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Heterogeneous beliefs, monetary policy, and stock price volatility

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Abstract

In this paper, I build a two-agent New Keynesian model in which households with subjective and objective beliefs about capital gains from stock prices exist. The former type of households constructs their beliefs about expected capital gains by Bayesian learning from observed growth rates of stock prices. In a homogenous agent model with only subjective beliefs, the effect of the interest rate on stock prices tends to be unrealistically strong. I show how the presence of heterogeneity improves second moments of stock prices with realistic moments of business cycle properties. This quantitative improvement in stock price behaviors allows me to conduct a realistic analysis of how the stance of monetary policy affects stock price volatilities. Strong inertia of monetary policy provides the stability of stock prices. This is because the near-term real interest rate has dominant effects on stock prices under the presence of subjective beliefs since the presence limits the forward-looking nature in pricing stocks. However, because output depends on the expected path of the real interest rate in the forward-looking manner, strong monetary policy inertia does not necessarily provide stabilities of stock prices and output at the same time.

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Notes

  1. See Challe and Giannitsarou (2014) for a summary of empirical studies about the effect of monetary policy shock on stock prices including Bernanke and Kuttner (2005).

  2. Srour (2001) lists an argument that the large surprises in short-term interest rates can cause volatility in financial markets as one reason for smoothing interest rates. Rudebusch (2006) examines a discussion about a rationale for policy gradualism, which is a desire to reduce the volatility in asset prices. González-Páramo (2006) argues that a gradual monetary policy could reduce the likelihood of financial market disruptions. In actual policymaking, FOMC Secretariat (1994) records that there were discussions led by chairman Alan Greenspan on whether a 25 basis point policy tightening was preferable to a 50 basis point tightening because some members considered the larger move to have a higher probability of cracking financial markets. To this question, Bernanke (2004) gives no decisive conclusion on whether gradualism of monetary policy provides stability of financial market or asset prices.

  3. De Paoli et al. (2010) and Challe and Giannitsarou (2014) assume 5 for relative risk aversion rate. The prominent work by Jermann (1998), which is a real model, also assumes 5.

  4. Caines and Winkler (2018) study housing prices in a New Keynesian model with learning about housing price capital gains. Because housing stock quantity is directly included in households’ utility, housing price has wealth effects on business cycles.

  5. The belief structure in Winkler (2019) is also similar. However, Winkler (2019) assumes that agents have “conditionally model-consistent expectations”. Conditionally model-consistent expectations are consistent with all equilibrium conditions of the model, except those that would convey knowledge of the price that clears the asset market, when agents solve for the perceived law of motion. On the other hand, this study follows Adam and Merkel (2018) in which market clearing conditions are known by agents.

  6. On the other hand, external rationality postulates that agents’ subjective probability belief equals the objective probability density of external variables as they emerge in equilibrium.

  7. Both adjustment costs have positive values when the stock and bond holdings deviate from steady state levels. I assume these costs to incorporate demand for stock and bond from the two types of households.

  8. These representations of stock and bond adjustment costs are parsimonious ways to aggregate two types of agents by having a mathematical representation similar to that obtained by the mean-variance optimization under constant absolute risk aversion utility. For example, see Brock and Hommes (1998) and Hanson and Stein (2015). Intuitively speaking, \(\zeta ^S\) and \(\zeta ^B\) represent the variance of expected returns of stocks and bonds, respectively.

  9. The reason why a small value is necessary for the optimal Kalman gain g is that when g is not small enough, the Blanchard-Kahn condition is not satisfied in general equilibrium, as mentioned later. From (15) and (16) shown shortly, g becomes large if I do not assume (13). The calibrated value for g in this model is \(\frac{1}{150}\).

  10. The linearized equation of (50) is given by \(\hat{\pi }_t = \delta E_t \hat{\pi }_{t+1} + \frac{1}{\zeta ^{P}(\mu -1)} \hat{\varOmega }_t + \frac{1}{\zeta ^{P}(\mu -1)} \kappa u_t\).

  11. Winkler (2019) uses the second-order perturbation method to solve his New Keynesian model. I use the first-order perturbation method because the excess return of the stock is out of my interest.

  12. This quarterly value implies 0.5 at an annual rate.

  13. The effect on consumption of the sizes of two parameters does not appear in the homogenous cases with subjective beliefs, \(\alpha =1\). This is because \(S_{s,t}\) is always equal to 1 (or \(S_{ss}\)) in Eq. (18) and \(B_{s,t}\) is always equal to 0 (or \(B_{ss}\)) in Eq. (8) from the market clearing conditions of stock and bond markets, (27), (28), and (30). The same observation applies to the homogenous cases with objective beliefs, \(\alpha =0\).

  14. The sample period almost corresponds to the periods during which the U.S. central bank targeted the interest rate rather than money growth.

  15. In Smets and Wouters (2007), the productivity shock is estimated as 0.45%, monetary policy shock is estimated as 0.24%, preference shock is estimated as 0.24%, and investment-specific shock is estimated as 0.45% in standard deviation on a quarterly basis. The sample period in their study is 1966–2004. Smets and Wouters (2007) use an ARMA(1,1) process for the markup shock process which is different from this study. Levin et al. (2005) estimate that standard deviation of markup shock is 0.2%. The sample period in their study is 1955–2001.

  16. The definition of dividend in the model shown in (53) is not the same as that in the statistics of U.S. Bureau of Economic Analysis because of my model structure. The data are constructed here to be consistent with the definition of dividend in the model.

  17. Due to data accessibility to subjective stock price growth expectations to cover the same data periods with other variables, I did not show \(Corr[\log P^s/d, \log (\alpha m_t + (1-\alpha )E_t[p^s_{t+1}/p^s_{t}])]\) in Table 3.

  18. Castelnuovo and Nistico (2010) claim that incorporating sticky wages helps generate the procyclicality of dividends and their study shows positive responses of stock prices to preference shocks. However, their model does not include investment and capital, which could be crowded out under positive preference shocks and reduce the long-run production capacity. I examined how sticky wages change properties of responses to preference shocks in the model. However, it does not change responses of stock prices qualitatively from what the model without sticky wages presents in this study.

  19. I experimented lower values for g than \(\frac{1}{1000}\). However, results are similar to what I discuss here.

  20. I set \(\alpha =0.95\) instead of 0.94 in this study to match the moments.

  21. In addition, in “Appendix A.2”, I investigate the effects on stock price volatilities under the habit formation parameter \(\phi \) with different values.

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    Katsuhiro Oshima

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I would like to thank Tomoyuki Nakajima, Akihisa Shibata, Kazuhiro Yuki, Munechika Katayama, and Kosuke Aoki. This work is supported by the Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research (No. 16H02026).

Appendix

Appendix

1.1 Data Sources

The actual data formations are conducted by procedures as follows. The actual data used in this paper is of the United States. Real consumption, real investment, and real wage data are from the Federal Reserve Bank of St. Louis FRED economic database (https://fred.stlouisfed.org/). FRED series IDs for these variables are PCECC96, GPDIC1, and LES1252881600Q, respectively. These are seasonally adjusted quarterly data. For real output data, I took a sum of seasonally adjusted nominal consumption and investment, and divided it by the implicit price deflator for gross domestic purchases. The corresponding FRED series ID are PCEC, GPDI, and A712RD3A086NBEA, respectively. Real capital stock data is real net stock (private fixed assets) from the Bureau of Economic Analysis. The total nominal compensation of employee data is from FRED and its FRED series ID is A4102C1Q027SBEA. This is seasonally adjusted. To retrieve the labor amount, I divided the total nominal compensation data by real wage and implicit price deflator for gross domestic purchases. For dividend for which development is consistent with the model, I subtracted nominal total compensation and nominal investment (FRED series ID: GDPI) from the nominal output, which I define as the sum of nominal consumption (FRED series ID: PCEC) and nominal investment, and divided it by the implicit price deflator for gross domestic purchases. The stock price data is S&P 500 index data. Real stock price data are deflated by implicit GDP deflator (FRED series ID: GDPDEF). As the above dividend data covers all U.S. companies, including unlisted companies, I constructed dividend data, which is a product of the S&P stock index level and its dividend yield for the price dividend ratio in Sect. 4.2.

Interest rate data is based on the Federal Reserve Bank of New York’s treasury term premia database (https://www.newyorkfed.org/research/data_indicators/term_premia.html). I use one-year fitted zero coupon market yield of U.S. treasury for the nominal interest rate. I deflated it by the actual inflation rates of the implicit gross domestic product deflator. In Sect. 4.2, I used gross yields data. Both rates in Sect. 4.3 are of the natural log of gross yields.

In Sect. 4.3, I take the natural log and de-trend by the third-order time polynomial regression, except interest rate data. I chose third-order because the Akaike information criterion shows the lowest value for output when I examined the fit up to the fourth order. For consistency, I used the same order in time polynomial regressions for other variables, too. I chose the time polynomial regression to detrend among other candidates of detrending methods for the following reasons. One candidate is to take the first difference. The distance of the stock price level from the long-run trend and persistence of the deviation of the stock price, which implies the cyclical momentum of stock prices, are what this study wants to explain using subjective beliefs. The first difference is not necessarily the best candidate for detecting the cyclical momentum of the stock price in terms of the distance of the stock price level from the long-run trend. Given this, a candidate to detect momentum or level deviation from the trend is the Hodrick-Prescott filter. However, this method is not necessarily a good candidate to extract momentum or level deviations from the long-run trend, either. Hence, I avoid using the Hodrick-Prescott filter and use the time polynomial regression in this study. Real capital data is real net private fixed assets provided on annual basis. I translated them to a quarterly basis by interpolation for de-trending.

1.2 Habit formation parameter and stock price

This appendix section shows how parameters other than those of monetary policy affect stock price volatilities. I discuss the effects of the habit formation parameter on stock price volatility. I show the high habit formation case, \(\phi =0.8\), and the low habit formation case, \(\phi =0\), in Fig. 11. Under the productivity shock, a high habit formation parameter amplifies stock price volatilities. The intuition behind this is that the strong need for consumption smoothing by high habit formation generates large volatilities of real interest rates. In addition, when \(\alpha \) is close to 1, the shape of dividend flows over time matters because \(\hat{d}\) at time \(t+1\) has a strong impact in (33). When consumption smoothing is strong and the stochastic discount factor increases in response to a positive productivity shock, firms are inclined to increase their dividends far into the future. Therefore, \(\hat{d}_{t+1}\) becomes small by firms’ optimization in case of strong consumption smoothing. This implies small stock price reactions from the second term of the right hand side in (33). However, these dividend effects on stock price have a much smaller magnitude than the real interest rate effects, because the second term has less weight than the first term. Therefore, the stock price shows a larger increase in \(\phi =0.8\) case than \(\phi =0\) case.

Under a positive monetary policy shock, the high habit formation parameter case does not differ much from the low habit formation parameter case, as shown in the right-hand side chart in Fig. 11. A monetary policy shock directly changes the real interest rate given the price rigidity. Therefore, stochastic discount factors that affect stock prices do not show notable differences between two habit formation parameter cases.

Fig. 11
figure 11

Stock price response with different habit formation parameters to a 1.0% positive productivity shock, a 0.25% (1.0% at an annualized rate) positive monetary policy shock, a 1.0% positive markup shock, a 1.0% positive preference shock, and a 1.0% positive investment-specific technology shock. These sizes are the same as those in Sect. 4.4. The value of \(\alpha \) is 0.94. 1 on the vertical axis scale amounts to 1% deviation from steady state levels

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Oshima, K. Heterogeneous beliefs, monetary policy, and stock price volatility. Ann Finance 17, 79–125 (2021). https://doi.org/10.1007/s10436-020-00379-9

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