Term Structure of Interest Rates

Interest rates are often considered to be constant/fixed for all time periods or time instants. Nevertheless, we know that in practice this is not the case. In this chapter we analyze the behavior of interest rates and try to capture their determinants as well as their evolution/ movements, realizing that they can be a source of uncertainty. We deploy the theories of the interest rate term structure, explain its behavior and examine a series of interest rate models. This chapter assists the reader in comprehending the way interest rates move.

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Author information

Authors and Affiliations

1. Department of Economics, Democritus University of Thrace, Komotini, Greece Thomas Poufinas

1. Thomas Poufinas

Corresponding author

Exercises

Exercises

3.1.1 Exercise 1

An investor knows the particulars of three bonds and wishes to find the interest rate term structure. The bonds are:

• A zero coupon bond with a maturity of 1 year which sells at a price of EUR 980.4.
• A bond with a maturity of 2 years, annual coupon payments at a coupon rate of 3% with a price of EUR 1,009.8.
• A zero coupon bond maturing in 3 years with a price of EUR 915.2.

All bonds have a face value of EUR 1,000.

1. a. What are the spot, forward and short interest rates for the first 3 years?
2. b. Draw the graph of the interest rate term structure.
3. c. What is the price of a bond that expires in 3 years, has a face value of EUR 1,000 and pays an annual coupon at a coupon rate of 4%?

3.1.2 Exercise 2

Assume that there is a zero coupon bond with a maturity of 1 year at a price of EUR 99, a zero coupon bond with a maturity of 2 years at a price of EUR 96 and a coupon-bearing bond that pays coupons annually at a rate of 4% per annum with a maturity of 3 years and a price of EUR 101. All bonds have a face value of EUR 100.

1. a. What are the spot, forward and short interest rates for the first 3 years?
2. b. Draw the graph of the interest rate term structure on a maturity date—interest rate diagram using linear interpolation/extrapolation where/as necessary.
3. c. What is the price of a bond that expires in 3 years, has a face value of EUR 100 and pays an annual coupon at a coupon rate of 3% per annum?

3.1.3 Exercise 3

The future spot rates for a period of time are equal to the corresponding forward rates. This statement:

1. A. Is always correct.
2. B. Is never correct.
3. C. Is correct if we accept the Expectations Hypothesis Theory.
4. D. Is correct if we accept the Liquidity Preference Theory.

Justify your answer.

3.1.4 Exercise 4

The future spot rates for a period of time are lower than the corresponding forward rates. This sentence:

1. A. Is always correct.
2. B. Is never correct.
3. C. Is correct if we accept the Expectations Hypothesis Theory.
4. D. It is correct if we accept the Liquidity Preference Theory.

Justify your answer.

3.1.5 Exercise 5

The interest rate term structure includes:

1. A. All spot, forward and short rates.
2. B. Only spot rates.
3. C. Only forward rates.
4. D. Only short rates.

Justify you answer.

3.1.6 Exercise 6

Consider a zero-coupon bond, with a face value of EUR 1,000 that matures in 2 years. The one-year spot rate is 2% and the two-year spot rate is 4%. The price of the bond today is:

1. A. EUR 961.17
2. B. EUR 1,000
3. C. EUR 924.56
4. D. EUR 942.68

Justify you answer.

3.1.7 Exercise 7

Let us assume that a = 0.08 and b = 0.08 in Vasicek’s model and in the Cox, Ingersoll, Ross model. Suppose that in both models, the initial short rate is 4% and the initial standard deviation of the short rate is 1%. Use both models to price a zero-coupon bond that matures in 8 years. Compare the outputs of the two models. What do you observe?

3.1.8 Exercise 8

Indicate which models of the interest rate curve you know.

1. a. What are their similarities and what are their differences?
2. b. Outline how they are used to value bonds.
3. c. What is the difficulty?

3.1.9 Exercise 9

Let us consider a bond that matures in 4 years, has a face value of EUR 1,000 and makes annual coupon payments at a coupon rate of 4% per annum. The spot rate for a one-year period is 2%, for a two-year period 3%, for a three-year period 3.5% and for a four-year period is 3.8%.

1. a. Find the present value of the bond.
2. b. What is the yield to maturity of the bond if the bond price is equal to the present value you calculated in question (a)?
3. c. What conclusions can you draw about potential weaknesses of the yield to maturity?
4. d. If interest rates are not fixed, i.e. the term structure is not horizontal, then what is the added value of the yield to maturity?

3.1.10 Exercise 10

Let us assume that an investor believes that the liquidity preference theory holds true. He or she wants to take a bet on the evolution of the future spot rates so as to potentially gain from their move.

1. a. What positions could he or she take in order to exploit such potential opportunities?
2. b. What risk is he or she exposed to?