Moments of Returns

In this section I focus on the moments of returns on a one-period riskless asset and on levered equity. To simplify the resulting expressions, I assume that zi+1 = xt+l as is standard in applications of the Lucas fruit-tree model.12 This assumption implies that fux = juz= jj. and a] = axz =

The aim of this section is to understand how the unconditional means and variances of asset returns depend on the underlying preference parameters and payoff characteristics. I derive exact expressions for these moments of asset returns but some of the resulting expressions are too cumbersome to clearly reveal the effects of parameter values and payoffs on moments of returns. A common approach when facing such cumbersome analytic expressions is to calibrate the model and simulate it for various choices of parameter values. Rather than pursue a numerical strategy, I will pursue an analytical strategy by deriving first-order approximations that clearly illustrate the relationships between the underlying parameters and the moments of returns. It is worth emphasizing that I do not derive the solution to a problem in which the objective function and/or constraints have first been linearized.13 Instead, I first obtain exact solutions to the nonlinear problem, derive closed-form expressions for the moments of equilibrium returns, and then approximate the moments. The quality of these approximations is shown to be excellent in Table 1. www.easyloans-now.com

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