Beyond the Local Mean-Variance Analysis in Continuous Time: The Problem of Non-Normality

Abstract

The paper investigates the effects of deviations from normality on the estimates of risk premiums and the real equilibrium, short-term interest rate in the conventional rational expectations equilibrium model of Lucas (1978). We consider a time-continuous approach, where both the aggregate consumption process as well as cumulative dividends from risky assets are assumed to be jump-diffusion processes. This approach allows for random jumps in the fundamental underlying processes at random time points. Preferences are time separable and additive. We derive testable expressions for these quantities, and confront these with 20. century sample estimates. Since there are non-linear components in the formulas for the risk premiums and the interest rate, we can readily explore what effect deviation from normality has on these quantities. Our results test the boundaries of the conventional model.

Elements of Economics of Uncertainty and Time with Recursive Utility

“…Since the multinormal distribution has moments of all orders, and the conditional joint probability distribution is fully characterized by the conditional mean vector and conditional covariance matrix, our expressions depend on the first two moments of the various random variables involved. In Aase and Lillestøl (2015) it is demonstrated that deviations from normality may matter to some degree, but can in no way explain the equity premium puzzle. The results of this approach are as follows: The risk premium of any risky asset, denoted R, is given at any time t by the formula…”

“…Notice that this table is not a mere transformation of Table 3, but developed from the the original data set used in the Mehra and Prescott (1985)-study, by taking logarithms of the relevant yearly quantities, and basing the statistical analysis on these transformed data points. 6 Expectat. Standard dev.…”

“…This non-indifference means, in our notation, that the expressions in (5) and (6) in Section 4.1 will, in general, not be equal.…”

“…Proof: Consider the following set of inequalities: where ξ t = E(V t+1 (c )|F 1 t ) and ξ t = E(V t+1 (c )|F 1 t ). The first equivalence follows from the development in Kreps and Porteus (1978) related to (5) and (6), which is what they mean by early resolution of uncertainty in the present setting, the second follows by taking conditional expectation across the inequality, given F 1 t , observing that f c t , E(V t+1 (c α )|F 1 t ) is F 1 t -measurable, the third follows from Jensen's inequality on conditional form, and the fourth follows from the rule of iterated expectations.…”

“…There are of course newer data by now, but these retain the same basic features. If we can explain the data inTable 1, we can most likely explain any of the newer sets as well 6. We have obtained the original data set from Professor R. Mehra.…”