# Term Structure of Interest Rates

The **term structure of interest rates** is the variation of the yield of bonds of similar risk profiles with the terms of the bonds. In most cases, bonds with longer maturities have higher yields. However, sometimes the yield curve becomes inverted, with short-term notes and bonds having higher yields than long-term bonds.

The **yield curve** is the relationship of the yield to maturity of bonds to the time to maturity. There are 3 characteristics of the term structure of interest rates:

- The change in yields of different term bonds tends to move in the same direction.
- The yields on short-term bonds are more volatile than long-term bonds.
- The yields on long-term bonds tend to be higher than short-term bonds.

The expectations hypothesis has been advanced to explain the 1^{st} 2 characteristics and the premium liquidity theory have been advanced to explain the last characteristic.

## Expectations Hypothesis

The **expectations hypothesis** states that different term bonds can be viewed as a series of 1-period bonds, with yields of each period bond equal to the **expected short-term interest rate** for that period. For example, compare buying a 2-year bond with buying 2 1-year bonds sequentially. If the interest rate for the 1^{st} year is 4% and the *expected* interest rate, which is often referred to as the **forward rate**, for the 2^{nd} year is 6%, then one can be either buy a 1-year bond that yields 4%, then buy another bond yielding 6% after the 1^{st} one matures for an average interest rate of 5% over the 2 years, or he can buy a 2-year bond yielding 5%—both options are equivalent: (4%+6%) / 2 = 5%. Hence, the sequential 1-year bonds are equivalent to the 2-year bond. (Actually, the geometric mean gives a slightly more accurate result, but the average is simpler to calculate and the argument is the same.)

Note that this relationship must hold in general, for if the sequential 1-year bonds yielded more or less than the equivalent long-term bond, then bond buyers would buy either one or the other, and there would be no market for the lesser yielding alternative. For instance, suppose the 2-year bond paid only 4.5% with the expected interest rates remaining the same. In the 1^{st} year, the buyer of the 2-year bond would make more money than the 1^{st} year bond, but he would lose more money in the 2^{nd} year—earning only 4.5% in the 2^{nd} year instead of 6% that he could have earned if he didn’t tie up his money in the 2-year bond. Additionally, the price of the 2-year bond would decline in the secondary market, since bond prices move in opposition to interest rates, so selling the bond before maturity would only decrease the bond's return.

Note, however, that expected future interest rates are just that – expected. There is no reason to believe that they will be the actual rates, especially for extended forecasts, but, nonetheless, the expected rates still influence present rates.

According to the expectations hypothesis, if future interest rates are expected to rise, then the yield curve slopes upward, with longer term bonds paying higher yields. However, if future interest rates are expected to decline, then this will cause long term bonds to have lower yields than short-term bonds, resulting in an **inverted yield curve**. The inverted yield curve often results when short-term interest rates are higher than historical averages, since there is a greater expectation that rates will decline, so long term bond issuers would be reluctant to issue bonds with higher rates when the expectation is that lower rates will prevail in the near future.

The expectations hypothesis helps to explain 2 of the 3 characteristics of the term structure of interest rates:

- The yield of bonds of different terms tend to move together.
- Short-term yields are more volatile than long-term yields.

However, the expectations hypothesis does not explain why the yields on long-term bonds are usually higher than short-term bonds. This could only be explained by the expectations hypothesis if the future interest rate was expected to continually rise, which isn’t plausible nor has it been observed, except in certain brief periods.

## Liquidity Premium Theory

The liquidity premium theory has been advanced to explain the 3^{rd} characteristic of the term structure of interest rates: that bonds with longer maturities tend to have higher yields.

There are 2 risks with holding bonds that increases with the term of the bond: inflation risk and interest rate risk. Both the inflation rate and the interest rate become more difficult to predict farther into the future. **Inflation risk** reduces the real return of the bond. **Interest rate risk** is the risk that bond prices will drop if interest rates rise, since there is an inverse relationship between bond prices and interest rates. Of course, interest rate risk is only a real risk if the bondholder wants to sell before maturity, but it is also an opportunity cost, since the long-term bondholder forfeits the higher interest that could be earned if the bondholder’s money was not tied up in the bond. Therefore, a longer term bond must pay a higher **risk premium** to compensate for the bondholder for the greater risk.

Another factor that affects the liquidity of a bond is its **marketability** — the easier it is to sell a bond in the secondary marketplace, the more liquid it will be, thus reducing liquidity risk. This explains why long-term Treasuries have such low yields, because they are the easiest to sell. However, there may also be reinvestment risk, since if interest rates decline, then the purchase of any new securities will be at a lower yield.

A bond’s yield can theoretically be divided into a risk-free yield and the risk premium. The risk-free yield is simply the yield calculated by the formula for the expectation hypothesis. The risk premium is the **liquidity premium** that increases with the term of the bond. Hence, the yield curve slopes upward, even if future interest rates are expected to remain flat or even decline a little, and so the **liquidity premium theory of the term structure of interest rates** explains the generally upward sloping yield curve for bonds of different maturities.

Besides the liquidity premium theory, 2 other factors also explain the upward sloping yield curve. The 1^{st} factor is that both the credit risk and default risk of corporate bonds increases with time. While it is generally accepted that there is no credit or default risk for Treasuries, most corporate bonds do have a credit rating that can change in time because of changing business or economic conditions that can increase default risk. Indeed, during recessions, the yield spread between Treasuries and corporate bonds increases, because of the increased credit risk during recessions, as can be seen in the graph below.

The 2^{nd} reason why bonds with longer maturities pay a higher yield is that most issuers would rather issue long-term bonds than a series of short-term bonds, since it costs money to issue bonds regardless of maturity.

## Inverted Yield Curve

As already stated, short-term bonds may actually pay a higher yield than long-term bonds when short-term rates are expected to decline sharply. This was the situation in 1980-1982, when interest rates were much higher than normal. Because they were so high, it was expected that they would decline to more normal values. Hence, no bond issuer is going to issue long-term bonds at a low price when they can fetch a higher price later, when interest rates are lower. Although long-term bond yields will be higher under a high interest rate environment than under low interest rates, long-term bonds will still be priced higher than short-term bonds in the secondary market, because investors will be willing to pay more for the longer-term bonds to lock in the higher yields over the long term to reduce their reinvestment risk.