The Canonical Asset

Rather than proceed with separate derivations for the prices and rates of return for different assets such as equity, short-term bills and long-term bonds, I will introduce a canonical asset that includes all of these assets as special cases. The canonical asset is an «-period asset. From the standpoint of period t, the terminal period of an «-period asset is period f+л. This asset pays ау-^Дя_ • in the period that is j periods before the terminal period for j = 0,. . и-l, where yt+n.j > 0 is a random variable, aQ > 0 is a constant, a. > 0, j = 1,…, л -1 are constants, and Я is a constant that indexes the variability of future payoffs. Thus, for instance, in the terminal period, the asset pays a0yf+n; and in period /+1 the asset pays an_Ay^x.

The canonical asset introduced here includes equities and fixed-income securities of all maturities. Fixed-income securities such as bonds and bills are represented by Я = 0 which implies that the payoff in period t+n-j is the known amount a}. A standard coupon bond with face value F and coupon d is represented by a0 =F + d, and a\ =• • • = Д„-1 = d > 0; in this formulation, a pure discount bond is represented by d = 0. Securities with risky payoffs have nonzero values of Я.

For instance, in the Lucas (1978) fruit-tree model, the dividend (per capita) on unlevered equity equals consumption per capita Cr In terms of our canonical asset, unlevered equity in the Lucas model is an infinite-period asset that pays Ct in period t. Using the notation for canonical assets, n = oo, a. = 1 for all j > 0, yt s Ct, and Я = 1. As discussed in section IV, levered equity can be represented by values of Я greater than one.

Let pt(n9X) be the ex-payment price of the canonical «-period asset in period t.

The price of this asset also depends on the sequence of constants a. J = 0,. . ., и-l, and on the stochastic specification ofy, but this dependence is not reflected in the notation. The gross rate of return on the canonical asset between period t and period t+1 is

The Stochastic Structure

Recall that x,+1 = C;+1 / Ct is the growth rate of aggregate consumption per capita.

Define z,+1 = yl+l / yt to be the growth rate of 7,. As noted earlier, in the specification of unlevered equity in the Lucas fruit-tree model, yt = C( so that zt+l =xl+l. More generally, and znl are distinct random variables and are elements of a random vector of growth rates W^x which is observable at the beginning of period f+1. I assume that Wt+{ is i.i.d. Electronic Payday Loans Online