A New Solution to the Equity Premium Puzzle andthe Risk-Free Rate Puzzle: Theory and Evidence
Hideaki Tamura
Yoichi Matsubayashi

August 2014
Discussion Paper No.1422

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANA New Solution to the Equity Premium Puzzle and the Risk-Free Rate
Puzzle: Theory and Evidence

Hideaki Tamura
Graduate School of Economics
Kobe University
Yoichi Matsubayashi
Graduate School of Economics
Kobe University

Abstract
This paper develops a new method for solving both equity premium and risk free rate puzzles basedon the standard utility function. The method for solving the equity premium puzzle in accordancewith Mehra and Prescott (1985) needs to be simultaneously consistent with the method for solving
the risk-free rate puzzle presented by Weil (1989). That is, the reasonable estimated values for the
degree of relative risk aversion in the former solution and for the subjective discount rate in the latter
solution need to plausibly fall within experiential bounds. This study indicates that a consistentsolution is possible for the equity premium and risk-free rate puzzles even when there is a standard
constant relative risk aversion (CRRA) type utility function. This solution is possible byformularizing the Euler equation for consumption, considering the precautionary saving effect.

JEL Classification Number: E21, E44, G12
Keywords: equity premium puzzle, risk-free rate puzzle, uncertainty, Euler equation

Correspondence to:
Yoichi Matsubayashi
Graduate School of Economics, Kobe University
Rokkodai, Nada-ku, Kobe, 657-8501, Japan
TEL +81-78-803-6852
E-mail myoichi@econ.kobe-u.ac.jp

11. Introduction

A central issue in the verification of the permanent income hypothesis since Hansen andSingleton (1982) has been the estimation of parameters in the Euler equation for consumption (thesubjective discount rate and degree of relative risk aversion) based on the consumption-based capital
asset pricing model (C-CAPM). However, uncertainty has arisen in the estimation of the degree ofrelative risk aversion using U.S. data that depends on the estimation period, making it difficult tohave confidence in these analysts.
Hansen and Singleton (1983, 1984) reported on the negative degree of relative risk aversion
when using monthly U.S. data. In addition, results often have been dismissed due to the overidentification restriction of J-statistics. Another issue is the equity premium puzzle shown by Mehra
and Prescott (1985), who demonstrated that the general equilibrium model using stock return, rate of
return on treasury bills, and sample annual consumption growth rate from 1890 to 1979 maximizes
the restrictions imposed on the stock price index and average rate of return on treasury bills. Mankiw
and Zeldes (1991) addressed this problem using the relational expression derived by applying the
Taylor approximation to the Euler equation, but they also reported that the degree of relative risk
aversion calculated using Mehra and Prescott’s (1985) sample is 26.3. In particular, they reported an
unusually high value of 89 when taking the post-war sample from 1948 to 1988.
Weil (1989) pursued the values of the risk premium and risk-free rate for each combinationof degree of relative risk aversion (γ) and elasticity of inter-temporal substitution (1/ρ) in the nonindependent and identically distributed (i.i.d) dividend growth process using Kreps–Porteus typepreferences. Specifying a constant relative risk aversion (CRRA)-type preference (ρ＝γ), he derived
a risk premium of 6.37%. Although that specification approximates the actual level wherein β = 0.98,
γ = 20, and 1/ρ = 0.05, the 15.01% risk-free rate at that time was high. He also derived a risk premiumof 5.72% and a risk-free rate of 0.85%, which are close to actual levels, wherein β = 0.95, γ = 45,
and 1/ρ = 0.1, but noted this result depends on a high value for γ. This inconsistency is referred to as
the risk-free rate puzzle.
According to the retrospective review by Mehra and Prescott (2003), the existing solution
for the equity premium puzzle based on traditional theory is largely split among the alternative2

preference structure (the time non-separable model that does not assume separability in relation totime and habit formation), the model that incorporates idiosyncratic and uninsurable income risk andmodels incorporating a disaster state and survivorship bias. Solutions not based on traditional theory
are affected by borrowing constraints, liquidity premiums, taxes and regulation.
Of the above solutions, the solution for both the equity premium and the risk-free rate puzzles
in accordance with Mehra and Prescott (2003) is limited to the alternative preference structure (thetime non-separable model that does not assume separability in relation to time and habit formation).
An evaluation is as follows.
First, the model that does not assume separability in relation to time mitigates the severe(reciprocal) relationship between the degree of relative risk aversion and inter-temporal elasticity of
substitution, which comes into existence under the traditional CRRA-type utility function, and also
introduces the Kreps–Porteus type utility function that has been partitioned respectively, as is
exemplified in Epstein and Zin (1989, 1991). This model expresses the inter-temporal elasticity of
substitution as the weighted average of the marginal rate of substitution (MRS) for consumption andmarket portfolio. This builds the CAPMthat generalizes both the C-CAPM and Sharp–Lintner CAPM.
However, the inadequacy of any special device to influence an estimation of the degree of relativerisk aversion implies it will not solve the equity premium puzzle, although it is believed that the riskfree rate puzzle can be solved by making the inter-temporal elasticity of substitution and degree ofrelative risk aversion independent of each other.
Next, habit formation determines the utility of current consumption relative to past
consumption that is considered to have become habitual or consumption when benchmarked toneighbours. Constantinides (1990) defines utility as the difference between current and past
consumption (habit) with a lag, indicating that the equity premium puzzle can be solved when theweighting of past consumption (habit) is increased, even when the coefficient for the degree ofrelative risk aversion is low according to the internal habit model that calculates a high degree ofeffective risk aversion. In addition, Campbell and Cochrane (1995) presented a model that wasconsistent for both consumption and asset market data incorporating the possibility of an economic
downturn using the state variable that risk aversion changes in a non-linear way. Abel (1990)
presented a model that can solve the equity premium puzzle while also avoiding the risk-free rate
puzzle modelled in accordance to the proportion of consumption (habit). These habit formation3

models can solve the equity premium puzzle, anticipating a consumer with an extreme dislike of
consumption risk even when the degree of risk aversion is small. In addition, it is believed that thedislike of consumption risk boosts demand for safe assets and reduces the risk-free interest rate andwill be used to solve the risk-free rate puzzle. Although these habit formation models can solve boththe equity premium and risk-free rate puzzles given limited assumptions, all are based on methods
that complicate the standard CRRA-type utility function. Thus, they can hardly be referred to as
generic solutions.
The Euler equation for consumption that uses the standard CRRA-type utility function hasfeatured only two explanatory variables till date: the consumption growth rate and return on assets.
On the other hand, future expected marginal utility rises under a theoretical model of precautionarysaving due to the increased income uncertainty when U"'＞0, and the maximisation of household
first-order condition of expected utility. However, this suggests that an increase in income uncertainty
influences the MRS for the inter-temporal indifference curve assumed for estimating the utility
function parameters. That is, the MRS for an indifference curve will always change during
optimization of multiple periods when there is a relative change in income uncertainty between thepresent and future due to the change in relative status of a utility function that considers present andfuture uncertainty. Thereafter, if estimate employs only the consumption growth rate and return on
assets without correction of that influence, there will be an incorporated bias for the estimated valueof the parameters themselves, distorting the estimate for the degree of relative risk aversion.
The Euler equation for consumption that is expanded to include income uncertainty as an
explanatory variable needs to be formularized to estimate parameters with a greater degree of
information to correct for this precautionary saving effect and thereby estimate parameters correctly.
This study formularizes the Euler equation for consumption (the coefficient of variation (CV) model)
with three explanatory variables (consumption growth rate, return on assets, and growth rate ofincome CV). Using the Mankiw and Zeldes (1991) method, the estimated degree of relative risk
aversion can be explained using the two covariances between the return on assets and consumption
growth rate and the return on assets and growth rate of the income CV while clarifying the featuresby comparing results of the analysis using the Euler equation for consumption (the “normal model”)
that uses the two traditional explanatory variables of the consumption growth rate and return on
assets using U.S. data.

4

This paper is organized as follows. Section 2 identifies the features of the CV model throughanalysis of a model that uses the same Taylor expansion as Mankiw and Zeldes (1991) while also
formularizing the Euler equation for consumption under income uncertainty. Section 3 calculates thedegree of relative risk aversion and subjective discount rate for the normal model and the CV modelusing the U.S. monthly real consumption growth rate, real return on assets, and the income
uncertainty index for February 1978 to December 2010. Section 4 considers the correction effect forthe degree of relative risk aversion under the CV model from the perspective of a two-period model.
Section 5 presents our conclusions.

２.

Model

２．１

Euler equation for consumption under income uncertainty

This section specifies the method for setting an optimal consumption model under income
uncertainty using the marginal utility that is influenced by the income CV under the standard CRRAtype utility function to derive the Euler equation for consumption that includes the income CV.
The individual’s expected marginal utility function under income uncertainty can be
expressed as follows from equation (A.6) in Appendix A:[]

U * ' (Ct ) = Ct−γ 1 + 0.5(γ + γ 2 )CVt 2 .

(1)However, Ct represents an individual’s real consumption for period t, CVt 2 represents the squarevalue of the CV for consumption at period t: CVt 2 = ( ht / C t ) 2 where ht represents the standard
deviation of consumption at period t due to income uncertainty and γ is a parameter representing
a constant degree of relative risk aversion.
From equation (1), the individual’s expected marginal utility under income uncertainty is the
expected marginal utility of the normal model multiplied by 1 + 0.5(γ + γ 2 )CVt 2 . The expectedmarginal utility rises in proportion to the square value of CVt
increases. 1
1This analysis is a modification referred to in Skinner (1988).

5

when the CV for consumption

The multi-period optimal consumption model that uses the individual’s expected utility
function (equation (A.5) in Appendix A) under income uncertainty is set as follows:∞Et [∑ β iU * (Ct + i )] ,

max

(2)

i =0

N

s.t.

∑q A
jt

N

jt +1

j =1

+ Ct = ∑ (q jt + d jt )A jt + Yt .

(3)

j =1

However, β is the subjective discount rate (0 < β < 1) , q jt is the price of asset j at period ｔ(j=1,2,…,N), d jt is the dividend from asset j at period ｔ (j=1,2,…,N), A jt is the quantity ofasset j held at period t, Yt is the non-asset income for period t and Et ・[ ] is the conditional
expectation operator based on information available at time t.
There are N assets in the economy, and individuals select cash flow for each asset andconsumption that maximizes the present discounted value of their expected utility derived throughconsumption from the present (t = 0) to the future.
Solving the above optimization problem results in the following first-order condition for
maximization:

U * ' (Ct +1 ) q jt +1 + d jt +1
Et [ β *()] − 1 = 0 .
U ' (Ct )q jtHere, the rate of return(4)

r jt +1 on asset j is defined as r jt +1 = ( q jt +1 + d jt +1 ) / q jt − 1 , so

(q jt +1 + d jt +1 ) / q jt in equation (4) can be substituted for (1 + r jt +1 ) . Therefore, by replacing thisthen substituting equation (1) into equation (4), the individual’s Euler equation for consumption
under income uncertainty when there is a CRRA-type utility function can be expressed as follows:C 
E t [ β  t +1  Ct 

−γ

1 + 0.5(γ + γ 2 )CVt +21(1 + r jt +1 )] − 1 = 0
1 + 0.5(γ + γ 2 )CVt 2（j=1,2,…,N）.

(5)

Transforming 1 + 0.5(γ + γ 2 )CVt 2 in the expected marginal utility of equation (1) using a6

linear approximation of the exponential function for the Taylor expansion formula results in

1 + 0.5(γ + γ 2 )CVt 2 ≅ exp[ 0.5(γ + γ 2 )CVt 2 ] . Therefore, the middle term of equation (5) can betransformed as follows:

1 + 0.5(γ + γ 2 )CVt +21  exp(CVt +21 ) ≅
1 + 0.5(γ + γ 2 )CVt 2  exp(CVt 2 ) 

0.5 ( γ +γ 2 )Applying the transformed middle term to equation (5) leads to the following for the Euler
equation for consumption. The transformation also adds the growth rate of the exponential of thesquared value of the CV for consumption as an explanatory variable while the coefficient for the
degree of relative risk aversion 0.5(γ + γ 2 ) is applied as an index of that growth rate of the CV for
consumption.

C β  t +1  Ct 

−γ

 exp(CVt +21 ) 
2  exp(CVt ) 

0.5 ( γ +γ 2 )(1 + r jt +1 ) = 1 .

(6)This is the Euler equation for consumption under income uncertainty formularized with the three
explanatory variables (consumption growth rate, return on assets and the growth rate of the CV for
consumption).

２．２

Analyzing the model for determining the degree of relative risk aversion
and subjective discount rate

Mankiw and Zeldes (1991) applied the Taylor expansion of the two variable functions to theEuler equation with two variables from equation (7) as explanatory variables (consumption growthrate and return on assets). This led to the relational expression among the equity premium, degree ofrelative risk aversion and covariance between return on assets and consumption growth rate whenconstantly abbreviated as expressed as equation (8).

E[(1 + r i )(1 + g C ) −γ ] = 1 + ρ ,(7)E[ri ] − r ≅ γCov(r i , g C ) .

(8)7

However, g C = (Ct +1 / Ct ) − 1 and the subscript for time have been abbreviated. In addition, r i
represents the rate of return on risk asset i , and r represents the rate of return on the risk-free
asset. E[ r i ] − E[ r ] represents the equity premium, and ρ represents the time preference rate(equivalent to (1 / β ) − 1 ).
In accordance with equation(8), the degree of relative risk aversion for the normal model

is defined as the equity premium divided by the covariance between the return on assets and theconsumption growth rate. That is,*
γ NM≅ ( E[ri ] − r ) / σ ic ,(9)

σ ic = Cov ( r i , g C ) .

In addition, the relational expression for the risk-free asset that can be used in the results of equation(8) can be expressed as follows:r ≅ ρ + γE[ g C ] − 0.5(γ + γ 2 )Var ( g C ) .

(10)

Substituting ρ = (1 / β ) − 1 into equation (10), and solving for the subjective discount rate β resultsin the following equation:β ≅ 1 /(1 + E[r ] − γE[ g C ] + 0.5(γ + γ 2 )σ c2 ) ,(11)

σ c2 = Var ( g C ) .
*The subjective discount rate for the normal model is obtained by substituting γ NMasdetermined from equation (9) and substituting the average, variance and covariance for each variable
into equation (11).
On the other hand, the CV model formularized in the preceding paragraph uses three
explanatory variables (consumption growth rate, return on assets and the growth rate of income CV).
This can be expressed in the same way as equations (7) and (8) for the normal model as follows fromequation (B.6) in Appendix B:

8E[(1 + r i )(1 + g C ) −γ (1 + g eCVSQ ) 0.5(γ +γ ) ] = 1 + ρ ,(12)

E[ r i ] − r ≅ γCov (r i , g C ) − 0.5(γ + γ 2 )Cov ( r i , g eCVSQ ) .

(13)

2

However, g eCVSQ = (exp(CVt +21 ) / exp(CVt 2 )) − 1 , and the subscript for time has been abbreviated.
In the normal model shown in equation (8), the degree of relative risk aversion was explained
only through the covariance between the return on assets and consumption growth rate. The degreeof relative risk aversion when extended to the CV model is explained by both the covariance between
the return on assets and the consumption growth rate and the covariance between return on assets
and the growth rate of income CV.
Equation (13) can be rewritten as the function for degree of relative risk aversion γ, resulting
in the following:

Cov ( r i , g C )γ ≅ E[r i ] − r + 0.5Cov ( r i , g eCVSQ )(γ + γ 2 ) .

i

In addition, applying Cov ( r , g

eCVSQ

(14)

) = 0 to equation (14) results in the following equation, whichis the same as the equation for determining the degree of relative risk aversion for the normal model(equation (8)):

Cov ( r i , g C )γ ≅ E[ r i ] − r .

(15)Under normal market conditions when the equity premium exceeds zero, the covariance
between return on assets and the consumption growth rate is positive and the covariance between
return on assets and the growth rate of the income CV is positive or negative, the relationship for
determining the degree of relative risk aversion under the normal model and the CV model based onequation (14) and equation (15) is shown in Figure 1.

【

Figure 19】Figure 1 determines the degree of relative risk aversion for the normal model by the level of*that results in the straight line with the gradient of Cov ( r i , g C ) on the left-hand side ofγ NMequation (15) being consistent with the equity premium of E[ r i ] − r . On the other hand, in the CVmodel, when the covariance between the return on assets and growth rate of income CV is negative,
the right-hand side of equation (14) will resemble (a) with a curve that shifts

to the right of segment

E[ r i ] − r due to the monotonic increase function of γ + γ 2 when γ > 0 . Thus, the degree ofrelative risk aversion for the CV model will be determined by the level of γ CV _ a that intersects*iCwith the straight line with the gradient of Cov ( r , g ) from the left-hand side of equation (14). In
addition, when the covariance between return on assets and the growth rate of the income CV ispositive, the right-hand side of equation (14) will resemble (b) with a curve that shifts to the rightfrom segment E[ r i ] − r . Therefore, it will be determined by the level of γ CV _ b that intersects with*the straight line with the gradient of Cov ( r i , g C ) from the left side of equation (14).
The equity premium puzzle notes that the covariance between return on assets andconsumption growth rate obtained from actual data is extremely low relative to the equity premium.
*
derived from equation (15) is an extremely large
Therefore, the degree of relative risk aversion γ NM

value exceeding what is considered normal (within 10), or the theoretical model only partiallyexplains the equity premium within what is considered normal for the degree of relative risk aversion(within 10).
On the other hand, under the CV model using equation (14), the estimated result can begreatly improved over the normal model, even when covariance between return on assets and theconsumption growth rate is extremely small because the degree of relative risk aversion can be*when covariance between return on assets
determined with the γ CV _ a being much lower than γ NM*

and growth rate of the income CV is a large negative value.
Solving equation (14) for γ means the determinant level for the degree of relative risk
aversion for the CV model can be defined as follows:

γ

*
CV

≅

2(σ ic − 0.5σ iv ) ± 4(σ ic − 0.5σ iv ) 2 − 8σ iv ( E[ri ] − r )2σ iv

σ ic = Cov ( r i , g C ) , σ iv = Cov ( r i , g eCVSQ ) .

10

(16)

The condition for solving the degree of relative risk aversion under a positive risk premiumbythediscriminant

D = 4(σ ic − 0.5σ iv ) 2 − 8σ iv ( E[ri ] − r )fromequation(16)

isσ iv = Cov ( r i , g eCVSQ ) ≤ 0 . Return on assets and the income uncertainty index must conditionallyhave zero or negative covariance. That is, a requirement for solving the CV model is choosing theappropriate choice of income uncertainty index—i.e. income uncertainty falls when return on assets
rises and rises when return on assets falls.
Substitutingρ = (1 / β ) − 1 into equation (B.5) in Appendix B and solving for the

subjective discount rate β results in the following equation:β ≅ 1 /(1 + E[r ] − γE[ g C ] + 0.5(γ + γ 2 )σ c2 + 0.5(γ + γ 2 ) E[ g eCVSQ ]{}+ 0.25(γ + γ 2 ) 0.5(γ + γ 2 ) − 1 σ v2 − 0.5γ (γ + γ 2 )σ cv )(17)

σ c2 = Var ( g C ) , σ v2 = Var ( g eCVSQ ) , σ cv = Cov ( g C , g eCVSQ ) .
*Substituting the γ CVdetermined from equation (16) and the average, variance and covariance foreach variable into equation (17), we obtain the subjective discount rate for the CV model.

３ . Empirical analysis using the U.S. income uncertainty index and concurrent
solution of the equity premium and risk-free rate puzzles

３ .１
１

Data and processing methods
For calculating the degree of relative risk aversion and subjective discount rate we usedmonthly data for the real consumption growth rate, real return on assets (for equities, the S&P 500
Index©; for U.S. government bonds, the secondary market rate for 90-day Treasury bills) and theincome uncertainty index from February 1978 to December 2010. Real consumption uses theaggregate amount of nondurable goods and services per person.
The following four indices are used for the income uncertainty index data.
・ Labour Share (spline conversion to monthly) (CV1).
・ Reciprocal of Unit Profit (spline conversion to monthly) (CV2).
・ Unemployment rate UNRATE (CV3).
・ The reciprocal of the University of Michigan consumer confidence index UMCSENT (CV4).

11

Labour Share and Unit Profit are not released monthly, so quarterly data are converted to monthly
data through non-linear interpolation using the cubic spline function.
It is best to use the figures for each income uncertainty index adjusted for the average value
and standard deviation benchmarked to the CV for consumption. However, no continuous long-term
U.S. data exist for the CV for consumption. We have used original data that has not been adjusted
for the average value and standard deviation. For the observation period, Figure 2 demonstrates the
movement of CV1–CV4 indexed to 100 at the beginning of the period.

【

Figure 2】

Figure 2 shows that CV4 moves almost three months to more than six months ahead of CV3
and at about the same time or much earlier than CV1 and CV2. In addition, the extent of movement
is much smaller.

３ .２
２ Calculations of the degree of relative risk aversion and subjective discount
rate

3.2.1

Trends in Risk premium
The average value for the risk premium is needed to calculate the degree of relative risk

aversion, and a period displaying a positive average value must be chosen to achieve a stable result.
Figure 3 illustrates the movement in average value of the risk premium calculated for each month ofthe observation period. We set February 1978 as the first month and set the end month as each monthafter the next month of the first month up until December 2010. Figure 3 illustrates that the averagevalue of the risk premium was steadily positive for each period when the end month was on or afterJanuary 1985.
【

3.2.2

Figure 3】

Calculations
The calculation of the degree of relative risk aversion considers the observation period andcalculates the covariance for each variable and the average of risk premium for each period. To obtain12the degree of relative risk aversion, the results are substituted into equation (9) for the normal modeland into equation (16) for the CV model. Table 1 presents the descriptive statistics for each variable
used in this calculation.

【

Table 1】

In addition, the calculation of the subjective discount rate considers the observation period
and calculates the average, the variance and the covariance for each variable for each period. [Remark
2] Together with the results from calculating the degree of relative risk aversion for the normal model,
these calculations are substituted into equation (11) to obtain the subjective discount rate of thenormal model. In a similar fashion, together with the results of calculating the degree of relative risk
aversion for the CV model, these are substituted into equation (17) to obtain the subjective discountrate of the CV model. Table 2 presents the descriptive statistics for each variable used in thiscalculation.
【

Table 2】

Figures 4–7 illustrate the results of calculating the degree of relative risk aversion andsubjective discount rate for the observation period for each income uncertainty index (CV1–CV4).
The movement in the result of the degree of relative risk aversion and subjective discount rate
calculated for the normal model and CV model (considering the case that the average values for theincome uncertainty index from CV1 to CV4 are 0.1, 0.3 [Editor1] and 0.5) are illustrated in the graph.

【

Figure 4】【

Figure 5】【

Figure 6】【

Figure 7】

These graphs illustrate the results of the solution for each period wherein the equity premium has astable positive value for the end months of January 1988 and beyond. Average values of the solution
for each period are shown below the graph.

13

Comparing Figures 4–7, the graph for the CV model is not shown in Figure 6, which uses
CV3 because σ iv = Cov ( r i , g eCVSQ ) > 0 , and the real root condition for the solution in equation (16)is not met. Therefore, CV3is noncompliant for the income uncertainty index. As is evident in Figure
2, since there is a change three to six months after the other income uncertainty indices (CV1, 2 and4), the lag in this timing can be attributed to the collapse of the normally anticipated negative
correlation.
The degree of relative risk aversion for the CV model is lower than the normal model due tothe upper graph in Figures 4, 5 and 7 and the average value of the solution. In addition, an even lower
degree of relative risk aversion is obtained if the average values of the income uncertainty index are
increased to 0.1, 0.3 and 0.5. This indicates the CV model that considers the negative covariancerelationship for the return on assets and the income uncertainty index can solve the equity premium
puzzle that points to an abnormal value for the degree of relative risk aversion.
In addition, the subjective discount rate for the normal model repeatedly rises and falls
substantially and holds the level beyond 1 after August 2001 (bottom graphs in Figures 4, 5 and 7)
and the average value of the solution, whereas the subjective discount rate for the CV model has
levelled at slightly below 1. In addition, it has fallen due to a rise of 0.1, 0.3 and 0.5 in the averagevalues of the income uncertainty index. This levelling is pronounced for CV4 in Figure 7. This
indicates the CV model that considers the negative covariance for the return on assets and income
uncertainty index can improve the risk-free rate puzzle that indicatesan abnormal value for the
subjective discount rate.
Comparing Figures 4, 5 and 7 and examining the relative merits of income uncertainty indicesindicate that the degree of relative risk aversion and the subjective discount rates are more suitablevalues for CV4 than CV1 and CV2, and the best as an income uncertainty index. This is attributed to
the negative covariance of the return on assets and income uncertainty index in Table 1 being thelargest for CV4.

４

Explanation of the CV model using the two period model based on Mehra and
Prescott (1985) data

Aggregating the utility function in equation (2) as i=0, 1 for just the two periods, and14

replacing the total utility with Z results in the following equation:

Z = U * (C t ) + β U * (C t +1 ) .

(18)

The Euler equation, equation (6), represents the first-order condition of expected utility
maximization derived from the multi-period optimal consumption model. That condition is the
MRS of the inter-temporal indifference curve − dCt +1 / dCt which is the same as the gradient of thebudget constraint line (1 + rt +1 ) . To obtain this MRS, replace the complete differential of equation(18) with zero to arrive at dZ = U * ' (Ct ) dCt + β U * ' (Ct +1 )dCt +1 = 0 . Doing so generates thefollowing equation:−

dCt +1
U * ' (Ct )Ct−γ [1 + 0.5(γ + γ 2 )CVt 2 ]==.
dCt dZ = 0 βU * ' (Ct +1 ) βCt−+γ1[1 + 0.5(γ + γ 2 )CVt 2+1 ](19)Transforming the CV item for equation (19) using the Taylor expansion formula for the same index
function as in Section 2 means the MRS for the indifference curve under the two-period model for
the CV model can be represented as follows:2

dC t +1
1 C 
−=  t 
dC t dZ =0 β  C t +1 −γ exp(CVt 2 ) 
2 exp(CVt +1 ) 

0. 5 ( γ + γ 2 )(20)

On the other hand, the MRS for the indifference curve under the two-period model for the normal
model can be represented by the following equation that eliminated the CV item for equation (20) by
substituting exp( 0) = 1 for the numerator and the denominator in equation (20) for the CV for
consumption CVt to remain zero.

1 C 
dC− t +1=  t dCt dZ = 0 β  Ct +1 

2

−γ

.

(21)

When the MRS for equation (20) is the same as the gradient for the budget constraint line, it can be transformed to derive
the consumption Euler equation under income uncertainty（equation (6)).

15

Now, in the case that income uncertainty rises from period ｔ to period t+1 so thatCVt +1 > CVt , the MRS for the indifference curve under the normal model will not change due toequation (21). However, in the CV model under equation (20), the third item on the right side, whichis the growth rate for the consumption CV exp(CVt 2 ) / exp(CVt 2+1 ) , will be below 1. Therefore, thedecline in the MRS produces a change in the focus on future consumption for the entire indifference
curve. On the other hand, if income uncertainty falls from period ｔ to period t+1 so that

CVt > CVt +1 , the MRS for the indifference curve under the normal model will not change due toequation (21). However, in the CV model under equation (20), the third item on the right side, whichis the growth rate for the consumption CV exp(CVt 2 ) / exp(CVt 2+1 ) , will exceed 1. Therefore, the risein the MRS prompts a change in the focus on present consumption for the entire indifference curve.
Figure 8 shows the correction effect using the two-period model in the estimated value of thedegree of relative risk aversion due to the CV model based on the Mehra and Prescott (1985) data
used to indicate the U.S. equity premium puzzle.

【

Figure 8】

In Figure 8, the solid line represents the budget constraint that corresponds to the maximum value(1.40649) and the minimum value (0.72354) for the annualized rate of the return on stocks from 1949
to 1979. The dotted line rising rightward shows the extent of the maximum value (1.04080) andminimum value (0.99650) for the annualized consumption growth rate (cons) fixed at thecorresponding subjective equilibrium point. Thus, the magnification of the standard deviation for thestock

return relative

to the standard deviationforthe consumption growthrate is0.13555/0.01227=11.05x. The extent of change in the consumption growth rate is very small incomparison to the large change in the budget constraint line. The indifference curve for the normal
model under this constraint is illustrated by the extremely large convexity in the dotted line in Figure
8. The degree of relative risk aversion for the corresponding CRRA-type utility function is also
extremely large.
On the other hand, under the CV model that considers the negative covariance relationshipfor return on assets and income uncertainty index, a rise in return on assets coincides with a decline16

in future income uncertainty under household sentiment. Therefore, the increase in the MRS shifts
the focus on present consumption for the entire indifference curve. The indifference curve indicatinga rise in MRS is not constrained by the budget constraint corresponding to a minimum value(0.72354) for return on stocks. Because it is possible to gently approach something like the solid line
in Figure 8, there is no increase in convexity of the indifference curve even when the consumption
growth rate fluctuates only slightly. The degree of relative risk aversion for the CRRA-type utility
function can be estimated within the normally anticipated realm.
Return on assets and income uncertainty generally trend towards a negative correlation.
When an excessive return degree of relative risk aversion is estimated because the fluctuation in
consumption is small relative to the fluctuation in return on assets, applying the CV model improves
estimation results.

５Conclusions

A typical attempt to solve the equity premium puzzle uses a non-CRRA alternative preference
structure (the time non-separable model that does not assume separability in relation to time andhabit formation). However, the generalization and specialization of these utility functions have notyielded generic solutions to the puzzle. Thus, from the viewpoint of parsimony, it would be preferable
if the equity premium puzzle could be explained using a simple CRRA utility function.
To this point, data indicates that our model is useful in solving the equity premium puzzle
and risk-free rate puzzle with the CRRA utility function. Moreover, it could act as one of the genericsolution because there is no need to assume particular household behaviours such as habit formation.
This study performed two tasks to solve the puzzle. First, in addition to the return on assets
when deciding the optimal consumption path over multiple periods, we assumed that households
accounted for income uncertainty. Similar to the extensive acceptance of precautionary savings
arising from income uncertainty, our assumptions have already been accepted by several researchers.
Second, we formulated the effect of precautionary savings and withdrawal of these savings that arise
from the change in income uncertainty by incorporating them into the Euler equation. Doing so
enabled us to estimate the CRRA utility function parameters for households, simultaneously
accounting for both the profitability of financial assets and income uncertainty.

17

Until now, dynamic optimization of consumption across periods and precautionary savings
have been discussed as separate topics. We believe that unifying them within one Euler equation isessential to the development of consumption theory and to stable estimations of generalized methodof moments (GMM). This unification forges a close and compatible relationship between theory andreality and emerges as a principal way to solve the stagnation seen in applying consumption theories
such as C-CAPM, which has been affected by this puzzle over the past 30 years.

18

Appendix A: Derivation of the individual’s expected marginal utility function underincome uncertainty
First, we take Ct to represent real consumption for an individual’s period t, and take U (Ct )to represent the instantaneous utility function that has additive separability for that point in time.
Next, we represent the level of wavering in consumption due to income uncertainty as standard
deviation ht , and assume an uncertain situation in which case there is a 50% probability that
individual consumption will increase by only ht and a 50% probability it will decrease by only ht .
The utility that takes account of the uncertainty in the individual consumption level Ct at this time
is as follows.

U * (Ct ) = 0.5U (Ct − ht ) + 0.5U (Ct + ht )(A.1)

Here, taking the extent of reduction in the utility level for the consumption level Ct due tothe income uncertainty as

ρ (Ct , ht ) , ρ (Ct , ht ) = U (Ct ) − U * (Ct ) and equation (A.1) result in thefollowing equation.

U (Ct ) − ρ (Ct , ht ) = 0.5U (Ct − ht ) + 0.5U (Ct + ht )(A.2)

Taking the Taylor expansion up to the second order term of the U (Ct − ht ) and U (Ct + ht )on the right hand side of equation (A.2) and substituting them respectively into equation (A.2) once
again means that

ρ (Ct , ht ) can be expressed as follows.

ρ (Ct , ht ) = −0.5U '' (Ct ) ht2(A.3)

Now, we specify the utility function as the following constant degree of relative risk aversion(CRRA) type.

U (Ct ) = Ct1−γ /(1 − γ ),γ ≠ 1,= ln(Ct ),γ = 1.

19

However, γ is a parameter representing a constant degree of relative risk aversion, and 1 / γ
represents the inter-temporal elasticity of substitution. The second derivative of the CRRA typeutility function is − γ / Ctγ +1 , so substituting this into equation (A.3) means that

ρ (Ct , ht ) can be

expressed as follows.

ρ (Ct , ht ) = 0.5γCt1−γ (ht / Ct )2(A.4)

However, ( ht / Ct ) 2 represents the square value of the coefficient of variation for consumption atperiod t, so hereinafter we express this as CVt 2 .
Substituting

the

CRRA

type

utilityfunctionandequation

(A.4)

into

U * (Ct ) = U (Ct ) − ρ (Ct , ht ) means the individual expected utility function under income
uncertainty can be expressed as follows.

[

U * (Ct ) = Ct1− γ /(1 − γ ) 1 − 0.5(γ − γ 2 )CVt 2

]

(A.5)

Taking the derivative of equation (A.5), the individual’s expected marginal utility function
under income uncertainty can be expressed as follows. 3[

U * ' (Ct ) = Ct−γ 1 + 0.5(γ + γ 2 )CVt 2

]

(A.6)

Appendix B: The Taylor expansion for the three variable’s Euler equation of thecoefficient of variation model
From equation (12) , target equation for the Taylor expansion can be expressed as follows.

f (r i , g C , g eCVSQ ) = (1 + r i )(1 + g C ) −γ (1 + g eCVSQ) ) 0.5(γ +γ )
2

(B.1)

On equation (B.1), by taking the Taylor expansion of the three variable functions up to the second
order with r = gi3C= g eCVSQ = 0 or thereabouts and substituting the derived derivatives from a∂CVt / ∂Ct = − ht / Ct2 is used in the development from equation (A.5) to equation (A.6).
20

separate calculation into it results in the following approximation equation.

f (r i , g C , g eCVSQ ) = 1 + r i − γg C − γr i g C + 0.5(γ + γ 2 )( g C ) 2+ 0.5(γ + γ 2 ) g eCVSQ + 0.25(γ + γ 2 )[0.5(γ + γ 2 ) − 1]( g eCVSQ ) 2− 0.5γ (γ + γ 2 ) g C g eCVSQ + 0.5(γ + γ 2 )r i g eCVSQ(B.2)

Substituting the result of equation (B.2) into equation (12) leads to the following.

E[r i ] − γE[ g C ] − γ {E[r i ]E[ g C ] + Cov(r i , g C )} + 0.5(γ + γ 2 ){( E[ g C ]) 2 + Var ( g C )}+ 0.5(γ + γ 2 ) E[ g eCVSQ ]+ 0.25(γ + γ 2 ){0.5(γ + γ 2 ) − 1}{( E[ g eCVSQ ]) 2 + Var ( g eCVSQ )}− 0.5γ (γ + γ 2 ){E[ g C ]E[ g eCVSQ ] + Cov( g C , g eCVSQ )})+ 0.5(γ + γ 2 ){E[r i ]E[ g eCVSQ ] + Cov(r i , g eCVSQ )} ≅ ρIgnoring each item

E[r i ]E[ g C ] ,( E[ g C ])2 ,( E[ g eCVSQ ]) 2 ,(B.3)

E[ g C ]E[ g eCVSQ ] ,E[r i ]E[ g eCVSQ ] with a comparatively small value, replacing with 0, and solving for E[r i ] resultsin the following equation.

E[r i ] ≅ ρ + γE[ g C ] + γCov (r i , g C ) − 0.5(γ + γ 2 )Var ( g C )− 0.5(γ + γ 2 ) E[ g eCVSQ ] − 0.25(γ + γ 2 ){0.5(γ + γ 2 ) − 1}Var ( g eCVSQ )+ 0.5γ (γ + γ 2 )Cov( g C , g eCVSQ ) − 0.5(γ + γ 2 )Cov(r i , g eCVSQ )(B.4)

The rate of return on the risk free asset is determined to have no relationship to theconsumption growth rate and the growth rate of income coefficient of variation, so substituting

Cov(r i , g C ) = 0 and Cov(r i , g eCVSQ ) = 0 into equation (B.4) results in the following equation.

r ≅ ρ + γE[ g C ] − 0.5(γ + γ 2 )Var ( g C ) − 0.5(γ + γ 2 ) E[ g eCVSQ ]− 0.25(γ + γ 2 ){0.5(γ + γ 2 ) − 1}Var ( g eCVSQ )+ 0.5γ (γ + γ 2 )Cov( g C , g eCVSQ )

21

(B.5)

Subtracting equation (B.5) from equation (B.4) leads to the following equation relating to theequity premium, the degree of relative risk aversion, the covariance between the return on assets andthe consumption growth rate, as well as the growth rate of income coefficient of variation.

E[ r i ] − r ≅ γCov (r i , g C ) − 0.5(γ + γ 2 )Cov ( r i , g eCVSQ )

22

(B.6)

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26

Table 1
Descriptive statistics for the calculation of the degree of relative risk aversion

27

Table 2
Descriptive statistics for the calculation of the subjective discount rate28

Figure 1
Determination of the degree of relative risk aversion

E[ r i ] − r + 0.5Cov ( r i , g eCVSQ )(γ + γ 2 ) , Cov (r i , g C )γ

γ +γ2

(b)

E[ r i ] − r

Cov (r i , g C )

(a)

γ
*
γ CV
_a

CV modeli（ Cov ( r , g

eCVSQ

*
γ NM

Normal model) < 0）*γ CV_b

CV modeli（ Cov (r , g

29

eCVSQ

) > 0）

Figure 2
Movement of the US income uncertainty index（CV1-4, Feb-1978=100）

Figure 3
Movement in the average value of the risk premium

30

Figure 4
The results of the calculation of the degree of relative risk aversion and subjective discount rate
(Income uncertainty index: Labor Share CV1)

31

Figure 5
The results of the calculation of the degree of relative risk aversion and subjective discount rate
(Income uncertainty index: The reciprocal of Unit Profit CV2)32

Figure 6
The results of the calculation of the degree of relative risk aversion and subjective discount rate
(Income uncertainty index: The Unemployment Rate UNRATE CV3)33

Figure 7
The results of the calculation of the degree of relative risk aversion and subjective discount rate
(Income uncertainty index: The reciprocal of the University of Michigan consumer confidence index
UMCSENT CV4)

34

Figure 8
The correction effect in the estimated value of the degree of relative risk aversion

Ct +1

Yt +1
Normal modelCV modeli（ Cov(r , g

eCVSQ

Ct
max 1.04080 ～min 0.99650

Yt

max 1.40649 ～min 0.72354(st.dev. 0.01227)(st.dev. 0.13555)

cons

stocks35) < 0）