A?e?0?05?0?5?e?0?06?1?0?e?0?065?1?5?e?0?07?2?0?3?6935

The formula in the text gives the par yield as

(100?100?0.8694)?2?7.0741

3.6935 To verify that this is correct we calculate the value of a bond that pays a coupon of 7.0741% per year (that is 3.5370 every six months). The value is

3.537e?0.05?0.5?3.537e?0.06?1.0?3.537e?0.065?1.5?103.537e?0.07?2.0?100

verifying that 7.0741% is the par yield.

Problem 4.14.

Suppose that risk-free zero interest rates with continuous compounding are as follows: Maturity( years) 1 2 3 4 5 Rate (% per annum) 2.0 3.0 3.7 4.2 4.5 Calculate forward interest rates for the second, third, fourth, and fifth years.

The forward rates with continuous compounding are as follows: Year 2: 4.0% Year 3: 5.1% Year 4: 5.7% Year 5: 5.7%

Problem 4.15.

Use the risk-free rates in Problem 4.14 to value an FRA where you will pay 5% (compounded annually) and receive LIBOR for the third year on $1 million. The forward LIBOR rate (annually compounded) for the third year is 5.5%.

We value the FRA by assuming that the forward LIBOR will be realized. The value of the FRA is

1,000,000×(0.055?0.050)e?0.037×3 = $4,474.69

Problem 4.16.

A 10-year, 8% coupon bond currently sells for $90. A 10-year, 4% coupon bond currently sells for $80. What is the 10-year zero rate? (Hint: Consider taking a long position in two of the 4% coupon bonds and a short position in one of the 8% coupon bonds.)

Taking a long position in two of the 4% coupon bonds and a short position in one of the 8% coupon bonds leads to the following cash flows

Year 0?90?2?80??70

Year 10?200?100?100because the coupons cancel out. $100 in 10 years time is equivalent to $70 today. The 10-year rate,R, (continuously compounded) is therefore given by

100?70e10R

The rate is

1100ln?0?0357 1070or 3.57% per annum.

Problem 4.17.

Explain carefully why liquidity preference theory is consistent with the observation that the term structure of interest rates tends to be upward sloping more often than it is downward sloping.

If long-term rates were simply a reflection of expected future short-term rates, we would expect the term structure to be downward sloping as often as it is upward sloping. (This is based on the assumption that half of the time investors expect rates to increase and half of the time investors expect rates to decrease). Liquidity preference theory argues that long term rates are high relative to expected future short-term rates. This means that the term structure should be upward sloping more often than it is downward sloping.

Problem 4.18.

“When the zero curve is upward sloping, the zero rate for a particular maturity is greater than the par yield for that maturity. When the zero curve is downward sloping, the reverse is true.” Explain why this is so.

The par yield is the yield on a coupon-bearing bond. The zero rate is the yield on a zero-coupon bond. When the yield curve is upward sloping, the yield on an N-year

coupon-bearing bond is less than the yield on an N-year zero-coupon bond. This is because the coupons are discounted at a lower rate than the N-year rate and drag the yield down below this rate. Similarly, when the yield curve is downward sloping, the yield on an N-year coupon bearing bond is higher than the yield on an N-year zero-coupon bond.

Problem 4.19.

Why are U.S. Treasury rates significantly lower than other rates that are close to risk free?

Two reasons given in the chapter are

1. The amount of capital a bank is required to hold to support an investment in Treasury bills and bonds is substantially smaller than the capital required to support a similar investment in other very-low-risk instruments.

2. In the United States, Treasury instruments are given a favorable tax treatment compared with most other fixed-income investments because they are not taxed at the state level.

Problem 4.20.

Why does a loan in the repo market involve very little credit risk?

A repo is a contract where an investment dealer who owns securities agrees to sell them to another company now and buy them back later at a slightly higher price. The other company is providing a loan to the investment dealer. This loan involves very little credit risk. If the borrower does not honor the agreement, the lending company simply keeps the securities. If the lending company does not keep to its side of the agreement, the original owner of the securities keeps the cash.

Problem 4.21.

Explain why an FRA is equivalent to the exchange of a floating rate of interest for a fixed rate of interest?

A FRA is an agreement that a certain specified interest rate, RK, will apply to a certain

principal, L, for a certain specified future time period. Suppose that the rate observed in the market for the future time period at the beginning of the time period proves to beRM. If the FRA is an agreement that RK will apply when the principal is invested, the holder of the FRA can borrow the principal at RM and then invest it atRK. The net cash flow at the end of the period is then an inflow of RKL and an outflow ofRML. If the FRA is an agreement that RK will apply when the principal is borrowed, the holder of the FRA can invest the borrowed principal atRM. The net cash flow at the end of the period is then an inflow of

RML and an outflow ofRKL. In either case we see that the FRA involves the exchange of a fixed rate of interest on the principal of L for a floating rate of interest on the principal.

Problem 4.22.

A five-year bond with a yield of 7% (continuously compounded) pays an 8% coupon at the end of each year.

a) What is the bond’s price? b) What is the bond’s duration?

c) Use the duration to calculate the effect on the bond’s price of a 0.2% decrease in its yield.

d) Recalculate the bond’s price on the basis of a 6.8% per annum yield and verify that the result is in agreement with your answer to (c).

a) The bond’s price is

8e?0?07?8e?0?07?2?8e?0?07?3?8e?0?07?4?108e?0?07?5?103.05

b) The bond’s duration is

1??0?07?0?07?2?0?07?3?0?07?4?0?07?5??2?8e?3?8e?4?8e?5?108e ?8e? ??103.05 ?4?3235years

c) Since, with the notation in the chapter ?B??BD?y

the effect on the bond’s price of a 0.2% decrease in its yield is

103.05?4?3235?0?002?0?89

The bond’s price should increase from 103.05 to 103.94

d) With a 6.8% yield the bond’s price is

8e?0?068?8e?0?068?2?8e?0?068?3?8e?0?068?4?108e?0?068?5?103.95

This is close to the answer in (c).

Problem 4.23.

The cash prices of six-month and one-year Treasury bills are 94.0 and 89.0. A 1.5-year Treasury bond that will pay coupons of $4 every six months currently sells for $94.84. A two-year Treasury bond that will pay coupons of $5 every six months currently sells for $97.12. Calculate the six-month, one-year, 1.5-year, and two-year Treasury zero rates.

The 6-month Treasury bill provides a return of 6?94?6?383% in six months. This is 2?6?383?12?766% per annum with semiannual compounding or 2ln(1?06383)?12?38% per annum with continuous compounding. The 12-month rate is 11?89?12?360% with annual compounding or ln(1?1236)?11?65% with continuous compounding. For the 11 year bond we must have 24e?0?1238?0?5?4e?0?1165?1?104e?1?5R?94?84 where R is the 11 year zero rate. It follows that 2

3?76?3?56?104e?1?5R?94?84

e?1?5R?0?8415

R?0?115or 11.5%. For the 2-year bond we must have

5e?0?1238?0?5?5e?0?1165?1?5e?0?115?1?5?105e?2R?97?12

where R is the 2-year zero rate. It follows that

e?2R?0?7977

R?0?113

or 11.3%.

Problem 4.24.

“An interest rate swap where six-month LIBOR is exchanged for a fixed rate 5% on a

principal of $100 million for five years is a portfolio of nine FRAs.” Explain this statement.

The first exchange of payments is known. Each subsequent exchange of payments is an FRA where interest at 5% is exchanged for interest at LIBOR on a principal of $100 million. Interest rate swaps are discussed further in Chapter 7.

Further Questions

Problem 4.25.

When compounded annually an interest rate is 11%. What is the rate when expressed with (a) semiannual compounding, (b) quarterly compounding, (c) monthly compounding, (d) weekly compounding, and (e) daily compounding.

We must solve 1.11=(1+R/n)n where R is the required rate and the number of times per year the rate is compounded. The answers are a) 10.71%, b) 10.57%, c) 10.48%, d) 10.45%, e) 10.44%

Problem 4.26.

The following table gives Treasury zero rates and cash flows on a Treasury bond: