A Note on Wealth Effect under CARA Utility
Dmitry Makarov
New Economic School
Nakhimovsky pr. 47
Moscow 117418
Russia
Tel: 7 495 129-3911
Fax: 7 495 129-3722
E-mail: [email protected]
Astrid Schornick
Insead
Boulevard de Constance
77300 Fontainebleau Cedex
France
Tel: 33 (0)1 60 72 49 92
Fax: 33 (0)1 60 72 40 45
E-mail: [email protected]
April 2010
Electronic copy available at: http://ssrn.com/abstract=1350225
Abstract
There is a simple but overlooked way of capturing the wealth effect under CARA utility
via making the absolute risk aversion parameter wealth-dependent. Focusing on the setting of
Verrecchia (1982), we compare our approach with that of Peress (2004) who instead changes preferences. We demonstrate that implementing our approach leads to a straightforward, tractable
analysis, while Peress has to resort to approximate methods. Importantly, our closed-form solution reveals that the relation between wealth and wealth share invested in a risky asset can
be negative, while Peress’s main result is that this relation is uniquely positive.
JEL Classifications: D31, D51, D82, D83.
Keywords: CARA utility, wealth effect, information acquisition, asset pricing, portfolio choice.
Electronic copy available at: http://ssrn.com/abstract=1350225
1.
Introduction
In various economic settings, assuming normally distributed returns and exponential utility is
necessary for tractability. This is often the case for models with asymmetric information, pioneered
by Grossman and Stiglitz (1980) and Hellwig (1980). A well-known criticism of these CARA-normal
environments is the absence of wealth effect – no matter what an investor’s wealth is, she invests
the same dollar amount in a risky stock. This implies that the share of risky wealth decreases with
wealth, while empirical studies typically document the opposite result (Vissing-Jorgensen (2002)).
If wealth were distributed evenly across the population, the absence of wealth effect would probably
not be of much concern. However, it is well known that the degree of wealth heterogeneity in most
developed countries is quite substantial. Existing models with CARA utility are largely silent
on how such heterogeneity affects portfolio choice, stock prices, volatility, and other important
parameters.
The purpose of this paper is twofold. First, we point out a simple way to introduce the wealth
effect in models with CARA investors. The idea is, while keeping CARA preferences, to directly link
investors’ risk aversion coefficients to their contemporary wealth. To demonstrate the usefulness
of our approach, we apply it in the context of CARA-normal information acquisition model of
Verrecchia (1982) which results in a straightforward, tractable extension of his model. Peress
(2004) follows a different route to incorporate the wealth effect – he abandons CARA utility and
switches to preferences exhibiting decreasing absolute risk aversion, with CRRA utility being the
leading example. As a result, he has to resort to approximate techniques to solve the model.
The second point of this paper is on economic intuition. Our closed-form solution lucidly
identifies the effect that is lost through log-linearization and that, importantly, changes the main
conclusion of Peress (2004). Specifically, while Peress finds that the average share of wealth invested into the risky asset increases with wealth, we show that the relation is indeed ambiguous.
We uncover that the relation becomes negative when the risky asset (random) supply has a sufficiently high variance. This difference in results is due to the nature of log-linearization as a
non-affine transformation — it distorts the correlation between random equilibrium demand and
price, thus leading to the putative unambiguously positive relationship between wealth and investment. Given our closed-form expressions, one can explicitly see the parameter conditions under
which the negative relation emerges. A general argument often made for log-linearization is that
this approximation affects second and higher moments, and thus is not likely to change qualitative
1
results. However, several studies find that approximation errors from log-linearization can be as
large as to reverse the qualitative predictions (see Kim and Kim (2003) and references therein). Our
paper contributes to this literature by identifying a similar issue in an asymmetric information asset
pricing setting. All in all, we believe that assuming away the impact of correlations, particularly
in an informational setting which by construction relies on the covariances between fundamentals,
signals and equilibrium quantities, is a somewhat inconsistent route to take.
A caveat is in order. For tractability, we assume that investors are myopic, in a sense that they
do not anticipate that their absolute risk aversion will change in the subsequent periods, and it
is only after the new wealth materializes that they update their risk attitude. As discussed later,
assuming myopia does not affect our main result pertaining to the relation between wealth and
fraction of wealth invested in risky assets (see footnote 3). Without this assumption, the argument
of an investor’s utility function would be a product of two random variables, absolute risk aversion
and wealth, and so in general would not be normally distributed, which destroys the tractability.
The rest of the paper is organized as follows. In Section 2, we describe the economic set-up
and provide the details on how we incorporate the wealth effect into our setting. Section 4 solves
for the equilibrium price and characterizes the optimal level of information acquisition. Section 5
investigates the relation between the initial wealth and wealth share invested in the risky asset.
Section 6 concludes. The Appendix contains all proofs.
2.
Economic Setting
Our setting is essentially as in Verrecchia (1982). There are two assets in the market, a risky
stock and a riskless bond. At t = 0, upon observing their endowments, investors decide how much
information to buy. At t = 1 the information is revealed and investors trade but do not consume.
At t = 2 the asset payoffs are realized and the traders consume their terminal wealth. The bond
serves as a numeraire and returns one unit of wealth at t = 2. The realized payoff on the risky
asset, denoted by ũ, is not known to traders until t = 2. Investors share the common prior belief
that ũ is normally distributed with mean µ0 and variance σ02 . Denote by h0 the precision of ũ:
h0 =
2
1
.
σ02
(1)
At time 0, each investor i is able to buy a signal ỹi about the true realization of ũ:
ỹi = ũ + ǫ̃i ,
where ǫ̃i is normally distributed with mean 0 and precision si . Buying information in this context
is modeled in the form of receiving a signal with higher precision si . The information quality si
comes at a cost c(si ). It is assumed that c(·) is a continuous, twice-differentiable function with
c′ > 0 and c′′ ≥ 0. The functional form of c(·) implies that a more precise signal comes at a higher
cost, and also that the marginal cost of a signal is increasing with its precision.
There are N investors in the economy, where N is large. At t = 0, investor i has one unit of
the bond and x̃i units of the risky stock, where x̃i is drawn from a normal distribution with mean
x0 and variance N × V . We assume that x̃i are independent across investors and also independent
of signals ỹi . The x̃i are independent across investors, and also independent of the signals ỹi . The
per-capita supply of the risky asset, denoted by X̃, is given by
X̃ ≡
PT
i=1 x̃i
N
,
implying that X̃ is normally distributed with mean x0 and variance V .
If one wants to account for the wealth effect, the widespread perception seems to be that one
needs to consider a “proper” utility function, such as CRRA. One of the key messages of this paper
is that the wealth effect can be incorporated in a CARA-normal setting via making the absolute
risk aversion parameter wealth-dependent. In particular, we assume investor i’s utility Ui defined
over her terminal wealth wiT is given by:
Ui (wiT ) = − exp −wiT /ri
where ri denotes the absolute risk tolerance. Depending on the economic environment, one can
entertain different functional forms linking an investor’s absolute risk tolerance to her wealth. We
consider the functional form that leads to CRRA-type behavior so as to enable the comparison of
our results with that of Peress. In particular, we posit that an investor’s absolute risk tolerance is a
linearly increasing function of current wealth, as seen for CRRA utility. The details are as follows.
At time 0, the market value of an investor’s wealth is not yet determined due to no trading.
In this case, as in Peress, we treat an investor’s endowment as a natural proxy for her wealth and
3
assume that
ri = xi /a, a > 0.
(2)
For notational brevity, in (2) we ignore bond endowment when computing investor i’s absolute
risk tolerance ri . This does not affect any of our results since all investors start with identical
bondholdings, and so ranking the investors by initial wealth is equivalent to ranking them by stock
endowment.
At time 1, after trading has taken place, the market value of investor i’s portfolio wi is known,
and so we reset the absolute risk tolerance parameter ri accordingly:
ri = wi /a, a > 0.
(3)
We assume that the investors are myopic in that, sitting at time 0, they do not realize that their
future absolute risk aversion will change. Assuming myopia is needed for tractability, and does not
necessarily reflect the actual behavior of investors.1 As we argue later, the key result of this paper
is not driven by investors myopia (see footnote 3).
3.
Rational Expectations Equilibrium
This Section characterizes the competitive equilibrium price and portfolio choice at t = 1, after
the investors have observed the signals purchased at t = 0. The formation of the equilibrium takes
into account that the realized market price reflects the beliefs held and, conversely, the beliefs
reflect the information portrayed by the price. The expression for the equilibrium price presented
in Proposition 1 is the same as in Lemma 2 of Verrecchia, and so is given without proof.
Proposition 1. The equilibrium price P̃ converges in probability to
P̃ → α + β ũ − γ X̃,
(4)
1
We are aware of only informal arguments in favor of investors myopia, such as Brandt’s (2009) reasoning that
“The myopic portfolio choice is an important special case for practitioners and academics alike. There are, to my
knowledge, few financial institutions that implement multi-period investment strategies.” Despite the lack of hard
empirical evidence, there is a large literature which assumes that investors are myopic (e.g., Ait-Sahalia and Brandt
(2001), Jagannathan and Ma (2003), Bansal, Dahlquist and Harvey (2004), Acharya and Pedersen (2005), Hong,
Scheinkman and Xiong (2006), Campbell, Serfaty-de Medeiros and Viceira (2007)).
4
where
α =
β =
γ =
E[r]V h0 µ0 + x0 E[r]E[rs]
,
E[r]V h0 + E[rs]V + E[r](E[rs])2
E[rs]V + E[r](E[rs])2
,
E[r]V h0 + E[rs]V + E[r](E[rs])2
V + E[r]E[rs]
.
E[r]V h0 + E[rs]V + E[r](E[rs])2
(5)
(6)
Before choosing the portfolio portfolio at time 1, each investor i updates her prior beliefs about
the mean and the variance of the stock payoff after observing her signal ỹi and the market price P̃ .
The prior beliefs are characterized in Lemma 1.
Lemma 1. The vector (ũ,ỹi ,P̃ ) of mean payoff, signal and t = 1 stock price has, as of t = 0, a
jointly normal distribution with mean (µ0 , µ0 , P0 ), where
P0 = µ0 −
V x0
,
E[r]V h0 + E[rs]V + E[r](E[rs])2
(7)
and variance-covariance matrix

σ02
σ02
βσ02





 σ 2 σ 2 + s−1
.
2
βσ
 0

0
0
i


2
2
2
2
2
βσ0
βσ0
β σ0 + γ V
The joint normality of the risky payoff ũ, the signal ỹi , and the market price P̃ allows us to find in
closed form the updated mean µi and variance σi2 of the risky payoff ũ (see Gelman, Carlin, Stern,
and Rubin (2004) for Bayesian updating under normality). The updated mean and variance, and
the ensuing optimal portfolio, are given in the next Proposition.
Proposition 2. After observing her signal ỹi = yi and the market price P̃ = P1 , investor i’s
optimal number of stocks held, denoted by Di , is given by2
Di = ri
µi − P1
σi2
2
(8)
We do not present explicitly the demand for the bond as we will not need it in the subsequent analysis. It is
easily obtained from the budget constraint.
5
where µi and σi2 are the posterior mean and variance of payoff ũ, given by
si γ 2 V (yi − µ0 ) + β (P1 − P0 )
,
(h0 + si )γ 2 V + β 2
1
.
h0 + si + (β 2 /γ 2 V )
µi = µ0 +
σi2 =
(9)
(10)
We now analyze the information acquisition problem of investors at time t = 0. An investor i
chooses the precision of her signal, si , by maximizing their expected utility from terminal wealth,
which depends on her portfolio choice to be made at t = 1. Proposition 3 characterizes investor i’s
optimal choice of the precision si , and is given without proof since it is straightforwardly obtained
by combining Verrecchia’s Lemma 2 with (2).
Proposition 3. There exists a unique competitive information acquisition equilibrium. Investor
i’s optimal choice of the precision si is given by max[0, ŝi ], where ŝi is implicitly given by
(E[rs])2
2ac′ (ŝi )
ŝi + h0 +
= 1.
xi
V
(11)
Proposition 3 implies that: i) there exists a wealth threshold such that only agents whose initial
wealth lies above the threshold acquire information about the stock, ii) when the initial wealth is
above the threshold, increasing initial wealth leads to more information being purchased.
We have now fully characterized the solution of our model. Notice that our analysis is a
straightforward tractable extension of Verrecchia. Most of his proofs go through with no or minimal
modifications. Peress follows an alternative way of incorporating the wealth effect in Verrecchia’s
setting via changing the utility specification. As a result, he has to resort to log-linearization and
solve the model from scratch. In terms of the optimal level of information acquisition, Peress’s
approximate approach leads to the same conclusion as ours.3 Namely, he finds that only those
investors whose wealth is above a threshold optimally choose to purchase information, and that,
when above the threshold, wealthier investors acquire more information. However, what appears
to be the main result of Peress – the unambiguously positive relation between wealth and wealth
share invested in the risky asset – no longer holds once an analytic solution considered, as discussed
in the next Section.
3
Note that investors face a truly multi-period optimization problem only at time 0 when choosing the signal
precision, while the choice of optimal portfolio at time 1 is obtained by solving a single-period problem. This means
that our assumption of myopia can only affect the information acquisition equilibrium at time 0. Since our time-0
equilibrium is qualitatively the same as in Peress, we have that investors myopia does not drive our main prediction.
6
4.
Wealth and Portfolio Shares
We now turn from a methodological point to an economic question of how the level of investor i’s
signal precision si affects the share of wealth invested into the risky security θi , defined as
θi ≡ (P Di )/wi .
(12)
Proposition 4 presents the main result of this Section.
Proposition 4. The effect of the purchased signal’s precision on the expected share of wealth allocated to stocks is given by
P0 (µ0 − P0 ) + β σ02 − βσ02 − γ 2 V
dE0 [θi (si )]
=
.
dsi
a
(13)
The sign of dE[θi (si )]/dsi is ambiguous.
In the region of low wealth, in which investors do not buy any information, marginally increasing
the initial wealth has no effect on the signal and, hence, on the wealth share allocated to stocks.
On the other hand, when the initial wealth is high enough so as to make investors buy a positive
amount of information, the sign of dE[θi (si )]/dsi is the same as the sign of dE[θi (si )]/dxi , as follows
from Proposition 3. Proposition 4 establishes that the effect of a higher precision si on the expected
fraction invested into the stock is ambiguous. Indeed, while the first two terms therein are positive
(due to µ0 > P0 and β < 1), the total effect can be negative if the term γ 2 V is large enough. That
is, when the dispersion of the stock supply is sufficiently high, the wealthier investors put a lower
share of their wealth in the risky asset than the poorer ones.
To understand the intuition behind Proposition 4, let us first decompose the expectation of θi
as follows:
P1 (µi − P1 )
µi − P1
µi − P1
E[θi ] = E
= E[P1 ]E
+ cov P1 ,
,
aσi2
aσi2
aσi2
(14)
where the first equality in (14) follows from plugging (8) into (12) and using the “wealth effect”
relation ri = wi /a. From (9), the covariance between the posterior mean stock return µi and the
equilibrium price P1 decreases as si increases. The reason is that when investor i has a more precise
private signal, she pays less attention to the information conveyed by the market price P1 and trusts
his own signal more. In the limit, when si is very high, the expected value of the payoff ũ, µi ,
depends only on the private signal. Thus, the covariance term in (14) decreases with si . The first
7
term in (14) increases in the signal’s precision si since the expected value of the random numerator
is constant (determined from (7)), while from (10) the non-random denominator decreases in si .
The combined effect of the two terms is ambiguous – as stated in Proposition 4.
The result of Proposition 4 is at odds with the conclusion of Peress who establishes an unambiguous positive relation between wealth and wealth share invested in the risky stock. The
difference stems from the fact that the log-linearization technique employed by Peress distorts
the correlations between random variables, leading to qualitatively different results. If we were
to use the log-linearization in our context and take the logarithm of θi – assuming away for the
sake of argument the possibility of negative θi – then from the first equality in (14) we would get
ln(P1 ) + ln(µi − P1 ) − ln(σi2 ). The covariance term from (14) would no longer be present, leading
to a unambiguously positive relation between wealth and wealth share.
Finally, we describe the exact numerical solution of Peress’s model in the trading period, so as
to demonstrate that our main result described in Proposition 4 remains valid in his setting.
ΑR -ΑP
0.1
0
ΣΘ
1
5
-0.1
Figure 1: Share of risky wealth: rich versus poor investors. The difference between the
average share of wealth invested in the risky stock by rich and poor investors, αR − αP , depending
on the variance of stock supply σθ . In what follows, the notation is as in Peress, whereby a
superscript R or P means that the variable corresponds to rich or poor investors, respectively. The
P
2
f
parameters are: W0R = 6, W0P = 4, xR
0 = 2, x0 = 1, σπ = E(θ) = z = 1, E(π) = γ = 2, r = 0.05,
nd = 2.
Example. Exact solution of Peress’s model. To derive the exact numerical solution, we use
the same technique as implemented by Peress in his analysis of a calibrated version of the model
(see his Appendix F).4 However, in that analysis he assumes that all investors have the same levels
of wealth and information precision, and so his exact solution is silent about the main result of
his paper which crucially relies on wealth heterogeneity. We extend Peress’s setting by considering
two types of investors (with the same mass): “rich” investors (higher wealth, higher information
4
The technique is the projection approach developed by Bernardo and Judd (2000). We are very grateful to Joel
Peress for sharing with us his matlab code which implements this approach.
8
precision) and “poor” investors (lower wealth, lower information precision). To be consistent with
Peress’s notation, we denote by αR and αP the average share of wealth put into the risky stock by
rich and poor investors, respectively, and by σθ the standard deviation of the stock supply (V in
our setting). While from Peress’s approximate solution it follows that αR > αP , from Proposition
4 we conjecture that solving his model exactly may lead to the opposite relation αR < αP when σθ
is sufficiently high. This conjecture is confirmed by Figure 1 which illustrates that the difference
αR − αP may well be negative.
5.
Conclusion
We propose a simple way to account for the wealth effect in models with CARA utility. The idea
is to explicitly link an investor’s absolute risk aversion to her contemporaneous wealth. We apply
this approach in the context of Verrecchia (1982), which results in a straightforward and tractable
analysis. The alternative approach of changing the utility function, implemented in Peress (2004),
relies on the approximate methods and leads to the conclusion that does not survive once a closedform solution is considered. Namely, we obtain that the relation between initial wealth and wealth
share invested into risky assets can be negative, while an approximate solution suggests that the
relation is uniquely positive.
9
Appendix
Proof of Lemma 1.
First, we derive the means of the random variables. By assumption, E[ũ] = µ0 . For the signal, we
have E[ỹi ] = E[ũ] + E[ǫ̃i ] = µ0 . For the equilibrium price, we have
E[P̃ ] = E[α + β ũ − γ X̃] = α + βµ0 − γx0
(E[r]V h0 µ0 + x0 E[r]E[rs]) + (E[rs]V µ0 + E[r](E[rs])2 µ0 ) − (V x0 + E[r]E[rs]x0 )
E[r]V h0 + E[rs]V + E[r](E[rs])2
µ0 (E[r]V h0 + E[rs]V + E[r](E[rs])2 ) − V x0
=
E[r]V h0 + E[rs]V + E[r](E[rs])2
V x0
= µ0 −
.
E[r]V h0 + E[rs]V + E[r](E[rs])2
=
Now we derive the variance-covariance matrix. By assumption,
V ar[ũ] = h−1
0 ,
−1
V ar[ỹi ] = V ar[ũ] + V ar[ǫ̃i ] = h−1
0 + si .
For the equilibrium price, we have
2
V ar[P̃ ] = V ar[β ũ] + V ar[γ X̃] = β 2 h−1
0 + γ V.
Finally,
Cov[ũ, ỹi ] = Cov[ũ, ũ + ǫ̃i ] = h−1
0 ,
Cov[ũ, P̃ ] = Cov[ũ, α + β ũ − γ X̃] = βh−1
0 ,
Cov[ỹi , P̃ ] = Cov[ũ + ǫ̃i , α + β ũ − γ X̃] = βh−1
0 .
Q.E.D.
Proof of Proposition 2.
We could present a full-blown proof here, but in principle the expressions are the same as in
Verrecchia, p. 1420, with the only difference being the average equilibrium price: in his setting
it equals µ0 and so the posterior mean has the term β(P − µ0 ), while in our setting it equals
µ0 − V x0 /(E[r]V h0 + E[rs]V + E[r](E[rs])2 ) and so we have this expression instead of µ0 .
10
Q.E.D.
Proof of Proposition 4.
Plugging (9) and (10) into (8), after some algebra, yields:
Di = ri
β2
β
β2
β
µ0 h0 + 2
− h0 − 2
+ s(yi − P ) + P
− P0 2
.
γ V
γ2V
γ V
γ V
Because the fraction of wealth invested into the risky stock is defined as θi ≡ (P Di )/wi and because
ri = wi /a, we have
θi =
P
a
β2
β
β2
β
µ0 h0 + 2
+ s(yi − P ) + P
−
h
−
−
P
.
0
0 2
γ V
γ2V
γ 2V
γ V
We have
dE[θi (si )]
dsi
= E [dθi (si )/dsi ]
= (1/a)E[P (yi − P )] = (1/a)(E[P ]E[yi ] + cov[P, yi ] − (E[P ])2 − var[P ]).
Now we make use of the fact that P and yi are jointly normal with known mean and covariance
matrix derived in Lemma 1. This allows us to write the last expression as
dE[θi (si )]
dsi
=
=
P0 µ0 + βσ02 − P02 − β 2 σ02 − γ 2 V
a
P0 (µ0 − P0 ) + β σ02 − βσ02 − γ 2 V
.
a
(15)
Q.E.D.
11
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