Revisiting asset pricing under habit formation in an overlapping-generations economy

Journal of Banking & Finance 37 (2013) 132–138

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Revisiting asset pricing under habit formation in an overlapping-generations economy Sei-Wan Kim a, Joshua Krausz b, Kiseok Nam b,⇑ a b

Department of Economics, Ewha Womans University, Seoul, Republic of Korea Sy Syms School of Business, Yeshiva University, NY, USA

a r t i c l e

i n f o

Article history: Received 23 May 2010 Accepted 19 August 2012 Available online 30 August 2012 JEL classification: E21 E27 G12 Keywords: Equity premium Habit formation preference Overlapping-generations economies Consumption asset pricing model Calibration

a b s t r a c t By incorporating habit formation into an overlapping-generations economy, we show that the middleaged consumers’ savings decision has a substantial impact on the equity premium. The higher incentive for savings for the middle-aged, resulting from the habit formation preference, causes an even higher demand for bonds and a lower demand for equity, which eventually generates a lower risk-free rate and a higher required return for holding equity than does the framework of non-habit forming models. Calibration results verify that the habit formation setting, together with an OLG framework is capable of yielding lower bond returns and higher equity returns than the standard CRRA utility models, and the borrowing constraint imposed on the young-aged consumers amplifies the positive effect of habit formation on the equity premium. The findings imply that habit formation preferences within the overlapping-generations framework under the borrowing-constrained economy can provide a more improved explanation of the equity premium puzzle. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction In their seminal work, Mehra and Prescott (1985) identify the phenomenon that the historical real returns of stock over government bonds are anomalously high. They show that the historical equity premium, which is defined as equity returns less government bond returns, exhibits an abnormally high level not only in the United States but also in many other industrialized countries, over long time periods.1 Since the equity premium is supposed to reflect the relative risk of stocks compared to risk-free government bonds, the unexpectedly large percentage of the risk premium for equity implies an implausibly high level of risk aversion among consumers.2 The ⇑ Corresponding author. Tel.: +1 212 960 0845; fax: +1 212 960 0824. E-mail addresses: [email protected] (S.-W. Kim), [email protected] (J. Krausz), [email protected] (K. Nam). 1 They demonstrate that it is difficult to reconcile the empirical fact of a suspiciously high level of equity premium and the process of consumption growth with a reasonable assumption about the relative rate of risk aversion and the pure rate of time preference, in a conventional infinite-horizon model with an additively separable, constant relative rate of risk aversion (CRRA) utility function. 2 By looking at the disparity from a different perspective, Weil (1989) raises an issue, known as the risk-free rate puzzle, on why bond returns are lower than equity returns. Ebrahim and Mathur (2001) suggest an equilibrium model reflecting investor heterogeneity, market segmentation and leverage to resolve the two puzzles. 0378-4266/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jbankfin.2012.08.018

problem, known as the equity premium puzzle, is that the magnitude of the equity premium is too large to reflect a reasonable level of compensation justified under the standard neoclassical equilibrium asset pricing model. Due to the importance of its economic implications, the equity premium puzzle has spawned extensive research efforts to resolve the puzzle in the macroeconomics and finance literature.3 In general, most of the papers explaining the puzzle take the approach of either finding factors requiring adjustment to the empirical side of the puzzle, or exploring alternative theoretical frameworks. The studies focusing on the empirical side of the puzzle include the question of sample time periods and mean reversion or aversion by Siegel (1992a,b). On the other hand, the studies attempting to modify the theoretical features of the Mehra and Prescott (1985) model propose alternative assumptions about preference (Constantinides, 1990; Abel, 1990; Epstein and Zin, 1991; Meyer and Meyer, 2005; Giordani and Söderlind, 2006), disaster states and survivorship bias (Reitz, 1988; Brown et al., 1995; Barro, 2006), incomplete markets

3 The excessive magnitude of the equity premium has many important economic implications, such as those for resource allocation, social welfare, and economic policy, other than financial market implications. See Grant and Quiggin (2006) for more details.

S.-W. Kim et al. / Journal of Banking & Finance 37 (2013) 132–138

(Constantinides and Duffie, 1996; Heaton and Lucas, 1996; Storesletten et al., 1999), and market imperfection (Bansal and Coleman, 1996; Alvarez and Jermann, 2000; Constantinides et al., 2002). Also, more recent studies of the puzzle attempt to provide different rationales for explaining the equity premium, such as investor prospects (Benartzi and Thaler, 1995; Durand et al., 2004; Fielding and Stracca, 2007), macroeconomic influences (Campbell and Cochrane, 1999), and changes in tax rates (McGrattan and Prescott, 2001, 2005).4 Despite a great deal of literature suggesting a wide range of useful theoretical and empirical tools, the puzzle has not been completely resolved. In this paper, we extend the framework of Constantinides et al. (2002). Incorporating habit formation into an overlapping-generations (hereafter OLG) economy with the borrowing constraint imposed on the young generation, we verify that there is a positive impact of habit formation on the savings levels of middle-aged consumers. The higher incentive for savings for the middle-aged, resulting from the habit formation preference, causes an even higher demand for bonds and a lower demand for equity, which eventually generates lower risk-free rates and higher required returns for holding equity than does the framework of non-habit forming models. Calibrating our model, we confirm that our model yields a lower risk-free rate and a higher equity return than do other general non-habit forming models. We thus argue that habit formation preferences within the overlapping-generations framework under the borrowing-constrained economy can provide a more improved explanation of the equity premium puzzle. The rest of the paper is organized as follows: In Section 2, we discuss the related works on habit formation in the equity premium puzzle. In Section 3, we derive the optimum savings of the habit-forming middle-aged consumers under a borrowing-constrained economy, and confirm a positive effect of the habit formation preference on the middle-aged consumers’ savings. In Section 4, we discuss the calibration of our model and its results. Section 5 concludes the paper.

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explicitly captures the saving and dissaving behavior of consumers subject to a borrowing constraint. CDM show that with a simple time separable utility function and a borrowing constraint, consumers in a three-period overlapping-generations economy have an incentive to hold a diversified portfolio for different stages, over their life cycle. That is, a borrowing constraint prevents the youngaged generation from holding equity, and that equity prices are assumed to be exclusively determined by the middle-aged consumers. Knowing that their future retirement income is either zero or deterministic, and that their future consumption is highly correlated with equity income, the middle-aged consumers will save more by holding more bonds and less equity. Therefore, the middle-aged consumers’ savings decision has a dominant impact on the level of the equity and the bond return. In this paper, we extend CDM’s work by incorporating habit formation into the OLG economy, such that the habit-forming consumers’ optimal savings decision is derived from an overlappinggenerations framework. Under the habit formation utility and the OLG economy, the impact of the middle-aged consumers’ savings decision on the demand for equity and bonds is affected not only by the presence of a borrowing constraint, but also by the habit formation process. A habit-forming middle-aged consumer will have a much higher incentive to smooth consumption over time, and he/she will have a lower incentive to bear risk in order to guarantee a stable consumption for the next period, by demanding more bonds and less equity than a non-habit-forming consumer. Thus, the habit formation utility causes an even higher demand for bonds (yielding a lower risk-free rate) and a lower demand for equity (yielding a higher required return for holding equity), and thereby yielding a higher equity premium, than does a nonhabit formation utility such as the CRRA utility suggested by CDM (2002). Also, we show that the effect of habit formation on the demand for equity and bonds is more profound under the borrowing-constraint economy than under the borrowing-unconstraint economy. In sum, the combination of the habit formation utility, and an OLG economy with a borrowing constraint, yields better results which can be used in explaining the equity premium.

2. Related works Habit formation has been widely used in recent studies of financial economics as an important assumption in explaining the dynamic equilibrium path of consumption. For example, Constantinides (1990) and Abel (1990) show that habit-forming consumption, with its flexibility in modeling risk aversion and consumption paths, can partially resolve the equity premium puzzle posed by Mehra and Prescott (1985).5 This finding has indeed motivated a line of habit formation approaches in dynamic modeling of optimal consumption, savings and portfolio decisions (Sundaresan, 1989; Jermann, 1998; Campbell and Cochrane, 1999; Lettau and Uhlig, 2000; Guvenen, 2009).6 On the other hand, Constantinides et al. (2002) (hereafter CDM) propose an overlapping-generations (hereafter OLG) model that 4 McGrattan and Prescott (2001) suggest that the changes in tax rates can explain the equity premium puzzle. They show that the large reduction in individual income tax rates and the increased income from tax shelter opportunities have led to a dramatic increase in equity prices between income in 1960 and 2000. In turn, these increased equity prices generate much higher ex post returns on equity than on debt, such that they argue that at least for the post-WWII period, the equity premium is not puzzling. 5 See Cochrane and Hansen (1992) and Kocherlakota (1996) for surveys on the equity premium puzzle. 6 Guvenen (2009) proposes an asset pricing model focusing on the limited stock market participation and heterogeneity in the elasticity of intertemporal substitution in consumption. His model is partially successful in calibrating major features of asset pricing such as high equity premiums, smooth interest rates, procyclical stock prices, and countercyclical variations in the equity premium.

3. Habit formation and optimum savings under borrowing constraint In this section, we present a habit formation exchange economy in the OLG framework and derive the optimum savings of the middle-aged generation with the borrowing constraint imposed on the young generation. Under the borrowing-unconstrained economy, the young will borrow to purchase equity, thereby raising bond returns. The increase in bond returns induces the middle-aged investors to shift their portfolio holdings from equity to bonds, thereby reducing equity return. The increase in the demand for equity by the young will be overweighed by the decrease in equity demand by the middle-aged, such that the net effect is an increase in equity returns. The increase in equity returns and the increase in bond returns together shrink the equity premium. Under a borrowing-constrained economy, however, the young are prevented from borrowing for equity investments, so that both the equity and bond returns are exclusively determined by the habit-forming middle-aged investors. Due to the inability of the young to hold equity, resulting from the borrowing constraint together with high fluctuations in equity income, the demand for equity is reduced, and consequently, the net demand for bonds by middle-aged consumers is raised. Thus, the middle-aged consumers’ savings decision has a substantial impact on the high equity premium (i.e., a high risk premium and a lower risk-free rate) under the borrowing-constrained economy.

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To derive the optimum savings decision under the borrowingconstrained economy, we consider a utility-maximizing representative agent in an OLG economy, where each generation lives within three discrete generational periods: as members of the young, middle, and old aged generations. The representative consumer born at t = 0 with no endowment assets receives labor income w0 in period t = 0, w1 in period t = 1, and zero labor income in period t = 2. In the first period, the consumer receives a relatively low labor income sufficient only to satisfy his or her first period consumption requirements. In the second period, the consumer receives increased wage income, and seeks to accumulate sufficient assets for third period consumption. The consumer retires in the third and last period, and consumes the assets accumulated during the second period. Savings for smoothing lifetime consumption is done by holding a diversified portfolio of equity and bonds. Following Sundaresan (1989) and Constantinides (1990), we assume that the representative consumer’s utility exhibits habit formation preferences, such that the habit level of consumption at time t, Xt, is a positive fraction of the consumer’s own previous consumption level, i.e., Xt = dCt1.7 The parameter d is the constant habit persistence parameter and it is assumed to have a value between 0 and 1, which characterizes the consumption of non-durable goods and services.8 Since the representative consumer in the first period does not have the previous period consumption for habit formation, the consumer is assumed to have a habit formation utility function from the second period on. Consequently, the consumer has the following sum of discounted utility flows over three periods:

U¼

½C 0 1c ½C 1  X 1 1c ½C 2  X 2 1c þb þ b2 ; 1c 1c 1c

ð1Þ

where C0, C1 and C2 are consumption at t = 0, t = 1 and t = 2, respectively, and all are assumed to be positive. Habit level at time t is determined by Xt = dCt1. b is the constant subjective discount factor, and c > 0 is the constant RRA coefficient. The representative consumer faces the following budget constraints over his life cycle:

C 0 6 w0  S0 ; 1

ð2Þ

V¼

½w0  S0 1c ½ðw1 þ R1 S0  S1 Þ  dðw0  S0 Þ1c þb 1c 1c þ b2

½R2 S1  dðw1 þ R1 S0  S1 Þ1c : 1c

ð5Þ

Assuming a constant RRA in the overlapping generation economy, CDM (2002) show that, imposing the borrowing constraint on the young-aged generation reduces the risk-free rate and increases the equity return. Under the borrowing constraint, the young generation bears a restriction on borrowing against future labor income, which is realistic in that human capital alone does not collateralize major loans in modern economies for reasons of moral hazard and adverse selection. In addition to this constraint, the young-aged generation’s labor income (w0) is assumed to be at a lower level than the middle-aged generation’s labor income (w1), so that it is enough only for the consumption at t = 0. These two assumptions together rationalize the zero savings of the youngaged generation, i.e., S0 = 0. Since the young-aged generation is excluded from participating in the equity markets, the equity price (and, thus, the equity premium) is exclusively determined by the middle-aged consumers’ savings decision. Using comparative statics on the optimum savings decision of middle-aged consumers, we examine the effect of habit-formation preferences on the optimum savings level. First, we derive the optimum savings level of the middle-aged generation by solving the maximization problem of the discounted utility over the life cycle. Then, we show the positive impact of the habit formation preferences on the optimum savings level of the middle-aged. Differentiating the value function V (with S0 = 0) of Eq. (5) with respect to S1 yields the first order condition Q as follows:

Q : b2 ðR2 þ dÞ½R2 S1  dðw1  S1 Þc  b½ðw1  S1 Þ  dw0 c ¼ 0; ð6Þ S1

where is the optimal savings level of the middle-aged generation. The second order condition for the maximization problem is also satisfied as follows:

cb½ðw1  S1 Þ  dw0 c1  cb2 ðR2 þ dÞ2 ½R2 S1  dðw1  S1 Þc1 < 0:

C 1 6 w þ R1 S0  S1 ; and

ð3Þ

C 2 6 R 2 S1 ;

ð4Þ

where S0 and S1 are savings of young and middle generations, R1 and R2 are the gross rates of return for the middle and old generations, and w0 and w1 represent the labor income of the young and

ð7Þ Given that the first and second order conditions are satisfied, the effect of habit formation on the optimum savings level of the middle-aged generation can be examined by the sign of  .  dS1 @Q can be expressed as follows: ¼  @Q @d @S dd 1

b2 ½R2 S1  dðw1  S1 Þc þ b2 cðR2 þ dÞðw1  S1 Þ½R2 S1  dðw1  S1 Þc1  cbw0 ½w1  S1  dw0 c1

cb½w1  S1  dw0 c1 þ b2 cðR2 þ dÞ2 ½R2 S1  dðw1  S1 Þc1

middle generations. With Eqs. (2)–(4), the objective function U becomes the following value function:

7 Abel (1990) proposes another type of habit formation, i.e., Catching up with the Joneses, where the habit-forming behavior is based upon the consumption of other consumers. Comparing his or her own consumption to that of others, a consumer could get utility from knowing that he or she is consuming more than others. 8 The standard CRRA utility is a special case of the habit formation utility with d = 0. d < 0 implies negative habit formation, which is applied to the consumption of durable goods.

dS1 . dd

ð8Þ

:

Using the budget constraints, Eq. (8) can be rewritten as follows: dS1 b2 ½C 2  dC 1 c þ b2 cðR2 þ dÞC 1 ½C 2  dC 1 c1  cbC 0 ½C 1  dC 0 c1 : ¼ dd cb½C 1  dC 0 c1 þ b2 cðR2 þ dÞ2 ½C 2  dC 1 c1 

ð9Þ dS

We calibrate Eq. (9) to determine the sign of dd1 under a plausible range of parameters and the consumption set. Since it is always the case that C1 > dC0 and C2 > dC1 (in order to have positive utility dS levels), the denominator of Eq. (9) is positive. Thus, the sign of dd1 is determined by the sign of the numerator of Eq. (9).

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S.-W. Kim et al. / Journal of Banking & Finance 37 (2013) 132–138 Table 1 Parameters set on 20-year basis. Coefficient of relative risk averseness: c Subjective discount rate: b Habit formation parameter: d Average share of income going to labor: E[w0 + w1]/E[y] Average share of income going to the labor of the young: w0/E[y] Average share of income going to interest on government debt: b/E[y] Coefficient of variation of the 20-year wage income of the middle aged: r(w1)/E[w1] Coefficient of variation of the 20-year aggregate income: r(y)/E[y] a

2, 4 and 6 0.44a 0.0, 0.1, 0.2, . . . , 0.8 0.6–0.75 0.10–0.25 0.03 0.20–0.70 0.10–0.30

Implies b = 0.96/year.

Table 2 Calibration results of the equity premium with the equity and bond returns. Habit formation parameter d = 0.0 Without borrowing constraint c = 2.0 re:8.01% rf:7.76% rp:0.25%

d = 0.1

d = 0.2

d = 0.3

d = 0.4

d = 0.5

d = 0.6

d = 0.7

d = 0.8

re:8.31% rf:7.85% rp:0.46%

re:8.44% rf:7.22% rp:1.22%

re:8.44% rf:7.21% rp:1.23%

re:8.51% rf:6.75% rp:1.76%

re:8.68% rf:6.05% rp:2.63%

re:8.78% rf:5.64% rp:3.14%

re:8.78% rf:5.63% rp:3.15%

re:8.79% rf:5.35% rp:3.44%

c = 4.0

re:8.60% rf:7.77% rp:0.63%

re:8.79% rf:7.26% rp:1.53%

re:8.83% rf:6.90% rp:1.93%

re:8.88% rf:6.16% rp:2.72%

re:8.91% rf:6.19% rp:2.72%

re:8.93% rf:6.02% rp:2.91%

re:8.99% rf:5.67% rp:3.32%

re:9.10% rf:5.49% rp:3.61%

re: 9.25% rf:5.53% rp:3.97%

c = 6.0

re:10.05% rf:8.93% rp:1.12%

re:10.21% rf:8.21% rp:2.00%

re:10.23% rf:7.88% rp:2.35%

re:10.26% rf:7.06% rp:3.20%

re:10.30% rf:6.67% rp:3.63%

re:10.42% rf:6.62% rp:3.80%

re:10.48% rf:6.57% rp:3.91%

re:10.56% rf:5.78% rp:4.72%

re:10.56% rf:5.83% rp:4.73%

Average

re:8.89% rf:8.15% rp:0.67%

re:9.10% rf:7.77% rp:1.33%

re:9.17% rf:7.33% rp:1.83%

re:9.19% rf:6.81% rp:2.38%

re:9.24% rf:6.54% rp:2.70%

re:9.34% rf:6.23% rp:3.11%

re:9.42% rf:5.96% rp:3.46%

re:9.48% rf:5.63% rp:3.83%

re: 9.53% rf:5.57% rp:4.05%

With borrowing constraint re:8.00% rf:7.46% rp:0.54%

re:8.22% rf:7.35% rp:0.87%

re:8.23% rf:6.67% rp:1.56%

re:8.30% rf:6.63% rp:1.67%

re:8.52% rf:6.20% rp:2.32%

re:8.58% rf:5.56% rp:3.09%

re:8.79% rf:5.19% rp:3.60%

re:9.02% rf:5.05% rp:3.97%

re:9.02% rf:5.00% rp:4.02%

c = 4.0

re: 8.21% rf:7.44% rp:0.77%

re: 8.23% rf:6.59% rp:1.64%

re: 8.32% rf:6.18% rp:2.14%

re: 8.33% rf:5.56% rp:2.77%

re: 8.38% rf:5.62% rp:2.74%

re: 8.61% rf:5.29% rp:3.32%

re: 8.71% rf:5.11% rp:3.60%

re: 8.98% rf:4.86% rp:4.12%

re:9.01% rf:4.72% rp:4.29%

c = 6.0

re: 8.47% rf:7.00% rp:1.47%

re:8.55% rf:6.42% rp:2.13%

re: 8.58% rf:5.96% rp:2.62%

re: 8.65% rf:5.36% rp:3.29%

re: 8.70% rf:4.99% rp:3.71%

re: 8.78% rf:4.91% rp:3.87%

re: 8.93% rf:4.76% rp:4.17%

re: 9.03% rf: 4.17% rp:4.86%

re: 9.07% rf: 4.16% rp:4.91%

Average

re: 8.23% rf:7.30% rp:0.93%

re: 8.33% rf:6.79% rp:1.55%

re:8.38% rf:6.27% rp:2.11%

re:8.43% rf:5.85% rp:2.58%

re:8.53% rf:5.60% rp:2.92%

re:8.66% rf:5.25% rp:3.43%

re:8.81% rf:5.02% rp:3.79%

re: 9.01% rf:4.69% rp:4.32%

re: 9.03% rf: 4.63% rp:4.41%

c = 2.0

Given permissible parameter values for b, d, c, and R2 under several combinations of consumption paths over the life cycle, calibrating the model readily confirms the following inequality:9

b2 ½C 2  dC 1 c þ b2 cðR2 þ dÞC 1 ½C 2  dC 1 c1 > cbC 0 ½C 1  dC 0 c1 : ð10Þ dS1 dd

The above inequality indicates that > 0, which implies that habit formation has a positive impact on the optimum savings level, thereby showing a higher incentive to save more than with the CDM framework. Solving Eq. (6) for the equilibrium savings decision, we also calculate the optimum savings for the middle-aged agents for various levels of habit persistence under three different levels of risk aversion (c = 2, 4, 6), with and without borrowing 9

Boundaries of parameters and variables are determined as follows. For the value of discount factor (b) we assign around 0.955 per year. By recalculating b in terms of 20 years (one generation period), we get 0.3982. For the habit persistence parameter (d) we use the value 0.615 following Otrok et al. (2002). The value of the relative risk aversion parameter c is set between 1 and 10 following Mehra and Prescott (1985). For the consumption over different age cohorts, we use values between 20,000 and 40,000, which are consistent with the Consumer Expenditure Survey (conducted by the Bureau of Labor Statistics) from 1984 to 1996.

constraints. There are two notable findings. (a) The average growth rate in aggregate savings increases monotonically as habit persistence increases, for both the borrowing-constrained and the borrowing-unconstrained economies. (b) For a given level of habit persistence, the growth rate in aggregate savings is greater under the borrowing-constrained economy than under the borrowingunconstrained economy. Indeed, our result is consistent with Lahiri and Puhakka (1998) and Carroll et al. (2000), who also show a positive impact of habit formation on the aggregate savings – though they do not consider borrowing constraints in their models. 4. Calibration results for equity premium In this section we calibrate the habit formation preference in the OLG framework for both the borrowing-constrained and the borrowing-unconstrained economies. The only difference between the two economies is that while the young agents are able to save under the borrowing-unconstrained economy, they are not allowed to save under the borrowing-constrained economy. Under the borrowing-constrained economy the young agents are prevented from issuing bonds to buy equity, such that their future

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consumer adjusts the bond holding to zb1 . Then, with a no bequests assumption, the consumer liquidates all of his/her bonds in period t = 2 (when old). Thus, the bond market clearing condition is that the total demand for bonds by the young and middle-aged consumers must equal the fixed supply of bonds:

zb0 þ zb1 ¼ b

ð10Þ

Equity is a claim on the dividend stream and pays net dividends dt in period t. Like bonds, equity is supplied at the beginning of each period and its supply is fixed at one unit in perpetuity. The equity value after the dividend payment in period t is denoted by qet , which is the claim to the net dividend stream in perpetuity in period t + 1. With no initial endowment of equity, the representative consumer purchases ze0 equity in period t = 0 (when young), adjusts the equity holding to ze1 in period t = 1 (when middle aged), and sells his/her entire portfolio in period t = 2 (when old). The equity market clearing condition is that the demand for equity by the young and middle-aged consumers must equal the fixed supply of equity:

ze0 þ ze1 ¼ 1

ð11Þ 0

The representative consumer receives fixed wage income (w ) in period t = 0, stochastic wage income (w1) in period t = 1, and zero wage income (w2 = 0) in period t = 2. As mentioned by CDM (2002), the assumptions on the income process reflect three important aspects of reality; first, the condition that w1 > w0 > w2 reflects the incentive that the middle-aged are willing to save; second, due to future wage uncertainty, the young would like to borrow against future income and invest in equity. Under the borrowing constraint, however, the young cannot borrow for equity investment. Third, due to the no wage uncertainty, the saving middle-aged consumers have more flexibility to invest in a diversified portfolio of equities and bonds. With the consumption in period t denoted as Ct (t = 0, 1, 2), we specify the dynamic budget constraints for the representative consumer, as follows:

When young : C 0 þ zb0  qb0 þ ze0  qe0 6 w0 ;

ð12Þ

When middle aged : C 1 þ zb1  qb1 þ zet;1  qe1 6 w1 þ zb0 ðqb1 þ bÞ þ ze0 ðqe1 þ d1 Þ; When old : C 2 6 zb1 ðqb2 þ bÞ þ ze1 ðqe2 þ d2 Þ;

Fig. 1. Equity premium in different economies.

income is determined by their forthcoming wages in their middle age, while the future income of the middle-aged agents is derived from their savings in bonds and equity. In our calibration, a bond represents a risk free asset and is supplied at the beginning of the period. The supply of bonds is fixed at b units in perpetuity. We consider the bond to be a representative asset for long-term government bond, i.e., the 20 year US Treasury bond. Each bond pays a fixed amount of coupon (b > 0) every period permanently, which is financed out of the economy’s capital income payments. The bond price after the coupon payment in period t is denoted by qbt . This bond price can be interpreted as the value of a claim to the coupon b paid in perpetuity from period t + 1. The representative consumer born in period t = 0 has a zero endowment of this bond and purchases zb0 of the bond in period t = 0 (when young). In period t = 1 (when middle aged), the

ð13Þ ð14Þ

where C0, C1, and C2 are the consumption levels for the young, middle-aged, and old consumers, respectively. ze0 and ze1 are the equity levels held by the young and middle-aged consumers, respectively. zb0 and zb1 are the demand levels for bonds held by the young and middle-aged consumers, respectively. qe0 ; qe1 , and qe2 are the ex-dividend prices for equity at each period, qb0 ; qb1 , and qb2 are the ex-coupon prices of bonds at each period. d1 and d2 are the dividends paid at period t = 1 and t = 2, respectively. Note that under the borrowing constraint, the young cannot issue a bond for equity investment, and hence zb0 ¼ ze0 ¼ 0. To assess the effect of habit formation in the three-generation OLG framework, a total of 54 different versions of the economy are considered, with different ranges of parameters set for both the borrowing-constraint and borrowing-unconstraint economies. These economies range from the conventional ‘constant relative risk averse (CRRA) utility economy’, i.e., d = 0, to the strong ‘habit formation utility economy’, i.e., d = 0.8, with an increment of 0.1. We also set three different level of the risk aversion parameter, i.e., c = 2, 4, 6. In each economy, a representative consumer maximizes his/her utility by choosing the optimal consumption and investment policies under the constraints of non-negativity consumption and the budget constraints specified in Eqs. (12)–(14). To obtain dynamic equilibrium asset returns, we define the

S.-W. Kim et al. / Journal of Banking & Finance 37 (2013) 132–138

equilibrium condition of each economy, with the following first order conditions, as specified in the following definition: Definition 1. A stationary rational expectation equilibrium in a three-generation economy is a pair of price functions, qe(k) and qb(k) that satisfy:

(i) (ii) (iii) (iv) (v) (vi)

P e U 0 ðC 0 Þqe ¼ b 4k¼1 U 1 ðC 1 Þ  ðqek þ dk ÞPjk P4 b b U 0 ðC 0 Þq ¼ b k¼1 U 1 ðC 1 Þ  ðqk þ bÞPjk P e U 1 ðC 1 Þqe ¼ b 4k¼1 U 2 ðC 2 Þ  ðqek þ dk ÞPjk P U 1 ðC 1 Þqb ¼ b 4k¼1 U 2 ðC 2 Þ  ðqbk þ bÞPjk ze1 ¼ 1  ze0 zb1 ¼ b  zb0 ;

where Ut(Ct) is the t period’s marginal utility against consumption, k represents four different states of the economy (k = 1, 2, 3, 4), and Pjk is the Markov transition matrix. Substituting the dynamic budget constraints from Eqs. (12)–(14) into consumption, the above equilibrium conditions can be expressed as follows:

P e (i) U 0 ðw0  qe ze0  qb zb0 Þqe ¼ b 4k¼1 U 1 ð½qek þ dk ze0 þ ½qbk þ bzb0 þ e 1 e e b b e wk  ½qk z1  qk z1 Þ  ðqk þ dk ÞPjk P e (ii) U 0 ðw0  qe ze0  qb zb0 Þqb ¼ b 4k¼1 U 1 ð½qek þ dk ze0 þ ½qbk þ bzb0 þ 1 e e b b b wk  ½qk z1  qk z1 Þ  ðqk þ bÞPjk P e (iii) U 1 ð½qe þ dk ze0 þ ½qe þ bzb0 þ w1  qe ze1  qb zb1 Þqe ¼ b 4k¼1 U 2 e e e e b b e ð½qk þ dk z1 þ ½qk þ bz1 Þðqk þ dk ÞPjk P e (iv) U 1 ð½qe þ dk ze0 þ ½qe þ bzb0 þ w1  qe ze1  qb zb1 Þqb ¼ b 4k¼1 U 2 e ð½qek þ dk ze1 þ ½qbk þ bzb1 Þðqbk þ bÞPjk Instead of specifying the joint process of the wage income of the middle-aged generation and the dividends, (w1, dt), we specify the joint process of the aggregate income and the wages of the middleaged generation, (yt, w1).10 As in CDM (2002), the aggregate income in period t to all generations is specified as yt ¼ w0 þ w1t þ b þ dt . In the calibration, yt and w1t have two values for the good and bad states for each variable, such that four possible realizations of the pair (yt, w1t ) are represented by the state variable st = k, where k = 1, 2, 3, 4.11 After log linearization, the equilibrium conditions in the Definition 1 can be rewritten as the following eight equations: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

he1 qe1 þ he2 qe2 þ he3 qe3 þ he4 qe4 þ hb1 qb1 þ hb2 qb2 þ hb3 qb3 þ hb4 qb4 ¼ 0 ue1 qe1 þ ue2 qe2 þ ue3 qe3 þ ue4 qe4 þ ub1 qb1 þ ub2 qb2 þ ub3 qb3 þ ub4 qb4 ¼ 0 je1 qe1 þ je2 qe2 þ je3 qe3 þ je4 qe4 þ jb1 qb1 þ jb2 qb2 þ jb3 qb3 þ jb4 qb4 ¼ 0 ke1 qe1 þ ke2 qe2 þ ke3 qe3 þ ke4 qe4 þ kb1 qb1 þ kb2 qb2 þ kb3 qb3 þ kb4 qb4 ¼ 0 pe1 qe1 þ pe2 qe2 þ pe3 qe3 þ pe4 qe4 þ pb1 qb1 þ pb2 qb2 þ pb3 qb3 þ pb4 qb4 ¼ 0

-e1 qe1 þ -e2 qe2 þ -e3 qe3 þ -e4 qe4 þ -b1 qb1 þ -b2 qb2 þ -b3 qb3 þ -b4 qb4 ¼ 0 qe1 qe1 þ qe2 qe2 þ qe3 qe3 þ qe4 qe4 þ qb1 qb1 þ qb2 qb2 þ qb3 qb3 þ qb4 qb4 ¼ 0 we1 qe1 þ we2 qe2 þ we3 qe3 þ we4 qe4 þ wb1 qb1 þ wb2 qb2 þ wb3 qb3 þ wb4 qb4 ¼ 0;

where each of h, u, j, k, p, -, q, and w is the vector consisting of 10 The joint process of (yt, w1t ) is modeled as a Markov chain with the following transition matrix.

2 ðY 1 ; w11 ðY 1 ; w11 Þ 6 6/ ðY 1 ; w12 Þ 6 6p þ D Pjk ¼ ðY 2 ; w11 Þ 6 6 4r ðY 2 ; w12 Þ H

3 ðY 1 ; w12 Þ ðY 2 ; w11 Þ ðY 2 ; w12 Þ 7 p r H 7 7 7 /D H r 7 7 H /D /þD 5

r

p

/

11 Following CDM (2002), we set the correlation between yt and w1t as 0.1. We also confirmed that the main results are robust with respect to the different levels of the correlation.

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the combination of eight parameters (b, c, d, w1gs , w1bs , ygs, ybs, w0). Note that ygs and ybs are the aggregate incomes for the good and bad states, respectively. Also, w1gs and w1bs are the wage incomes of the middle-aged agent for the good and bad states, respectively. The above dynamic equilibrium equations can be solved for the eight target price variables (qe1 ; qe2 ; qe3 ; qe4 ; qb1 ; qb2 ; qb3 ; qb4 ) by employing the Method of Undetermined Coefficients. For robustness of calibration results, we calibrate the model over 54 different economy sets, for both the borrowing-constrained and the borrowing-unconstrained economies respectively, by changing the coefficient of relative risk aversion (c) to 2, 4, and 6, and the habit forming parameter (d) from 0.0 to 0.8 with an increment of 0.1. The set of parameters employed for calibration in this study is reported in Table 1. Note that, since one-period implies one-generation in our OLG model, all parameters are converted to 20-year values, such that the annualized return is defined as the geometric average over a 20-year holding period return, i.e., (1 + 20-year hold period return)1/20  1. Table 2 presents the annualized mean equity return, risk-free bond return, and equity premium under each economy. For comparison, we test 54 different economies defined by three different values of the CRRA parameter (c = 2, 4, and 6), and eight different values of the habit formation parameter (d = 0.0 through 0.8 with an increment of 0.1) with and without the borrowing constraint. Economies with d = 0.0 represent the simple (non-habit forming) CRRA utility and are used as benchmark economies for comparison to the other habit formation economies in each group of the CRRA parameters. Likewise, each of the economies with d = 0.8 is characterized by the highest level of habit formation preference in each CRRA group. Note that we calibrate the model both with and without borrowing constraints. There are several notable observations on bond returns (rf) from the calibration results. First, regardless of the borrowing constraint, incorporation of habit formation into the OLG framework, consistently reduces bond returns (rf). Except for the economy with c = 0.2 and d = 0.1, for all three values of the risk averse parameter c, bond returns under a habit formation utility (d = 0.1 through 0.8) are all less than those under a non-habit formation utility (d = 0.0). Second, for a given value of the CRRA parameter, bond returns on average decline as the degree of habit formation increases. For example, for c = 0.2 with no borrowing constraint imposed, bond returns decrease from 7.76% under d = 0.0 to 7.72% under d = 0.2, 6.75% under d = 0.4, and 5.35% under d = 0.8 (7.46% under d = 0.0 to 6.67% under d = 0.2, 6.20% under d = 0.4, and 5.00% under d = 0.8, with the borrowing constraint imposed). Also, for c = 0.6 with no borrowing constraints imposed, bond returns decrease from 8.93% under d = 0.0 to 7.88% under d = 0.2, 6.67% under d = 0.4, and 5.83% under d = 0.8 (7.00% under d = 0.0 to 5.96% under d = 0.2, 4.99% under d = 0.4, and 4.16% under d = 0.8, with the borrowing constraint imposed). The results imply that as habit persistence increases, bonds are relatively more attractive than equity, so that bond returns decrease as the demand for bonds increases. Third, for a given coefficient level of c and d, the magnitude of bond returns is relatively smaller under the borrowing-constrained economy than under the borrowing-unconstrained economy. Calibration results on equity returns (re) also show consistent patterns. First, habit formation on average raises equity returns. For a given level of the CRRA parameter, equity returns under a habit-forming utility are all greater than the equity returns under a non-habit CRRA utility for both the borrowing-constraint and the borrowing-unconstraint economies. Also, the level of equity return increases as habit persistence increases. For example, with no borrowing constraint imposed, the average equity returns increase from 8.89 under d = 0.0 to 9.17% under d = 0.2, 9.24% under d = 0.4, and 9.53% under d = 0.8 (8.23% under d = 0.0 to 8.38% under d = 0.2, 8.53% under d = 0.4, and 9.03% under d = 0.8, with the bor-

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rowing constraint imposed). Second, the calibration results show that imposing the borrowing constraint on average reduces the level of equity returns for all levels of habit formation for a given CRRA coefficient level of c. Third, the equity returns increase as the CRRA parameter increases. The above patterns in the equity and bond returns provide four important observations about equity premiums, namely: (a) habit formation raises equity premiums, (b) equity premiums increases as the degree of habit formation increases, (c) imposing the borrowing constraint raises the level of equity premiums, for all levels of habit formation, for a given c, and (d) for a given level of the CRRA parameter, the highest level of equity premium occurs under the borrowing-constrained economy with the highest level of habit persistence. This confirms that habit formation preferences within the overlapping-generations framework under the borrowing-constrained economy can provide the most favorable results in our setting. Fig. 1 also shows the above patterns in the calibration results. The habit-forming middle-aged agent becomes more risk averse to consumption variation than those with a simple non-habit preference, thereby having a stronger incentive to choose a safer asset. Thus, bonds become relatively more attractive than equity to the habit-forming middle-aged agent. Consequently, the net increase in the demand for bonds and the decrease in equity demand reduce the bond returns and increase the equity returns, thereby raising the equity premium.12 This pattern is more profound under a no borrowing constraint imposed on the young. In sum, our calibration results show that (a) the habit formation setting together with an OLG framework is capable of yielding lower bond returns and higher equity returns than the standard CRRA utility models, and (b) the borrowing constraint imposed to the young-aged consumers amplifies the positive effect of habit formation preference on the equity premium.

5. Summary and conclusions In this paper, as an attempt to explain the equity premium puzzle, we incorporate habit formation into an overlapping-generations economy. Using comparative static analyses and calibrations, we show that incorporating habit formation preferences into the three-period OLG model has a positive impact on the savings level of the middle-aged consumers. When compared to the non-habit formation preferences, the explicit inclusion of habit formation within an overlapping-generations model results in a stronger incentive for agents to secure their future consumption, so that the habit-forming middle-aged consumers will save even more than the middle-aged consumers in the non-habit formation case. Our calibration results verify that the higher incentive to save, causes a higher demand for bonds, and a lower demand for equity, thereby yielding a lower risk-free rate and a higher required return for holding equity, than do any other non-habit forming models. Therefore, we argue that the habit formation preferences in the OLG framework, with the borrowing constraint im-

12 Habit formation preference results in the income effect and the substitution effect on the demand for equity and bonds. Due to stronger incentives to save more, habit formation increases the portion of wealth to be invested in assets as savings. Thus, under the income effect, habit formation increases the demand for equity and bonds, thereby decreasing both equity and bond returns. In contrast, habit formation causes the representative agent to have a stronger incentive to invest in a safer asset, in that the agent prefers bonds to equity. Under the substitution effect, habit formation reduces the bond returns, but increases the equity returns. Our calibrations show a decreasing pattern of bond returns and an increasing pattern in the equity returns, under habit formation, which implies that the substitution effect outweighs the income effect in the portfolio choice between equity and bonds for the middle-aged consumers.

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