ریسک غیر قابل بیمه شدن و پازل بازار مالی

We compare the empirical performances of three risk-sharing arrangements involving idiosyncratic skill shocks: (a) where individuals are unable to directly insure their consumption against individual-specific shocks, (b) where agents strike long-term insurance contract with financial intermediaries involving a truth revelation constraint as in Kocherlakota and Pistaferri (2009), (c) full risk sharing. Based on the widely accepted assumption of cross-sectional log-normality of individual consumption levels, we work out closed form expressions of the pricing kernels for (a) and (b). We put these three models to test four financial market anomalies, namely the equity premium, currency premium, risk-free rate, and consumption-real exchange rate puzzles simultaneously in an integrated framework. We find that the pricing kernel associated with (a) outperforms the other two models in terms of the produced estimates of the agent’s preference parameters and the model ability to predict the equity and currency premia, the risk-free rate, and the log growth in the exchange rate. However, the predictive ability is still far from satisfactory for all three models under scrutiny.

There are four well known puzzles in the macro-finance literature. Two of these puzzles are home based. Mehra and Prescott (1985) and Hansen and Jagannathan (1991), among others, observe that the covariance of aggregate per capita consumption growth with the excess return on the market portfolio over a risk-free asset is very low, so that the representative-agent consumption CAPM can explain the observed market premium only if the typical investor is extremely risk averse. This is known as the equity premium puzzle. In addition, Weil (1989) observes that, given the lack of variability of aggregate consumption growth, the representative-agent must have a negative rate of time preference for the model to match the observed mean risk-free rate. This anomaly is known as the risk-free rate puzzle. The other two puzzles appear on the international front. Economic theory predicts that in a complete market setting, the log real exchange rate growth between any two countries equals the difference in the logs of the foreign and domestic stochastic discount factors (Brandt et al., 2006). With a representative agent in each country, the log real exchange rate growth must be perfectly correlated with the difference in the log growth rates of marginal utilities of aggregate per capita consumption of respective countries. This implies that, under the standard assumption of power utility, the log real exchange rate and log relative consumption should be perfectly correlated. In practice, however, it is observed that the correlation between relative consumption and the real exchange rate is close to zero or even negative. The real exchange rates are more volatile and persistent than the log relative consumption. This is the consumption-real exchange rate puzzle documented by Kollmann, 1991 and Kollmann, 1995 and Backus and Smith (1993). The fourth anomaly with the representative-agent model is that it is unable to reconcile the highly volatile excess return on currency with the smooth aggregate consumption growth rate unless the agent is assumed to bear an implausibly high level of risk aversion. This is the currency premium puzzle, illustrated by Lustig and Verdelhan (2007), for example. The representative-agent consumption CAPM is implicitly based on the assumption of market completeness. In the absence of complete contingent claims markets, agents are not able to completely insure their consumption against idiosyncratic risks they face and hence realized IMRS can differ across individuals. Bewley, 1982, Mehra and Prescott, 1985, Mankiw, 1986, Constantinides and Duffie, 1996, Brav et al., 2002, Semenov, 2004 and Basu and Wada, 2006, and Balduzzi and Yao (2007), among others, argue that consumers’ heterogeneity induced by market incompleteness may be relevant for asset pricing. To assess the potential of the incomplete market hypothesis in explaining the Backus and Smith (1993) puzzle and the Mehra and Prescott (1985) equity premium puzzle, Kocherlakota and Pistaferri, 2007 and Kocherlakota and Pistaferri, 2009 assume that markets are complete with respect to country-specific shocks (individuals can fully insure their consumption against cross-country shocks), but domestic markets are incomplete (individuals cannot completely insure themselves against idiosyncratic skill shocks). This partial insurance creates heterogeneity across individuals making it necessary to relax the assumption of a representative agent within each country. Kocherlakota and Pistaferri, 2007 and Kocherlakota and Pistaferri, 2009 consider two forms of partial insurance against idiosyncratic skill shocks. The first KP call domestically incomplete (DI) markets. Under this formulation, individuals are unable to directly insure their consumption against individual-specific shocks. The second form of partial insurance they label as Private Information Pareto Optimal (PIPO). Here, the agents are able to sign insurance contracts, which allow them to completely insure themselves against idiosyncratic shocks, subject to the incentive constraint that agents reveal truth about their private skill shocks to the financial intermediary. For each of these two forms of partial insurance, KP (2007) derive a restriction relating the growth rate of the real exchange rate to the difference in the growth rates of the moments of the cross-sectional distributions of consumption in two countries. Using household-level consumption data for the US and the UK, they show that the asset pricing model associated with the PIPO insurance scheme fits the data on real exchange rates with the relative risk aversion coefficient of around five, while the complete risk-sharing (CRS) model and the DI model both perform poorly. In another companion paper, KP (2009) demonstrate the superior performance of the PIPO model to explain the observed mean equity premium in the US when households have a relative risk aversion coefficient between five and six.1 Although both papers of KP make a major methodological contribution to model consumer heterogeneity in the presence of uninsurable risk, there are two problems with their approach. The first problem is about the robustness of this approach to attack various puzzles. Ideally, one expects that the same economic fundamentals should be responsible for understanding home and international financial puzzles described earlier. Kocherlakota and Pistaferri, 2007 and Kocherlakota and Pistaferri, 2009 focus on only one puzzle in isolation and the proposed pricing kernels do not work properly if two or more asset pricing anomalies are addressed together in an integrated framework. For example, the stochastic discount factors proposed by Kocherlakota and Pistaferri, 2007 and Kocherlakota and Pistaferri, 2009 fail to jointly explain the observed mean equity premium and risk-free rate.2 The second problem is about the robustness of KP’s estimation results to the sample design. As documented by Kollmann (2009), the estimation and testing results for the PIPO pricing kernel proposed in Kocherlakota and Pistaferri, 2007 and Kocherlakota and Pistaferri, 2009 are highly sensitive to the presence of outliers. Kollmann (2009) shows that discarding outliers and minor specification changes may overturn the KP (2007) findings. He reestimates the KP (2007) PIPO model and finds that the real exchange rate anomaly continues to persist. As in KP (2009), we assume that international markets are complete, while domestic markets are incomplete, and consider two market structures: (a) where agents are unable to insure their consumption against idiosyncratic skill shocks (the DI markets) and (b) where idiosyncratic shocks to their skills can be partially insured by striking long-term insurance contract with truth revelation constraint (the PIPO form of partial insurance against idiosyncratic shocks). For each of these two market structures, we derive the associated stochastic discount factor. Throughout the paper, we refer to these pricing kernels as the DI and PIPO stochastic discount factors, respectively. Although our PIPO and DI frameworks are somewhat similar to those in Kocherlakota and Pistaferri, 2007 and Kocherlakota and Pistaferri, 2009, they differ from them in several fundamental respects. When deriving our PIPO and DI stochastic discount factors, we address the four asset pricing anomalies (the equity premium, risk-free rate, consumption-real exchange rate, and currency premium puzzles) simultaneously in an integrated framework and not only one anomaly in isolation (for example, the consumption-real exchange rate puzzle as in KP, 2007 or the equity premium puzzle as in KP, 2009). Since we address the currency premium puzzle, in contrast with Kocherlakota and Pistaferri, 2007 and Kocherlakota and Pistaferri, 2009, this requires careful modeling of international currency trades by motivating the transaction demand for currency in terms of a cash-in-advance constraint. In this respect, our approach is theoretically more elegant than the approach in Kocherlakota and Pistaferri, 2007 and Kocherlakota and Pistaferri, 2009. Following the tradition of Constantinides and Duffie, 1996, Sarkissian, 2003 and Basu and Wada, 2006, and Semenov (2008), we use the assumption of log-normality of the cross-section consumption process to work out closed form expressions for the pricing kernels for alternative market environments.3 The derived specifications of the DI and PIPO pricing kernels are different from those in Kocherlakota and Pistaferri, 2007 and Kocherlakota and Pistaferri, 2009 and are new to the literature. To assess the empirical performance of the incomplete risk-sharing stochastic discount factors proposed in this paper, we test them empirically using data for the US and the UK treating the US as the home country. If the results in KP (2007) are so sensitive to sample design, as documented in Kollmann (2009), the issue arises whether the superior performance of the PIPO pricing kernel will survive when more than one asset pricing anomalies are addressed together in an integrated framework. In this present paper, we precisely address this issue. For each of the considered stochastic discount factors, we jointly estimate the Euler equations for the equity premium, the risk-free rate, and the currency premium, as well as the linear equation for the real exchange rate. This is for the first time in the literature, when the above-mentioned four puzzles are addressed empirically in an integrated framework using the same pricing kernel. Thus, we propose to get a unique set of estimates for the preference based fundamentals, namely risk aversion and the subjective time discount factor, which could reconcile all four puzzles simultaneously. When estimating the model parameters, we use the quarterly data set that ranges up to 2008, while the KP data set is limited only up to 1999. We compare the results for our data set with the results obtained when the KP data are used in estimation. This can be viewed as a check whether inferences about the model performance are robust to the sample period considered. The Generalized Method of Moments (GMM) estimation and testing results show that, when the four asset pricing anomalies are considered jointly in an integrated framework, the model with the DI pricing kernel outperforms the CRS and PIPO models. In contrast with the CRS and PIPO pricing kernels, the DI stochastic discount factor allows us to jointly explain the observed equity premium, risk-free rate, currency premium, and real exchange rate with an economically plausible value of the relative risk aversion coefficient and the value of the subjective time discount factor that is close to but lower than one. This is a substantial contribution in view of the fact that many models may succeed in domestic dimension, but fail in international dimension and vice versa. We also put these three models to further scrutiny by testing their predictive abilities. We find that the DI model outperforms the CRS and PIPO models in predicting the equity premium, the risk-free rate, the log growth in the exchange rate, and the currency premium. However, the predictive ability is still far from satisfactory for all three models under scrutiny. We find these results to be robust to the used measure of consumption, sample design, and proxy for the market portfolio. The rest of the paper is organized as follows. In Section 2, we describe the DI and PIPO environments and derive the associated stochastic discount factors. Section 3 addresses the empirical implementation of the models derived in Section 2. In Section 4, we report the empirical estimation and testing results for these models. Conclusions and suggestions for further research are presented in Section 5.

This paper addresses four extant financial market anomalies, namely (i) the equity premium puzzle, (ii) the risk-free rate puzzle, (iii) the consumption-real exchange rate puzzle, and (iv) the currency premium puzzle. We investigate the potential of two candidate pricing kernels, which allow incomplete risk sharing in economies with consumer heterogeneity, to resolve these anomalies in an integrated framework. The first pricing kernel corresponds to the market structure with domestically incomplete (DI) financial markets, where idiosyncratic privately observed shocks are uninsured, while sequential trade in assets enables agents to partially hedge publicly observed shocks. We refer to this stochastic discount factor as the DI pricing kernel. The second stochastic discount factor is the pricing kernel that describes the market environment in which both private and public shocks are insured subject to truth revelation constraint by agents. Kocherlakota and Pistaferri (2007) call it the Private Information Pareto Optimal (PIPO) stochastic discount factor. Based on the widely accepted assumption of cross-sectional log-normality of individual consumption levels, we work out closed form expressions for these two pricing kernels. Our derived stochastic discount factors are new to the literature. We test empirically both these stochastic discount factors using household-level data on consumption expenditures from the US and the UK. To investigate whether the assumption of incomplete risk sharing plays an important role in explaining asset returns, we compare the results for the PIPO and DI pricing kernels with the results obtained under the assumption of complete risk sharing (the complete risk-sharing (CRS) model). In contrast with Kocherlakota and Pistaferri, 2007 and Kocherlakota and Pistaferri, 2009, when assessing the empirical performance of each candidate stochastic discount factor, we address the four financial puzzles in an integrated framework with the same pricing kernel in all the equations. Using the GMM estimation technique, we find that, when the Euler equations for the equity premium, the risk-free rate of return, and the currency premium, as well as the regression equation for the log growth in the exchange rate are estimated jointly, in contrast with the CRS and PIPO models, the DI model yields economically plausible estimates of both the relative risk aversion coefficient and the subjective time discount factor. The DI model is also found to outperform the PIPO and CRS models in predicting the equity premium, the risk-free rate, the log growth in the exchange rate, and the currency premium. It must, however, be mentioned that, although the DI model outperforms the CRS and PIPO models, this model does not allow precise estimation of the relative risk aversion coefficient and the predictive ability of this model is still far from satisfactory. Our results are robust to the used measure of consumption, sample design, proxy for the market portfolio, and variables in the agents’ information set. These findings suggest that the asset pricing anomalies considered in this paper may be even deeper than one usually thinks. Although it is often possible to find a pricing kernel that enables to explain a puzzle of interest in isolation, it is much more difficult to find the stochastic discount factor that is able to jointly explain several asset pricing anomalies. The conclusion about the relative performance of pricing kernels in solving an asset pricing anomaly in isolation may change substantially when several anomalies are considered jointly in an integrated framework. We conjecture that the forecasting performances of these models can be further improved by using a richer utility specification. For example, one can employ the Garcia et al. (2006) utility function with a reference consumption level or the Epstein and Zin, 1989 and Epstein and Zin, 1991 non-expected utility. Since in both specifications, risk aversion can be disentangled from the elasticity of intertemporal substitution as opposed to the conventional CRRA utility framework, we believe that the forecasting performances of the models with these preferences could be better. We leave this investigation for our future work.