发布时间 : 星期二 文章HullOFOD8eSolutionsCh04更新完毕开始阅读
(100?100?0?8694)?23?6935?7?072
To verify that this is correct we calculate the value of a bond that pays a coupon of 7.072% per year (that is 3.5365 every six months). The value is
3?536e?0?05?0?5?3?5365e?0?06?1?0?3?536e?0?065?1?5?103?536e?0?07?2?0?100
verifying that 7.072% is the par yield.
Problem 4.14.
Suppose that 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 rates in Problem 4.14 to value an FRA where you will pay 5% for the third year on $1 million.
The forward rate is 5.1% with continuous compounding or e0?051?1?1?5?232% with annual compounding. The 3-year interest rate is 3.7% with continuous compounding. From equation (4.10), the value of the FRA is therefore
[1?000?000?(0?05232?0?05)?1]e?0?037?3?2?078?85
or $2,078.85.
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??70Year 10?200?100?100
because 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
110ln10070?0?0357
or 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?
There are three reasons (see Business Snapshot 4.1).
1. Treasury bills and Treasury bonds must be purchased by financial institutions to fulfill a
variety of regulatory requirements. This increases demand for these Treasury instruments driving the price up and the yield down.
2. 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.
3. 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 11% (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 10.8% per annum yield and verify that the result is in agreement with your answer to (c).
a) The bond’s price is
8e?0?11?8e?0?11?2?8e?0?11?3?8e?0?11?4?108e?0?11?5?86?80
b) The bond’s duration is
186?80???8e?0?11?2?8e?0?11?2?3?8e?0?11?3?4?8e?0?11?4?5?108e?0?11?5???
?4?256years
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
86?80?4?256?0?002?0?74
The bond’s price should increase from 86.80 to 87.54.
d) With a 10.8% yield the bond’s price is
8e?0?108?8e?0?108?2?8e?0?108?3?8e?0?108?4?108e?0?108?5?87?54
This is consistent with 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 bond that will pay coupons of $4 every six months currently sells for $94.84. A two-year 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 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 2
4e?0?1238?0?5?4e?0?1165?1?104e?1?5R?1?5R?1?5R?94?84
where R is the 11 year zero rate. It follows that 23?76?3?56?104e
?94?84?0?8415
eR?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?7977R?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.
Each 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 (Excel file)
A five-year bond provides a coupon of 5% per annum payable semiannually. Its price is 104. What is the bond's yield? You may find Excel's Solver useful.
The answer (with continuous compounding) is 4.07%
Problem 4.26 (Excel file)
Suppose that LIBOR rates for maturities of one month, two months, three months, four months, five months and six months are 2.6%, 2.9%, 3.1%, 3.2%, 3.25%, and 3.3% with continuous compounding. What are the forward rates for future one month periods?
The forward rates for the second, third, fourth, fifth and sixth months are (see spreadsheet) 3.2%, 3.5%, 3.5%, 3.45%, 3.55%, respectively with continuous compounding.