## ASSET PRICING MODEL: The yield curve 3

Long-term bonds and the Hirshleifer effect

In order to obtain some preliminary insights into the effects of liquidity on the term structure, let us again return to the example of section 2.21 Assume that the government at date 0 issues two types of bonds: I short-term bonds yielding one unit of the good at date 1, and L (zero coupon) long-term bonds yielding в units of the good at date 2. We allow for a coupon risk on long-term bonds, so let в be a random variable with support [0, oo), density h(6), cumulative distribution Н(в), and mean Eq(9) = 1. As discussed above, 9 can be interpreted as the date-2 price of money in terms of the good. The case of a deterministic inflation rate (which can be normalized to 0) corresponds to a spike in the distribution at 9=1. We let q and Q denote the date-0 prices of short- and long-term bonds. A long-term price premium corresponds to q > Q. Treasury bonds are the only liquid assets in the economy.

In the example the date-1 income x is assumed to be perfectly correlated across firms. Let g(x) and G(x) denote the density and the cumulative distribution of income x. Assume that 9 and x are independent. In this economy, firms have no liquidity demand past date l. Therefore, if there is no coupon risk (9 = 1) or if there is no signal about the realization of 9 before date 2 (so that there is a coupon risk, but no price risk at date 1), short-term and long-term bonds will be perfect substitutes and so q = Q. Suppose instead that the realization of 9 is learned at date 1. Now the price at which long-term bonds can be disposed of at date 1, namely 9, will vary. The coupon risk in this case induces a price risk.

Let £ and L denote the number of short-term and long-term bonds purchased at date 0 by the corporate sector (in equilibrium, £ = £ if q > 1 and L = L if Q > 1.) The corporate sector solves

Let ц again denote the shadow cost of the investors’ break-even constraint and let y* = b ^ [i denote the optimal unconstrained reinvestment level. Because 1 = 1 and L = L in equilibrium, equilibrium prices are characterized by:

gt; 1 and L = L if Q