发布时间 : 星期四 文章HullFund8eCh04ProblemSolutions更新完毕开始阅读
can borrow the principal at RMand 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 thatRKwill 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
ofRMLand an outflow ofRKL. In either case, we see that the FRA involves the exchange of a fixed rate of interest on the principal ofLfor a floating rate of interest on the principal.
Problem 4.22.
Explain how a repo agreement works and why it involves very little risk for the lender.
The borrower transfers to the lender ownership of securities which have a value
approximately equal to the amount borrowed and agrees to buy them back for the amount borrowed plus accrued interest at the end of the life of the loan. If the borrower defaults, the lender keeps the securities. Note that the securities should not have a value significantly more than the amount borrowed. Otherwise the borrower is subject to the risk that the lender will not honor its obligations.
Further Questions
Problem 4.23
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 4.24
The following table gives Treasury zero rates and cash flows on a Treasury bond: Maturity (years Zero rate Coupon payment 0.5 2.0% $20 1.0 2.3% $20 1.5 2.7% $20 2.0 3.2% $20 Zero rates are continuously compounded (a) What is the bond’s theoretical price? (b) What is the bond’s yield?
Principal $1000 The bond’s theoretical price is 20×e-0.02×0.5+20×e-0.023×1+20×e-0.027×1.5+1020×e-0.032×2 = 1015.32
The bond’s yield assuming that it sells for its theoretical price is obtained by solving 20×e-y×0.5+20×e-y×1+20×e-y×1.5+1020×e-y×2 = 1015.32 It is 3.18%.
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.
Problem 4.27
A bank can borrow or lend at LIBOR. The two-month LIBOR rate is 0.28% per annum with continuous compounding. Assuming that interest rates cannot be negative, what is the arbitrage opportunity if the three-month LIBOR rate is 0.1% per year with continuous compounding. How low can the three-month LIBOR rate become without an arbitrage opportunity being created?
The forward rate for the third month is 0.001×3 ? 0.0028×2 = ? 0.0026 or ? 0.26%. If we assume that the rate for the third month will not be negative we can borrow for three months, lend for two months and lend at the market rate for the third month. The lowest level for the three-month rate that does not permit this arbitrage is 0.0028×2/3 = 0.001867 or 0.1867%.
Problem 4.28
A bank can borrow or lend at LIBOR. Suppose that the six-month rate is 5% and the
nine-month rate is 6%. The rate that can be locked in for the period between six months and nine months using an FRA is 7%. What arbitrage opportunities are open to the bank? All rates are continuously compounded.
The forward rate is 0.06?0.75?0.05?0.50?0.08
0.25or 8%. The FRA rate is 7%. A profit can therefore be made by borrowing for six months at 5%, entering into an FRA to borrow for the period between 6 and 9 months for 7% and lending for nine months at 6%.
Problem 4.29.
An interest rate is quoted as 5% per annum with semiannual compounding. What is the
equivalent rate with (a) annual compounding, (b) monthly compounding, and (c) continuous compounding.
a) With annual compounding the rate is 1?0252?1?0?050625 or 5.0625%
b) With monthly compounding the rate is 12?(1?0251?6?1)?0?04949 or 4.949%. c) With continuous compounding the rate is 2?ln1?025?0?04939or 4.939%.
Problem 4.30.
The 6-month, 12-month. 18-month,and 24-month zero rates are 4%, 4.5%, 4.75%, and 5% with semiannual compounding.
a) What are the rates with continuous compounding?
b) What is the forward rate for the six-month period beginning in 18 months
c) What is the value of an FRA that promises to pay you 6% (compounded semiannually) on a principal of $1 million for the six-month period starting in 18 months?
a) With continuous compounding the 6-month rate is 2ln1?02?0?039605 or 3.961%. The 12-month rate is 2ln1?0225?0?044501 or 4.4501%. The 18-month rate is
2ln1?02375?0?046945 or 4.6945%. The 24-month rate is 2ln1?025?0?049385 or 4.9385%.
b) The forward rate (expressed with continuous compounding) is from equation (4.5)
4?9385?2?4?6945?1?5
0?5or 5.6707%. When expressed with semiannual compounding this is 2(e0?056707?0?5?1)?0?057518 or 5.7518%.
c) The value of an FRA that promises to pay 6% for the six month period starting in 18 months is from equation (4.9) 1?000?000?(0?06?0?057518)?0?5e?0?049385?2?1124?or $1,124.
Problem 4.31.
What is the two-year par yield when the zero rates are as in Problem 4.30? What is the yield on a two-year bond that pays a coupon equal to the par yield?
The value, A of an annuity paying off $1 every six months is
e?0?039605?0?5?e?0?044501?1?e?0?046945?1?5?e?0?049385?2?3?7748
The present value of $1 received in two years, d, is e?0?049385?2?0?90595. From the formula in Section 4.4 the par yield is
(100?100?0?90595)?2?4?983
3?7748or 4.983%.
Problem 4.32.
The following table gives the prices of bonds Bond Principal ($) 100 100 100 100 Time to Maturity (yrs) 0.5 1.0 1.5 2.0 Annual Coupon ($)* 0.0 0.0 6.2 8.0 Bond Price ($) 98 95 101 104 *Half the stated coupon is paid every six months
a) Calculate zero rates for maturities of 6 months, 12 months, 18 months, and 24 months.
b) What are the forward rates for the periods: 6 months to 12 months, 12 months to 18 months, 18 months to 24 months?
c) What are the 6-month, 12-month, 18-month, and 24-month par yields for bonds that provide semiannual coupon payments?
d) Estimate the price and yield of a two-year bond providing a semiannual coupon of 7% per annum.
a) The zero rate for a maturity of six months, expressed with continuous compounding is
2ln(1?2?98)?4?0405%. The zero rate for a maturity of one year, expressed with continuous compounding is ln(1?5?95)?5?1293. The 1.5-year rate is Rwhere
3?1e?0?040405?0?5?3?1e?0?051293?1?103?1e?R?1?5?101
The solution to this equation isR?0?054429. The 2.0-year rate is R where
4e?0?040405?0?5?4e?0?051293?1?4e?0?054429?1?5?104e?R?2?104
The solution to this equation isR?0?058085. These results are shown in the table below Maturity (yrs) 0.5 1.0 1.5 2.0 Zero Rate (%) 4.0405 5.1293 5.4429 5.8085 Forward Rate (%) Par Yield (s.a.%) 4.0405 4.0816 6.2181 5.1813 6.0700 5.4986 6.9054 5.8620 Par yield (c.c %) 4.0405 5.1154 5.4244 5.7778 b) c)
d) e) f)
The continuously compounded forward rates calculated using equation (4.5) are shown in the third column of the table
The par yield, expressed with semiannual compounding, can be calculated from the formula in Section 4.4. It is shown in the fourth column of the table. In the fifth column of the table it is converted to continuous compounding
The price of the bond is
3?5e?0?040405?0?5?3?5e?0?051293?1?3?5e?0?054429?1?5?103?5e?0?058085?2?102?13
The yield on the bond, y satisfies
3?5e?y?0?5?3?5e?y?1?0?3?5e?y?1?5?103?5e?y?2?0?102?13
The solution to this equation isy?0?057723. The bond yield is therefore 5.7723%.