Prospect Theory and Asset Pricing in an RBC Framework

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Prospect Theory and Asset Pricing in an RBC Framework Powered By Docstoc
					    Prospect Theory and Asset Pricing in an RBC
                    Framework
                   Julia Lendvai and Rafal Raciborski

                           First Draft: June 2006
                              PRELIMINARY

                                      Abstract
         We construct a fully-fledged production economy model with Kahne-
     man and Tversky’s Prospect Theory features. The agents’ objective func-
     tion is a weighted sum of the usual utility over consumption and leisure
     and the utility over the relative changes of the agents’ wealth. It is also
     assumed that the agents are more sensitive to wealth losses than to gains.
     Apart from the changes in the utility, our RBC model is standard. We
     study prices of different assets in our economy. It is demonstrated that
     under plausible parametrizations of the objective function our model ex-
     plains all first and second moments of the returns on risky assets, short
     bonds and long bonds. In particular, we are able to match the empirical
     Sharpe Ratio, the equity premium and the volatilities of the risky asset
     and the riskless bond.


1    Introduction
In his survey on behavior of financial asset prices Campbell (1999) documents
a host of stylized facts about the stock markets. The average real return on
stocks is high and volatile. The returns on Treasury Bills are low and not
very volatile. Additionally, real consumption growth is very smooth and it is
only mildly correlated with real stock returns. These findings seem robust both
across countries (see Campbell, 1999) as well as in time (Siegel, 1992a,b). Yet,
explaining them within the framework of the consumption-based asset pricing
model, an approach that has become a standard in the field, constitutes a veri-
table challenge for the economists. As it is clearly stated by Campbell, there are
two fundamental problems arising from the application of this approach, when
confronted with the data. First, for what is believed to be a plausible utility
parametrization, it is hard to explain the magnitude of the premium of average
stock returns over the risk-free rate. This problem, arguably the best known
of the financial puzzles, was termed by Mehra and Prescott (1985) the ”equity
premium puzzle”. Second, it is difficult to reconcile the high volatility of stock
returns with the low volatility of the short term real interest rate. This is what
Campbell calls the ”stock market volatility puzzle”.

                                         1
    For a long time all attempts to explain the aforementioned anomalies had
been notoriously unsuccessful (for a list of works trying to challenge the puz-
zles see Constantinides et al, 2002; see also the surveys: Kocherlakota, 1996,
Campbell, 1999 and Campbell, 2000). It is only relatively recently that the
economists have proposed some modifications to the baseline consumption-
based, representative-agent asset pricing model that in the endowment-economy
framework seem to be helpful in resolving the asset market puzzles. Among
them, the Campbell-Cochrane habit persistence model (1999) is perhaps con-
sidered the most successful, as except the historical means and volatilities of the
asset returns it is able to match a number of other financial statistics.
    The task of explaining the puzzles in the fully-fledged production economy
framework occurs even more demanding. The challenge comes from the fact that
the consumers in this framework possess multiple means to alleviate the impact
of an unfortunate investment. Consequently, their price for risk is low and thus
are the risk premia. An early example of the difficulties one encounters when
passing from an endowment economy setting to an endogenous-consumption
and dividends model was provided by Rouwenhourst (1995). One simple way
to match the equity premium in the endowment economy is to assume that the
degree of agents’ risk aversion is considerably higher than what was traditionally
considered plausible1 (Kandel and Stambaugh, 1991). However Rouwenhourst
observed that in the standard real business cycle model very risk-averse agents
are able to substantially reduce their consumption stream fluctuations, which
provides them with a good insurance against the consumption risk. Hence,
an extreme degree of risk-aversion must be assumed in order to obtain non-
negligible risk premia. Then, the model also produces counterfactually low
volatility of consumption.
    Also attempts to carry over the success of ’habits’ to production economy
models encountered serious problems. Just like in the case of increasing risk
aversion, introducing the habit persistence to a standard real business cycle
model pushes the agents to smooth excessively their consumption paths, leaving
the equity premium unaffected. It is only when additional frictions on capital
and labor mobility are added to the model that the favorable properties of the
habit persistence are restored (Jermann 1998, Boldrin et al 2001). In fact,
Boldrin et al argue that their two sector economy model with habit persistence
is also able to significantly improve the business cycle properties of the standard
RBC model.
    Even so, the RBC model with habits does not seem satisfactory in all di-
mensions. As it is shown in Jermann, the long term premium of long bond
returns over the risk-free rate is of the same size as the equity premium in this
type of models, a clearly counterfactual prediction. Also, the model consider-
ably overshoots the risk-free rate volatility (Jermann, 1998; Boldrin et al, 2001).
   1 But see Campbell (1999) who argues that it does not resolve the volatility puzzle. The

assumption of high risk aversion in these models leads also to what Weil (1989) dubbed the
”risk-free rate puzzle”.
  For a short discussion of ”what a plausible degree of risk aversion is” see Kocherlakota
(1996).



                                            2
In fact both problems had been known already before and seem to be insepa-
rably related to the habit persistence. Although in their endowment economy
habit persistence model Campbell and Cochrane (1999) were able to engineer
the habit stock process in such a way that the risk-free rate becomes virtually
constant, they themselves admit that this is a knife-edge result. Moreover, Let-
tau and Uhlig (1997) show that in the Campbell-Cochrane environment under
some parametrizations consumption bunching is desirable, a certainly disturb-
ing finding. Finally, the empirical evidence for the strength of habit persistence
seems mixed (for a short discussion see Guvenen, 2003). In particular, K. Dynan
(2000) does not find any evidence of habit formation in the food consumption
data from the Panel Study on Income Dynamics.
    This paper attempts to extend the standard endogenous production set-
ting in another direction. We construct an RBC model with Kahneman and
Tversky’s (1979) Prospect Theory features. Prospect Theory emerges from the
experiments on the process of decision making under risk and is recently widely
recognized among psychologists as a good descriptive model of such decisions. In
their model Kahneman and Tversky posit that people assign value to gains and
losses rather than to final absolute position of their assets. They also introduce
the concept of loss aversion: In their theory the individuals are substantially
more sensitive about the losses than the gains they experience. Kahneman and
Tversky’s theory is able to explain a set of anomalies observed in the individ-
uals’ decision making that contradict the expected utility paradigm. Although
originally based on laboratory experiment results on simple gambles, since the
date of its publication the theory has gained a wealth of empirical support (see
for instance Camerer, 1998 and List, 2004). Evidence consistent with this the-
ory was recently found also in the case of the professional asset traders behavior
(Coval and Shumway, 2001).
    Our work is a continuation of a strand of the behavioral finance literature
that seems promising in resolving the financial market puzzles. In a partial equi-
librium model, Benartzi and Thaler (1995) show that loss-averse agents demand
a high average risk premium in order to be willing to invest in stocks. Barberis,
Huang and Santos (2001, BHS for future reference) examine whether the same
will be true in a general equilibrium, endowment economy framework. Their
answer is negative as far as it concerns a standard optimizing-agents model
with the loss aversion as the only extra feature. However, once additional mod-
ifications based on psychological theory developments are added, their model
is shown to explain the mean and volatility of stock returns. Moreover, it re-
produces the predictability of the returns on risky assets. In a follow-up paper
(2001) Barberis and Huang further develop this approach.
    The goal of this paper is to check whether Prospect Theory helps explaining
the two previously discussed financial puzzles also in a fully-fledged production
economy framework. To this end we extend the agents’ preferences in the other-
wise standard RBC model to account for the utility derived from changes in the
value of the financial wealth they hold and assume that agents are loss-averse.
Contrary to BHS, we do not consider any other modifications suggested by the
psychological evidence. Hence, our setting corresponds to the simplest version

                                        3
of the model considered by BHS.
    Our methodology is similar in spirit to the rapidly developing literature
aiming at reproducing the financial facts in models with nontrivial production
sectors. In particular, we draw on the works of Jermann (1998) and Boldrin
et al (2001). Other important papers in the field are Guvenen (2003), Uhlig
(2006) and Guvenen and Kuruscu (2006). There is one major difference be-
tween these papers and our approach. The modifications of the baseline RBC
model introduced by these authors (habit persistence in the case of Jermann
and Boldrin et al, heterogeneity of agents in the case of Guvenen and Guvenen
and Kuruscu and rigidity of wages in the case of Uhlig) in order to reproduce the
asset market statistics usually do not leave unaffected the real side of the model
economies. Therefore, a lot of attention is paid to ensure that the outcome of
these modifications could be reconciled also with the stylized macroeconomic
facts2 . This type of problems does not appear in the setting with loss-averse
agents. As it will be shown later in the paper, the loss-aversion term that enters
the agents’ utility in an additively separable way does not have any effect on
the dynamics and the volatility of real variables like consumption, output or
investment. Hence, in our setting we are able to fully disentangle the financial-
and the real side of the model economy. Since there exists reach literature on
reproducing macroeconomic facts in the RBC framework (see for instance King
and Prescott, 1995 or King and Rebelo, 2000), here we focus solely on matching
the asset market statistics.
    There are six statistics we attempt to fit: The Sharpe Ratio, the equity pre-
mium, the long-term bond premium, the real interest rate and the volatilities of
the stock returns and the real interest rate returns. When a standard bare-bone
RBC model with Prospect Theory features is considered, we are able to match
only the Sharpe Ratio, and the short-term real interest rate moments. This is
not to be neglected, as fitting the empirical Sharpe Ratio alone has proven dif-
ficult even in the endowment economy framework (see Lettau and Uhlig, 2002).
In order to fit the other statistics we introduce capital adjustment costs and a
moderate degree of financial leverage to our model. This version of the RBC
model allows us to reproduce with a high precision all asset market statistics
of interest. Hence, none of the financial puzzles discussed before emerge in our
framework.
    The presence of the Prospect Theory features in the consumers’ utility func-
tion is neutral as regarded the real side of the economy. However the intro-
duction of the adjustment costs is shown to deteriorate the performance of the
model in this dimension. Hence more work is needed in order to reconcile the
capital adjustment costs with the business cycle facts.
    The rest of the paper is organized as follows. In the next section some
characteristics of the Prospect Theory are displayed. Then the setting of the
model is explained and the first order conditions derived. The calibration issues
and the numerical solution technique are also presented in that section. We
   2 See for instance Lettau and Uhlig (2000) on the problem of consumption oversmoothing

in the presence of habit persistence.



                                           4
discuss our results in section 3. The last section concludes.


2     The model
In this section we describe our simple RBC model with loss-averse agents. We
start with a short discussion of the Prospect Theory.

2.1     Loss aversion
The Kahneman and Tversky’s Prospect Theory (1979) is a model of decision
making under risk. More precisely, it aims at describing the behavior of an
agent facing a choice of risky prospects, or gambles. There are four main char-
acteristics that distinguish the Prospect Theory from the more widely applied
Expected Utility Theory:

    1. The agents assign the value to wealth reductions and increases rather than
       to the absolute magnitude of the final wealth. These reductions (losses)
       or increases (gains) are measured as compared to some initial reference
       level.
    2. The agents are loss-averse: They are more sensitive to losses than to gains
       (their objective function has a kink at the reference point and is steeper
       for losses than for gains).
    3. The agents’ objective function is concave for gains and convex for losses.
    4. The agents overweigh low probabilities.

    As it was noted by Campbell (2000), the Theory was tested in experiments
necessarily involving only relatively small payoffs. Therefore it needs not be
readily applicable to decisions with higher risks at stake. For this reason, despite
their apparent attractivity for our purposes, we decided to disregard the last
two features of the Prospect Theory and focus solely on the loss aversion of the
agents. In doing so we closely follow BHS (2001), who also concentrate on loss
aversion.
    The Kahneman and Tversky’s original formulation of the Theory postulates
that the individuals care solely about the relative changes of their wealth. As in
BHS, we consider a more general setting in which we allow the agents’ objective
function to be a weighted sum of the standard utility over consumption stream
and leisure and the utility over the relative changes of the wealth (further re-
ferred to as a Loss Aversion or LA term). In order to specify the latter we need
to define three concepts (see Campbell 2000): the time horizon, the reference
level and the type of wealth the variations of which enter the LA term. We start
the description from the last.
    The issue of specifying the argument of the LA term is closely related to the
psychological concept of the narrow framing (Thaler, 1980; Thaler et al, 1997).
The standard economical theory predicts that what matters for the agents in

                                         5
a given moment is their total wealth, regardless of its particular partition on
different types of assets. Contrary to this viewpoint, the psychological evidence
suggests that when doing their mental accounting, people tend to focus on each
narrowly defined component of their wealth separately: They engage in narrow
framing. This controversy has a direct impact on our work. In order to comply
with the standard theory we would need to consider the loss aversion over the
total wealth. Alternatively, we could define it over the portfolio of financial
assets. An extreme position (although not an unreasonable one, see Barberis
et al, 2003) is to assume that the agents are exposed to the loss aversion over
fluctuations in the value of individual assets they hold. Under this assumption,
the investors create separate mental accounts for stocks of every single firm.
Barberis and Huang (2001) demonstrate that this last approach is especially
successful in reproducing observed asset market statistics.
    In order to single out the influence of the pure loss aversion we do not
make a distinction between individual stocks. All firms in the economy will be
assumed identical so that their equities are homogeneous. Hence, we closely
follow BHS (2001) who also consider only one type of stock3 . In addition, we
consider a second type of risky asset, a long-term bond. We postulate an ad hoc
degree of narrow framing by positing that the LA terms are separately defined
for the stocks and the long bonds. There are two other types of assets in our
framework: one period riskless bonds and one period firms’ discount bonds.
Since both assets guarantee a certain payout next period, we do not apply the
Prospect Theory in their case4 .
    The next question concerns the definition of the reference level. The most
basic formulation (Kahneman and Tversky, 1979) posits that individuals mea-
sure their gains and losses in reference to the status quo, which in our case is
the value of the initial investement in a risky asset. Let ar denote the number of
                                                            t
shares or long bonds purchased by an agent in t and held until t+1 and Ptr their
current price. Then the initial investment value and the status quo reference
level is simply ar Ptr . BHS suggest instead that we should take into account the
                 t
opportunity cost of the risky investment that is, the wealth one could obtain
                                                         f
from investing ar Ptr in a riskless asset. Let denote Rt the gross return on the
                  t
                                             r
one period (riskless) discount bond and Rt+1 the stochastic return on the risky
asset, both over a t to t + 1 period. Then we assume that the reference level of
                          f                            r
an individual is ar Ptr Rt . Hence, her gain or loss Xt+1 is:
                    t

                                                         f
                            r              r
                           Xt+1 := ar Ptr Rt+1 − ar Ptr Rt+1
                                    t             t

The form of the loss aversion term will be then defined as:
                                        r                 r
                         r            Xt+1               Xt+1 ≥ 0
                     V (Xt+1 ) =         r        for     r                            (1)
                                      φXt+1              Xt+1 < 0

In order to capture the property of the loss aversion, we set φ > 1.
   3 Note that the return on this single stock may be interpreted as a return on the market

portfolio.
   4 We do not consider a default risk in our model.




                                            6
    The last issue to be dealt with is the timing horizon. The question to be
asked here is how frequently the agents evaluate their asset positions. Benartzi
and Thaler (1995) argue that it is natural to assume that a serious evaluation is
carried out once a year. It would then correspond to the frequency the taxes are
filed, the fund reports are published and the money managers are most thor-
oughly scrutinized (Benartzi and Thaler 1995). Based on this argumentation,
BHS calibrate their model on the annual basis. Instead, we decided to work with
a quarterly calibration. We do so since the host of the RBC models (including
all production economy models listed in the introduction) are using quarters as
their basic time unit. Experimentation with yearly calibration does not seem to
alter the results of our model.
    As it was mentioned in the introduction, BHS find that loss aversion alone
is insufficient to generate high equity premia. Guided by another strand of
the psychological research (Thaler and Johnson 1990) BHS relate the intensity
with which the agents experience gains and losses from an investment to their
prior investment outcomes. So modified model performs very well in terms of the
generated average equity premium. Nonetheless, we do not follow their strategy.
Just as in the BHS case, the results of a baseline RBC model with loss averse
agents do not come even close to the empirical equity premium. However, we are
able to remedy this problem by enriching the model with measures considered
standard extensions to the simple RBC setting. These extensions are capital
adjustment costs and a moderate degree of the financial leverage.
    As a next step, we pass now to specifying our benchmark RBC setting.

2.2    The economy
The economy is populated with an infinite number of identical, infinitely-lived
firms and households. The only good in the economy is produced with a stan-
dard Cobb-Douglas production function, subject to random productivity shocks.
It may be consumed or serve to replenish the capital stock.

   Households.      The representative household maximizes:
                                                                    
                    ∞         1−γ             1−χ
                            C        (1 − Nt )
         max E0        βt  t     +θ              + βBt         r
                                                            V (Xt+1 )       (2)
        Ct ,Nt ,at
                   t=0
                            1−γ         1−χ
                                                         r=s,l

As usually, β denotes the subjective time discount factor. The first two terms
in the brackets are standard in the RBC model: The consumers maximize their
utility over consumption Ct and leisure Lt = 1 − Nt , where Nt is the time spent
on working. The total amount of time available to households in a single period
is normalized to one. The last part of the expression in brackets introduces the
                     r
two LA terms V (Xt+1 ). The superscript r takes the value ’s’ for shares and
                                                   r
’l’ for long bonds. As it is discussed in BHS, V (Xt+1 ) must be properly scaled
so that it does not dominate the utility function as the aggregate wealth grows




                                       7
over time. We follow BHS in specifying the scaling factor as:
                                                    ¯ −γ
                                             Bt = B Ct
       ¯
where Ct is the aggregate per capita consumption in a given period. The choice
of this particular exogenous variable is neutral for the main message of the
                                                             ¯
paper. Without any influence on our results, instead of using Ct we could apply
any exogenous variable whose growth in the steady state is equal to the growth
of the economy, for instance the aggregate per-capita wealth or the aggregate
capital.
    The households face a standard budget constraint:

                   Ct +            ai Pti = Wt Nt +
                                    t                       ai    i    i
                                                             t−1 Pt + Dt        (3)
                               i                        i

In a period t they assign a part of their wealth to the current consumption,
while the remainder is invested in a vector of assets at = {as , af , al , aq } held
                                                                  t  t   t t
until the period t + 1, where the superscripts i = s, f , l or q stand respectively
for shares, risk-free bonds, long-term bonds and firms’ discount bonds. A given
asset i may be purchased in a period t at a price Pti . The payout of an asset
                                              i
held from period t − 1 to t is denoted by Dt . The agents receive also a labor
income Wt Nt , where Wt denotes the date t wage.

    Firms. Firms maximize the present value of their future dividend flow
discounted at the marginal rate of substitution β i ΛΛt of their owners:
                                                     t+i



                                   ∞
                                            Λt+i
               max        Et           βi         s
                                                 Dt+i (Kt−1+i , Kt+i , Nt+i )   (4)
             Kt+i ,Nt+i
                               i=0
                                             Λt

where
                                            Λt+i  C −γ
                                                 = t+i
                                                    −γ
                                             Λt   Ct
is the stochastic discount factor of the economy. The dividends are defined as:

         Dt (Kt−1 , Kt , Nt ) ≡ Dt = Yt − Wt Nt − It + Ptq µKt − µKt−1
          s
                                                                                (5)

Yt is the current output, It - investment and Kt - end of period capital stock.
The constant µ measures the degree of financial leverage of the firms. When
µ = 0 the new capital of the firm is fully financed through retained earnings.
    The maximization is subject to the Cobb-Douglas production function:
                                                  1−α          α
                                   Yt = At Ntd           d
                                                        Kt−1       ,            (6)
               d
where Ntd and Kt−1 denote respectively labor and capital demand of the firm.
The technology coefficient At is growing at a constant growth rate g in the



                                                  8
steady state. The deviation of At from its steady state follows an AR(1) process.
         ˜    At                       ˜
Denote At ≡ Ass . It is assumed that At follows:
               t


                                 ˜        ˜
                             log At = log At−1 +                     t


with < 1 and t i.i.d such that E t = 0, and std( t ) = σε . By this definition,
               ˜                ˜
At ∼ Ass elog At = Ass eg+ρ log At−1 + t .
   = t               t−1
   It is assumed that the transformation of the investment It into capital Kt is
costly. As it is argued in Jermann (1998), assuming positive capital adjustment
cost allows the shadow price of installed capital to diverge from the price of
an additional unit of capital, which brings the model closer to the reality. In
specifying the capital adjustment cost we follow Uhlig (2006). Hence, our capital
accumulation equation takes the form:

                                                           It
                        Kt = (1 − δ) Kt−1 + G                        Kt−1       (7)
                                                          Kt−1

with the function G(·) such that
                                                                 1−1/ξ
                           It               a1          It
                    G                ≡                                   + a2   (8)
                          Kt−1           1 − 1/ξ       Kt−1

where a1 , a2 are two positive constants. For ξ < ∞ the adjustment cost becomes
strictly positive.

   Returns on assets.           The gross return in t + 1 on an asset i is:
                                               i       i
                                    i         Pt+1 + Dt+1
                                   Rt+1 =
                                                   Pti
                                                   s
In the case of the risky asset we will use Rt+1 ≡ Rt+1 . Consider the vector of
purchase prices in t:

                         {Pts , Ptf , Ptl , Ptq } = {Pt , Ptf , Ptl , Ptq }

One period later, after the dividends have been paid and the coupons stripped,
the prices of these assets are:
                     s      f      l      q                   l
                   {Pt+1 , Pt+1 , Pt+1 , Pt+1 } = {Pt+1 , 0, Pt+1 , 0}

while the vector of payouts:
                            f             q
                     s             l
                   {Dt+1 , Dt+1 , Dt+1 , Dt+1 } = {Dt+1 , 1, Dl , 1}

Therefore the risky assets returns are:

                            l
                                                         l
                                            Pt+1 + Dt+1 Pt+1 + Dl
                   {Rt+1 , Rt+1 } = {                  ,          }             (9)
                                                 Pt        Ptl

                                                 9
with Dl > 0 being a constant payoff of the long bond and Dt+1 defined as in
(5). The vector of risk-free asset returns is:

                                  f    q          1         1
                                {Rt , Rt } = {         ,       }                (10)
                                                 Ptf       Ptq

Note our convention to use the superscript t for a riskless asset return realized
in t + 1. It is to stress the fact that these returns are known in advance.

2.3    The solution
   Clearing conditions.         In the equilibrium the transversality condition holds.
Also, all markets clear:

   • Goods market: Ct + Kt − (1 − δ)Kt−1 = Yt
                           d
   • Capital market: Kt = Kt ;
   • Labor market: Nt = Ntd ;

   • Risk-free bonds market: af = 0;
                              t

   • Long-term bonds market: al = 0;
                              t

   • Firms’ bonds market: aq = µKt
                           t

   • Equity market: as = 1
                     t


   First order conditions. Let xs denote the equity premium and xl
                                    t+1                          t+1
the long bond premium over the risk-free rate:
                                      f
                               r
                        xr := Rt+1 − Rt
                         t+1                           for r = s, l

Define v(xr ) as:
         t+1

                                     xr                       xr ≥ 0
                    v(xr ) =          t+1
                                                  for          t+1
                       t+1           φxr
                                       t+1                    xr < 0
                                                               t+1

From the consumers’ problem (2) and (3) and taking into account the clearing
conditions we obtain the following set of Euler Equations (EE):
                                  −γ    f     −γ
                                 Ct = βRt Et Ct+1                               (11)

the EE on the riskless asset;
                                  −γ    q     −γ
                                 Ct = βRt Et Ct+1                               (12)

the EE on the firms’ discount bond;
                      −γ        −γ
                     Ct = βEt {Ct+1 [Rt+1 + Bv(xs )]}
                                                t+1                             (13)

                                          10
the EE on the firms’ shares;
                           −γ        −γ    l
                          Ct = βEt {Ct+1 [Rt+1 + Bv(xl )]}
                                                     t+1                                 (14)
the EE on the long-term bond. The returns are defined as in (9) and (10).
Note that the first two Euler Equations have a form typical for the standard
RBC models. They imply the usual intertemporal trade-off between consuming
goods for one more dollar today or investing this dollar in an asset with return
  i
Rt in order to enjoy a bit higher consumption tomorrow. On the other hand
the EEs on shares and long bonds contain an additional term implied by the
formulation of our model in the spirit of the Prospect Theory: Consuming a bit
less today and investing the proceeds in a risky asset brings a slightly higher
expected consumption utility tomorrow, but it also exposes the consumer to a
higher degree of the loss-aversion risk.
    The first order condition with respect to labor gives the standard expression:

                              θ(1 − Nt )−χ
                                    −γ     = Wt                                          (15)
                                  Ct
Finally from the clearing conditions we have:
                                       Ct + It = Yt                                      (16)
The maximization problem (4), (6) and (7) of the firm gives the following stan-
dard expressions for the price of capital and labor respectively5 :
                                   −γ        −γ k
                                  Ct = βEt {Ct+1 Rt+1 }                                  (17)
where
                                                                                 It+1
                                                                                                 
                    It
              G    Kt−1                At+1 Nt+1
                                                    1−α          1−δ+G             Kt        It+1 
 k
Rt+1 =                            α                       −µ+                           −
         1−G        It
                           µPtq           Kt                         G     It+1               Kt
                   Kt−1                                                     Kt
                                                                                         (18)
and                                                    α
                                        Kt−1
                             (1 − α)A1−α
                                     t          = Wt                     (19)
                                         Nt
These, together with the capital accumulation equation (7) and the production
function (6) end the description of the main equations of the model.

    The balanced growth path. As it was already noted in King et al (1988),
in a standard RBC model with additively separable consumption and leisure and
a positive productivity growth rate g, the existence of the balanced growth path
is assured only for the special case γ = 1. This observation is also true in our
model. There are several ways to deal with this problem (see Lettau and Uhlig
1997 or Lettau et al 2001), but we simply choose to keep g = 0. This complies
with most of the RBC literature.
   5 Note that firms’ managers are supposed not to be loss-averse. Therefore Rk , the return
                                                                                t
on capital, is determined by the Euler Equation of the standard form. An interesting question
emerges how the economy would be affected, had also the managers were loss-averse.


                                             11
    Calibration. The baseline RBC parameter values are set in line with the
standard RBC literature (Cooley and Prescott, 1995; King and Rebelo, 2000).
We choose γ = 1 so that the elasticity of intertemporal substitution of the
consumption good is also 1. The leisure utility is parametrized as follows. The
power parameter χ is set equal to 1. Then the parameter θ is picked so that
the proportion of time spent by the consumers on working in the steady state is
equal to 0.3. We set β equal 0.995. This, combined with the zero productivity
growth rate will give the equilibrium risk-free interest rate of around 2%, which
is slightly above its true value6 .
    For the production technology, the parameter governing the persistence of
the technology shock ρ = 0.95. Then, the value of the standard deviation of
the shock t is set in such a way that std[(1 − α) t ] ≈ 0.0072 as in Lettau et al
(2001)7 . The capital share α is set to the standard value of 0.33.
    In calibrating the functional form of the cost-of-adjustment we closely stick
to the parametrization used in Uhlig (2006). The constants a1 and a2 are chosen
so that G(δ) = δ and G (δ) = 1, where δ = 0.025 is the quarterly depreciation
rate. We focus on two values of the parameter ξ that often appear in the
literature: ξ = ∞ (no adjustment cost) and ξ = 0.23.
    For the loss-aversion parameters φ and B, BHS (2001) use φ = 2.25, a
magnitude based on the experimental findings of Kahneman and Tversky (1979),
and B = 2 as their benchmark. Our baseline parametrization of the LA term is
identical to theirs. We also provide a sensitivity analysis for these parameters.
    Finally, the payout of the long term-bond is normalized to 1%, Dl = 0.01,
which is paid every period ad infinitum. We allow for different values of the
financial leverage ratio, varying from µ = 0 to µ = 0.5.

    Details of the solution approximation. The LA term in the consumers’
utility is a kinked function and is hence not continuously differentiable. The
model can therefore not be linearized. Instead, we use a non-linear solution
method based on Coleman (1991) and following Davig (2004a and 2004b).8 .
See the appendix for details.


3     The results
3.1     The data.
The statistics a successful model of financial markets should be able to repro-
duce are reported in Table 1. We focus on the Sharpe Ratio which, given its
   6 The papers on the asset pricing in the RBC framework assume usually a positive produc-

tivity growth rate g. In most cases it demands having β very close to 1 in order to match the
risk-free interest rate (see for instance Boldrin et al, 2001). The choice of g does not affect
other financial market statistics we intend to match.
   7 See also King and Rebelo (2000)
   8 We are grateful to Troy Davig for having sent us sample codes for the solution of a

bare-bone RBC model.




                                             12
interpretation of the price for risk, is argued by Uhlig (2006) to be a more re-
vealing statistics than a simple equity premium. Its logarithmic unconditional
version is defined as:
                                                    f
                                  E[log Rt+1 − log Rt ]
                           SR =
                                           σlog r
         2
where σlog r = var[log Rt+1 ]. The annualized logarithmic Sharpe Ratio was
found by Lettau and Uhlig (2002) to be around 0.27. Uhlig (2006) finds SR
only slightly higher. Other authors consider rather the arithmetic version of the
Ratio:
                                        E[xs ]
                                            t+1
                                SRa =
                                          σr
with σr being the volatility of the stock returns. Based on the stock return
moments estimated by Cecchetti et al (1993), Boldrin et al (2001) calculated
the arithmetic Ratio to be 0.34. A yet larger number is given in Campbell
(2004), SRa = 0.46. As it is suggested by Uhlig (2006), a part of the difference
between his and others’ estimates is probably due to different definitions of the
Sharpe Ratio, although it is not a full story.
    In accordance with the tradition initiated by Mehra and Prescott (1985), we
attempt also to match the equity premium, which is estimated to be about 7%
on the yearly basis. In order to account for the volatility puzzle identified by
Campbell (1999) two other financial statistics we want to reproduce are the risk-
free rate volatility and the equity return volatility. The latter one is around 19%
in the data. There is some kind of controversy regarding the former. According
to Cecchetti et al (1993), the standard deviation of the risk-free rate is around
5% in the data. However Campbell 2000 finds the volatility of the returns on
the US Treasury Bills to be 1.76%. Additionally, he argues that half of it could
be due to ex-post inflation shocks.
    Finally, we also check how well our model performs in terms of the mean
risk-free rate and the mean premium on the long bonds. The risk free premium
is about 1%, while the premium on the long bonds over the risk free rate is
found in the data to be below 2%.
    Table 1 Standard asset statistics
      moment value a
      SR          0.27b
          f
      Ert         1.19
      Exs  t      6.63
      σr f        5.27
      σr          19.4
      Exl  t      1.70c
    a
        The estimates from this column are taken from Cecchetti et al (1993) as reported by Boldrin
et al (2001). They are based on the U.S. data from the period 1892-1987.
     b
       The Sharpe Ratio estimate is taken from Lettau and Uhlig 2002.
     c
       The estimate of the premium on the mean long term bond is taken from Jermann (1998).
   Table 2 reports the main business cycle facts. We do not attempt to match
these statistics and they are provided solely for the comparison purposes. We


                                                13
can see that the quarterly volatility of output is around 2%. The consumption
is less volatile than the output while the investment is three times as much
volatile. The hours worked vary about as much as the output. Finally, the
consumption, the investment and the labor are strongly procyclical.
    Table 2 Standard business cycle statistics
      moment value a
      σY         1.81
      σC /σY     0.74
      σI /σY     2.93
      σN /σY     0.99
      ρY C       0.88
      ρY I       0.80
      ρY N       0.88
   a
       The estimates from this column are taken from King and Rebelo (2000).


3.2      Results: The Basic Framework
In this section we consider a basic framework with loss-aversion and capital
adjustment cost but without leverage. In order to compute the moments of
variables of interest we simulated the economy for 10200 periods. The moments
reported are based on time series consisting of last 10000 observations.
    The first row of Table 3 presents the statistical asset moments for the sim-
plest RBC model. It demonstrates the well-known fact that in the classical
RBC framework with moderately risk averse agents, one is unable to reproduce
any of the most important financial statistics. In particular, the Sharpe Ratio
produced by the model appears virtually zero. As it was argued in the intro-
duction, even a significant increase in the parameter of risk aversion does not
resolve the puzzle. The addition of the capital adjustment cost by itself cannot
do the job either, see the second row of the table (compare also Jermann, 1998).
In contrast, as we will see shortly, the introduction of the loss-aversion terms
has the potential to significantly improve the performance of the model.
    We consider first a bare-bone RBC model (no adjustment costs nor leverage)
with LA term parameters b = 2 and φ = 2.25. This is the baseline parametriza-
tion of BHS (2001). The addition of the LA features to the model change the
things radically: The Sharpe Ratio becomes 0.25, which matches the estimate
given by Uhlig (0.27). Hence, the usefulness of the loss-aversion in explaining the
financial market puzzles appears hard to be questioned also in the production
economy framework.
    Intuitively, the mechanism that pushes up the Sharpe Ratio in our model
may be explained as follows. The agents in this framework care not only about
the consumption level, but also about the changes in their risky asset positions.
Since they are loss-averse, the feeling of pain they experience in the case of losses
is relatively stronger than the satisfaction they achieve in the case of gains. This
implies that the risk premium the households demand is high. Hence, for a given
amount of risk (as measured by the standard deviation of the risky asset return),
the Sharpe Ratio increases.

                                              14
    Note that we were able to obtain this result without constraining the labor
or the capital mobility in the model. This is in contrast to those production
economy models in which the explanation of the financial puzzles is based on the
habit persistence (Jermann 1998, Boldrin et al 2001), which demand a consid-
erable amount of frictions on both, the labor and the consumption-capital good
markets in order to match the Ratio. For instance Boldrin et al (2001) report a
Sharpe Ratio of 0.002 for their parametrization of a production economy habit
formation model without labor and capital frictions.
    The intuition behind this stark difference between the two types of models is
straightforward. The performance of the habit persistence models relies heavily
on the high consumers’ aversion to the consumption risk. They are capable to
reproduce easily the high empirical Sharpe Ratio in the endowment economy
framework where the consumption volatility, even low, is given exogenously.
However, in the models with fully-fledged production economies the degree of
consumption volatility depends on the actions taken by consumers. They are
able to smooth their consumption paths by varying the labor effort and the rate
of capital accumulation and thus establish an insurance against unexpected
changes in consumption. It turns out that if at least one of these channels is
easily accessible, the consumers are able to substantially reduce their exposure
to the consumption risk, even if their risky asset position is high. What follows,
they do not deem investing in risky assets truly risky. It is only when the
labor-leisure choice and the capital-consumption choice are both considerably
constrained that the habit persistence models are able to generate the observed
risk premium.
    The models based on the Prospect Theory do not demand a high degree
of aversion to the consumption risk. It is because, as it was pointed out by
Campbell, ”they generate risk-averse behavior [...] also from direct aversion to
wealth fluctuations” (Campbell 2000, p. 1554). In other words, the consumers
exhibit what is called ”first-order risk aversion”9 . Clearly, the households cannot
easily insure against the risk related to the aversion to the wealth fluctuations.
Neither working more nor investing less (and thus consuming more) will alleviate
the pain from an unfortunate investment. It is because a loss of wealth is hurtful
independently of the level of consumption chosen. In fact the only way to avoid
the pain is to refrain from investing in risky assets. Since in the equilibrium
all assets must be owned by the households, the implied price for risk must be
high.
    Although this version of the model comes close to matching the empirical
Sharpe Ratio, it fails to reproduce other asset markets statistics. In particular
it produces only a tiny equity premium (Exs = 0.03%). It is not very hard to
                                               t
find the reason for this failure. The model implies a very low volatility of stock
returns (σr = 0.33%) - two orders smaller than the observed volatility. The
introduction of the loss-aversion to the model affected the intensity with which
the agents experience the exposure to the risky asset return fluctuations, and
hence the price for risk, but did not have an impact on the amount of the risk
  9 See   Epstein and Zin (1990) for a different model of first-order risk aversion.



                                               15
as measured by the asset return volatility. Hence, in order to match the equity
premium we need to ratchet up the standard deviation of the returns.
    In the fourth row of Table 3 the results of the model with loss-aversion and
capital adjustment costs are reported. As expected, the volatility of the stock
returns is now much higher, σr = 4.19%. Still, it does not reach the true stock
return volatility of almost 20%. Therefore, the implied equity premium still
does not come close to the level of 7%, Exs = 1.06%.
                                             t
    We can check the sensitivity of our results to changes in the loss-aversion
parameters b and φ (two last rows of Table 3). The choice of b = 100 implies
that the consumers’ utility is fully dominated by the LA term. Despite this
fact, the increase of the stock return volatility and of the equity premium is
only moderate. Setting the parameter φ to 4 makes the losses almost doubly as
painful as in the baseline parametrization. This can be measured by the Sharpe
Ratio, that has increased almost twice. Yet, this change does not lead either to
more realistic stock return moments. On the other hand it helps in matching
the premium on the long term bonds over the risk-free rate (the last column of
Table 3).
    Table 3 Financial statistics (no leverage)
                                                                 f
                    model                  SR Exs   t   σr    Ert     σr f  Exlt
               bare-bone RBC               0.00 0.00 0.32 2.03 0.29 0.01
            RBC with adj. costs            0.01 0.10 3.80 1.99 0.52 0.07
          bare-bone RBC with loss
                                           0.25 0.03 0.33 2.02 0.31 0.18
       -aversion (baseline calibration)
        RBC with adj. costs and loss
                                           0.24 1.06 4.19 2.01 0.49 0.74
       -aversion (baseline calibration)
          RBC with adj. costs and
                                           0.32 1.45 4.31 2.01 0.54 0.96
           loss-aversion (b = 100)
          RBC with adj. costs and
                                           0.43 2.01 4.55 2.00 0.46 1.29
            loss-aversion (φ = 4)
   Baseline calibration:
   loss-aversion term: b=2, phi=2.25; adjustment costs: ksi=0.23


3.3    Leverage
So far we have assumed that the firms finance their activity through retained
earnings. Yet, in reality a non-negligible part of firms’ capital is financed by
borrowing. Allowing for a non-zero financial leverage in our model will obviously
increase the riskiness of holding stocks. In this subsection we examine whether
a plausible degree of financial leverage may be helpful in matching the stock
return volatility and the equity premium. The financial moments generated in
the model with leverage are reported in Table 4. The following results stand
out.
    The first row of the table shows that a combination of a plausible degree
of leverage with capital adjustment costs alone is not sufficient to produce a
reasonable values of the financial statistics. As expected, it does considerably
increase the volatility of risky returns, which may be interpreted as an increase of

                                             16
the quantity of risk. However the price of risk in the economy is low. Therefore
the equity premium hardly reaches the mere 0.5%.
     The leverage ratio used was µ = 0.5, which implies that the firms’ finance
mix is composed in 50% of stocks and in 50% of credits10 . Assuming a much
higher leverage ratio would certainly do the job of further ratcheting up the
premium. It would not however do the trick with the Sharpe Ratio, which in
this version of the model is extremely low (0.02). Clearly, in order to explain
both high equity premium and high Sharpe Ratio, the model must be capable
of generating not only a non-negligible quantity, but also a high price of risk.
This last feature is guaranteed if the agents are loss-averse.
     Indeed, adding the loss-aversion to the setting with adjustment cost and
leverage allows for matching basically all statistics of interest. Take first the
BHS baseline calibration with b = 2 and φ = 2.25. As we have seen already
before, the model with thus calibrated LA terms produces the Sharpe Ratio of
about 0.25. Due to the leverage ratio of 0.5, the stock holders bear now twice
as much risk as they did before. This translates into the volatility σr = 16.85,
a number that is only slightly lower than the empirical equity return volatility
of 19.4%. This combination of the high price for risk and the high quantity of
risk results in a very reasonable equity premium Exs = 5.16. We find an even
                                                        t
better match with the data, if we depart from the BHS baseline calibration of
the LA term. Assuming φ = 3 (and decreasing the leverage ratio µ to 0.4) while
still keeping b = 2 causes a further increase of the equity premium by 0.5%.
Finally, increasing the parameter b also to 3 produces the premium of 6.5%,
which almost perfectly matches its empirical value (6.6%).
     Note that this undeniable success of matching the empirical equity pre-
mium did not come at the expense of other statistics of interest. In partic-
ular, our approach does not give rise to a too high long bond risk premium or
too volatile short-term real interest rate returns, two problems that plague the
habit-formation based models (see for instance Boldrin et al, 2001 and Jermann,
1998). The reason why the short-term real interest rate does not vary much in
our model is that the expected consumption stream the model produces is not
very volatile either. For the value of the power utility parameter we are using
(γ = 1) this implies that the volatility of the expected marginal utility and hence
also of the expected stochastic discount factor (SDF) is modest. Since the Euler
                                                                f
Equation (11) determining the short-term real interest rate rt is standard and
                                                           f
depends only on this last expectation, the volatility of rt must also be low.
     It is also intuitive why we did not obtain a too high mean long bond risk
premium, despite the fact that the agents in our economy are loss-averse over
the returns on these bonds. It is because the introduction of the leveraged
firms’ ownership does not influence the risk of holding long bonds. Compare
the mean levels of the long bond risk premium in the case of the BHS baseline
calibration with capital adjustment costs but no leverage (Table 3, row 4) and in
  10 Boldrin et al (2001) also apply the leverage ratio of 0.5. Some authors consider slightly

higher values of this ratio (see Jermann 1998). However in the light of the empirical research
we do not find it legitimate to allow for much higher numbers. For instance Masulis’ (1988)
market values estimates of the leverage ratio for the US do not surpass 0.44.


                                             17
the case of identical calibration but with µ = 0.5 (Table 4, row 2). In both cases
Exl t    0.7511 . For our preferred calibration (b = 3, φ = 3, ξ = 0.23, µ = 0.4,
see Table 4, row 5) the mean level of the long bond premium is 1.1%. It is of
the same order of magnitude as the empirically observed value Exl = 1.7%.
                                                                        t
    Our preferred calibration implies the logarithmic Sharpe Ratio of 0.36, which
is larger than the number SR = 0.27 given by Lettau and Uhlig (2002). As it
was argued earlier, in order to compare our results to other studies, we need to
compute the Sharpe Ratio based on the arithmetic returns. For the calibration
under discussion (b = 3, φ = 3, ξ = 0.23, µ = 0.4) it gives SRa = 0.42, which is
almost the same as the value reported by Campbell (0.46). We conclude that
our result for the Sharpe Ratio lies well within the bounds of what is considered
a plausible value for this statistics.
    It is also important to stress that in our model the elasticity of intertemporal
substitution is still equal to the inverse of the power utility parameter γ. Hence,
the assumption that γ = 1 implies that the consumers do not have a strong desire
to smooth their consumption over time. This suppresses the mean value of the
short-term real interest rate, which in our model does not differ much from its
steady state value. For our calibration (discount factor β = 0.995, productivity
                            f
growth g = 0) it gives rt         2%, but could even be lower, should we choose
a higher β. Hence the risk-free rate puzzle (Weil 1989) is not an issue in our
approach.
    Table 4 Financial statistics
                                                                               f
                          model                       SR Exs     t    σr     Ert   σr f    Exl
                                                                                             t
             RBC with adjustment cost and
                                                     0.02 0.46 7.56 1.93 0.51              0.13
                leverage, no loss-aversion
                RBC, baseline calibration            0.24 5.16 16.85 1.99 0.48             0.76
        RBC with adj. costs, leverage µ = 0.4
                                                     0.31 5.56 15.01 2.01 0.51             1.00
                and loss-aversion (φ = 3)
        RBC with adj. costs, leverage µ = 0.4
                                                     0.36 6.48 15.63 2.00 0.48             1.10
            and loss-aversion (b = 3, φ = 3)
              RBC with leverage and loss
                                                     0.23 0.05 0.37 2.04 0.29              0.16
             -aversion , no adjustment cost
          RBC with adj. costs ξ = 2, leverage
                                                     0.48 2.97 5.91 2.01 0.34              1.07
       µ = 0.4 and loss-aversion (b = 4, φ = 4)
   Baseline calibration:
   loss-aversion term: b=2, phi=2.25; leverage: mu=0.5; adjustment costs: ksi=0.23


3.4       The macroeconomic performance of the model.
As far as the real variables statistics reported in Table 2 are regarded, the bare-
bone RBC model reproduces the qualitative characteristics of the true economies
(see Table 5, row 1). On the other hand, although the capital adjustment costs
may be useful in explaining some other dimensions of the data (for instance the
sluggishness of the inflation and the output), when combined with the simplest
 11 The   tiny difference follows from the fact that the results are based on simulations



                                              18
RBC framework, they lead to an overall deterioration of the model macroe-
conomic performance. In order to see it, regard the second row of Table 5,
where real variables statistics for the bare-bone model with adjustment-costs
(ξ = 0.23) are reported. Clearly, the standard deviation of output is too low,
the volatility ratios are reversed, with consumption being the most volatile and
the volatility of investment being only a third of that of output, and the cor-
relations of consumption and investment with output are too extreme. The
model generates also a countercyclical behavior of labor. This last feature is
perhaps not of a concern however, as the procyclicality of this variable in the
data has been recently hotly debated (compare Uhlig, 2006 and the references
given therein).
    The reason why the capital adjustment costs have adverse effects, when
added to a simple RBC model is intuitive. When the agents face adjustment
costs, they prefer the stock of capital to adjust only smoothly in response to
unexpected changes in the productivity. This implies that the stock of capital
does not get a significant boost in the times of prosperity. This in turn leads to
a lower volatility of output. The fall of the variability of the investment stream
is even more pronounced. On the other hand the volatility of consumption con-
siderably increases: If not invested, the extra amount of the consumption good
produced due to a positive productivity shock will be immediately consumed.
And when the shock is negative, the agents are willing to give up an important
part of their consumption in order to keep the stock of capital at its stable level.
Finally, since an immediate increase of the scale of production is not profitable,
a positive productivity development leads the agents to work less hard. In effect,
the hours worked behave countercyclically.
    This situation does not change when we assume the households to be loss-
averse. It is because the introduction of the loss-aversion to the model does not
affect the dynamics nor the volatilities of variables like consumption, output
or investment. In fact, its only effect is to shift the stochastic steady states of
these variables. The third and the fourth row of Table 5 report the statistical
moments of these variables for the calibrations of the LA term that we found
the most successful in resolving the asset market puzzles. Clearly, the reported
statistics are virtually identical to the ones coming from the model with only
adjustment costs.
    What happens if we switch off the adjustment costs? Consider an RBC
model with leverage (µ = 0.5) and the baseline loss-aversion (b = 2, φ = 2.25),
but without adjustment costs. The real economy statistics for this version of
the model are reported in the fifth row of Table 5. Clearly, they do not differ
at all from the ones obtained in the simplest RBC framework, and hence are
crudely in line with the data. The question now arises whether this version of
the model does also generate a considerable equity premium. As it is shown in
the fifth row of Table 4, the answer is negative. The reason is that the leverage
ratio of 0.5 alone is not enough to reproduce the high volatility of risky asset
returns12 . And as we already learned, when the standard deviation of the equity
 12 It   is worth stressing again that allowing for a higher µ would increase the return volatility,



                                                 19
returns is low, adding the loss aversion terms to the utility helps in explaining
the empirical Sharpe Ratio, but does not make it easier to match the mean risky
asset premia. In particular, setting very high values of the parameters b and
φ is of no help: It raises the Sharpe Ratio excessively, but increases the equity
premium only marginally.
    Finally, one might wonder whether setting an intermediate value of the ad-
justment cost parameter would not allow to match the empirical statistics in
both dimensions. To answer this question, we pick the adjustment cost param-
eter ξ = 2 and consider the leverage ratio µ = 0.4 and the Prospect Theory
parameters b = 4 and φ = 4. The last row of Table 5 shows that, as far as the
real economy statistics are concerned, the chosen value of ξ appears too low.
The output volatility σY = 1.01 is too small of order of two as compared to the
data (σY = 1.81). Also the volatility ratios σY = 1.45 and σN = 0.12 must
                                                σI
                                                                  σY
                                        σI
be considered too low (in the data σY = 2.93 and σN = 0.99). Yet, as it is
                                                       σY
demonstrated in the last row of Table 4, this value of ξ is still too high to be of
much help in matching the asset market statistics. With high values of b and
φ chosen we obtain the Sharpe Ratio SR = 0.48. It is much above its reliable
estimates. Still, the mean equity premium Exs = 3% is only a half of what we
                                                t
find in the data. As in all previously discussed cases, the origin of the problem
is the same: The stock returns are not sufficiently volatile. Unless we assume
an implausible degree of financial leverage, the only easy way to increase this
volatility is to consider higher capital adjustment costs.
    Table 5 Real variables statistics
                                                           σC       σI    σN
                          model                     σY     σY       σY    σY     ρY C   ρY I   ρY N
                     bare-bone RBC                 1.41 0.29 2.99 0.51 0.90             0.99    0.99
                  RBC with adj. costs              0.83 1.25 0.33 0.17 1.00             1.00   -1.00
               RBC, baseline calibration           0.81 1.26 0.33 0.17 1.00             1.00   -1.00
        RBC with adj. costs, leverage µ = 0.4
                                                   0.85 1.25 0.33 0.17 1.00             1.00   -1.00
            and loss-aversion (b = 3, φ = 3)
              RBC with leverage and loss
                                                   1.37 0.29 3.00 0.51 0.90             0.99   0.99
             -aversion , no adjustment cost
         RBC with adj. costs ξ = 2, leverage
                                                   1.01 0.83 1.45 0.12 1.00             1.00   1.00
       µ = 0.4 and loss-aversion (b = 4, φ = 4)
    Baseline calibration:
    loss-aversion term: b=2, phi=2.25; leverage: mu=0.5; adjustment costs: ksi=0.23



4     Conclusions
In this paper we have attempted to provide a resolution to several famous fi-
nancial market puzzles. In contrast to the gross of the classic literature on
the subject, we have taken up the challenge to work in a fully-fledged general
equilibrium model, where the consumption and dividend streams are generated
endogenously. In this framework we have shown that the assumption that the
and hence also the risk premium.




                                              20
agents act in line with the behavioral Prospect Theory of Kahneman and Tver-
sky, and in particular that they are loss-averse over the fluctuations of the value
of the risky assets they hold allows to easily match the empirical Sharpe Ratio.
Further, when combined with capital adjustment costs and a moderate degree
of financial leverage, it also helps to exactly reproduce the US economy mean
equity premium.
    The consumers’ elasticity of intertemporal substitution in our model is dis-
entangled from the time-varying degree of risk aversion and constant. For the
parametrization of the utility function considered in the paper, this elasticity is
equal to one. Therefore the agents are not urged to borrow from the future, as it
is often the case in other models aiming at explaining the high equity premium.
Thus, the risk-free rate puzzle does not emerge. Additionally, our model gener-
ates a low volatility of the risk-free rate and a moderate premium on long-term
bonds. This is in contrast to the popular habit-formation based models, usually
flawed with too high long bond returns and an implausible variability of the
short-term real interest rate.
    The Prospect Theory, with the loss-aversion as one of its central elements,
provides us with a mechanism that allows to generate a high price of risk in the
economy. Simply put, the agents with the Prospect Theory utility are extremely
risk averse on the border between the gains and the losses. Due to this local
risk sensitivity, the agents demand high premium for bearing a given amount of
risk. When applied to financial markets, this implies that the agents expect a
high average return on risky assets, as compared to the quantity of risk these
assets bear.
    In absolute terms however, the equity premium the model generates does not
need to be large. The reason is that the simple RBC model does not produce
high variation in risky returns. The loss-aversion by itself does not provoke a
notably higher variation in the risky asset returns either. Therefore, in order to
obtain a sizeable equity premium, one has to rely on an additional mechanism
that would produce a considerable volatility of stock returns. In our paper we
obtain a high standard deviation of the equity returns by incorporating costly
capital adjustment to the model. The rationale behind adding the cost-to-adjust
to the macroeconomic models is that it generates a time-varying Tobin’s q. It is
also widely used as a tool to enhance the propagation properties of the output
and inflation dynamics. Nevertheless, in a simple RBC framework, it leads to
several counterfactual results. The question therefore remains, how to generate
a plausible variability of stock returns, without deteriorating the real economy
properties of the RBC model. In this light, the observation of Campbell (2000)
that the volatility puzzle seems more robust than the equity premium puzzle
finds an additional confirmation.




                                        21
5    References
References
 [1] Barberis, Nicholas, Ming Huang (2002): ”Mental Accounting, Loss Aver-
     sion, and Individual Stock Returns”, NBER Working Paper 8190

 [2] Barberis, Nicholas, Ming Huang, Tano Santos (2001): ”Prospect Theory
     and Asset Prices”, Quarterly Journal of Economics, CXVI / 1, 1-53.
 [3] Barberis, Nicholas, Ming Huang, Richard Thaler (2003): ”Individual Pref-
     erences, Monetary Gambles and the Equity Premium”, NBER Working
     Paper 9997
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6     Technical Appendix.
The complete model is described by the equations (6), (7) and (9) to (17).
The solution conjectures candidate decision rules for Kt , Nt and It denoted
                    ˜              ˜                   ˜
respectively by hk Kt−1 , At , hn Kt−1 , At , and hi Kt−1 , At and conjec-
                                                      f
tured pricing functions for Rt , Pts and Ptl : hr            ˜               ˜
                                                             Kt−1 , At , hps Kt−1 , At ,
      ˜
hpl Kt−1 , At respectively. The conjectured rules and pricing functions are
substituted into the model’s equations.
    For the solution, we descretize the state space. For any given state (Kt−1 , At ) ,
the decision rules and pricing functions are taken as given. This allows us to
                    f
compute Ptq = 1/Rt ; Yt by the production function (6), Ct by the resource con-
straint (16) and G by its definition (8) for the given state conjecturing agents
decisions for capital, labor and investment and the pricing rules.
    For the expectational terms, we first compute the conjectured decisions and
                                                            f         f
                                ˜              ˜
prices for the period t + 1: Kt+1 = hk Kt , At+1 , Rt+1 = hr Kt , At+1 ,  ˜
          ˜           ˜         ˜           ˜S         ˜           ˜L
Nt+1 = hn Kt , At+1 , Xt+1 = hx Kt , At+1 , Pt+1 = hps Kt , At+1 , Pt+1 =
    ˜                            ˜                                      q   ˜
hpl Kt , At+1 for     εt+1 given K and At . From these, we can compute Pt+1 Kt , At+1 ,
˜    ˜           ˜    ˜                                      ˜                ˜
Yt+1 Kt , At+1 , Ct+1 Kt , At+1 , G (t + 1), G (t + 1), Dt+1 Kt , At+1 , Rt+1 Kt , At+1 ,
 x   ˜            l   ˜
Rt+1 Kt , At+1 , Rt+1 Kt , At+1 again for                      ˜
                                                    εt+1 given K and At .
                                   1                  1                           f
    The expectations for Et        ˜
                                eg Ct+1
                                          , Et        ˜
                                                   eg Ct+1
                                                                Rt+1 + B˜(Rt+1 , Rt )
                                                                        v               ,
                                                                     x
         1       ˜L         L                f                      Rt+1
Et       ˜
      eg Ct+1
                 Pt+1  +D +        v L
                                  B˜(Rt+1 , Rt )  and Et          e
                                                                     are computed
                                                                     ˜t+1
                                                                    gC

by numerical quadrature.
    The solution technique treats the conjectured rules as six unknowns. It
solves the sytem of the four Euler equations (11, 12, 13, 14), the consumption-
leisure trade-off and the capital accumulation equation for the unknowns in any
set of state variables defined over the discrete partition of the state space. This
procedure is repeated until the iteration improves the current decision rule at
any given state vector by less than some tolerance level which we set to 10−12 .




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