Appendix Describing the Numerical Method Used in 'When Can Life-Cycle Investors Benefit from Time-Varying Bond Risk Premia?'

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Abstract
We rigorously explain the numerical approach used in the above-mentioned paper. The methodology is based on Brandt, Goyal, Santa-Clara, and Stroud (2005) (Review of Financial Studies) and Carroll (2006) (Economics Letters). In addition to combining these numerical techniques, we suggest two extensions. First, the approach of Brandt, Goyal, Santa-Clara, and Stroud (2005) approximates the conditional expectations encountered in optimizing the utility function via polynomial expansions in the state variables. The coefficients in these expansions are estimated using cross-sectional regressions across a set of simulated trajectories of returns and state variables. We develop an accurate approximation of these regression coefficients to facilitate fast optimization over the portfolio weights. This allows us to deal with a large number of decision variables without relying on iterative procedures. Second, to approximate the conditional expectations that lead to the optimal consumption strategy, we ensure that the approximation remains strictly positive, while keeping the approximation computationally tractable.