Agents are assumed to have a power risk aversion utility function in an otherwise standard asset pricing model. These preferences are shown to be capable of eliminating one version of the equity premium and risk free rate puzzles when they display decreasing relative risk aversion.asset pricing,equity premium,risk aversion

M. J. Roche

English

Research Papers in Economics

 The equity premium puzzle and decreasing relative risk aversion 
 
Maurice J. Roche 
 
The National University of Ireland, Maynooth, Co. Kildare, Ireland. 
 
 
 
 
Abstract 
Agents are assumed to have a power risk aversion utility function in an otherwise standard asset 
pricing model.  These preferences are shown to be capable of eliminating one version of the 
equity premium and risk free rate puzzles when they display decreasing relative risk aversion.  
 
 
Keywords: asset pricing; equity premium; risk aversion 
 
JEL classification:  G10, G12  
 
 
 
 
 
 
 
 
  
1. Introduction 
It has been twenty years since the seminal paper of Mehra and Prescott (1985) that first 
articulated the equity premium puzzle.  In a recent critical review of the literature Mehra and 
Prescott (2003) point out that many of the resolutions to solve this puzzle have failed.  The 
assumption that agents have preferences that exhibit decreasing relative risk aversion has 
recently been shown to eliminate one version of the equity premium and risk-free rate puzzles 
(see Meyer and Meyer (2005)).  However their utility function is too cumbersome to use in many 
other applications and lacks intuition.  In this paper we show that a power risk aversion utility 
function that displays decreasing relative risk aversion is also capable of eliminating the puzzles. 
 
2. A standard asset pricing model 
Mehra and Prescott (1985) derive two Euler equations from a standard asset pricing model 
where all agents choose consumption so as to maximize the present discounted value of future 
expected utility arising from random consumption streams.  The two equations are given by 
  ()
1
11
()
0
()
eb t
tt t
t
Uc
ER R
Uc
+
++
 ′
−  ′ 
=   (1) 
and 
 
1
1
()
1
()
b t
tt
t
Uc
ER
Uc
+
+
 ′
−=  ′ 
β 0   (2) 
where Et is the expectations operator conditional on information at time t,  is the marginal 
utility of real consumption per capita,
( ) t Uc ′
1
e
t R + is the gross real return on equity,  1
b
t R +  is the gross real 
return on bonds andβ is a constant discount factor.  Kocherlakota (1996) uses the law of iterated 
  1 
expectations to replace the conditional expectation in equations (1) and (2) with an unconditional 
expectation and estimates the population means of  
  (
1
11
()
()
ee t
tt
t
Uc
eR
Uc
+
++
 ′
=−  ′  ) 1
b
t R +   (3) 
and 
 
1
1
()
1
()
b t
t
t
Uc
e
Uc
+
+
 ′
=  ′ 
β 1
b
t R + −   (4) 
using annual U.S. data from 1889-1978.  Kocherlakota (1996) assumes that agents have utility 
functions that exhibit constant relative risk aversion and finds that one or both of individual null 
hypotheses  1 0
e
t e + =  and  1 0
b
t e + =  are rejected for the same parameter of constant relative risk 
aversion which he varied from 0 to 10.  The null hypothesis  1 0
e
t e + =  is not rejected, at the 95% 
significance level, for values of the parameter of constant relative risk aversion greater than 
seven, while the null hypothesis  1 0
b
t e + =  is not rejected, at the 95% significance level, for values 
of the parameter of constant relative risk aversion less than one.  This finding characterizes one 
version of the equity premium puzzle. 
 
3. Decreasing relative risk aversion 
Meyer and Meyer (2005) point out that a reason why preferences that allow habit 
formation, as in Campbell and Cochrane (1999), can “reduce or eliminate the equity premium 
puzzle” is because their utility function displays decreasing relative risk aversion.  Meyer and 
Meyer (2005) consider a set of utility functions where marginal utility is given by 
  () t
t
Uc
c
′ = λ
δ
λ
 (5) 
  2 
for  0 λ >  and  0. δ >   Relative risk aversion is given by / . t c
λ δ  The parameter λ  governs the rate 
of decrease in relative risk aversion.  Using the same data set as Kocherlakota (1996) they 
normalize the level of real consumption per capita in 1889 to unity and calculate relative risk 
aversion for average consumption as  /2.3.
λ δ  Meyer and Meyer (2005) choose values for λ  
ranging from 0.5 to 2 and adjust δ  so as relative risk aversion for average consumption is in the 
0.5-10 range.  A sample of their results is presented in Table 1.  They show that for a large range 
of values of λ  and δ  the t-statistics for testing whether the null hypotheses  1
e
t+ 0 e =  and  1
b
t e + 0 =  
are not rejected when relative risk aversion for average consumption is in the 6-10 range and 
0.99. β =  Assuming that  0.99 β =  we estimated the parameters λ  and δ  using equations (1) 
and (2) by generalized method of moments
1 using the data set from Kocherlakota (1996).   We 
estimatedλ  to be 2.18 with a standard error of 1.23 and estimatedδ to be 36.32 with a standard 
error of 17.02.  The probability value of the J-test for overidentifying restrictions is 0.45.  The 
parameter estimates of λ  and δ  are significant at the 10% and 5% levels respectively.  Most 
values of λ  and δ  used by Meyer and Meyer (2005) which were presented in Table 1 are within 
the 95% confidence interval of our estimates of λ  and  . δ  
One problem with Meyer and Meyer (2005) preference specification is that the 
underlying utility function is rather cumbersome.  They do not actually present the utility 
function in their paper and state “the exact form of the utility function is unknown”.  Using a 
mathematics software program such as Maple one can integrate equation (5) with respect to 
consumption and this gives the following utility function 
 
1
11
() ,
t
t
tt
c
Uc
cc
λ
λ
δ
λλ λ λ λ
    =− − Γ − −Γ − −    
   
λ
δ 
 

 (6) 
  3 
where 
1  Γ− 


 λ
is a gamma function and 
1
,
t c
λ
δ
λλ

− 


 Γ− is an incomplete gamma function.  We 
believe that using this function in other applications would be difficult and lack intuition. 
In this paper we offer an alternative functional form for utility, the power risk aversion 
utility function (see Xie (2000)).  This function has the property of decreasing relative risk 
aversion under certain parameter values.  The function is given by 
 
1 1
() 1e x p 1
1
t
t
c
Uc
−   
=−− −     −   
α
γ
γα
     (7) 
Note when  0 > α  and  0 = γ  then equation (7) is the commonly used constant relative risk 
aversion utility function.  Marginal utility is given by 
 
1
() e x p 1
1
t
tt
c
Uc c
−
−  
′ =− −   −  
α
α γ
α

   (8) 
and relative risk aversion is
1 . t c
− +
α αγ  Thus equation (7) exhibits decreasing relative risk 
aversion when  1 α >  and  0. γ >   We calculate the t-statistics for testing the asset pricing Euler 
equations (1) and (2).  Similar to Meyer and Meyer (2005) we show that for a large range of 
values of α  and γ  the t-statistics for testing whether the null hypotheses 1 0
e
t e + = and 1 0
b
t e + = are 
not rejected when relative risk aversion for average consumption is in the 6-10 range.  We 
present our results in Table 2.  Since α  governs the rate of decrease in relative risk aversion we 
choose values for α  ranging from 2 to 5.  Then is γ  calculated so that relative risk aversion for 
average consumption takes on values 6, 8 or 10. 
Assuming that  0.99 β =  we estimated the parameters α  and γ  using equations (1) and 
(2) by generalized method of moments.
2  We estimated α  to be 3.95 with a standard error of 
2.98 and estimated γ  to be 30.93 with a standard error of 16.14.  The probability value of the J-
  4 
test for overidentifying restrictions is 0.39.  The parameter estimates of α  and γ  are significant 
at the 20% and 10% levels respectively.  Most values used in Table 2 are within the 95% 
confidence interval of our estimates of α  and  . λ  
 
4. Conclusions 
In a recent paper Meyer and Meyer (2005) show that preferences that display decreasing 
relative risk aversion are capable of eliminating one version of the equity premium and risk free 
rate puzzles.  We suggest that their utility function is too cumbersome to use in many other 
applications and lacks intuition.  We show that a power risk aversion utility function that 
displays decreasing relative risk aversion is also capable of eliminating the puzzles.  This 
function is relatively straightforward to use and has been employed in other applications (see Xie 
(2000). 
 
  5 
References 
Meyer, D. and J. Meyer, 2005.  Risk preferences in multi-period consumption models, the equity 
premium puzzle, and habit formation utility.  Journal of Monetary Economics, forthcoming. 
Kocherlakota, N., 1996. The equity premium: It’s still a puzzle.  Journal of Economic Literature 
34:42-71. 
Mehra, R. and Prescott, E., 1985. The equity premium: A puzzle.  Journal of Monetary 
Economics 15:145-61. 
Mehra, R. and Prescott, E., 2003. The equity premium in retrospect.  The Handbook of the 
Economics of Finance, Eds. Constantinides, G., Harris, M., and Stulz, R., North-Holland: 
Amsterdam, 888-936. 
Xie, D.,  (2000). Power risk aversion utility functions. Annals of Economics and Finance 1, 265-
282.  
  6 
Table 1 
Testing asset pricing Euler equations using the Meyer and Meyer (2005) utility function 
Relative risk aversion at the 
average consumption level 
λ  δ  t-statistic 
01 :0
e
t He + =  
t-statistic 
01 :0
b
t He + =  
6 1  14.04  1.85  -1.92 
 1.5  21.48  1.64  -1.21 
 2  32.88  1.34  -0.34 
      
8 1  18.72  1.33  -1.42 
 1.5  28.64  1.07  -0.60 
  2  43.84  0.68    0.34 
        
10 1  23.40  0.09  -0.92 
 1.5  35.80  0.55  -0.03 
  2   54.80   0.09   0.84 
Source: Table 3 in Meyer and Meyer (2005) 
  7 
Table 2 
Testing asset pricing Euler equations using the power risk aversion utility function 
Relative risk aversion at the 
average consumption level 
α  γ  t-statistic 
01 :0
e
t He + =  
t-statistic 
01 :0
b
t He + =  
6 3  20.12  1.71  -2.10 
 4  24.43  1.63  -1.61 
 5  28.13  1.66  -1.50 
      
8 3  26.52  1.17  -1.16 
 4  48.86  0.97  -0.01 
 5  84.40  0.72    0.83 
        
10 3  37.12  0.69  -0.35 
 4  73.29  0.39    0.84 
 5  140.67  -0.52    1.06 
 
 
  8 
  9
                                                
Endnotes 
 
1 The instruments chosen were a constant, consumption lagged one period, the equity premium 
lagged one period and the real return on treasury bills lagged one period. 
2 The instruments were the same as in footnote 1. 

Power risk aversion utility functions.

Risk preferences in multi-period consumption models, the equity premium puzzle, and habit formation utility.

The equity premium in retrospect.

The equity premium: A puzzle.

The equity premium: It’s still a puzzle.

The equity premium puzzle and decreasing relative risk aversion

http://economics.nuim.ie/sites/economics.nuim.ie/files/working-papers/N1510205.pdf

The equity premium puzzle and decreasing relative risk aversion

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics