3. The Term Structure, its Estimation, and Smoothing 69
3.7.1 Theories of the Shape of the Term Structure
There are several theories which help describe the shape of the term struc- ture of interest rates. Some compete with one another. Others can be viewed more reasonably as complementary. Taken all together, they pro- vide a sensible description of the underlying determinants of the shape of the yield curve.
The Unbiased Expectations Theory
The unbiased expectations theory posits that investors are risk-neutral. In- vestors, regardless of their investment time horizon, will choose the instru-
ments with the highest return. They require no additional compensation for any perceived risk associated with the time frame involved.There is a direct relationship between the spot rates in the market place and the for- ward rates of interest. Letr1be the one-period spot interest rate (that is, the rate of interest which prevails from time period zero to time period one), and let r2 be the two-period spot interest rate (prevailing from time zero to time two). The corresponding forward rate, f12, is the one-period rate of interest which prevails from time one to time two which is implicit in these two spot rates. It can be calculated from the two spot rates as fol- lows: (1+r2)2
1+r1 −1. More generally, any forward rate can be calculated according to the following formula:
(1+rn)n (1+rn−1)n−1−1.
The expectations theory is based on the idea that these risk-neutral in- vestors set interest rates in a manner so that the forward rate is equal to the spot rate expected in the market one year from now. The theory is ex- pressed in terms of expected one-period spot rates. In terms of bonds, the yield on a two-year bond is set in such a way that the return on that two- year bond is equal to the return on a one-year bond plus the expected return on another one-year bond purchased in one year. This notion is not unique to one- and two-year bonds. If all investors in the marketplace operate this way, then prices will adjust until the expected return from holding a two- year bond is the same as the expected return from holding two one-year bonds.
According to the expectations theory of the term structure, the yield curve can be derived from a series of expected one-year spot rates. Con- sider, for example, that the one-year spot rate is 5 percent from year zero to year one, the expected spot rate from year one to year two is 6 percent, and the expected spot rate from year two to year three is 7 percent. A two-year bond would earn the spot rate over the period from time zero to time two, which, according to this theory, is the same as investing in a one-year bond from time zero to time one, and then investing in another bond for one year from time one to time two.
Thus, the two-year spot rate from time zero to time two is calculated as
1+r2
2 2
=
1+0.05
2 1+0.06 2
. The two-year spot rate is 5.5 percent. The three-year spot rate from time zero to time three can be cal- culated as
1+r3
2 3
=
1+0.05
2 1+0.06
2 1+0.07 2
. The three- year spot rate is 6 percent.Furthermore, the market’s belief about the future of one-year spot rates can be read easily from an observed yield curve.
Note that this theory assumes that the expected future spot rate is equal to the corresponding forward rate. This assumption does not hold for some of the other theories of the term structure.
The Liquidity Preference Theory
Under the liquidity preference theory of the term structure, investors again examine the returns from holding bonds of differing maturities. This the- ory does not assume that investors are risk-neutral. Investors are assumed to demand extra compensation to be induced to hold a bond of a rela- tively long maturity over a bond of relatively short maturity. Furthermore, the market is populated with relatively more short-term investors, which requires that investors receive additional inducement to hold long-term bonds.
In the example of the section which described the expectations theory of the term structure, we calculated that if the one-period rate from time zero to time one were 5 percent and the one-period rate from time one to time two were 6 percent, then the two-period rate prevailing from time zero to time two would be expected to be 5.5 percent. Under the liquidity preference theory, the two-period rate would have to be higher than 5.5 percent to induce investors to hold relatively longer-term instruments. For an investor with a one-year investment horizon, there is risk associated with the two-year investment.
The liquidity preference theory leads us to different conclusions about the shape of the term structure. Even if expectations are such that there will be no change in one-period rates, it would still be the case that the yield curve would be upward sloping in the presence of a liquidity premium.
Even if one-period spot rates were expected to decline, if the liquidity pre-
mium were sufficiently large, there could still be an upward sloping term structure. A flat or downward sloping yield curve, under the liquidity pref- erence theory, can only be possible in an environment of decreasing one- period spot rates.
The Market Segmentation Theory
The origin of the market segmentation theory stems from the observation that some investors apparently prefer debt of a particular maturity. This preference is so pronounced that these investors are insensitive to the yield differential of their preferred debt maturity over the debt of another matu- rity. The theory posits that investors are so risk-averse that they remain in their desired maturity spectrum and cannot be induced by yield differen- tials to change maturities. In this way, long-term rates are determined by the supply of and demand for long-term debt instruments, and similarly for short-term interest rates. Proponents of this theory closely watch the flow of funds into different segments of the bond market to determine changes in the yield curve.
Consider examples of investors who can reasonably be expected to have strong preference for debt of a particular maturity. An insurance company facing liabilities which are in the distant future will choose to invest for a long time horizon. There may be considerable risk to the insurance com- pany in investing in a series of short-term instruments, compared to the known return and the predictability of available long-term debt. Similarly, corporations may have strong preference for issuing debt of a particular maturity depending on the use to which the funds will be put. Corpora- tions will generally prefer to pay for long-term investment projects over a long period of time and so the corresponding debt issued for those projects is likely to be of long maturity.
The market segmentation theory of the term structure is popular with practitioners. Academics maintain that the market is more likely composed of both investors with definite maturity preferences and those who invest on the basis of relative yields. The predictions of the shape of the term structure based on the market segmentation theory will be offset if there are in fact enough investors who fall into the second category.