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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?0.500.25?0.08
or 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?50?5
or 5.6707%. When expressed with semiannual compounding this is 0?056707?0?52(e?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?1?124
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 $1 received in two years,d, is e?0?049385?2?0?90595. From the formula in Section 4.4 the par yield is
(100?100?0?90595)?23?7748?4?983
or 4.983%. By definition this is also the yield on a two-year bond that pays a coupon equal to the par yield.
Problem 4.32.
The following table gives the prices of bonds Bond Principal ($) 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 a) b) c) d)
*Half the stated coupon is paid every six months
Calculate zero rates for maturities of 6 months, 12 months, 18 months, and 24 months.
What are the forward rates for the periods: 6 months to 12 months, 12 months to 18 months, 18 months to 24 months?
What are the 6-month, 12-month, 18-month, and 24-month par yields for bonds that provide semiannual coupon payments?
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 is2ln(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?5The solution to this equation is
?3?1e?103?1e?101 R?0?054429. The 2.0-year rate is R?0?051293?1?R?1?5 where
4e?0?040405?0?5?4e?0?051293?1The solution to this equation isbelow
Maturity (yrs) 0.5 1.0 1.5 2.0 Zero Rate (%) 4.0405 5.1293 5.4429 5.8085 ?0?054429?1?5?R?2?4e?104e?104
R?0?058085. These results are shown in the table
Forward Rate (%) 4.0405 6.2181 6.0700 6.9054 Par Yield (s.a.%) 4.0816 5.1813 5.4986 5.8620 Par yield (c.c %) 4.0405 5.1154 5.4244 5.7778
b) The continuously compounded forward rates calculated using equation (4.5) are shown in the third column of the table
c) 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
d) The price of the bond is
?3?5eThe yield on the bond, y
3?5e?y?0?53?5e?0?040405?0?5?0?051293?1?3?5e?0?054429?1?5?103?5e?y?2?0?0?058085?2?102?13
satisfies
?y?1?0?3?5eThe solution to this equation is
?3?5e?103?5e?102?13
y?0?057723. The bond yield is therefore 5.7723%.
?y?1?5
Problem 4.33.
Portfolio A consists of a one-year zero-coupon bond with a face value of $2,000 and a 10-year zero-coupon bond with a face value of $6,000. Portfolio B consists of a 5.95-year zero-coupon bond with a face value of $5,000. The current yield on all bonds is 10% per annum.
(a) Show that both portfolios have the same duration.
(b) Show that the percentage changes in the values of the two portfolios for a 0.1% per annum increase in yields are the same.
(c) What are the percentage changes in the values of the two portfolios for a 5% per annum increase in yields?
a) The duration of Portfolio A is
1?2000e?0?1?1?10?6000e?6000e?0?1?102000e?0?1?1?0?1?10?5?95
Since this is also the duration of Portfolio B, the two portfolios do have the same duration.
b) The value of Portfolio A is
2000e2000e?0?1?6000e?6000e?0?1?10?4016?95 ?3993?18
When yields increase by 10 basis points its value becomes
?0?101?0?101?10The percentage decrease in value is
23?77?1004016?95?0?59%
The value of Portfolio B is
5000e5000e?0?1?5?95?2757?81 ?2741?45 ?0?59%
When yields increase by 10 basis points its value becomes
?0?101?5?95The percentage decrease in value is
16?36?1002757?81The percentage changes in the values of the two portfolios for a 10 basis point increase in yields are therefore the same.
c) When yields increase by 5% the value of Portfolio A becomes
2000e?0?15?6000e?0?15?5?95?0?15?10?3060?20
and the value of Portfolio B becomes
5000e956?754016?95709?662757?81?2048?15
The percentage reductions in the values of the two portfolios are:
PortfolioA?PortfolioB??100?23?82
?100?25?73Since the percentage decline in value of Portfolio A is less than that of Portfolio B, Portfolio A has a greater convexity.