EXAMPLE 5.2
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We can make this argument more general. For an investment of $1, consider the choice of holding, for two periods, a two-period bond or two one-period bonds.
Using the definitions
it = today’s (time t) interest rate on a one-period bond
iet+1 = interest rate on a one-period bond expected for next period (time t + 1) i2t = today’s (time t) interest rate on the two-period bond
the expected return over the two periods from investing $1 in the two-period bond and holding it for the two periods can be calculated as
(1 + i2t)(1 + i2t) - 1 = 1 + 2i2t + (i2t)2 - 1 = 2i2t + (i2t)2
After the second period, the $1 investment is worth (1 + i2t)(1 + i2t).
Subtracting the $1 initial investment from this amount and dividing by the initial $1 investment gives the rate of return calculated in the previous equation. Because (i2t)2 is extremely small—if i2t = 10% = 0.10, then (i2t)2 = 0.01—we can simplify the expected return for holding the two-period bond for the two periods to
2i2t
With the other strategy, in which one-period bonds are bought, the expected return on the $1 investment over the two periods is
(1 + it)(1 + iet+1) - 1 = 1 + it + iet+1 + it(iet+1) - 1 = it + iet+1 + it(iet+1) This calculation is derived by recognizing that after the first period, the $1 investment becomes 1 + it, and this is reinvested in the one-period bond for the next period, yielding an amount (1 + it)(1 + iet+1). Then subtracting the $1 initial invest- ment from this amount and dividing by the initial investment of $1 gives the expected return for the strategy of holding one-period bonds for the two periods. Because it(iet+1) is also extremely small—if it = iet+1 = 0.10, then it(iet+1) = 0.01—we can simplify this to
it + iet+1
Both bonds will be held only if these expected returns are equal—that is, when 2i2t = it + iet+1
Solution
The expected return over the two years will average 10% per year ([9% + 11%]/2 = 10%). The bondholder will be willing to hold both the one- and two-year bonds only if the expected return per year of the two-year bond equals 10%. Therefore, the interest rate on the two-year bond must equal 10%, the average interest rate on the two one-year bonds. Graphically, we have:
Today
0 9%
10%
Year 1
Year 11% 2
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Solving for i2t in terms of the one-period rates, we have i2t = it + iet+1
2 (1)
which tells us that the two-period rate must equal the average of the two one-period rates. Graphically, this can be shown as
Today 0
Year 1
Year
it iet11 2
i2t5 it 1iet11 2
We can conduct the same steps for bonds with a longer maturity so that we can examine the whole term structure of interest rates. Doing so, we will find that the interest rate of int on an n-period bond must be
int = it + iet+1 + iet+2 + g + iet+(n-1)
n (2)
Equation 2 states that the n-period interest rate equals the average of the one- period interest rates expected to occur over the n-period life of the bond. This is a restatement of the expectations theory in more precise terms.2
2The analysis here has been conducted for discount bonds. Formulas for interest rates on coupon bonds would differ slightly from those used here but would convey the same principle.
The one-year interest rates over the next five years are expected to be 5%, 6%, 7%, 8%, and 9%. Given this information, what are the interest rates on a two-year bond and a five-year bond? Explain what is happening to the yield curve.
Solution
The interest rate on the two-year bond would be 5.5%.
int = it + iet+1 + iet+2 + g + iet+(n-1) n
where
it = year 1 interest rate = 5%
iet+1 = year 2 interest rate = 6%
n = number of years = 2 Thus,
i2t = 5% + 6%
2 = 5.5%
The interest rate on the five-year bond would be 7%.
int = it + iet+1 + iet+2 + g + iet+(n-1) n
Expectations Theory
EXAMPLE 5.3
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The expectations theory is an elegant theory that explains why the term struc- ture 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 higher than 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 result 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 lower than the current short-term rate, imply- ing 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, which states 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 expec- tations 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, which states that yield curves tend to have an upward slope when short-term interest rates are low and are inverted when short-term 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, long-term interest rates will be substantially higher than 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
where
it = year 1 interest rate = 5%
iet+1 = year 2 interest rate = 6%
iet+2 = year 3 interest rate = 7%
iet+3 = year 4 interest rate = 8%
iet+4 = year 5 interest rate = 9%
n = number of years = 5 Thus,
i5t = 5% + 6% + 7% + 8% + 9%
5 = 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.
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would then drop below short-term rates because the average of expected future short-term rates would be lower than current short-term rates, and the yield curve would slope downward and become inverted.3
The expectations theory is an attractive theory because it provides a simple explanation of the behavior of the term structure, but unfortunately it has a major shortcoming: It cannot explain fact 3, which says that yield curves usually slope upward. The typical upward slope of yield curves implies that short-term interest rates are usually expected to rise in the future. In practice, short-term interest rates are just as likely to fall as they are to rise, and so the expectations theory suggests that the typical yield curve should be flat rather than upward-sloping.