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Unformatted text preview: WEB EXTENSION 5D The Pure Expectations Theory and Estimation of Forward Rates I n Chapter 5, we saw that the shape of the yield curve depends primarily on two factors: (1) expectations about future inflation and (2) the relative risk of securities with different maturities. We also saw how to calculate the yield curve, given inflation and maturityrelated risks. In practice, this process often works in reverse: Investors and analysts plot the yield curve and then use information embedded in it to estimate the market's expectations regarding future inflation and risk. This process of using the yield curve to estimate future expected interest rates is straightforward, provided (1) we focus on Treasury bonds, and (2) we assume that all Treasury bonds have the same risk (in other words, there is no maturity risk premium). Although this second assumption may not be reasonable, it enables us for the time being to take out the effects of risk and focus exclusively on how expectations about future interest rates affect the shape of the yield curve. Later on, we will show what happens when we once again assume that there is a maturity risk premium. In fact, even though most evidence suggests that there is a positive maturity risk premium, some academics and practitioners contend that this second assumption is reasonable, at least as an approximation. They argue that the market is dominated by large bond traders who buy and sell securities of different maturities each day, that these traders focus only on shortterm returns, and that they are not concerned with maturity risk. According to this view, a bond trader is just as willing to buy a 20year bond to pick up a shortterm profit as to buy a 3month security. Strict proponents of this view argue that the shape of the Treasury yield curve is therefore determined only by market expectations about future interest rates. This position has been called the pure expectations theory of the term structure of interest rates. The pure expectations theory (often simply referred to as the "expectations theory") assumes that (1) bond traders establish bond prices and interest rates strictly on the basis of expectations for future interest rates, and (2) they are indifferent to maturity because they do not view longterm bonds as being riskier than shortterm bonds. If this were true, then the maturity risk premium (MRP) would be zero, and longterm interest rates would simply be a weighted average of current and expected future shortterm interest rates. To illustrate this point, assume that 1year Treasury securities currently yield 5% while 2year Treasury securities yield 5.5%. Investors with a 2year horizon have two primary options, as follows. Option 1: Buy a 2year security and hold it for 2 years. Option 2: Buy a 1year security, hold it for 1 year, and then, at the end of the year, reinvest the proceeds in another 1year security.
1 2 Web Extension 5D: The Pure Expectations Theory and Estimation of Forward Rates If investors select Option 1, then for every dollar they invest today they will have accumulated $1.113025 by the end of Year 2: Funds at end of Year 2 $11:0552 $1:113025 Investors who select Option 2 must expect to end up with at least the same amount (else they would be indifferent between the options): Funds at end of Year 2 $11:051 X Here X is the expected interest rate on a 1year Treasury security one year from now. If the expectations theory is correct, then each option must provide the same amount of cash at the end of 2 years, which implies that 1:051 X 1:0552 We can rearrange this equation and then solve for X: 1 X 1:0552 =1:05 X 1:0552 =1:051 0:0600238 6:00238% Thus, if the expectations theory is correct, the current yield curve would indicate that the market expects the 1year rate to be 6.00238% 1 year from now. We can use yield curve data to help predict future shortterm interest rates. In the absence of maturity risk premiums, an upwardsloping Treasury yield curve implies that the market expects future shortterm rates to increase. Also, notice that the 2year rate of 5.5% is a geometric average of the current and expected 1year rates.1 not an average of expected shortterm yields. For example, suppose investors expected the 1year Treasury rate to be 6.00238% a year from now yet 2year bonds yielded only 5.25%. Bond traders would be able to earn a profit by adopting the following strategy: 1. Borrow money for 2 years at an annual cost of 5.25%. 2. Invest the money in a series of 1year securities. The annual expected return over the 2year period would be [(1.05)(1.0600238)]1/2  1 = 5.5%. Earning 5.5% when the cost is 5.25% is a good deal, so bond traders would rush to borrow money (demand funds) in the 2year market and invest (or supply funds) in the 1year market. Recall that an increase in the demand for funds raises interest rates whereas an increase in the supply of funds reduces interest rates. Therefore, bond traders would push up the 2year yield but reduce the yield on 1year bonds. The net effect would be a market equilibrium in which the 2year rate is a weighted average of expected future 1year rates. The preceding analysis is based on the assumption that the maturity risk premium is zero, but most evidence suggests that there is a positive maturity risk premium. For example, assume once again that 1 and 2year maturities yield 5.0% and 5.5%, respectively, so we have a rising yield curve. However, now assume that the maturity risk premium on the 2year bond is 0.2% versus 0% for the 1year bond. This premium means that in equilibrium the expected annual return on a 2year bond (5.5%)
1 The geometric average of the current and expected 1year rates can be expressed as [(1.05)(1.0600238)]1/21 = 0.055 or 5.50%. The arithmetic average of the two rates is (5% + 6.00238%)/2 = 5.50119%. The geometric average is theoretically superior, but the difference is only 0.00119%. With interest rates at the levels they have been in the United States and most other nations in recent years, the geometric and arithmetic averages are so close that many people simply use the arithmetic average, especially given the other assumptions that underlie the estimation of future 1year rates. Web Extension 5D: The Pure Expectations Theory and Estimation of Forward Rates 3 must be 0.2% higher than the expected return on a series of two 1year bonds (5% and X), so the expected return on the series must be 5.5%  0.2% = 5.3%: Expected return on 2year series Rate on 2year bond  MRP 0:055  0:002 0:053 5:3% Now recall that the annual expected return from the series of two 1year securities can be expressed as follows, where X is the 1year rate next year: 1:051 X 1 Expected return on 2 year series2 1:0532 1:05X 1:0532 1:05 0:0588090 X 1:05 0:0560086 5:60086% Therefore, in this scenario market participants must expect the 1year rate next year to be 5.60086%. Observe that the yield curve rises by 0.5% when the years to maturity increases from 1 to 2: 5.5%  5.0% = 0.5%. Of this 0.5% increase, 0.2% is attributable to the MRP and the remaining 0.3% is due to the expected increase in 1year rates next year. Putting all this together, we see that we can use the yield curve to predict the shortterm rates that will exist for next year, but to do this we must have an estimate of the maturity risk premium. If our estimate of the MRP is incorrect, then so will be our yield curvebased interest rate forecast. Thus, although the yield curve can be used to obtain insights into the direction of future interest rates, we cannot back our expected interest rates with precision unless either the pure expectations theory holds exactly or else we know with certainty the exact maturity risk premium. Since neither of these conditions holds, forecasts of future interest rates are only approximationsthough they may be good approximations.
SelfTest What key assumption underlies the pure expectations theory? Assuming that the pure expectations theory is correct, how are expected shortterm rates used to calculate expected longterm rates? According to the pure expectations theory, what would happen if longterm rates were not an average of expected shortterm rates? Most evidence suggests that a positive maturity risk premium exists. How does this affect your calculations when determining interest rates? Assume the interest rate on a 1year Tbond is currently 7% and the rate on a 2year bond is 9%. If the maturity risk premium is 0%, what is a reasonable forecast of the rate on a 1year bond next year? (11.04%) What would the forecast be if the maturity risk premium on the 2year bond were 0.5% and the maturity premium were 0% for the 1year bond? (10.02%) Problems
(5D1)
Expectations Theory Oneyear Treasury securities yield 5%. The market anticipates that 1 year from now, 1year Treasury securities will yield 6%. If the pure expectations theory is correct, what is the yield today for 2year Treasury securities? 4 Web Extension 5D: The Pure Expectations Theory and Estimation of Forward Rates (5D2)
Expectations Theory Interest rates on 4year Treasury securities are currently 7%, while 5year Treasury securities yield 7.5%. If the pure expectations theory is correct, what does the market believe that 2year securities will be yielding 4 years from now? Suppose 2year Treasury bonds yield 4.5% while 1year bonds yield 3%. The maturity risk premium is 0% and r* is 1%. a. Using expectations theory, what is the yield on a 1year bond 1 year from now? b. What is the expected inflation rate in Year 1? Year 2? Assume that the real riskfree rate is 2% and that the maturity risk premium is 0%. If the 1year bond yield is 5% and a 2year bond (of similar risk) yields 7%, then what is the 1year interest rate expected for Year 2? What inflation rate is expected during Year 2? Comment on why the average interest rate during the 2year period differs from the 1year interest rate expected for Year 2. (5D3)
Expectations Theory and Inflation (5D4)
Expectations Theory ...
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 Spring '10
 MR.Wroshr

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