LONG-TERM BONDS: The Demand for Indexed Bonds 2

In order to find optimal savings and the optimal allocations to the two bonds, we adopt an approximate analytical solution method. The first step is to characterize аП)£, the optimal allocation to the n-period bond, by combining a second-order log-linear approximation to the Euler equation with a first-order approximation to the intertemporal budget constraint. We then guess a form for the optimal consumption and portfolio policies and show that policies of this form satisfy the approximate Euler equation and budget constraint. Finally we use the method of undetermined coefficients to identify the coefficients of the optimal policies from the primitive parameters of the model. By using a second-order expansion of the log Euler equation we account for second-moment effects in the model. Electronic Payday Loans Online

Following Campbell (1993, 1996), Campbell and Viceira (1996), and Restoy (1992), we first log-Iinearize the Euler equation (14) for г = n and г — 1, where asset 1 is the short-term riskless asset. Subtracting the log-linearized Euler equation for the riskless asset from the log-linearized equation for asset n, we find:
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where lowercase letters denote variables in logs and A is the first-difference operator. This expression obtains from (14) by using both a second-order Taylor approximation around the conditional mean of {rPtt+i, Ac£+i} and the approximation log(l -fa:) ~ x for small x. It holds exactly if consumption growth and the return on wealth have a joint conditional lognormal distribution. We show later that this is indeed the case along the optimal path: the approximate optimal policies imply that the log return on wealth and log consumption growth are jointly normal.

We can log-linearize (15) in a similar fashion. After reordering terms, we obtain the well-known equilibrium linear relationship between expected log consumption growth and the expected log return on wealth:
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