Chapter 05 Solutions - Interest Rates resources PDF

Preview

Summary

Interest Rates resources ...

Description

Chapter 5 Interest Rates Questions 5. What does the term risk-free interest mean, and why do we usually use the Treasury bill yield as the risk-free rate? Risk-free interest is really a guaranteed interest rate that is paid with certainty at the end of the contract period. Risk means that there is more than one potential outcome, absence of risk (risk-free) means that there is one and only one possible outcome. We assume the government will not default on any issued Treasury bills and the face value of the bill will always be paid at maturity. Thus, it has one and only one possible outcome or payoff. 6. Why does a mortgage typically have a lower interest rate than a car loan? A mortgage has a lower interest rate compared to a car loan because it has a lower default premium. The default premium is made up of two components: the frequency of default and the loss if a default should happen. The collateral of a house is better than that of a car and so the potential loss in a default is less for a mortgage than a car loan. 8. Since 1950, the interest rates for corporate bonds have averaged a higher interest rate than long-term government bonds. Why? In general, we attribute the higher rate on corporate bonds to the larger default premium for corporate bonds over government bonds. We assume a very low probability of default for government bonds.

1

Problems 1. Periodic interest rates. In the following table, fill in the periodic rates and the effective annual rates. Period

APR

Compounding Per Year

Periodic Rate

Effective Annual Rate

Semi-Annual

8%

2

Quarterly

9%

4

Monthly

7.5%

12

Daily

4.25%

365

APR

Compounding Per Year

Periodic Rate

Effective Annual Rate

Period Semi-Annual

8%

2

4.0%

8.16%

Quarterly

9%

4

2.25%

9.31%

Monthly

7.5%

12

0.625%

7.76%

Daily

4.25%

365

0.01164%

4.34%

ANSWER

Periodic Rate = APR / (C/Y) = 0.08 / 2 = 0.04 = 4.0% Periodic Rate = APR / (C/Y) = 0.09 / 4 = 0.0225 = 2.25% Periodic Rate = APR / (C/Y) = 0.075 / 12 = 0.00625 = 0.625% Periodic Rate = APR / (C/Y) = 0.0425 / 365 = 0.0001164384 = 0.01164% EAR = (1 + Periodic Rate)C/Y – 1 = 1.042 – 1 = 1.0816 – 1 = 0.0816 = 8.16% EAR = (1 + Periodic Rate)C/Y – 1 = 1.02254 – 1 = 1.0931 – 1 = 0.0931 = 9.31% EAR = (1 + Periodic Rate)C/Y – 1 = 1.0062512 – 1 = 1.0776 – 1 = 0.0776 = 7.76% EAR = (1 + Periodic Rate)C/Y – 1 = 1.0001164365 – 1 = 1.0434 – 1 = 0.0434 = 4.34%

3. EAR. What is the effective annual rate of a mortgage rate that is advertised at 7.75% (APR) over the next twenty years and paid with monthly payments? ANSWER Periodic Rate = 0.0775 / 12 = 0.0064583333 EAR = (1 + Periodic Rate)C/Y – 1 = 1.0064583312 – 1 = 1.0803 – 1 = 0.0803 = 8.03% 2

13. Inflation, nominal interest rates, and real rates. Given the information below, estimate the nominal interest rate with the approximate nominal interest rate equation and the true nominal interest rate equation. Real Rate

Inflation Rate

3%

5%

8%

15%

1%

4%

2.5%

3.5%

Approximate Nominal Rate

True Nominal Rate

ANSWER Real Rate

Inflation Rate

Approximate Nominal Rate

True Nominal Rate

3%

5%

8%

8.15%

8%

15%

23%

24.20%

1%

4%

5%

5.04%

2.5%

3.5%

6%

6.09%

Approximate Nominal Rate = 3% + 5% = 8% Approximate Nominal Rate = 8% + 15% = 23% Approximate Nominal Rate = 1% + 4% = 5% Approximate Nominal Rate = 2.5% + 3.5% = 6% True Nominal Rate = 1.03 x 1.05 – 1 = 1.0815 – 1 = 0.0815 or 8.15% True Nominal Rate = 1.08 x 1.15 – 1 = 1.2420 – 1 = 0.2420 or 24.20% True Nominal Rate = 1.01 x 1.04 – 1 = 1.0504 – 1 = 0.0504 or 5.04% True Nominal Rate = 1.025 x 1.035 – 1 = 1.0609 – 1 = 0.0609 or 6.09% 16. Inflation, nominal interest rates, and real rates. From 1991 to 2000, the U.S. economy had an annual inflation rate of around 2.93%. The historical annual nominal risk-free rate for this same period was around 5.02%. What is the real interest rate using the approximate nominal interest rate equation and the true nominal interest rate equation for that decade? ANSWER Approximate Real Rate = 5.02% – 2.93% = 2.09% True Real Rate = 1.0502 / 1.0293 – 1 = 1.0203 – 1 = 0.0203 or 2.03%

3

21. Interest premium. The U.S. government offers two bonds: one selling to yield 6.5% and the other to yield 8.5%. Why would one bond sell for a lower yield if the originator is the same on both bonds? ANSWER If the bonds have different maturity dates the difference in yields is a reflection of the maturity premium where bonds with longer maturities have higher rates.

4...