قیمت گذاری دارایی در موضوع اقتصادی تولید برای شوک های پولی

This paper derives approximate analytical solutions for various financial assets in the production economy with monetary shocks. Both technology and monetary shocks drive the dynamics of various financial assets. Special cases of permanent and transitory shocks are considered. The solutions based on the loglinear approximation framework allow for a decomposition of risk that comes from real and monetary sides of the economy. Equity premium, volatility of the risk-free rate, Sharpe ratio, and inflation risk premium are calibrated to quarterly historical U.S. data. The model produces a realistic Sharpe ratio and inflation risk premium for empirically reasonable values of the relative risk aversion parameter, but results in the low equity premium. Overall, the results suggest that qualitatively the real business cycle model with monetary shocks has an advantage over the real business cycle model with respect to matching the key asset pricing facts.

In this paper I study the asset pricing implications of the real business cycle (RBC) model with monetary effects. The goal is to reexamine the most notorious puzzles in the asset pricing theory, namely, the equity premium puzzle, the fact that stock market returns consistently outperform Treasury bills by nearly six percent on average, and the risk-free puzzle, the fact that Treasury bills offer too low returns on average. Up to date research on the RBC models has mostly focused on the models that did not allow for any role for money. Plosser (1989) has shown that such models are empirically problematic.1 Several studies show that standard RBC models have also problems matching important features of the dynamics of U.S. asset prices Lettau, 2003, Lettau and Uhlig, 2002 and Rouwenhurst, 1995. Other authors document some success by introducing various frictions such as capital adjustment costs (Jermann, 1998), habit formation Boldrin et al., 2001 and Jermann, 1998, labor market frictions (Danthine & Donaldson, 2002), limited stock market participation (Guvenen, 2009), and idiosyncratic risk (Cárceles-Poveda, 2005). Despite some progress has been made towards studying asset pricing implications in the RBC models, it is clear that explaining asset prices in production economies remains challenging. One of the omitted economic variables in the analysis is money. Some researchers, including King and Plosser (1984), Eichenbaum and Singleton (1986) and Cooley and Hansen (1995) build RBC models with money, but they do not study implications for asset pricing. On the other hand, research that explores key asset pricing facts in the RBC framework, does not include monetary variables. This paper aims to fill this significant gap in the literature. One exception is a recent paper by Buraschi and Jiltsov (2005), who consider a structural monetary version of a real business cycle model with taxes and endogenous monetary policy. However, they examine bond pricing implications only. Monetary variables play important role in the asset pricing. In particular, Marshall (1992) shows that real returns are positively correlated with the growth rate of money in the economy, while Patelis (1997) finds that monetary policy variables are significant in predicting future excess returns. Recent empirical evidence suggests that unexpected monetary policy shocks have a significant impact on equity prices. Bernanke and Kuttner (2005) show that a hypothetical 25-basis point cut in the Federal Reserve funds target rate leads to about a 1% increase in broad stock indices. Challe and Giannitsarou (2010) provide similar empirical evidence of the adverse effect of the unexpected monetary shock on the stock market: they show that 1% unexpected increase in the nominal short rate results in the negative impact ranging from −2% to −9% in US and European countries. My work is also related to Bakshi and Chen (1996) who solve simultaneously for the price level, inflation, asset prices, and derive real and nominal term structures in the monetary asset pricing model. However, neither of the above papers discusses the direct effect of the monetary shocks to the equity premium and the risk-free rate puzzles, which is the focus of this paper. Money can be introduced in the RBC model in two ways. One can model it via defining cash-in-advance economy where money is held to facilitate consumption transactions (see, e.g., Alvarez et al., 2002, Christiano and Eichenbaum, 1992, Christiano and Eichenbaum, 1995, Fuerst, 1992, Giovanni and Labadie, 1991, Grossman and Weiss, 1983 and Marshall, 1992). Another way to model money is to define preferences over consumption and real monetary holdings (see, e.g., Brock, 1974, Brock, 1975 and Bakshi and Chen, 1996). The rationale for the latter approach is the assumption that money is held to provide a transaction service to consumption.2Feenstra (1986) shows that including real cash balances in the utility function is equivalent to including money as liquidity cost in the budget constraint (cash-in-advance economies). I follow the second approach in the paper. Mechanically, money-in-the-utility modeling is beneficial for dealing with asset prices. In the model, a representative agent has non-separable preferences over consumption and real cash balances.3 A representative agent invests in nominal bonds, real bonds, and equity shares. Firms are maximizing their profits subject to the capital accumulation constraint. Government supplies money into the economy according to some exogenously specified stochastic process. There are two imperfectly correlated shocks in the economy: technology shock and monetary shock. I contribute to the growing literature on asset pricing implications in the RBC models in two ways. First, I provide approximate closed-form solutions of the RBC model monetary effects (henceforth, MRBC model) by using log-linearization technique developed in Campbell (1994). Because money growth is uncertain and not perfectly correlated with stock returns, it is impossible to solve exactly the set of Euler equations in my model, so the budget constraint, the law of motion for capital and Euler equations are log-linearized. Then I solve for the elasticities of macroeconomic variables with respect to the state variables using the method of undetermined coefficients. On this front, my model extends the monetary asset pricing model of Bakshi and Chen (1996), who endogenously determine the price level, inflation, asset prices, nominal and real term structures of interest rates, but leave consumption process exogenous. In contrast, I solve for consumption level, which is a function of macroeconomic elasticities and state variables.4 The advantage of loglinear approximations lies in the possibility to show the direct effect of technology and monetary shocks on the endogenous variables. Other examples of the application of Campbell’s log-linearization technique to study the asset prices in the RBC models include Jermann (1998), Lettau (2003), and Cárceles-Poveda (2005). Second, I derive asset pricing implications of the MRBC model. I reexamine the equity premium puzzle, the risk-free rate puzzle, compute the maximal attainable Sharpe ratio, and derive approximate analytical solution for the inflation risk premium. It is possible to solve for the latter because of the presence of nominal and real bonds in the economy. While it is standard in the literature to evaluate the model performance against historical equity premium and volatility of the risk-free rate, estimation of the inflation risk premium has received a lot of attention rather recently (see, e.g., D’Amico et al., 2008, Ang et al., 2008, Grishchenko and Huang, 2010 and Haubrich et al., 2009). This is an additional dimension in which I extend the study of Bakshi and Chen (1996) who do not consider asset pricing implications of their model. Third, I calibrate the model using the standard set of parameters in the related literature. The MRBC model produces equity premium, which is low compared with historical one, although higher than in the standard RBC models. However, the model yields plausible estimates of Sharpe ratio and inflation risk premium for reasonable values of relative risk aversion (RRA), which vary between 6 and 10.5 When consumption and money holdings have equal weight in the utility function, the equity premium is 0.10% per quarter. As the weight on consumption decreases and on money increases, the equity premium increases: for example, the equity premium is 0.20% per quarter when consumption weight in the utility is 30%. Such an effect is also present when calibrating the risk-free rate volatility. Fixing γ = 10, volatility of the risk-free rate is roughly 0.15% per quarter when consumption and money holdings enter into the utility function with equal weights; it is 0.5% per quarter when the weight on consumption in the utility is around 30%. These estimates are lower than historically observed values of the equity premium (2% per quarter) and the volatility of the risk-free rate (0.8% per quarter), yet they compare favorably with the asset pricing quantities implied by the standard RBC model. Next, I check the performance of the model regarding Sharpe ratio to see how well the model can fit the second moments, or the equity return volatility. I show that Sharpe ratio is reasonable in my model for reasonable values of γ and/or relative weight of consumption and money in the utility function. When γ = 6, Sharpe ratio is equal to 0.26 for a permanent technological shock and equal weight on consumption and money in the utility function, which is in the ball-park with historical Sharpe ratio of 0.27. In the same situation, for higher γ (around 10), Sharpe ratio becomes close to 0.30 for a fairly persistent technological shock. When consumption weight converges to one, Sharpe ratio decreases so the model does not have the ability to match the moments of aggregate asset prices, judging by the magnitude of the maximal Sharpe ratio. Lastly, the model allows to use nominal and real bonds in the model to calibrate the short-term inflation risk premium. Thus, three-month inflation risk premium estimate is about 5 basis points (on the annual basis). These estimates indicate that the short-term inflation risk is indeed quite low. These estimates are roughly consistent with other estimates in the literature. Grishchenko and Huang (2010) report that the long-term inflation risk premium fluctuates between 8 and 13 basis points in the late 2000s, while shorter-term premium is around zero. Although with no general consensus yet, several recent studies (especially those with a focus on the TIPS market) come to a conclusion that the inflation risk premium has become rather low lately, (on average between zero and 30 basis points, see, e.g., Ang et al. (2008)) and so my estimates definitely agree with this evidence. Overall, monetary channel in the real business model results in favorable time-series implications for aggregate asset pricing. Another related literature studies the relevance of the monetary factor in the exchange-type economies for cross-sectional variation in expected asset returns, as opposed to considered time series dimension in my study. In particular, Chan, Foresi, and Lang (1996) and Balvers and Huang (2009) examine cross-sectional asset pricing implications of the (consumption-based) capital asset pricing model(s) augmented with money. In both studies, money serve to lower transaction costs or complete transactions. In those settings, money-based CAPM models outperform CAPM and C-CAPM models using various criteria. Thus, monetary factor has some explanatory power for the cross-sectional variation of asset returns as well. Of course, a natural question arises how accurate loglinear approximations are. There is a long tradition in this literature to use loglinear approximations in the RBC framework (see, e.g., Kydland and Prescott, 1982, King et al., 1988, Campbell, 1994, Lettau, 2003 and Cárceles-Poveda, 2005). Taylor and Uhlig (1990) compare various solution methods for RBC models. Jermann (1998) compares loglinear approximations with numerical solutions and concludes that the former ones are quite accurate. In addition, Lettau (2003) notes that the loglinear approximations perform well enough when the variance of the technology shock is not too high, which is the case in his and my papers. The variance of the monetary shock is quite small too. This lends credibility to loglinear solution in the MRBC model. Although numerical solutions might be desirable because some of the higher order effects could be incorporated in the dynamics of asset prices, loglinear approximations have advantage over numerical methods because of the possibility to evaluate a direct effect of some channels to the quantities in question.

This paper derives approximate analytical solutions for the real business cycle model with monetary shocks. It reexamines then the aggregate asset pricing quantities such as, the equity premium, the risk-free rate volatility, Sharpe ratio, and inflation risk premium in the context of the model. The model extends growing literature on exploring asset pricing models in production economies by adding a monetary channel into the real business cycle model. The paper aims to close the gap between macroeconomic literature that studies real business cycle models with monetary effects and the literature that studies asset pricing implications of real business models without role of money in them. I contribute to the literature on the real business cycle models by showing that incorporating monetary effects provides additional important dimension that makes certain progress towards reconciling the dynamics of aggregate asset prices within full-fledged real business cycle framework. In the model households value not only consumption but also real monetary holdings. In the data, aggregate consumption and monetary aggregates are positively correlated. For this reason, increased volatility of the stochastic discount factor leads to a higher equity premium with realistic values of risk aversion than in otherwise standard real business cycle model. The driving force behind the result is non-separability of consumption and real monetary holdings in the utility function. I derive approximate analytical solutions for consumption, capital stock, price level, and asset prices using the log-linearization technique of Campbell (1994). The macroeconomic variables and asset prices are functions of the deep parameters of the model and the elasticities of endogenous variables with respect to state variables. The elasticities are computed using an extended Campbell’s method. The model is calibrated to the historical US quarterly data. Higher weight on real monetary holdings increases the importance of the monetary channel in the model and results in the higher equity premium. As expected, lower weight on real monetary holdings results in the lower equity premium, yet the latter one is twice as large than in the real business cycle model without monetary effects. The equity premium is driven not only by the covariance of consumption growth with real risky returns, but also by the covariance of monetary growth with asset returns and the covariance of inflation with asset returns. The real risk-free rate is the sum of four ARMA(2,1) processes and AR(1) process. I discuss special cases of permanent and transitory shocks. When both shocks are permanent, the risk-free rate reduces to the sum of AR(1) processes, representing the sum of technology and monetary shocks. Alternatively, when both shocks are transitory, the risk-free rate reduces to the sum of two white noise processes. Joint effect of two shocks depends on the parameter value of the relative risk aversion coefficient. It is important to note that Sharpe ratio in the model is consistent with the historical one for fairly persistent technology shock, equal weight on consumption and real money holdings, and for reasonable levels of risk aversion, that is, between 6 and 10. I also solve for the spread between nominal and real rates and obtain the estimate of the three-month inflation risk premium. Inflation risk premium depends on covariance of inflation with consumption and monetary growth. Its value obtained in the paper is close to zero, indicates a low short-term inflation risk, and in the ball park with the estimates of inflation risk premia in recent literature. Overall, these results suggest that the monetary real business cycle model has advantage over standard real business cycle models in explaining the key asset pricing facts, and therefore, the monetary channel in real business models should not be ignored.