A Stochastic Gordon-Shapiro Formula with Excess Volatility

Abstract

Share prices fluctuate far more than dividends. In contemporary lit- erature, this excess volatility is usually discussed involving the Camp- bell-Shiller present value identity. In our view, it is more appropriate to model future dividends and prices explicitly as random variables. We refer to excess volatility if the coefficient of dispersion for share prices is higher than for dividends. It is often presumed that excess volatility could be properly expli- cated by time-varying discount factors. However, we will show that this idea is logically inconsistent as long as one uses deterministic dis- count factors. This even holds if one assumes more complex stochastic structures of the dividends, such as AR(2) processes. We therefore propose stochastic discount factors and show that our model is free of arbitrage, the transversality condition is met and prices are unique. Finally, we try to consolidate our approach with the Lucas Model. Here, it is shown on the one hand that in this model the cost of capital cannot be stochastic. Moreover, the equity premium puzzle can no longer be replicated, but rather a realistic value for risk aversion is obtained. Finally, we change the Lucas model such that stochastic capital costs are also possible.