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Erschienen in: Annals of Finance 4/2019

03.06.2019 | Research Article

Business-cycle pattern of asset returns: a general equilibrium explanation

verfasst von: Qiang Kang

Erschienen in: Annals of Finance | Ausgabe 4/2019

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Abstract

I develop an analytical general-equilibrium model to explain economic sources of business-cycle pattern of aggregate stock market returns. With concave production functions and capital accumulation, a technology shock has a pro-cyclical direct effect and a counter-cyclical indirect effect on expected returns. The indirect effect, reflecting the “feedback” effect of consumers’ behavior on asset returns, dominates the direct effect and causes counter-cyclical variations of expected returns. I show that the conditional mean, volatility, and Sharpe ratios of asset returns all vary counter-cyclically and they are persistent and predictable, and that stock market behavior has forecasting power for real economic activity.

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1
For example, Fama and French (1989) use the term spread, default spread, and dividend yield; Schwert (1989) enlists short-term interest rates, yields on corporate bonds, and growth rates of industrial production; Whitelaw (1997) employs dividend yield, default spread, commercial paper-Treasury spread, and one-year Treasury yield; Lettau and Ludvigson (2010) utilize consumption-wealth ratio and some of these conditioning variables. In contrast, Brandt and Kang (2004) and Binsbergen et al. (2013) adopt a VAR approach without relying on predictors. Overall, the empirical studies of the business-cycle behavior of asset returns are often related to a large literature documenting that excess returns are time-varying and predictable. See, e.g., Cochrane (1999) and Koijen and Nieuwerburgh (2011) for two surveys of the return predictability literature.
 
2
Lettau (2003) uses a similar decomposition of asset price responses to technology shocks to study why the premium of equity is small over the risk-free rate and over a real long-term bond.
 
3
Analytical results would be difficult to obtain in an infinitely-lived-agent setting. There is also a caveat. The two-period OLG model suits the low-frequency phenomenon well and the business cycle is a relatively high-frequency phenomenon, but the economic intuition illustrated in this article carries over to a representative-agent model or a more realistic OLG model with many periods of life for agents.
 
4
Cuoco and Cvitanic (1998) examine an optimal consumption and investment problem for a ‘large’ investor whose portfolio choices affect the instantaneous expected returns on the traded assets. Basak (1997) studies in an exchange economy a consumption-portfolio problem of an agent who acts as a price-leader in all markets and the implications of his behavior on equilibrium security prices.
 
5
The reason is as follows. Typically, when firms do not face adjustment costs, the social planner can perfectly smooth the impact of shocks on consumption and risk premia should be zero. When firms pay infinite adjustment costs, then this production economy is reduced to an exchange economy.
 
6
The literature has shown that standard RBC models have counter-factual quantitative asset pricing implications (see., e.g., Jermann 1998; Boldrin et al. 2001). Generally, two additional features are needed to solve the quantitative failure: frictions at the household level like habit formation preferences and borrowing constraints that prevent inter-temporal consumption smoothing, and frictions at the firm level like capital adjustment costs, investment irreversibility, and multi-production sectors with limited inter-sectoral factor mobility.
 
7
As well understood, a production model typically requires incredibly high risk aversion to fit the data unless the model features frictions. Like a generic OLG structure, my model does inherently impose the inter-generational risk sharing restriction, which in theory should help lower the value of risk aversion coefficient to match the data. However, due to the oversimplified model setup for the sake of analytical tractability, I still need quite high risk aversion values to achieve a match in magnitude of model predictions with the real-world data. In other words, more frictions are needed in my model in order to improve its quantitative performance with more economically reasonable risk aversion values.
 
8
The firm faces a dynamic problem. If the state of the economy follows a Markovian process then, denoting by \(V\left( k_{t},s_{t}\right) \) the firm value at time t,  the Bellman equation for the firm’s problem is:
$$\begin{aligned} V\left( k_{t},s_{t}\right) =\max _{\left\{ h_{t},i_{t}\right\} }p_{t}\left( z_{t}k_{t}^{\alpha }h_{t}^{1-\alpha }-w_{t}h_{t}-i_{t}\right) +E_{t}\left[ V\left( k_{t+1},s_{t+1}\right) \right] . \end{aligned}$$
 
9
A caveat is in order. A typical general-equilibrium model endogeneizes the risk-free rate. In my model, because there are no inter-generational transfers and each agent lives for two periods and only cares about period 2 consumption, the risk-free rate is constant. Given the availability of a risk-free storage technology delivering a constant rate of return \(r^{f}\), the no-arbitrage condition requires the risk-free rate to equal the return on the storage technology that is exogenously given. As a tradeoff, this result greatly simplifies the analytical exercise of this paper. Moreover, as empirical studies show that the fluctuation in the risk-free rate is unlikely to be a main source of the business-cycle pattern of asset returns, the exogenously given risk-free rate in my model is an innocuous modeling feature.
 
10
Let X be a random variable with a stochastic volatility distribution so that X|V is distributed \(N(\mu ,\sigma ^{2}V)\) and V has density p(V). The size-biased volatility-adjusted distribution Q is given by \( q(V)=V_{p}(V)/E(V)\).
 
11
Note that \(\frac{\partial \phi _{t}}{\partial y_{t}}=\frac{1}{144} A^{-3}y_{t}^{-4}\gamma \sigma _{z,t}^{2}Q_{t}/\sqrt{\gamma ^{2}\sigma _{z,t}^{4}+48Ay_{t}^{2}\mu _{z,t}}>0.\)
 
12
In principle, \(\frac{\partial \phi _{t}}{\partial z_{t}}\) can be obtained by applying the implicit function theorem to Eq. (24), but it is not trivial to get its sign. When \(\alpha =\frac{1}{2}\), \(\frac{\partial \phi _{t}}{\partial z_{t}}=\frac{2\phi _{t}}{z_{t} \left( 2Ay_{t}+\gamma \sigma _{z,t}^{2}\right) }\left[ 2\rho Ay_{t}+\gamma \sigma _{z,t}^{2}\right] >0\).
 
13
When \(\alpha =\frac{1}{2}\), \(\frac{d\mu _{r,t}}{dz_{t}}=\frac{\gamma \sigma _{z,t}^{2}}{2z_{t}y_{t}}\left( \rho -1\right) \leqslant 0\) with equality if \(\rho =1\).
 
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Metadaten
Titel
Business-cycle pattern of asset returns: a general equilibrium explanation
verfasst von
Qiang Kang
Publikationsdatum
03.06.2019
Verlag
Springer Berlin Heidelberg
Erschienen in
Annals of Finance / Ausgabe 4/2019
Print ISSN: 1614-2446
Elektronische ISSN: 1614-2454
DOI
https://doi.org/10.1007/s10436-019-00347-y