Expectations Hypothesis

The fundamental economic theory which connects short and long term interest rates is the Expectations Hypothesis. The Expectations Hypothesis deals with a long term investment decision in the …xed income market. One possibility is to buy a bond that has a long term maturity. The other possibility is to buy a one-period bond in every successive period until the end of the investment horizon (Ross, Wester…eld and Ja¤e (2002)). The investor is indi¤erent between holding one long term bond until maturity and rolling over a sequence of one-period bonds, if the expected return of both investment strategies is the same. This is the foundation of the Pure Expectations Hypothesis which

11Wu (2006) describes the relationship between main macroeconomic variables and the long term interest rate based on the Fisher equation.

states that long term interest rates are the average of expected future short term interest rates. The Pure Expectations Hypothesis neglects risk aversion and liquidity preference of an investor. In contrast to that, the Expectations Hypothesis includes a term premium due to risk aversion, liquidity preference or preferred habitat of investors.

A large number of empirical articles researches on the Expectations Hypothesis. A survey is given by Cook and Hahn (1990) and Ichiue (2004). Even though the empirical validity of the Expectations Hypothesis is not generally accepted in the literature, the Expectations Hypothesis is the working assumption in Financial Economics. Lutz (1940) relates the following empirical characteristics of the yield curve to the Expectations Hypothesis: a higher variance of short term interest rates than of long term interest rates (table 2.2), a negative correlation of short and long term interest rates and an upward sloping yield curve.

As the Pure Expectations Hypothesis assumes that investors are risk neutral and do not demand a risk premium for their willingness to hold long term assets, the ex-pected excess return of long term bonds over short term bonds is equal to zero. The Pure Expectations Hypothesis has two forms, the one-period and the n-period form (Campbell, Lo and MacKinlay (1997)). The one-period Pure Expectations Hypothesis focuses on the return in the next period of a one-period bond and an n-period bond.

The one-period bond is bought at timet and held until maturity, whereas the n-period bond is bought at time t and sold as a bond with a maturity of n 1 at time t+ 1.

The one-period Pure Expectations Hypothesis states that at time t, the known return of a one-period bond is equal to the expected return in the next period of an n-period bond. In contrast to that, the n-period Pure Expectations Hypothesis focuses on the expected return of the next n periods and states that at time t, the expected return of rolling over one-period bonds during the next n periods is equal to the known return of buying ann-period bond at timet and holding it until maturity.¹²

The Pure Expectations Hypothesis can also be formulated in terms of the forward rate and the expected spot rate (appendix A.1) and is de…ned as the equality of the

12Campbell, Lo and MacKinlay (1997) state that the one-period andn-period Pure Expecta-tions Hypothesis cannot hold simultaneously, because interest rates are random variables and therefore Jensen’s Inequality applies, i.e. the expectation of the inverse of a random variable is di¤erent from the inverse of the expectation of a random variable.

one-period forward rate for timet+ at timet (f(t; )) and the expectations at timet of the future one-period spot rate at time t+ (Et[r_t+ ] ). As the Pure Expectations Hypothesis considers risk neutral investors, the forward rate at timetis only determined by the expectation at time t of the one-period spot rate at time t+ ,

f(t; ) = E_t[r_t+ ]: (1.2)

The Expectations Hypothesis is based on the Pure Expectations Hypothesis and aug-mented by a further term which takes account of risk aversion, liquidity preference or preferred habitat. The Expectations Hypothesis equates the one-period forward rate for time t+ at time t (f(t; )) and the sum of the expectation at time t of the future one-period spot rate at time t+ (Et[r_t+ ]) and a constant premiumb ,

f(t; ) =E_t[r_t+ ] +b : (1.3)

If the premium is positive, the expected return of a long term bond is higher than rolling over short term bonds because of the investor’s gain of the premium (Gibson, Lhabitant and Talay (2001)). If risk aversion or liquidity preference are taken into account, the constant b is strictly greater than zero for > 0. Hence, the one-period forward rate f(t; ) is higher than the expected one-period spot rate E_t[r_t+ ] (Hicks (1946)¹³).

Equation 1.3 implies thatb is increasing with and that the term structure of interest rates is always upward sloping. As there are other observed shapes of the yield curve (downward sloping, ‡at, inverse and hump-shaped), additional theories are necessary.

If the Market Segmentation Hypothesis or Preferred Habit theory are included in the Expectations Hypothesis, the sign of the constant b is not restricted for > 0.

According to the Market Segmentation Hypothesis, the price of a bond with a certain maturity only depends on its demand and supply and is independent of demand and supply of bonds with other maturities (Culbertson (1957)¹⁴). So, arbitrage is not taken

13Cox, Ingersoll and Ross (1985a) quote Hicks, J. R., 1946, Value and Capital, 2nd edition, Oxford University Press, London.

14Cox, Ingersoll and Ross (1985a) quote Culbertson, J. M., 1957, The Term Structure of Interest Rates, Quarterly Journal of Economics, 71, 485-517.

into account by the Market Segmentation Hypothesis. The Preferred Habit theory assumes that investors are willing to buy bonds of maturities other than their most preferred maturity, if they are compensated for it (Modigliani and Sutch (1966)¹⁵).

Both theories allow for a negative b , because investors are willing to accept a lower yield for a bond with their preferred time to maturity.

The implications of the Expectations Hypothesis for the bidirectional e¤ects between the macroeconomy and the yield curve are strong. The reason is that macroeconomic variables (in‡ation and output) a¤ect the decision of the central bank concerning the short term interest rate. Therefore, it is important to consider how the central bank reacts to the current and expected path of output and in‡ation when modelling the term structure of interest rates by macroeconomic theory (section 2.4.3). Furthermore, the current short term interest rate and expected short term interest rates (due to expected in‡ation and output) are translated by the Expectations Hypothesis into the current long term interest rate. The long term interest rate in‡uences the aggregate demand in an economy via savings and investments (Piazzesi (2003)). Therefore, the Expectations Hypothesis indicates how current monetary policy a¤ects the long term interest rate and the real economy in the future.¹⁶