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Chapter 6: Problem 15

Assume that the real risk-free rate is 2 percent and that the maturity risk premium is zero. If the 1 -year bond yield is 5 percent and a 2 -year bond (of similar risk) yields 7 percent, what is the 1 -year interest rate that is expected for Year \(2 ?\) What inflation rate is expected during Year \(2 ?\) Comment on why the average interest rate during the 2 -year period differs from the 1 -year interest rate expected for Year 2.

Short Answer

Expert verified
The expected 1-year interest rate for Year 2 is 9.038%, with an expected inflation rate of 7.038% for the same year.

Step by step solution

01

Define Given Values

We are provided with: 1. The real risk-free rate is 2%. 2. Maturity risk premium is 0%. 3. 1-year bond yield is 5%. 4. 2-year bond yield is 7%.
02

Use the Expectation Theory to Find Year 2's Rate

According to the Expectation Theory, the yield of a longer-term bond is an average of the short-term interest rates expected over the life of the longer-term bond. The equation for a 2-year bond yield based on the Expectation Theory is:\[ (1+y_2)^2 = (1+y_1) \times (1+y_{1,2}) \]Where:- \(y_2 = 0.07\) is the 2-year bond yield.- \(y_1 = 0.05\) is the 1-year bond yield.- \(y_{1,2}\) is the expected yield for the 1-year bond in Year 2.Plug in the known values:\[ (1+0.07)^2 = (1+0.05) \times (1+y_{1,2}) \]
03

Solve for the 1-Year Rate Expected in Year 2

Let's solve the equation:\[ 1.1449 = 1.05 \times (1+y_{1,2}) \]Divide both sides by 1.05:\[ 1+y_{1,2} = \frac{1.1449}{1.05} \approx 1.09038 \]Therefore, the expected 1-year rate for Year 2 is:\[ y_{1,2} \approx 0.09038 \text{ or } 9.038\% \]
04

Calculate the Expected Inflation Rate for Year 2

We know that the nominal interest rate can be approximated as:\[ \text{Nominal rate} \approx \text{Real risk-free rate} + \text{Inflation rate} \]For Year 2, the nominal rate is 9.038%, and the real risk-free rate is 2%. Solve for the inflation rate:\[ 9.038\% = 2\% + \text{Inflation rate} \]Therefore, the expected inflation rate for Year 2 is:\[ \text{Inflation rate} \approx 9.038\% - 2\% = 7.038\% \]
05

Discuss the Difference in Average Rates

The average interest rate during the 2-year period differs from the 1-year rate expected in Year 2 because long-term interest rates typically incorporate expectations about future interest rates over the duration of the bond. In this case, the expectations for higher inflation in the second year elevate the expected rate compared to the 5% first-year rate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Risk-Free Rate
The Real Risk-Free Rate is the return on an investment with no risk, and it is adjusted for inflation, reflecting the true earning potential of an investment. It essentially answers the question: "What would I need to earn on my money to ensure that my future purchasing power is equal to my current purchasing power?" In a simpler world where inflation wouldn鈥檛 erode value, returns could be assessed using this rate. In the example problem, the real risk-free rate is given as 2%. This means that, ignoring inflation, an investor can expect to earn 2% on a risk-free investment. However, in reality, we must add expected inflation and other premiums to this rate to determine nominal rates, which is crucial for assessing other interest-related concepts.
Maturity Risk Premium
The Maturity Risk Premium compensates investors for the risk associated with holding longer-term bonds. Longer maturities carry the risk that interest rates might rise, causing the value of fixed-income investments to drop. In the given problem, the maturity risk premium is stated as being zero. This assumption simplifies calculations, indicating that any additional yield offered by longer-term securities is not a compensation for maturity-related risks. In practice, however, maturity risk premiums can vary and are typically positive. They are one of many components that influence bond yields alongside rates like the real risk-free rate and expected inflation.
Nominal Interest Rate
The Nominal Interest Rate is the overall interest rate without any adjustment for inflation. It's seen as the rate of return investors require or earn over a specified period. Utilizing the nominal interest rate helps investors understand the total return on an investment. From earlier, we can see that the 1-year bond yield is 5%, and the 2-year bond yield is 7%. Using the Expectation Theory, which suggests that the yield on a longer-term bond is the average of expected short-term rates over the bond's life, we determine this nominal rate. In the exercise example, we calculated that the expected nominal rate for the second year is 9.038%, highlighting expectations of higher returns as reflected by higher future rates.
Inflation Rate
The Inflation Rate represents the percentage increase in the general level of prices over a specified period and is a critical factor when determining real versus nominal interest rates. Incorporating expected inflation is crucial to understand the real returns an investor might achieve. For Year 2, given a nominal rate of 9.038% and a real risk-free rate of 2%, we estimate the inflation rate to be approximately 7.038%. This difference signifies expectations that costs might rise substantially, affecting purchasing power and justifies the higher nominal rate for the second year. Inflation expectations are pivotal in investment decisions, impacting how future yields are perceived and ultimately how the market prepares for such changes.

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Most popular questions from this chapter

In late 1980 , the U.S. Commerce Department released new data showing inflation was 15 percent. At the time, the prime rate of interest was 21 percent, a record high. However, many investors expected the new Reagan administration to be more effective in controlling inflation than the Carter administration had been. Moreover, many observers believed that the extremely high interest rates and generallytight credit, which resulted from the Federal Reserve System's attempts to curb the inflation rate, would lead to a recession, which, in turn, would lead to a decline in inflation and interest rates. Assume that at the beginning of 1981 , the expected inflation rate for 1981 was 13 percent; for 1982,9 percent; for 1983,7 percent; and for 1984 and thereafter, 6 percent. a. What was the average expected inflation rate over the 5-year period \(1981-1985 ?\) (Use the arithmetic average.) b. What average nominal interest rate would, over the 5 -year period, be expected to produce a 2 percent real risk-free return on 5 -year Treasury securities? c. Assuming a real risk-free rate of 2 percent and a maturity risk premium that equals \(0.1 \times(t) \%,\) where \(t\) is the number of years to maturity, estimate the interest rate in January 1981 on bonds that mature in \(1,2,5,10,\) and 20 years, and draw a yield curve based on these data. d. Describe the general economic conditions that could lead to an upward- sloping yield curve. e. If investors in early 1981 expected the inflation rate for every future year was 10 percent (that is, \(\mathrm{I}_{\mathrm{t}}=\mathrm{I}_{\mathrm{t}+1}=10 \%\) for \(\mathrm{t}=1\) to \(\infty\) ), what would the yield curve have looked like? Consider all the factors that are likely to affect the curve. Does your answer here make you question the yield curve you drew in part c?

One-year Treasury securities yield 5 percent. The market anticipates that 1 year from now, 1 -year Treasury securities will yield 6 percent. If the pure expectations theory is correct, what is the yield today for 2-year Treasury securities?

An investor in Treasury securities expects inflation to be 2.5 percent in Year 1,3.2 percent in Year \(2,\) and 3.6 percent each year thereafter. Assume that the real risk-free rate is 2.75 percent, and that this rate will remain constant. Three-year Treasury securities yield 6.25 percent, while 5 -year Treasury securities yield 6.80 percent. What is the difference in the maturity risk premiums (MRPs) on the two securities; that is, what is \(\mathrm{MRP}_{5}-\mathrm{MRP}_{3} ?\)

The real risk-free rate is 3 percent, and inflation is expected to be 3 percent for the next 2 years. A 2 -year Treasury security yields 6.2 percent. What is the maturity risk premium for the 2 -year security?

The real risk-free rate, \(r^{*}\), is 2.5 percent. Inflation is expected to average 2.8 percent a year for the next 4 years, after which time inflation is expected to average 3.75 percent a year. Assume that there is no maturity risk premium. An 8 -year corporate bond has a yield of 8.3 percent, which includes a liquidity premium of 0.75 percent. What is its default risk premium?
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