Asset Pricing in the Presence of Background Risk

Andrei Semenov (York University)

2

Introduction

The standard consumption CAPM:

Problems:

a) The equity premium puzzle

b) The risk-free rate puzzle

1=1,1

ti

t

tt R

C

CE

3

Generalizations:

a) Preference modifications

b) State-dependent parameters

c) Psychological models of preferences

d) Incomplete consumption insurance

Brav et al. (2002), Balduzzi and Yao (2007), and Kocherlakota and Pistaferri (2009)

4

Outline Consumption-Based Model with Background Risk

The stochastic discount factor (SDF) Risk vulnerability and the asset pricing puzzles Risk aversion and the EIS under background risk

Empirical Investigation The data The estimation procedure (calibration, conditional

HJ volatility bounds) Estimation results

Concluding Remarks

5

1. A Consumption-based asset pricing model with background risk1.1 The Stochastic Discount Factor (SDF)

The agent faces (Franke et al. (1998) and Poon and Stapleton (2005)): The financial investment risk An independent, non-hedgeable, zero-mean background risk (loss of employment, divorce, etc.)

6

In the presence of background risk, the representative agent maximizes:

is the agent i’s hedgeable consumption in period

. The non-hedgeable consumption

is independent of both optimal hedgeable consumption and the risky payoff and has a zero expected value.

t

tititi ,,,

0= ,, )]([[=

titit CuEEU

tiC ,

7

One of the first-order conditions:

or

This is the consumption CAPM with background risk. The SDF:

In the absence of background risk,)]([

)]([

,,

1,1,1

titi'

titi'

t CuE

CuEM

1=)]([

)]([1,

,,

1,1,

tjtiti

'

titi'

t RCuE

CuEE

])]([[=)]([ 1,1,1,,, tjtiti

'ttiti

' RCuEECuE

)(

)(

,

1,1

ti'

ti'

t Cu

CuM

8

1.2 The precautionary premium

Following Kimball (1990):

where is an equivalent precautionary premium.

Assume , then

),(=)]([ ,,,, titi'

titi' CuCuE

),)(,(= ,,, tititi uC

)(

)(

2

1),)(,(=

,

,2,,,,

ti''

ti'''

titi'

titi Cu

CuuC

2,

*,

1=

12,,

2, )(=]])[[(= titis

S

stititi CCSEE

0>= ,,*

, tistitis CC

9

We can write the SDF as:

Assume that the utility function is CRRA:

The precautionary premium for the agent with CRRA utility is hence

)(

)(=

,,

1,1,1

titi'

titi'

t Cu

CuM

1

1=

1,tiC

u

.1

2

1

,

2,

titit C

10

This implies that, with CRRA utility, for any t

Where is the normalized variance of :

We need for marginal utility to be well-defined.

The SDF is then

2

,,2,,,, 2

)1(exp

2

11=)( titititititi

' CCCu

2

,

*,

1=

12

,,

2,2

, 1=])[(

=

ti

tisS

stiti

titi C

CS

EC

2,ti ti ,

12/< 2, ti

21,

,

1,1 2

)1(exp= ti

ti

tit C

CM

11

1.3 Risk vulnerability and the asset pricing puzzles

As introduced by Kihlstrom et al. (1981) and Nachman (1982), define the following indirect utility function:

Gollier (2001) argues that, in the case of the background risk with a non-positive mean preferences exhibit risk vulnerability if and only if the indirect utility function is more concave than the original utility function, i.e.,

)]([=)( ,,, tititi CuECg

)(

)(

)(

)(

,

,

,

,

ti'

ti''

ti'

ti''

Cu

Cu

Cg

Cg

12

As shown by Gollier (2001), this inequality holds if at least one of the following two conditions is satisfied:

(i) ARA is decreasing and convex and

(ii) both ARA and AP are positive and decreasing in wealth (standard risk aversion (Kimball (1993)).

13

Risk vulnerability and the equity premium puzzleThe consumption CAPM with background risk in terms of :

Assume

Then

If , then is less concave than utility and hence .

)(u

1

1=

1,tiC

g

1=1,,

1,

tj

ti

tit R

C

CE

0][ , tiE

)(g

)(g1=

)(

)(1,

,

1,

tj

ti'

ti'

t RCg

CgE

14

1.4 Risk aversion and the EIS under background risk

Suppose that at time t the agent faces the background risk and a lottery with an uncertain payoff .

For any distribution functions and :

The RRA coefficient

tZ

ZF G

)]][([)]([ ,,,,,,,,

tititititititiZ ZECuEZCuE

)]([

)]([

2

1

,,

,,2,,

titi'

titi''

titi CuE

CuE

tititi

'

titi''

ti CCuE

CuE,

,,

,,, )]([

)]([=

15

Since is independent of ,

and then we may suppose

Denote as the risk premium we would observe if the utility curvature parameter were .

The proportion of the risk premium due to the background risk in the total risk premium is

ti, tiC ,

)]([=)( ,,)(

,)(

titin

tin CuECg

titi

'ti

''

ti CCg

Cg,

,

,, )(

)(=

ti ,

~

titi

titi

,,

,, 1~

16

Kimball (1992) defines the temperance premium by the following condition:

By analogy with the risk premium,

The conditions and (the necessary conditions for risk vulnerability) imply that .

ti ,

)][(=)]([ ,,,,, tititi''

titi'' ECuCuE

])[(

])[(

2

1

,,

,,2,,

titi'''

titi''''

titi ECu

ECu

0<)(''''u0, ti

0)( '''u

17

We have

With power utility

titititi

'

tititi''

t CECu

ECu,

,,,

,,,

)][(

)][(=

][

2

2

1

,,

2,

tititit EC

2,,

2,

1

2,,

2,

,,

,

])[(2

)1(1

])[(2

)2(1

][=

titi

ti

titi

ti

titi

tit

EC

ECEC

C

18

19

The EIS in the model with an independent non-hedgeable background risk

Since the representative agent with utility facing background risk has the same optimal consumption plan as the representative agent with utility in the no background risk environment, we can suppose that

where and

The model with background risk enables us to disentangle the curvature parameter and the EIS in the expected utility framework.

)(g

11

=][

= 1,1,

1,1,

tg

tF

tittu r

cE

)(u

)/(= ,1,1, tititi CClnc )(= 1,1, tFtF Rlnr

20

2. Empirical investigation2.1 The data

The consumption data

Quarterly consumption data (consumption of nondurables and services (NDS)) from the CEX (the US Bureau of Labor Statistics) from 1982:Q1 to 2003:Q4.

We drop householdsa) that do not report or report a zero value of consumption of food, consumption of nondurables and services, or total consumption,

21

b) nonurban households, households residing in student housing, households with incomplete income responses, households that do not have a fifth interview, and households whose head is under 19 or over 75 years of age.

We consider four sets of households based on the reported amount of asset holdings at the beginning of a 12-month recall period in constant 2005 dollars:

a) all households,

b) households with total asset holdings > $ 0,

c) households with total asset holdings $ 1000,

d) households with total asset holdings $ 5000.

22

The returns data

a) The nominal quarterly value-weighted market capitalization-based decile index returns (capital gain plus all dividends) on all stocks listed on the NYSE, AMEX, and Nasdaq are from the CRSP.

b) The nominal quarterly value-weighted returns on the five and ten NYSE, AMEX, and Nasdaq industry portfolios are from Kenneth R. French's web page.

c) The nominal quarterly risk-free rate is the 3-month US Treasury Bill secondary market rate from the Federal Reserve Bank of St. Louis.

d) The real quarterly returns are calculated as the quarterly nominal returns divided by the 3-month inflation rate based on the deflator defined for NDS.

2.2 The empirical SDF

Under Background risk:

Recall that

In any state s, . Since the background risk has a zero-mean,

23

21,

11 2

)1(exp= ti

t

tt C

CM

2

,

*,

1=

12, 1=

ti

tisS

sti C

CS

tistitis CC ,,*

, =

S

s titisti CCSCE1 ,

*,

1*, =][

Denote as

24

2

1

*,

1

*,

1=

12, 1=

S

s tis

tisS

sti

CS

CS

*1,

*,, / tsitistsis CC

2

1 ,1

,

1=

12, 1=

S

s tsis

tsisS

sti

SS

2

,

1=

1

2

1 ,1

,

1=

12 11

t

tiN

iS

s tss

tssS

st q

qN

SS

ti

N

it qNq ,1=

1=

25

2.3 The estimation

We candidate SDFs are1. Standard SDF:

2. The SDF in Brav et al. (2002):

3. The SDF in the consumption CAPM with background risk:

t

tt C

CM 1

1 =

3

1

1,

1=

2

1

1,

1=11 1

6

)2)(1(1

2

)1(1=

t

tiN

it

tiN

itt q

q

Nq

q

NqM

21

11 2

)1(exp= t

t

tt C

CM

26

For each of the above SDFs, we test the conditional Euler equations for the excess returns on risky assets

and the risk-free rate

Denote the error terms in the Euler equations as

Thus, at the true parameter vector,

where is a variable in the agent's time t information set.

0Z =][ 11 ttt ME

1=][ 11, ttFt MRE

,= 111 ttt MZε 1= 11,1, ttFtF MR

0,ε =][ 1 tt xE 0=][ 1, ttF xE

tx

Calibration and HJ volatility bounds

We calculate the statistic

Denote as the utility curvature parameter in .

The optimal value of the estimate of the curvature parameter is

Denote as the instrument , for which

27

,)(~

)( 1,

1

0 1,11

ttF

T

t tMtt xRRMTx

1

1 =~

t

t

MM

1

~tM

)(ˆ=ˆ}{

ttx

opt xmaxopttx tx opt

tx ˆ=)(ˆ

Since

or equivalently

we estimate the subjective time discount factor as

and the RRA coefficient as

28

0=])1([ 11,opttttF xMRE

],[=]~

[ 11,optt

opttttF xExMRE

optt

optttF

T

t

optt

T

t

xMR

x

)ˆ(~

=ˆ

11,

1

0=

1

0=

)ˆ)1ˆexp((ˆˆ 2t

optoptt

29

A lower volatility bound for admissible SDFs , which have unconditional mean m and satisfy

where is the unconditional variance-covariance matrix of . We look for the values of the SDF parameters, at which a considered SDF satisfies the volatility bound, i.e.,

where

0ε =][ 1 tt xE

)(1 mM at

1/21

11

21 ])[][(=))(( tt

'tt

at xExEmmM

ZΣZ

tt x1ZΣ

1>])[][~(

))~(~

(=

))((

))((1/2

11

12

1

1

1

tt'

tt

tat

t

xExEm

mM

mM

mM

ZΣZ

,=~ 1

1

t

t

MM 1

1

0=

1 ~=~

t

T

tMTm

30

31

32

33

34

Concluding remarks Empirical evidence is that, in contrast with the

previously proposed incomplete consumption insurance models, the asset-pricing model with the SDF calculated as the discounted ratio of expectations of marginal utilities over the non-hedgeable consumption states at two consecutive dates jointly explains the cross-section of risky asset excess returns and the risk-free rate with economically plausible values of the RRA coefficient and the subjective time discount factor.

The results are robust across different sets of stock returns and threshold values in the definition of asset holders.

35

The size of the background risk is an important component of the pricing kernel. This supports the hypothesis that the independent non-hedgeable zero-mean background risk can account for the market premium and the return on the risk-free asset.