain

( )

( )1 1 1 1 11

1 1, 1 .sc y m o

t s t s t s t s t ss f st t t s c y m o

t t t t t

N N N Nm R

N N N N

γθ

δθ

−

+ + + + + + + + + +++ + + +

⎛ ⎞+ + + ⋅⎜ ⎟=⎜ ⎟+ + + ⋅⎝ ⎠

(B8)

In the same manner, we approximate stm 1+ in equation (B8) around

( )omycaNN at

ast ,,,,1 ∈=++ and tst θθ =++1 using a Taylor series and take the first-order

to obtain

( )( )

( )

1 1 1 1 111 1, 1

1 1 11, 1

, , ,

1

sc y m ot s t s t s t s t ss f s

t t t s c y m ot t t t t

a af s t s t t s t

t t s s sa c y m o t t

N N N Nm R

N N N N

N NRN

γθ

δθ

θ θδ γ γθ

−

+ + + + + + + + + +++ + + +

+ + + + ++ + +

∈

⎛ ⎞+ + + ⋅⎜ ⎟=⎜ ⎟+ + + ⋅⎝ ⎠

⎧ ⎫⎛ ⎞ ⎛ ⎞− −⎪ ⎪≈ − −⎨ ⎬⎜ ⎟ ⎜ ⎟⎪⎪ ⎝ ⎠ ⎝ ⎠⎭⎩

∑

( )

( ) ( )11, 1 , 1, 1

, , ,1 1 1 ,a

af s s st

t t s s s tN ta c y m o t

NR g gN θδ γ γ+

+ + + ++∈

⎧ ⎫⎪ ⎪= − − − −⎨ ⎬⎪ ⎪⎩ ⎭

∑ (B9)

where at

asts

tN NNg a

+++= 1

1, , ( )omyca ,,,∈ and t

ststg

θθ

θ++

+ = 11, .



27

Putting equation (B9) into equation (B5), we can obtain

( )( )

( ) ( )( )

( )

( )( )

( )

( )( ) ( )

( )

, 1 , 1, 1, 1

, , ,, 1

, 1, 1, , ,

, 1 , 1 , 1

, 1, 1, 1

, , ,

var cov ,

var1 1 1

var cov ,

var1 1 1

aa

aa

a

ast ss N t i tN tt

i t saa c y m o s st N t

s tN ta c y m o t

s ss t t i t

ats st

s tN ta c y m o t

N g g RNE RgNE g g

N

g g R

gNE g gN

θ

θ θ

θθ

γ

γ

γ

γ

+ +++

∈+

++∈

+ + +

+++

∈

=⎡ ⎤

− − + −⎢ ⎥⎢ ⎥⎣ ⎦

+⎡ ⎤

− − + −⎢ ⎥⎢ ⎥⎣ ⎦

∑∑

∑ ( ).

s

It is the P-CCAPM with age groups shown in equation (2), where

( )( )s

tN

tis

tNsN

a

a

a gRg

1,

1,1,

var,cov

+

++=β , ( )omyca ,,,∈ ;

( )

( )( )

( )⎥⎦

⎤⎢⎣

⎡−+−−

≈

+∈

+

+

∑ 111

var

1,,,,

1,

1,

st

omyca

stN

t

at

s

stN

t

at

ssN

ggNNE

gNN

a

a

a

θγ

γλ , ( )omyca ,,,∈ ; and

( )

( )( ) ( )⎥

⎦

⎤⎢⎣

⎡−+−−

≈

++∈

+

∑ 111

var~

1,1,,,,

1,

st

stN

omyca t

at

s

stss

ggNNE

g

a θ

θθ

γ

γλ .

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29

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31

Table I Summary Statistics for Consumption Growth

This table reports the summary statistics for consumption growth. Consumption is measured by real per capita consumption expenditure on two sets of consumption goods, nondurables and services, and durables, nondurables and services. The consumption growth is given by 4,3,2,1,0%,100)1( 1,1 =×−=Δ ++ sgc s

tst θ

. Panel A reports one-year to five-year annual-annual consumption growth which is calculated using annual consumption data. Panel B reports one-year to five-year Q4-Q4 consumption growth which is calculated using Q4 consumption data.

Panel A: (s+1)-year Consumption Growth (Annual-Annual)(%) Nondurables and Services Durables, Nondurables and Services s 0 1 2 3 4 0 1 2 3 4 Mean 2.21 4.48 6.80 9.19 11.68 3.05 6.26 9.58 12.99 16.52 SD 1.19 1.98 2.56 3.05 3.45 1.85 3.06 3.99 4.74 5.42 Min -1.31 0.39 1.73 3.20 6.58 -0.72 -0.31 0.61 3.98 6.19 Max 4.24 8.26 12.57 15.57 19.46 6.13 12.27 17.90 22.35 26.63

Panel B: (s+1)-year Consumption Growth (Q4-Q4)(%) Nondurables and Services Durables, Nondurables and Services s 0 1 2 3 4 0 1 2 3 4 Mean 2.23 4.51 6.81 9.22 11.70 3.06 6.31 9.59 13.02 16.54 SD 1.41 2.24 2.69 3.14 3.58 2.14 3.30 4.12 4.82 5.53 Min -1.42 -1.01 1.12 3.80 6.11 -2.46 -0.62 0.96 4.07 5.12 Max 5.26 9.97 12.24 15.16 19.82 8.22 14.04 17.58 21.50 27.13



32

Table II Summary Statistics for Population Growth

This table reports the summary statistics for population growth. The population growth rates include the aggregate population growth given by 4,3,2,1,0%,100)1( 1,1 =×−=Δ ++ sgp s

tNst

and the population growth of age groups given by ( ) 4,3,2,1,0,,,,%,100)1( 1,1, =∈×−=Δ

++somycagp s

tNs

tN aa. Panel A reports the one-year to

five-year annual-annual aggregate population growth which is calculated using annual aggregate population data and the one-year to five-year Q4-Q4 growth which is calculated using Q4 aggregate population data. Panel B reports the one-year to five-year annual-annual population growth of the age groups which is calculated using the annual population data of the age groups.

Panel A: (s+1)-year Aggregate Population Growth (%) Annual-Annual Q4-Q4

s 0 1 2 3 4 0 1 2 3 4 Mean 1.22 2.46 3.71 4.97 6.24 1.22 2.45 3.69 4.94 6.20 SD 0.31 0.62 0.92 1.21 1.50 0.31 0.61 0.91 1.20 1.49 Min 0.88 1.78 2.71 3.63 4.58 0.88 1.78 2.71 3.63 4.57 Max 2.05 3.77 5.51 7.40 9.33 2.05 3.75 5.48 7.33 9.27

Panel B: (s+1)-year Population Growth of Age Groups (Annual-Annual) (%) Childhood Young-Aged s 0 1 2 3 4 0 1 2 3 4 Mean 0.91 1.83 2.77 3.71 4.63 1.14 2.34 3.60 4.93 6.34 SD 1.08 2.18 3.31 4.44 5.58 1.25 2.54 3.85 5.19 6.54 Min -0.55 -1.08 -1.58 -2.06 -2.50 -0.34 -0.64 -0.92 -1.17 -1.41 Max 3.32 6.27 9.26 12.26 15.28 3.38 6.84 10.39 13.97 17.62 Middle-Aged Old-Aged Mean 1.58 3.18 4.78 6.38 7.98 2.01 4.05 6.13 8.25 10.42 SD 1.07 2.17 3.29 4.43 5.58 0.75 1.50 2.25 2.99 3.69 Min -0.08 -0.02 0.06 0.19 0.29 0.58 1.20 1.87 2.56 3.28 Max 3.60 7.25 10.88 14.58 18.31 3.41 6.77 10.22 13.84 17.48



33

Table III Average Annual Excess Returns

This table reports the average annual excess returns on the 25 Fama-French portfolios for five sample periods. Equal-weighted annual excess returns are calculated from January to December.

Low Book-to-Market High Panel A: 1951-2005 (%)

Small 7.97 14.69 14.74 17.07 20.61

5.67 10.03 13.35 13.76 15.06

Size 6.72 10.74 10.87 13.48 14.29

7.79 8.74 11.69 12.21 13.03

Big 6.88 8.20 9.35 9.52 10.83

Panel B: 1951-2004 (%) Small 8.27 14.88 14.94 17.31 20.86

5.76 10.14 13.44 13.78 15.26

Size 6.83 10.82 10.89 13.67 14.39

7.79 8.72 11.68 12.14 13.09

Big 6.95 8.24 9.30 9.46 10.70

Panel C: 1951-2003 (%) Small 8.23 14.78 14.79 17.26 20.63

5.55 9.97 13.20 13.67 14.83

Size 6.70 10.60 10.72 13.58 14.29

7.73 8.58 11.45 12.02 12.98

Big 6.89 8.06 9.22 9.24 10.40

Panel D: 1951-2002 (%) Small 6.35 13.43 13.60 16.14 18.84

4.52 9.29 12.48 13.01 13.95

Size 5.90 9.99 10.29 12.89 13.41

7.03 7.99 10.83 11.23 12.43

Big 6.39 7.54 8.82 8.90 10.09

Panel E: 1951-2001 (%) Small 7.35 14.20 14.07 16.38 19.19

5.40 10.07 13.02 13.55 14.74

Size 6.73 10.62 10.90 13.45 14.04

7.82 8.44 11.29 11.94 13.19

Big 7.06 8.06 9.49 9.43 10.99



34

Table IV One-year Estimation of Linear Factor Models: CCAPM, P-CCAPM and

Fama-French Three-Factor Model This table includes three linear factor models: the CCAPM, the P-CCAPM, and Fama-French three-factor model. The estimation method is the Fama-MacBeth (1973) two-stage regression. The table reports the results of the second-stage (cross-sectional) regression:

[ ] βλλ ′+= 0,tiRE The betas are estimated by the first-stage (time-series) regression of excess returns, respectively, on the factors of three linear factor models. The coefficient estimates ( λ̂ ) are reported and the t-statistics appear in the parentheses. The last column in every panel gives the 2R and the adjusted 2R is in the parentheses.

Panel A: CCAPM and P-CCAPM (Nondurables and Services)

constant scΔ spΔ Rm-Rf SMB HML 2R

(adj 2R )

Annual-

Annual

12.03 (16.41)***

1.05 (1.91)^ 0.14

( 0.10) 14.59

(24.80)*** 0.50

(1.50) -0.51

(-6.53)*** 0.71 (0.68)

9.35 (2.21)*

0.64 (0.89)

-0.11 (-0.74)

-1.73 (-0.38)

3.37 (1.52)

5.64 (1.92)^

0.81 (0.76)

Q4-Q4

-1.15 (-0.73)

2.31 (8.25)*** 0.75

(0.74) 4.59

(1.70) 1.48

(3.50)** -0.24

(-2.49)* 0.80 (0.79)

8.51 (2.29)*

1.36 (1.99)^

-0.09 (-0.65)

-6.64 (-1.54)

3.15 (1.63)

1.30 (0.36)

0.84 (0.80)

Panel B: CCAPM and P-CCAPM (Durables, Nondurables and Services)

Annual-

Annual

13.82 (10.57)***

1.47 (2.07)* 0.16

(0.12) 15.49

(18.88)*** 0.75

(1.70) -0.51

(-6.54)*** 0.71 (0.69)

8.50 (2.34)*

2.14 (2.16)*

0.002 (0.01)

0.62 (0.15)

4.96 (2.32)*

3.68 (1.37)

0.84 (0.80)

Q4-Q4

1.52 (1.27)

3.56 ( 8.69)*** 0.77

(0.76) 5.96

(2.81)* 2.36

(3.84)*** -0.22

(-2.44)* 0.82 (0.80)

8.85 (2.71)*

2.48 (2.92)**

0.01 (0.08)

-7.76 (-1.98)^

4.24 (2.30)*

1.12 (0.39)

0.86 (0.83)

Panel C: Fama-French Model

- 10.15 (2.76)* -4.81

(-1.20) 3.92

(2.51)* 8.80

(4.64)*** 0.79

(0.76) Significance levels: ｀***＇ 0.001 ｀**＇ 0.01 ｀*＇ 0.05 ｀^＇ 0.1



35

Table V Cross-Sectional Regression Pricing Errors: CCAPM, P-CCAPM and Fama-French

Three-Factor Model This table compares the pricing errors of 25 portfolios generated by the CCAPM, P-CCAPM, and Fama-French three-factor model. The pricing errors are calculated by iiR βλλα ˆˆˆˆ 0 ′−−= . All numbers are annual percentages. The consumption set is nondurables plus services and the consumption and population growth rates are one-year growth rates.

Panel A: Pricing Errors CCAPM (annual-annual) P-CCAPM (annual-annual)

0.62 -5.80 -4.04 -5.48 -8.83 4.10 -4.76 0.55 -1.41 -3.213.21 0.50 -1.37 -1.44 -2.25 2.84 -0.05 -0.35 0.50 -2.113.79 1.01 1.75 -0.97 -1.1 0.06 -0.72 0.32 -1.37 2.33 2.46 2.82 -0.80 0.56 0.07 -0.55 -0.07 -2.35 1.48 1.23 4.38 3.54 2.33 2.94 2.09 0.28 1.98 -0.65 1.72 0.20

CCAPM (Q4-Q4) P-CCAPM (Q4-Q4) 2.10 -0.71 -2.62 -3.09 -3.7 3.65 -1.64 -0.73 -1.89 -2.752.08 -0.29 -1.30 -0.26 0.52 2.53 -0.18 -0.79 0.08 -0.451.64 0.79 1.22 -0.18 1.46 0.70 0.01 0.66 -0.76 2.06 -1.17 0.62 -2.29 0.22 3.43 -1.27 -0.01 -2.45 0.59 2.55 -0.26 -1.06 -0.79 0.96 2.68 -0.65 -0.34 -1.27 0.90 1.44

FF Three-Factor Model 3.83 -1.80 0.13 -1.96 -3.82 1.72 1.38 -1.07 -0.11 -0.22 0.09 0.69 1.61 -0.13 1.45 -1.28 1.36 -0.18 0.34 -0.86 -3.19 -0.50 0.21 1.27 1.03

Panel B: Summary of Pricing Errors

CCAPM (annual-annual)

P-CCAPM (annual-annual)

CCAPM (Q4-Q4)

P-CCAPM (Q4-Q4)

FF Three-Factor

Model Max |ˆ|α 8.83 4.76 3.70 3.65 3.83

Average |ˆ|α 2.57 1.41 1.42 1.21 1.21



36

Table VI Ultimate Estimation of Linear Factor Models (Annual-Annual):

CCAPM and P-CCAPM This table reports the (s+1)-year estimation (annual-annual) of the CCAPM and the P-CCAPM. The estimation method is the Fama-MacBeth (1973) two-stage regression. The table reports the results of the second-stage (cross-sectional) regression:

[ ] βλλ ′+= 0,tiRE The betas are estimated by the first-stage (time-series) regression of excess returns respectively, on the factors of the CCAPM and the P-CCAPM. The coefficient estimates ( λ̂ ) are reported and the t-statistics appear in the parentheses.

Panel A: Nondurables and Services

CCAPM P-CCAPM

S constant scΔ 2R

(adj 2R )constant scΔ spΔ

2R (adj 2R )

0 12.03 (16.41)***

1.05 (1.91)^

0.14 ( 0.10)

14.59 (24.80)***

0.50 (1.50)

-0.51 (-6.53)***

0.71 (0.68)

1 1.31 (1.04)

3.55 ( 8.52) ***

0.76 (0.75)

5.51 (2.19)*

2.42 (3.37)**

-0.38 (-1.89)^

0.79 (0.77)

2 6.94 (10.36)***

4.03 (8.07)***

0.74 (0.73)

10.00 (8.38)***

2.67 (4.22)***

-0.66 (-2.94)**

0.81 (0.80)

3 8.10 (14.00)***

4.44 (6.39)***

0.64 (0.62)

12.00 (8.84)***

2.62 (3.12)**

-1.13 (-3.08)**

0.75 (0.73)

4 10.19 (23.89)***

5.74 (6.74)***

0.66 (0.65)

12.46 (12.83)***

3.82 (3.56)**

-1.16 (-2.54)*

0.74 (0.72)

Panel B: Durables, Nondurables and Services

CCAPM P-CCAPM

S constant scΔ 2R


2R (adj 2R )

0 13.82 (10.57)***

1.47 (2.07)*

0.16 (0.12)

15.49 (18.88)***

0.75 (1.70)

-0.51 (-6.54)***

0.71 (0.69)

1 4.51 (5.19)***

5.23 (8.90)***

0.78 (0.77)

7.80 (5.25)***

3.55 (4.26)***

-0.42 (-2.60)*

0.83 (0.81)

2 9.25 (16.64)***

5.94 (6.41)***

0.64 (0.63)

12.13 (14.07)***

3.48 (3.61)**

-0.84 (-3.88)***

0.79 (0.77)

3 11.40 (19.61)***

5.00 (3.92)***

0.40 (0.37)

14.94 (17.17)***

2.30 (2.10)*

-1.57 (-4.65)***

0.70 (0.67)

4 12.80 (18.53)***

6.33 (3.63)**

0.36 (0.34)

15.08 (20.36)***

3.10 (2.06)^

-1.86 (-4.32)***

0.66 (0.63)

Significance levels: ｀***＇ 0.001 ｀**＇ 0.01 ｀*＇ 0.05 ｀^＇ 0.1



37

Table VII Ultimate Estimation of Linear Factor Models (Q4-Q4): CCAPM and P-CCAPM

This table reports the (s+1)-year estimation (Q4-Q4) of the CCAPM and the P-CCAPM. The estimation method is the Fama-MacBeth (1973) two-stage regression. The table reports the results of the second-stage (cross-sectional) regression:

[ ] βλλ ′+= 0,tiRE The betas are estimated by the first-stage (time-series) regression of excess returns respectively on the factors of the CCAPM and the P-CCAPM. The coefficient estimates ( λ̂ ) are reported and the t-statistics appear in the parentheses.


CCAPM P-CCAPM

S constant scΔ 2R


2R (adj 2R )

0 -1.15 (-0.73)

2.31 (8.25)***

0.75 (0.74)

4.59 (1.70)

1.48 (3.50)**

-0.24 (-2.49)*

0.80 (0.79)

1 -0.33 (-0.22)

2.97 (7.92)***

0.73 (0.72)

4.45 (1.25)

1.96 (2.54)*

-0.36 (-1.49)

0.76 (0.73)

2 5.61 (7.94)***

4.08 (9.33)***

0.79 (0.78)

8.22 (5.64)***

3.04 (4.59)***

-0.47 (-2.01)^

0.82 (0.81)

3 5.45 (6.58)***

4.84 (7.10) ***

0.69 (0.67)

9.83 (5.47)***

3.09 (3.47)**

-0.98 (-2.67)*

0.76 (0.74)

4 5.47 (8.26)***

6.12 (9.81)***

0.81 (0.80)

6.14 (3.44)**

5.75 (5.10)***

-0.19 (-0.41)

0.81 (0.79)


CCAPM P-CCAPM

S constant scΔ 2R


2R (adj 2R )

0 1.52 (1.27)

3.56 ( 8.69)***

0.77 (0.76)

5.96 (2.81)*

2.36 (3.84)***

-0.22 (-2.44)*

0.82 (0.80)

1 0.52 (0.52)

4.80 (11.64)***

0.85 (0.85)

1.28 (0.53)

4.52 (5.10)***

-0.06 (-0.35)

0.86 (0.84)

2 7.52 (11.74)***

6.32 ( 7.67)***

0.72 (0.71)

10.48 (9.38)***

4.11 (4.06)***

-0.68 (-3.04)**

0.80 (0.78)

3 9.47 (16.40)***

5.71 (4.44)***

0.46 (0.445)

13.79 (12.69)***

2.81 (2.39)*

-1.46 (-4.33)***

0.71 (0.68)

4 7.79 (14.60)***

9.93 (8.42)***

0.76 (0.74)

9.15 (6.15)***

8.33 (4.15)***

-0.50 (-0.98)

0.77 (0.74)

Significance levels: ***＇ 0.001 ｀**＇ 0.01 ｀*＇ 0.05 ｀^＇ 0.1



38

Table VIII The Robustness: Fama-French Value-Weighted Portfolios (Annual-Annual)

The test portfolios are 25 Fama-French value-weighted portfolios. Value-weighted annual returns are calculated from January to December. Consumption growth is calculated using annual consumption data.


CCAPM P-CCAPM

S constant scΔ 2R


2R (adj 2R )

0 11.03 (21.85)***

1.69 (3.47)**

0.34 (0.32)

13.65 (22.58)***

1.16 (3.33)**

-0.37 (-5.26)***

0.71 (0.68)

1 2.95 (2.03)^

2.73 (5.68)***

0.58 (0.57)

7.32 (2.22)*

1.67 (1.93)^

-0.38 (-1.47)

0.62 (0.59)

2 6.97 (11.06)***

3.38 (7.22)***

0.69 (0.68)

8.91 (5.78)***

2.63 (3.70)**

-0.38 (-1.37)

0.72 (0.69)

3 7.81 (16.00)***

4.01 (6.75)***

0.66 (0.65)

10.22 (6.86)***

2.99 (3.63)**

-0.65 (-1.71)

0.70 (0.68)

4 9.66 (27.11)***

5.27 (7.20)***

0.69 (0.68)

10.81 (10.09)***

4.42 (4.23)***

-0.52 (-1.14)

0.71 (0.68)


CCAPM P-CCAPM

S constant scΔ 2R


2R (adj 2R )

0 13.30 (14.10)***

1.91 (3.13)**

0.30 (0.27)

15.47 (21.53)***

1.41 (3.49)**

-0.39 (-5.73)***

0.72 (0.69)

1 4.81 (5.06)***

4.36 (6.90)***

0.67 (0.66)

7.57 (4.23)***

3.16 (3.511)**

-0.32 (-1.79)^

0.72 (0.69)

2 8.72 (18.19)***

5.24 (6.62)***

0.66 (0.64)

10.95 (10.77)***

3.71 (3.88)***

-0.56 (-2.43)*

0.73 (0.70)

3 10.73 (24.69)***

5.23 (4.98)***

0.52 (0.50)

13.56 (14.74)***

3.38 (3.27)**

-1.08 (-3.35)**

0.68 (0.65)

4 12.16 (23.34)***

6.96 (4.60)***

0.48 (0.46)

14.09 (18.55)***

4.71 (3.20)**

-1.27 (-3.14)**

0.64 (0.61)




39

Table IX The Robustness: Fama-French Value-Weighted Portfolios (Q4-Q4)

The test portfolios are 25 Fama-French value-weighted portfolios. Value-weighted annual returns are calculated from January to December. Consumption growth is calculated using Q4 consumption data.


CCAPM P-CCAPM

S constant scΔ 2R


2R (adj 2R )

0 0.26 (0.16)

1.94 (6.62)***

0.66 (0.64)

4.29 (1.28)

1.39 (2.79)*

-0.16 (-1.38)

0.68 (0.65)

1 2.47 (1.35)

2.17 (4.73)***

0.49 (0.47)

9.97 (2.17)*

0.66 (0.69)

-0.55 (-1.77)^

0.56 (0.52)

2 6.31 (8.65)***

3.27 (7.00)***

0.68 (0.67)

8.01 (4.28)***

2.66 (3.42)**

-0.30 (-0.99)

0.69 (0.67)

3 5.88 (7.70)***

4.11 (6.33)***

0.64 (0.62)

8.82 (4.21)***

3.00 (3.10)**

-0.64 (-1.50)

0.67 (0.64)

4 6.26 (8.31)***

4.92 (6.67)***

0.66 (0.64)

6.24 (2.71)*

4.93 (3.59)**

0.01 (0.01)

0.66 (0.63)


CCAPM P-CCAPM

S constant scΔ 2R


2R (adj 2R )

0 2.30 (1.81)^

3.07 (6.97)***

0.68 (0.66)

5.64 (2.29)*

2.24 (3.27)**

-0.16 (-1.57)

0.71 (0.68)

1 2.38 (1.88)^

3.80 (7.03)***

0.68 (0.67)

2.91 (0.87)

3.62 (3.08)**

-0.04 (-0.17)

0.68 (0.65)

2 7.68 (13.19)***

5.22 (6.76)***

0.67 (0.65)

9.81 (7.50)***

3.86 (3.66)**

-0.46 (-1.80)^

0.71 (0.68)

3 8.96 (19.43)***

5.70 (5.26)***

0.55 (0.53)

12.20 (10.50)***

3.70 (3.22)**

-1.00 (-2.97)**

0.68 (0.65)

4 7.85 (13.93)***

8.50 (6.51)***

0.65 (0.63)

8.38 (4.65)***

7.92 (3.46)**

-0.18 (-0.31)

0.65 (0.62)




40

Table X Other Portfolios

Test portfolios include 18 size-sorted, 18 book-to-market-sorted, 19 earning-to-price-sorted, 19 cashflow-to-price-sorted and 17 industries. Equal-weighted annual returns are calculated from January to December. Consumption betas are estimated using one-year Q4-Q4 consumption growth, and population betas are estimated using one-year Q4-Q4 population growth.

Panel A: Nondurables and Services CCAPM P-CCAPM

Constant 1=Δ sc 2R (adj 2R ) Constant 1=Δ sc 1=Δ sp 2R (adj 2R )

18 Size Portfolios -2.51

(-1.57) 2.49

(8.18)*** 0.81

(0.79) 10.69

(5.80)*** 0.66

(2.43)* -0.48

(-7.80)*** 0.96

(0.96)

18 B/M Portfolios -3.39

(-4.01)** 2.99

(20.01)*** 0.96

(0.96) 7.33

(3.26)** 1.27

(3.50)** -0.36

(-4.91)*** 0.99

(0.98)

19 E/P Portfolios -4.90 (-0.73)

-2.62 (-2.71)*

0.30 (0.26)

-10.10 (-1.26)

-4.00 (-2.61)*

-0.53 (-1.16)

0.36 (0.27)

19 CE/P Portfolios 4.54 (1.01)

-1.29 (-2.04)^

0.20 (0.15)

-18.89 (-2.28)*

-5.84 (-3.80)**

-1.01 (-3.14)**

0.50 (0.44)

17 Industry Portfolios 14.70 (11.39)***

-0.50 (-2.19)*

0.24 (0.19)

14.03 (9.88)***

-0.48 (-2.11)^

0.14 (1.10)

0.30 (0.20)

Panel B: Durables, Nondurables and Services CCAPM P-CCAPM

Constant 1=Δ sc 2R (adj 2R ) Constant 1=Δ sc 1=Δ sp 2R (adj 2R )

18 Size Portfolios 0.45

(0.46) 3.75

(10.41)*** 0.87

(0.86) 11.25

(5.48)*** 1.03

(1.90)^ -0.46

(-5.48)*** 0.96

(0.95)

18 B/M Portfolios 0.09

(0.12) 4.41

(18.64)*** 0.96

(0.95) 9.37

(4.88)*** 1.71

(3.03)** -0.38

(-4.98)*** 0.98

(0.98)

19 E/P Portfolios -5.27 (-0.83)

-3.89 (-2.92)**

0.33 (0.30)

-10.29 (-1.41)

-5.92 (-2.91)*

-0.56 (-1.30)

0.40 (0.32)

19 CE/P Portfolios 6.06 (1.77)^

-1.54 (-2.27)*

0.23 (0.19)

-10.04 (-2.05)^

-6.63 (-4.68)***

-1.03 (-3.85)**

0.60 (0.55)

17 Industry Portfolios 14.06 (12.58)***

-0.75 (-1.97)^

0.21 (0.15)

13.40 (10.43)***

-0.71 (1.86)^

0.14 (1.03)

0.26 (0.16)


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Table XI Ultimate Estimation of Linear Factor Models (Annual-Annual) in the P-CCAPM with

Age Groups This table reports the (s+1)-year estimation (annual-annual) of the P-CCAPM with age groups. The estimation method is the Fama-MacBeth (1973) two-stage regression. The table reports the results of the second-stage (cross-sectional) regression:

[ ] βλλ ′+= 0,tiRE The betas are estimated by the first-stage (time-series) regression of excess returns, respectively, on the factors of the P-CCAPM with age groups. The coefficient estimates ( λ̂ ) are reported and the t-statistics appear in the parentheses. The last column in every panel gives the 2R and the adjusted 2R is in the parentheses.


s constant scΔ sN cpΔ s

N ypΔ sN mpΔ s

N opΔ 2R

(adj 2R )

0 10.77 (5.11)***

1.04 (2.46)*

-5.26 (-3.07)**

-6.23 (-2.34)*

-0.37 (-0.30)

1.36 (1.36)

0.72 (0.65)

1 1.29 (0.52)

2.96 (5.21)***

-5.20 (-1.75)^

-6.27 (-1.37)

1.72 (0.80)

2.34 (1.52)

0.87 (0.84)

2 7.42 (5.49)***

2.93 (4.66)***

-10.47 (-2.09)^

-11.72 (-1.61)

1.86 (0.53)

4.21 (1.49)

0.86 (0.83)

3 9.97 (3.67)**

3.43 (2.50)*

2.95 (0.25)

3.46 (0.22)

-2.55 (-0.41)

-4.77 (-1.03)

0.76 (0.70)

4 7.78 (4.11)***

6.75 (4.62)***

23.52 (2.09)^

27.10 (1.80)^

-0.42 (-0.06)

-11.25 (-2.64)*

0.81 (0.76)


0 12.53 (6.38)***

1.46 (2.53)*

-5.65 (-3.23)**

-7.05 (-2.52)*

-0.88 (-0.68)

1.50 (1.49)

0.72 (0.65)

1 5.14 (2.97)**

3.91 (5.76)***

-7.90 (-3.03)**

-9.98 (-2.40)*

0.45 (0.23)

3.28 (2.29)*

0.89 (0.86)

2 9.85 (8.43)***

3.47 (3.74)**

-15.28 (-2.91)**

-18.41 (-2.40)*

-0.46 (-0.12)

5.68 (1.82)^

0.83 (0.79)

3 14.81 (7.74)***

1.99 (0.96)

-10.72 (-0.81)

-14.20 (-0.81)

-7.25 (-1.09)

-0.89 (-0.16)

0.70 (0.62)

4 12.57 (7.56)***

7.65 (2.44)*

17.56 (1.03)

20.84 (0.95)

-5.54 (-0.63)

-10.81 (-1.70)

0.69 (0.60)




42

Table XII Ultimate Estimation of Linear Factor Models (Q4-Q4) in the P-CCAPM with Age

Groups This table reports the (s+1)-year estimation (Q4-Q4) of the P-CCAPM with age groups. The estimation method is the Fama-MacBeth (1973) two-stage regression. The table reports the results of the second-stage (cross-sectional) regression:

[ ] βλλ ′+= 0,tiRE The betas are estimated by the first-stage (time-series) regression of excess returns respectively on the factors of the P-CCAPM with age groups. The coefficient estimates ( λ̂ ) are reported and the t-statistics appear in the parentheses. The last column in every panel gives the 2R and the adjusted 2R is in the parentheses.


s constant scΔ sN cpΔ s

N ypΔ sN mpΔ s

N opΔ 2R

(adj 2R )

0 -0.08 (-0.03)

1.98 (6.37)***

-0.96 (-0.82)

-0.54 (-0.35)

1.11 (1.38)

0.19 (0.30)

0.88 (0.85)

1 2.54 (0.76)

2.15 (3.38)**

-4.26 (-1.06)

-5.17 (-0.87)

0.52 (0.20)

1.31 (0.66)

0.81 (0.76)

2 6.02 (3.84)**

3.30 (4.73)***

-6.59 (-1.24)

-7.22 (-0.95)

2.03 (0.58)

2.65 (0.93)

0.87 (0.83)

3 6.73 (2.16)*

4.26 (3.21)**

5.75 (0.56)

6.81 (0.48)

-0.83 (-0.14)

-5.37 (-1.30)

0.80 (0.74)

4 3.46 (1.53)

6.77 (5.65)***

9.75 (1.26)

10.81 (0.96)

2.36 (0.37)

-4.57 (-1.43)

0.85 (0.81)


0 2.59 (1.16)

2.90 (5.67) ***

-1.50 (-1.22)

-2.30 (-1.42)

0.01 (0.01)

0.25 (0.37)

0.86 (0.83)

1 0.17 (0.07)

4.46 (6.16)***

-2.54 (-0.89)

-4.42 (-1.05)

0.42 (0.22)

1.11 (0.77)

0.90 (0.87)

2 8.55 (6.43)***

4.02 (3.96)***

-12.63 (-2.38)*

-15.79 (-2.07)^

-0.39 (-0.11)

4.62 (1.51)

0.84 (0.80)

3 12.84 (4.84)***

3.22 (1.43)

-4.67 (-0.34)

-7.58 (-0.43)

-6.97 (-1.08)

-3.24 (-0.56)

0.71 (0.64)

4 4.75 (1.95)^

12.00 (4.68)***

18.34 (1.79)^

17.56 (1.29)

-2.21 (-0.32)

-7.69 (-2.01)^

0.81 (0.76)




43

Table XIII Annual Excess Returns, Consumption Betas, and Population Betas

This table reports the betas of the first-stage (time-series) regression of Fama-MacBeth (1973) method. The betas are estimated by the first-stage (time-series) regression of excess returns, respectively, on the one-year factors (Q4-Q4; nondurables and services) of the CCAPM and the P-CCAPM.

Low Book-to-Market High Panel A: Average Annual Excess Return (%) High-Low

Small 7.97 14.69 14.74 17.07 20.61 12.64

5.67 10.03 13.35 13.76 15.06 9.39

Size 6.72 10.74 10.87 13.48 14.29 7.57

7.79 8.74 11.69 12.21 13.03 5.24

Big 6.88 8.20 9.35 9.52 10.83 3.95

Small-Big 1.09 6.49 5.39 7.54 9.78

Panel B: Consumption Betas High-Low Small 4.85 6.53 5.73 6.53 7.80 2.45

3.84 4.70 5.70 6.33 7.23 3.76 Size 4.11 5.48 5.72 6.24 7.30 2.22

3.36 4.54 4.56 5.86 7.60 -1.89 Big 3.36 3.58 4.19 5.02 6.33 -3.36

Small-Big 1.49 2.95 1.54 1.51 1.47 Panel C: Population Betas High-Low

Small 0.53 4.96 -4.04 -3.98 -7.42 -7.94 8.68 7.07 1.87 0.37 2.70 -5.97

Size 13.62 8.11 6.33 4.55 -4.18 -17.80 12.75 10.77 8.74 1.90 0.98 -11.77

Big 13.97 8.48 11.38 6.66 7.04 -6.93 Small-Big -13.44 -3.52 -15.43 -10.64 -14.46



44

Table XIV Population Betas of Age Groups

This table reports the betas of the first-stage (time-series) regression of Fama-MacBeth (1973) method. The betas are estimated by the first-stage (time-series) regression of excess returns on the one-year factors (Q4-Q4; nondurables and services) of the P-CCAPM with age groups.

Low Book-to-Market High Panel A: Population Betas of the Childhood High-Low

Small 1.30 2.07 -0.29 0.06 -0.38 -1.68

1.92 1.66 0.75 -0.01 1.71 -0.22

Size 2.79 1.71 1.43 1.60 -0.24 -3.02

2.74 2.37 2.39 1.52 0.42 -2.32

Big 3.17 1.33 2.43 1.76 1.92 -1.25

Small-Big -1.87 0.74 -2.72 -1.71 -2.30

Panel B: Population Betas of the Young-Aged High-Low Small -1.50 -1.80 -1.36 -1.34 -1.88 -0.39

-0.71 -0.49 -0.87 -0.01 -0.97 -0.26

Size -0.82 -0.67 -0.48 -0.90 -0.50 0.32

-1.59 -1.70 -1.45 -1.96 0.34 1.92

Big -1.73 -0.71 -1.45 -1.41 -0.73 0.99

Small-Big 0.23 -1.09 0.10 0.07 -1.15

Panel C: Population Betas of the Middle-Aged High-Low Small 0.85 1.46 2.65 2.07 3.64 2.79

0.04 -0.25 1.09 -0.15 0.30 0.27

Size 0.55 0.28 0.80 0.59 1.21 0.66

1.49 1.84 1.63 2.00 -1.21 -2.70

Big 1.75 0.99 1.23 1.89 0.65 -1.09

Small-Big -0.90 0.47 1.42 0.18 2.98

Panel D: Population Betas of the Old-Aged High-Low Small 0.88 2.47 -2.20 -1.51 -3.06 -3.94

4.49 3.92 0.98 1.79 2.59 -1.89

Size 4.86 3.42 2.94 2.50 -0.66 -5.52

2.41 2.70 2.80 -0.15 3.96 1.55

Big 2.21 2.39 3.39 1.54 3.50 1.28

Small-Big -1.33 0.08 -5.59 -3.05 -6.56



45

Table XV Cross-Sectional Regression Pricing Errors: CCAPM, P-CCAPM, and P-CCAPM with Age Groups

This table compares the pricing errors of 25 portfolios generated by the s (s=0,1,2)-year estimation (annual-annual) of the CCAPM, the P-CCAPM, and the P-CCAPM with age groups. The pricing errors are calculated by iiR βλλα ˆˆˆˆ 0 ′−−= . All numbers are annual percentages. The consumption set is nondurables plus services.

Panel A: Pricing Errors CCAPM (s=0) P-CCAPM (s=0) P-CCAPM with age groups (s=0)

0.62 -5.80 -4.04 -5.48 -8.83 4.10 -4.76 0.55 -1.41 -3.21 2.76 -3.89 0.73 -1.53 -1.87 3.21 0.50 -1.37 -1.44 -2.25 2.84 -0.05 -0.35 0.50 -2.11 2.37 -1.02 -0.19 -0.13 -3.05 3.79 1.01 1.75 -0.97 -1.10 0.06 -0.72 0.32 -1.37 2.33 -0.61 -0.57 -0.31 -1.88 0.63 2.46 2.82 -0.80 0.56 0.07 -0.55 -0.07 -2.35 1.48 1.23 -0.57 2.68 -2.38 2.57 0.31 4.38 3.54 2.33 2.94 2.09 0.28 1.98 -0.65 1.72 0.20 -0.46 2.59 1.48 2.57 -0.24

CCAPM (s=1) P-CCAPM (s=1) P-CCAPM with age groups (s=1) 3.47 -1.39 -1.32 -2.51 -3.3 4.55 -1.90 -0.05 -1.60 -2.60 3.56 -0.57 0.51 -1.43 -0.05 1.88 0.17 -0.57 0.31 0.28 2.37 0.21 -0.39 0.40 -0.60 1.74 -1.07 -0.25 -1.14 -1.06 1.68 1.28 1.29 -1.13 0.47 0.95 0.49 0.73 -1.41 1.08 1.00 0.54 0.46 -1.25 -0.13 -0.53 -0.18 -3.44 0.82 2.87 -0.64 -0.37 -3.23 0.81 2.07 -0.89 1.06 -2.64 1.43 0.40 0.44 -1.79 -1.40 0.50 2.10 -0.03 -0.80 -1.56 0.52 0.98 0.03 -1.35 -0.68 1.12 0.66

CCAPM (s=2) P-CCAPM (s=2) P-CCAPM with age groups (s=2) 0.86 -4.34 -1.37 -2.86 -4.11 3.13 -4.08 0.27 -1.32 -2.76 2.25 -2.95 0.53 -1.57 -0.83 0.67 -0.06 -0.52 1.15 0.61 1.41 0.00 -0.17 1.14 -0.63 0.76 -1.28 -0.35 -0.06 -1.70 -0.02 1.37 1.85 -0.86 0.56 -0.40 0.55 1.10 -1.25 1.34 -0.59 0.33 0.67 -0.84 0.62 -1.38 0.50 -1.47 1.48 2.12 -1.25 0.02 -1.99 1.37 1.23 -1.44 1.36 -1.43 2.26 0.53 0.39 -0.10 0.19 2.22 3.13 -0.23 0.32 -0.67 1.58 1.27 -0.18 0.14 0.39 2.17 1.20

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Table XV(cont’d.)

Panel B: Summary of Pricing Errors

S CCAPM P-CCAPM P-CCAPM with age groups

Max |ˆ|α 0 8.83 4.76 3.89 1 3.47 4.55 3.56 2 4.34 4.08 2.95

Average |ˆ|α

0 2.57 1.41 1.50 1 1.40 1.21 1.00 2 1.37 1.18 1.06



47

Table XVI Implied Coefficient of Relative Risk Aversion: CCAPM and P-CCAPM

This table reports the coefficients of relative risk aversion corresponding to the slope coefficients of the CCAPM and the P-CCAPM in the second-stage regression.


CCAPM P-CCAPM

Annual-Annual Q4-Q4 Annual-Annual Q4-Q4 s 1γ 1γ 2γ 3γ 2γ 3γ 0 28 32 16 31 21 33

1 18 16 12 17 11 17

2 12 12 8 11 8 11

3 9 9 6 8 6 8

4 7 7 5 6 5 16

Panel B: Durables, Nondurables and Services 0 19 23 11 25 16 26

1 12 12 9 13 9 40

2 8 8 6 8 6 8

3 6 6 4 6 4 6

4 5 5 3 5 4 5



48

Table XVII Implied Coefficient of Relative Risk Aversion: P-CCAPM with Age Groups

The table reports the coefficients of relative risk aversion corresponding to the slope coefficients of the P-CCAPM with age groups in the second-stage regression.


s Annual-Annual

4γ 5γ 6γ 7γ 8γ 0 21 30 30 36 29

1 12 15 15 13 14

2 8 10 10 9 9

3 6 6 6 8 7

4 5 5 5 33 6

Q4-Q4 0 22 34 38 27 27

1 11 15 15 11 14

2 8 10 10 9 9

3 6 6 7 11 7

4 5 5 5 5 6

Panel B: Durables, Nondurables and Services Annual-Annual 0 15 24 24 25 23

1 9 12 12 9 11

2 6 8 8 12 7

3 3 6 6 6 6

4 4 4 4 5 4

Q4-Q4 0 17 25 24 3 22

1 9 13 12 9 11

2 6 8 8 13 7

3 4 6 6 6 6

4 4 4 4 5 4



49

Figure 1: Realized and Fitted Excess Returns: CCAPM, P-CCAPM, and the Fama-French Three-Factor Model. This figure compares the realized annual excess returns and fitted annual excess returns of 25 Fama-French portfolios. Each two-digit number represents one portfolio. The first digit refers to the size quintile (1 smallest, 5 largest), and the second digit refers to the book-to-market quintile (1 lowest, 5 highest). The consumption set is nondurables plus services.



50

Figure 2: Realized and Fitted Excess Returns: CCAPM, P-CCAPM and P-CCAPM with Age Groups. This figure compares the realized annual excess returns and fitted annual excess returns of 25 Fama-French portfolios from the s (s=0,1,2)-year estimation (annual-annual) of the CCPM, the P-CCAPM and the P-CCAPM with age groups. Each two-digit number represents one portfolio. The first digit refers to the size quintile (1 smallest, 5 largest), and the second digit refers to the book-to-market quintile (1 lowest, 5 highest). The consumption set is nondurables plus services.

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études et Dossiers - Amazon S3 · 2013-05-07 · International Association for the Study of Insurance Economics études et Dossiers études et Dossiers No. 369 World Risk and Insurance