CHAPTER The Risk and Term Structure of Interest Rates 125 Thus i 5t * 5% + 6% + 7% + 8% + 9% * 7.0% Using the same equation for the one-, three-, and four-year interest rates, you will be able to verify the one-year to five-year rates as 5.0%, 5.5%, 6.0%, 6.5%, and 7.0%, respectively The rising trend in short-term interest rates produces an upward-sloping yield curve along which interest rates rise as maturity lengthens The expectations theory is an elegant theory that explains why the term structure of interest rates (as represented by yield curves) changes at different times When the yield curve is upward-sloping, the expectations theory suggests that short-term interest rates are expected to rise in the future, as we have seen in our numerical example In this situation, in which the long-term rate is currently above the short-term rate, the average of future short-term rates is expected to be higher than the current short-term rate, which can occur only if short-term interest rates are expected to rise This is what we see in our numerical example When the yield curve is inverted (slopes downward), the average of future short-term interest rates is expected to be below the current short-term rate, implying that short-term interest rates are expected to fall, on average, in the future Only when the yield curve is flat does the expectations theory suggest that short-term interest rates are not expected to change, on average, in the future The expectations theory also explains fact 1, that interest rates on bonds with different maturities move together over time Historically, short-term interest rates have had the characteristic that if they increase today, they will tend to be higher in the future Hence, a rise in short-term rates will raise people s expectations of future short-term rates Because long-term rates are the average of expected future short-term rates, a rise in short-term rates will also raise long-term rates, causing short- and long-term rates to move together The expectations theory also explains fact 2, that yield curves tend to have an upward slope when short-term interest rates are low and are inverted when shortterm rates are high When short-term rates are low, people generally expect them to rise to some normal level in the future, and the average of future expected short-term rates is high relative to the current short-term rate Therefore, longterm interest rates will be substantially above current short-term rates, and the yield curve would then have an upward slope Conversely, if short-term rates are high, people usually expect them to come back down Long-term rates would then drop below short-term rates because the average of expected future shortterm rates would be below current short-term rates and the yield curve would slope downward and become inverted.2 The expectations theory explains another important fact about the relationship between short-term and long-term interest rates As you can see looking back at Figure 6-4, short-term interest rates are more volatile than long-term rates If interest rates are mean-reverting that is, if they tend to head back down after they are at unusually high levels or go back up when they are at unusually low levels then an average of these short-term rates must necessarily have lower volatility than the short-term rates themselves Because the expectations theory suggests that the long-term rate will be an average of future short-term rates, it implies that the long-term rate will have lower volatility than short-term rates