The catching up with the Joneses feature of the utility function was originally introduced to help account for the high average value of the equity premium observed empirically. However, when this form of the utility function was specified to imply a realistic value of the equity premium in Abel (1990), the model produced a riskless rate of return that was far too volatile.
Campbell and Cochrane (1994) developed a form of catching up with the Joneses preferences that yielded, as in actual data, a large equity premium and low variability of the riskless rate. They achieved this low variability of the riskless rate by specifying a complicated recursive function for the determination of the benchmark level of consumption. Here I adopt a simpler formulation of the benchmark level of consumption that produces, with the inclusion of leverage, low variability of the riskless rate along with a large equity premium.
The analysis of leverage arises naturally from the formulation of the canonical asset introduced in this paper. The payoff in period t on the canonical asset is specified to be proportional to yt\ where yt is an observable random variable and Я is a constant. I use this formulation in order to include fixed-income securities and equities as special cases.
For fixed-income securities, Я = 0 so that payoffs are nonstochastic. Unlevered equity in the Lucas fruit-tree model is modeled by setting Я = 1 and yt equal to aggregate output (which equals consumption) per capita. Although the form of the canonical asset was initially developed with only the values of Я – 0 and Я = 1 in mind, it became apparent that much of the analysis can be conducted, and closed-form solutions derived, for arbitrary values of Я. Indeed, values of Я greater than one provide a good approximation to levered equity.
Although closed-form solutions are derived for various moments of asset returns, many of these expressions are too cumbersome to clearly illustrate the effects of various parameters on the moments of returns. To make these effects transparent, I derive first-order approximations to the exact solutions. Numerical calibration demonstrates that these approximations yield values close to the values calculated using the exact solutions. More importantly, I use these approximations to develop a simple algorithm for choosing parameter values that allow the approximate unconditional means and variances of the riskless rate and the rate of return on equity to match the corresponding historical sample values. Electronic Payday Loans Online