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Review questions

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- 1. Discuss the factors that determine the shape and level of a yield curve. How do term to maturity, credit risk, and tax treatment affect the interest rate on a particular asset?

- A yield curve is a graphic representation of the relationship between interest rates (yields) on a particular security and its term to maturity. The time to maturity is measured on the horizontal axis and the interest rate (yield) on the vertical axis. The relationship between the term to maturity and interest rate is what determines the shape and level of a yield curve.
- Term to maturity, credit risk, and tax treatment are all determinants of the interest rate on a particular asset. If credit risk, interest rates, or tax treatment change, the yield curve shifts.

- 2.Explain why a yield curve can be negatively sloped. Would interest rates be abnormally high or low? What would be the overall expectation of the direction of future short-term interest rates?

- A yield curve can be negatively sloped when the yield declines as the term to maturity increases. In this case, interest rates are generally abnormally high and short-term interest rates are expected to decline in the future.

- 3.According to the expectations theory, how is the long-term interest rate determined? Why is the geometric average used instead of the simpler arithmetic average?

- According to the expectations theory, the long‑term interest rate is the geometric average of the current short-term rate and the expected future short-term rate expected to prevail over the term to maturity of the longer‑term security. The geometric average is used instead of the simpler arithmetic average in order to take account of the effects of compounding.

- 4.BBB-rated corporate bonds are riskier than AAA-rated bonds. Explain where the two yield curves will lie relative to each other. What could cause the spread to widen?

- The yield curve for BBB-rated bonds would lie higher than the yield curve for AAA‑rated corporate bonds. The spread could widen if BBB-rated bonds became riskier relative to AAA corporate bonds or if AAA corporate bonds became safer relative to BBB bonds.

- 5.What determines expectations? Are expectations about future prices independent of expectations about future money supply growth rates? Why or why not?

- Interest rate expectations are determined by expectations about the money supply, national income or gross domestic product, and inflation. Expectations about future prices are not independent of expectations about future money supply growth rate because the larger the supply of money, the more prices will be expected to rise.

- 6.Could the yield curve for municipals ever lie above the yield curve for government securities? (Hint: Consider all tax rates.) What effect would an increase in marginal tax rates have on the position of the yield curve for municipals?

- Under normal circumstances, the yield curve for municipals lies below the yield curve for government securities because the interest earned on municipals is tax free. If the interest earned on municipals were taxed at the same rates as interest on government securities, then the yield curve for municipals could lie above the yield curve for government securities because municipals are less liquid (have less developed secondary markets) than government securities. An increase in marginal tax rates could lower the position of the yield curve for municipals and widen the spread between the yield curve for municipals and government securities.

- 7.Use the liquidity premium to give an explanation for why yield curves have most often been upward sloping over the past 50 years. Could a yield curve be upward sloping even if short-term rates were expected to remain constant? If interest rates are expected to fall dramatically, under what conditions would the yield curve still be upward sloping?

- Yield curves have most often been upward sloping over the past forty‑five years because lenders have required a higher return to lend long term rather than short term. This extra sweetener is called the liquidity premium.
- A yield curve could be upward sloping even if short‑term rates were expected to remain constant because of the liquidity premium that lenders require to lend long term rather than short term.
- If interest rates are expected to fall dramatically, the yield curve would be upward sloping only if the liquidity premium was large enough to offset the expectations of lower interest rates in the future.

- 8.Define preferred habitats. Explain how this modification affects the expectations theory. What could cause market segmentation based on preferred habitats to break down? How is the market segmentation hypothesis different from the expectations theory?

- "Preferred habitats" is the name given to the theory that borrowers and lenders have preferred maturities in which they wish to borrow and lend. While the expectation theory suggests that lenders and borrowers have no preference between long‑ and short‑term securities, preferred habitats suggests differently.
- Market segmentation based on preferred habitats could break down due to changes in liquidity premiums or if rate spreads widen enough to entice(诱惑） borrowers and lenders to leave their traditional borrowing and lending markets.

- 9.Discuss the following statements: Over a typical cycle, the movement of the yield curve is like the wagging of a dog’s tail. The entire tail wags, but short-term rates “wag” more than long-term rates.

- According to the expectations theory, the long‑run rate is the geometric average of the current short rate and the future short rates expected to prevail over the term to maturity of the longer‑term security. It is reasonable to conclude that short‑term rates will vary more than long‑term rates because long‑term rates are averaged so they don‘t tend to be scattered（散乱的，分散的）as much as short‑term rates.

- 10.If yield curves became flatter (steeper), what does this say about expectations of future interest rates?

- Flatter yield curves for certain years suggest that the spread between present short-term and expected future short‑term rates has narrowed. Flatter yield curves suggest that short-term interest rate expectations have been revised downward. If yield curves becomes steeper, this suggests that expectations about future short-term rates have been revised upward.

- 11.What would happen to the risk premium if the economy went into a strong expansion? A deep recession?

- If the economy went into a strong expansion, there would be less risk of default, and risk premiums would decrease. In this case, the spread between government default risk-free securities and other securities would narrow. If on the other hand the economy went into a deep recession, default risk would increase and so would risk premiums. The spread between government default risk-free securities and other securities would widen.

- 12.If the current short-term rate is 5 percent and the expected short-term rate is 8 percent, what is the long-term interest rate? (Use the expectations theory.)

- Based on the expectations theory, the long-term rate is the geometric average of the current short-term rate and the expected future short-term rate (il = [(1 + is)(1 + ise)]1/2 – 1). Solving for the geometric average in this case, we get that the long-term rate is 6.58 percent.

- 13.If the current short-term rate is 5 percent and the current long-term rate is 4 percent, what is the expected short-term interest rate? (Use the expectations theory.)

- Based on the expectations theory, if we know the current short-term rate and the current long-term rate, we can solve for the expected short‑term interest rate by plugging the long-term and short-term rates into the formula (il = [(1 + is)(1 + ise)]1/2 – 1) and solving for ise. In this case, the expected short-term rate is 3.01 percent.

- 14.Rework questions 12 and 13 assuming that there is no compounding. (Hint: Use the simple arithmetic average instead of the geometric average.)

- Using the simple arithmetic average that ignores the effects of compounding, the long-term rate is 6.5 percent ((5 percent + 8 percent)/2 = 13 percent/2 = 6.5 percent)).
- Using the simple arithmetic average, the expected short-term interest rate is 3 percent. This can be found by solving for the expected short-term rate in the following equation: ((is + ise)/2 = il). Substituting for the short and long-term rates and solving for ise, we get ise = 3 percent ((5 percent + ise)/2 = 4 percent)).

- 15.Assume that current interest rates on government securities are as follows: one-year rate, 5 percent; two-year rate, 6 percent; three-year rate, 6.5 percent; four-year rate, 7 percent. Graph the yield curve.

- 16.Given the yield curve in question 15, what is the expected direction of future one-year rates? Under what circumstances would one-year rates be expected to decline?

- According to the pure expectations theory, short‑term rates are expected to rise.
- However, we are not certain that this is the case because we don't know the magnitude of the liquidity premium. If the liquidity premium is larger than the difference between current long‑term rates and the current short‑term rates, then expected short‑term rates would actually be expected to fall. If the liquidity premium is equal to the difference between current long-term and current short-term rates, then short-term interest rates would be expected to remain the same. If the liquidity premium is smaller than the difference between current long and current short-term rates, then indeed short-term rates are expected to rise.

- 17.If a taxpayer’s marginal tax rate is 33 percent, what is the after-tax yield on a corporate bond that pays 5 percent interest? If the average marginal tax rate of all taxpayers is 50 percent, will the taxpayer with the 33 percent marginal tax rate prefer a corporate or a municipal security? Assume equivalent safety and maturity.

- If a taxpayer's marginal tax rate is 33 percent, the after tax yield on a corporate bond that pays 5 percent interest is 3.35 percent interest (5 percent – (5 percent x .33 percent) = 3.35 percent). The rate on municipal securities will gravitate to the rate where the average taxpayer is indifferent between a municipal security and a corporate security of equivalent safety and maturity. If the marginal tax rate of all taxpayers is 50 percent, the municipal security will pay 2.5 percent (5 percent – (5 percent x 50 percent) = 2.5 percent). A taxpayer with a 33 percent marginal tax rate will prefer a corporate security because it has a higher after-tax return of 3.35 percent.

- 18.Go to the Wall Street Journal and gather data on interest rates for government securities of various maturities for today. Graph the yield curve. (Hint: Check your answer by looking at the yield curve for Treasury securities that the Journal publishes daily in Part C.)
- This answer will vary depending on the dates the students choose.

- 19.What would happen to interest rates, given each of the following scenarios?
- a. The government increases marginal tax rates.
- b. The tax exemption on municipals is eliminated.
- c. Corporate profits fall severely.
- d. The federal government guarantees that the interest and principal on corporate bonds will be paid.
- e. A broader secondary market for government agency securities develops.

- a. The rate on municipal securities will fall so that the taxpayer in the average marginal tax bracket is indifferent between municipals and other securities of equivalent risk and maturity.
- b. The rate on municipal securities will increase until the after-tax return on municipals is the same as other securities of equivalent risk and maturity.
- c. The default risk for corporate securities will increase and the spread between the yield curves for corporate and other municipal and government securities will widen.
- d. The default risk for corporate securities will decrease. The spreads between corporate securities and other government and municipal securities will decrease.
- e. The spread between the yield curve for government securities and government agency securities will narrow.

- 20.Draw the yield curve assuming future short-term rates are expected to remain constant and the liquidity premium is positive. Now assume that net lenders increase their preference for short-term securities. Show what happens to the yield curve.

Observed Yield Curve = Expectations (ies = is) + Liquidity Premium

Yield Curve Based on Expectations (ies = is)

Liquidity Premium

Yield to Maturity (percent)

Term to Maturity

When net lenders increase their preference for short-term securities, the yield curve becomes steeper as short-term rate decrease relative to long term rates.

new yield curve

6.98

When net lenders increase their preference for short-term securities, the yield curve becomes steeper as long term rates increase relative to short-term rate.

Yield to Maturity (percent)

original yield curve

5.99

5

1

2

Term to Maturity