I extend the Epstein-Zin-lognormal consumption-based asset-pricing model to allow for general i.i.d. consumption growth processes. Information about the higher moments—equivalently, cumulants—of consumption growth is encoded in the cumulantgenerating function (CGF). I express four observable quantities (the equity premium, riskless rate, consumption-wealth ratio and mean consumption growth) and the HansenJagannathan bound in terms of the CGF, and present applications. Models in which consumption is subject to occasional disasters can be handled easily and flexibly within the framework. The importance of higher cumulants is a double-edged sword: those model parameters which are most important for asset prices, such as disaster parameters, are also the hardest to calibrate. It is therefore desirable to make statements which do not depend on a particular calibrated consumption process. First, I use properties of the CGF to derive restrictions on the time-preference rate and elasticity of intertemporal substitution that must hold in any Epstein-Zin-i.i.d. model which is consistent with the observable quantities. Second, I show that “good deal” bounds on the maximal Sharpe ratio can be used to derive restrictions on preference parameters without calibrating the consumption process. Third, given preference parameters, I calculate the welfare cost of uncertainty directly from mean consumption growth and the consumption-wealth ratio without having to estimate the amount of risk in the economy. Fourth, I analyze heterogeneous-agent models with jumps.