Implications of complete markets
We have allowed the investor to form a portfolio from only two assets, a short-term indexed bond and a single long-term indexed bond. Even with only two assets, however, markets are complete with respect to real-interest-rate risk because our real term-structure model has only one factor. This fact has several interesting implications fully.
First, with complete markets the investor can combine short- and long-term bonds so that the return on her bond portfolio is independent of the maturity of the longterm bond traded in the market. That is, she can synthesize her own optimal longterm bond, with the maturity optimal for her given her risk preferences. The return on the optimal bond portfolio is given by
which is the required result.
Third, Cox and Huang (1989) have proposed an alternative solution method for intertemporal consumption and portfolio choice problems with complete markets. They work in continuous time and show that with complete markets, optimally invested wealth must satisfy a partial differential equation (PDE). Unfortunately this PDE does not generally have a closed-form solution. We now show that our solution methodology is equivalent to a discrete-time version of the Cox-Huang approach; our loglinear approximation allows us to solve the discrete-time equivalent of the Cox-Huang PDE in closed form.
To keep the analysis simple, we will specialize the discussion to the power utility case (ф = I/7). The extension to recursive utility is discussed in the Appendix. We start by defining a new variable W* — Wt — Ct—invested wealth—and note that from the budget constraint (13), the portfolio return equals Rpj+1 = 1 + C*+i)/W*. Then the Euler equation (15) implies that the ratio W? jC% must satisfy the following
This nonlinear expect at ional difference equation is the discrete-time equivalent of the Cox-Huang PDE.