Asset Pricing with Long Run Risk and Stochastic Differential Utility: An Analytic Approach

The analytic method of Chen, Cosimano, and Himonas (CCH 2009) is extended to prove that the continuous time version of the long run risk model of Bansal and Yaron (2004) has an analytic solution. The long run risk model is dependent on the recursive utility introduced by Duffie and Epstein (1992a, 1992b) which leads to a nonlinear differential equation. The solution to this differential equation is an analytic function, so that the lifetime utility is represented by a power series near the stationary mean of the long run risk variable. The radius of convergence for this power series is determined to be at least seven standard deviations of the long run risk variable. Consequently, the lifetime utility can be quickly and accurately solved using a higher order Taylor polynomial approximation. With the solution to the lifetime utility the analytic method is then used to solve the linear ODE for the price-dividend function near the stationary mean of the long run risk variable with a radius of convergence at least one third of that for the lifetime utility. The short term properties of the long run risk model are compared with that of the external habit model of Campbell and Cochrane (1999). Finally, an alternative method from Hansen and Scheinkman (2009) is used to find the asymptotic rate of return on stocks for the long run risk model.