Notation. P τ. Let us assume that the prices and the interest rates follow a stationary stochastic process.

Transcription

1 Notation Let Pt τ denote the price at time t of a risk-free, pure-discount bond worth one dollar at its maturity in τ years, at time t + τ. Thus Pt 0 =. Let τ t denote the yield to maturity on this bond. By definition, P τ t = e ττ t. Let us assume that the prices and the interest rates follow a stationary stochastic process.

2 Expectations Theory The expectations theory says that the long-term interest rate is the average of current and expected future short-term rates. For example, the two-year interest rate is the average of the current one-year rate and the one-year rate expected for next year, 2 t = 2 [ t + E t t+ ]. 2

3 Variance Bounds Shiller [] shows how the expectations theory implies that the variability of the short-term interest rate sets an upper bound on the variability of the long-term interest rate, and he studies whether United States data satisfy these variance bounds. We work out the bounds in the context of the one-year and two-year pure-discount bonds. Shiller s analysis is analogous but more complex coupon-bearing, long-term bonds. We assume that expectations are rational. 3

4 Forecast Error Define the forecast error e t := t E t t. Since the forecast error is uncorrelated with the forecast, Var t = Var [ Et t ] + Varet. Hence Vare t Var t. 2 Also, since t is a possible forecast of t+, with forecast error Var t, Vare t Var t. 3 4

5 Ex Post ational Long-Term Interest ate The ex post rational long-term interest rate is 2 t := 2 t + t+. 4 Under perfect foresight, the long-term interest rate would equal this value. With uncertainty, however, 2 t = E t 2 t. 5

6 We have t 2 = 2 t + t+ = [ ] 2 t + E t t+ + 2 = t 2 + e t+, [ t+ E t t+ ] by. 6

7 A variance bound is Variance Bound Var t 2 Var 2 t. This inequality is comparable to [, p. 202], which is violated by the data: the variance of the ex post rational interest rate is low, and the variance of the long-term interest rate is higher. 7

8 Holding-Period eturn Define the one-period holding-period return on a two-year bond, Ht+ 2 := lnpt+ lnpt 2. 8

9 Hence H 2 t+ = t t = t+ + [ t + E t t+ ] by = t e t+. Of course this relationship expresses the basis of the expectations theory: all bonds have the same expected holding-period return. 9

10 The expected value E t H 2 t+ = t, and the error is e t+,so Var H 2 t+ = Var t + Varet+, since the error is uncorrelated with the forecast. Therefore Var t Var H 2 t 2Var t, by 2. The right inequality is comparable to [, I., p. 203], which is violated by the data: the holding-period return on the long-term bond fluctuates too much relative to the short-term interest rate. 0

11 Excess Holding-Period eturn Since the excess holding-period return on the two-year bond is therefore H 2 t+ t = e t+. Var H 2 t+ t Var t, by 3. This inequality corresponds to [, I.2, p. 204], which is satisfied by the data.

12 Averaging That the ex post rational interest rate is the average of the current and future short-term rates obtains another inequality: Var [ t 2 = Var 2 = + ρ 2 Var t Var t. t + t+ ] Here ρ is the first-order autocorrelation of t. 2

13 This inequality corresponds to [, p. 202]. Unlike the inequalities above, this inequality depends only on stationarity, but not on either the expectations theory or rational expectations. It is supported by the data. 3

14 eferences [] obert J. Shiller. The volatility of long-term interest rates and expectations models of the term stucture. Journal of Political Economy, 876:90 29, December 979. HBJ7. 4