Disasters and the Welfare Cost of Uncertainty - Martin, Ian W. R (American Economic Review, 2008)
The combination of power utility and i.i.d. lognormal consumption growth makes for a benchmark model in which asset prices and expected returns can be found in closed form. Introducing the consumption-based model, John H. Cochrane (2005, 12) writes, “The combination of lognormal distributions and power utility is one of the basic tricks to getting analytical solutions in this kind of model.” A message of this paper is that the lognormality assumption can be relaxed without sacrificing tractability. Working under two assumptions—that there is a representative agent with power utility and that consumption growth is i.i.d.—I introduce, in Section I, a mathematical object (the cumulantgenerating function, or CGF) in terms of which four fundamental quantities that are at the heart of consumption-based asset pricing can be simply expressed. Those quantities are the equity premium, riskless rate, consumption-wealth ratio, and mean consumption growth. The expressions derived relate the fundamentals directly to the cumulants (equivalently, moments) of consumption growth. The lognormal assumption is equivalent to the assumption that all cumulants above the second are zero. If one is in the business of making up stochastic processes, many suggest themselves most naturally in continuous time. Although there is an obvious discrete-time analogue of Brownian motion—a random walk with Normally distributed increments—it is less natural to map Poisson processes, say, into discrete time, and therefore harder to deal with the possibility of jumps in consumption. In Section II, I show that these results carry over to the continuous-time setting. The i.i.d. growth assumption is replaced by its continuous-time analogue: log consumption is a Levy process. Disasters and the Welfare Cost of Uncertainty