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Chapter 6: Problem 12

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

Short Answer

Expert verified
The difference in maturity risk premiums is 0.35%.

Step by step solution

01

Understand the Concepts

We need to calculate the difference between the maturity risk premiums of two Treasury securities with different terms to maturity. We need to use given yields, inflation expectations, and the real risk-free rate for this computation.
02

Calculate the Average Expected Inflation Rate

To find the expected average inflation rate over the three and five-year periods, we compute:\[ \text{Average Inflation}_{3} = \frac{2.5\% + 3.2\% + 3.6\%}{3} \]\[ \text{Average Inflation}_{5} = \frac{2.5\% + 3.2\% + 3.6\% + 3.6\% + 3.6\%}{5} \]
03

Perform Calculations for Step 2

Calculate the averages:\[ \text{Average Inflation}_{3} = \frac{2.5 + 3.2 + 3.6}{3} = 3.1\% \]\[ \text{Average Inflation}_{5} = \frac{2.5 + 3.2 + 3.6 + 3.6 + 3.6}{5} = 3.3\% \]
04

Use the Formula for Nominal Treasury Yield

The nominal yield on a Treasury security can be expressed as:\[ YR = r^* + \text{Average Inflation} + \text{MRP} \]Where:- \( YR \) is the Yield Rate.- \( r^* \) is the Real Risk-Free Rate (2.75%).- \( \text{MRP} \) is the Maturity Risk Premium.
05

Calculate the Three-Year MRP

Substitute values into the yield formula for the 3-year security:\[ 6.25\% = 2.75\% + 3.1\% + \text{MRP}_{3} \]Solving for \( \text{MRP}_{3} \), we get:\[ \text{MRP}_{3} = 6.25\% - 2.75\% - 3.1\% = 0.4\% \]
06

Calculate the Five-Year MRP

Substitute values into the yield formula for the 5-year security:\[ 6.80\% = 2.75\% + 3.3\% + \text{MRP}_{5} \]Solving for \( \text{MRP}_{5} \), we get:\[ \text{MRP}_{5} = 6.80\% - 2.75\% - 3.3\% = 0.75\% \]
07

Calculate the Difference in Maturity Risk Premiums

The difference in maturity risk premiums is:\[ \text{MRP}_{5} - \text{MRP}_{3} = 0.75\% - 0.4\% = 0.35\% \]

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Key Concepts

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

Treasury Securities
Treasury securities are financial instruments used by the government to raise funds. They are considered one of the safest investments because they are backed by the full faith and credit of the issuing government.

Treasury securities come in several forms, including Treasury bills, notes, and bonds. Each has different maturity periods and characteristics:
  • Treasury Bills: Short-term securities that mature in one year or less.
  • Treasury Notes: Medium-term securities with maturities ranging from two to ten years.
  • Treasury Bonds: Long-term securities with maturities over ten years.
These securities pay interest, making them attractive to investors seeking a safe return on investment. The yield on a Treasury security is determined by several factors, including the real risk-free rate, expected inflation, and the maturity risk premium.
Inflation Expectations
Inflation expectations refer to what investors predict will happen to inflation in the future. Understanding these expectations is crucial when investing, as inflation can erode the purchasing power of returns.

Investors base these expectations on:
  • Economic indicators
  • Historical inflation data
  • Government fiscal and monetary policies
In the example provided, the average expected inflation rates were calculated for three-year and five-year periods:
  • Three-Year Average: Aggregated for Years 1-3.
  • Five-Year Average: Includes inflation expectations for Years 1-5.
These rates are used in conjunction with the real risk-free rate to determine the nominal yield of a Treasury security.
Real Risk-Free Rate
The real risk-free rate is the return on an investment with zero risk, assuming no inflation. It reflects the pure time value of money.

This rate is crucial because it serves as a basic foundation for determining the interest rates on various financial instruments, including Treasury securities.
  • For investors, this rate indicates the minimum return needed to justify any investment.
  • It helps differentiate between inflation-driven and real-driven interests.
In our exercise, the real risk-free rate is constant at 2.75%. This value remains unchanged over the different maturity periods and is integral to calculating nominal yields and maturity risk premiums.
Nominal Yield
The nominal yield is the overall return on a financial instrument, without adjusting for inflation. It reflects the total interest an investor earns from holding a security.

The formula for the nominal yield includes:
  • The Real Risk-Free Rate
  • Inflation Expectations
  • Maturity Risk Premium (MRP)
The nominal yield equation used in the problem is: \[ YR = r^* + \text{Average Inflation} + \text{MRP} \] Where:
  • \( YR \): Yield Rate
  • \( r^* \): Real Risk-Free Rate (2.75% in the exercise)
  • \( \text{MRP} \): Maturity Risk Premium
Understanding this concept helps investors decipher the real returns they can expect and distinguish them from inflationary effects. It is especially important for determining long-term investment decisions.

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Most popular questions from this chapter

Interest rates on 4-year Treasury securities are currently \(7 \%\) while 6 -year Treasury securities yield \(7.5 \%\). If the pure expectations theory is correct, what does the market believe that 2-year securities will be yielding 4 years from now?

INFLATION Due to a recession, expected inflation this year is only \(3 \%\). However, the inflation rate in Year 2 and thereafter is expected to be constant at some level above \(3 \%\) Assume that the expectations theory holds and the real risk-free rate is \(\mathrm{r}^{*}=2 \% .\) If the yield on 3 -year Treasury bonds equals the 1 -year yield plus \(2 \%,\) what inflation rate is expected after Year 1?

The real risk-free rate is \(3 \%\). Inflation is expected to be \(3 \%\) this year, \(4 \%\) next year, and \(3.5 \%\) thereafter. The maturity risk premium is estimated to be \(0.05 \times(t-1) \%,\) where \(t=\) number of years to maturity. What is the yield on a 7 -year Treasury note?

Maria Juarez is a professional tennis player, and your firm manages her money. She has asked you to give her information about what determines the level of various interest rates. Your boss has prepared some questions for you to consider. a. What are the four most fundamental factors that affect the cost of money, or the general level of interests rates, in the economy? b. What is the real risk-free rate of interest \(\left(r^{r}\right)\) and the nominal risk-free rate \(\left(r_{R F}\right) ?\) How are these two rates measured? c. Define the terms inflation premium (IPP), default risk premium (DRP), liquidity premium (LP), and maturity risk ??????? (MRP). Which of these premiums is included in determining the interests rate on (1) short term Treasury securities, (2) long-term U.S. Treasury securities, (3) short-term corporate securities, and (4) lon term corporate securities? Explain how the premiums would vary over time and among the different securities listed d. What is the term structure of interests rates? What is a yield curve? e. Suppose most investors expect the inflation rate to be \(5 \%\) next year, \(6 \%\) the following year, and \(8 \%\) there after The real risk-free rate is \(3 \%\). The maturity risk premium is zero for bonds that mature in 1 year or less \(0.1 \%\) for 2 -year bonds then the MRP increases by \(0.1 \%\) per year there after for 20 years, after which it stable. What is the interest rate on 1.10 , and 20 -year Treasury bonds? Draw a yield curve with these data to What factors can explain why this constructed y yield curve is upward-sloping f. At any given time, how would the yield curve facing a AAA-rated company compare with the yield curve for US. Treasury securities? At any given time, how would the yield curve facing a BB-rated company compare with the yield curve for U.S. Treasury securities? Draw a graph to illustrate your answer g. What is the pure expectations theory? What does the pure expectations theory imply about the term structure of interest rates? h. Suppose you observe the following term structure for Treasury securities: $$\begin{array}{ll} \text { Maturity } & \text { Yield } \\ \hline \text { 1 year } & 6.0 \% \\ \text { 2 years } & 6.2 \\ \text { 3 years } & 6.4 \\ \text { 4 years } & 6.5 \\ \text { 5 years } & 6.5 \end{array}$$ Assume that the pure expectations theory of the term structure is correct. (This implies that you can use the yield curve provided to "back out" the market's expectations about future interest rates.) What does the market expect will be the interest rate on 1 -year securities 1 year from now? What does the market expect will be the interest rate on 3 -year securities 2 years from now?

In late 1980 , the U.S. Commerce Department released new data showing inflation was \(15 \%\). At the time, the prime rate of interest was \(21 \%,\) 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 generally tight 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 \(13 \%\); for \(1982,9 \%\); for \(1983,7 \% ;\) and for 1984 and thereafter, \(6 \%\) a. What was the average expected inflation rate over the 5-year period \(1981-1985 ?\) (Use the arithmetic average.) b. Over the 5 -year period, what average nominal interest rate would be expected to produce a \(2 \%\) real risk-free return on 5 -year Treasury securities? Assume \(\mathrm{MRP}=0\) c. Assuming a real risk-free rate of \(2 \%\) and a maturity risk premium that equals \(0.1 \times(\mathrm{t}) \%\) 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. 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 to be \(10 \%\) (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?
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