HullFund8eCh04ProblemSolutions
CHAPTER 4 Interest Rates
Practice Questions
Problem 4.8.
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 6946383%/=. in six months. This is 2638312766%?.=. per annum with semiannual compounding or 2ln(106383)1238%.=. per annum with continuous compounding. The 12-month rate is 118912360%/=. with annual compounding or ln(11236)1165%.=. with continuous compounding. For the 112
year bond we must have
012380501165115441049484R e e e -.?.-.?-.++=. where R is the 112
year zero rate. It follows that
15153763561049484
084150115
R R e e R -.-..+.+=.=.=.
or 11.5%. For the 2-year bond we must have 012380501165101151525551059712R e e e e -.?.-.?-.?.-+++=. where R is the 2-year zero rate. It follows that
207977
0113
R e R -=.=.
or 11.3%.
Problem 4.9.
What rate of interest with continuous compounding is equivalent to 15% per annum with monthly compounding?
The rate of interest is R where:
12
015112R
e .??=+ ??
?
i.e.,
01512ln 112R .??
=+ ???
01491=.
The rate of interest is therefore 14.91% per annum.
Problem 4.10.
A deposit account pays 12% per annum with continuous compounding, but interest is actually paid quarterly. How much interest will be paid each quarter on a $10,000 deposit?
The equivalent rate of interest with quarterly compounding is R where
4
012
14R e .??=+ ???
or
0034(1)01218R e .=-=.
The amount of interest paid each quarter is therefore:
01218
10000304554
.,?=. or $304.55.
Problem 4.11.
Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are 4%,
4.2%, 4.4%, 4.6%, and 4.8% per annum with continuous compounding respectively. Estimate the cash price of a bond with a face value of 100 that will mature in 30 months and pays a coupon of 4% per annum semiannually.
The bond pays $2 in 6, 12, 18, and 24 months, and $102 in 30 months. The cash price is 004050042100044150046200482522221029804e e e e e -.?.-.?.-.?.-.?-.?.++++=.
Problem 4.12.
A three-year bond provides a coupon of 8% semiannually and has a cash price of 104. What is the bond’s yield?
The bond pays $4 in 6, 12, 18, 24, and 30 months, and $104 in 36 months. The bond yield is the value of y that solves 05101520253044444104104y y y y y y e e e e e e -.-.-.-.-.-.+++++= Using the Goal Seek or Solver tool in Excel 006407y =. or 6.407%.
Problem 4.13.
Suppose that the 6-month, 12-month, 18-month, and 24-month zero rates are 5%, 6%, 6.5%, and 7% respectively. What is the two-year par yield?
Using the notation in the text, 2m =, 007208694d e -.?==.. Also 00505006100065150072036935A e e e e -.?.-.?.-.?.-.?.=+++=. The formula in the text gives the par yield as
(10010008694)2
707236935
-?.?=.. 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 0050500610006515007203536353653536103536100e e e e -.?.-.?.-.?.-.?..+.+.+.= verifying that 7.072% is the par yield.
Problem 4.14.
Suppose that zero interest rates with continuous compounding are as follows:
Calculate forward interest rates for the second, third, fourth, and fifth years.
The forward rates with continuous compounding are as follows: to
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.
.?-=.with annual The forward rate is 5.1% with continuous compounding or 0051115232
e% compounding. The 3-year interest rate is 3.7% with continuous compounding. From equation (4.10), the value of the FRA is therefore
00373
[1000000(005232005)1]207885
,,?.-.?=,.
e-.?
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 = ?70
Year 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
10
e
=
10070R
The rate is
1100
=.
ln00357
1070
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?
R, will apply to a certain
A FRA is an agreement that a certain specified interest rate,
K
principal, L, for a certain specified future time period. Suppose that the rate observed in the
R. If the market for the future time period at the beginning of the time period proves to be
M
R will apply when the principal is invested, the holder of the FRA FRA is an agreement that
K
can borrow the principal at M R and then invest it at K R . The net cash flow at the end of the period is then an inflow of K R L and an outflow of M R L . If the FRA is an agreement that K R will apply when the principal is borrowed, the holder of the FRA can invest the borrowed principal at M R . The net cash flow at the end of the period is then an inflow
of M R L and an outflow of K R L . 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.
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:
Zero rates are continuously compounded (a) What is the bond’s theoretical price? (b) What is the bond’s yield?
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 08.025
.050
.005.075.006.0=?-?
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 2102510050625.-=. or 5.0625%
b) With monthly compounding the rate is 1612(10251)004949/?.-=. or 4.949%. c) With continuous compounding the rate is 2ln1025004939?.=.or 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 2ln1020039605.=. or 3.961%. The 12-month rate is 2ln102250044501.=. or 4.4501%. The 18-month rate is
2ln1023750046945.=. or 4.6945%. The 24-month rate is 2ln10250049385.=. or 4.9385%.
b) The forward rate (expressed with continuous compounding) is from equation (4.5)
4938524694515
05
.?-.?..
or 5.6707%. When expressed with semiannual compounding this is 0056707052(1)0057518e .?.-=. 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) 004938521000000(0060057518)051124e -.?,,?.-.?.=, 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
003960505004450110046945150049385237748e e e e -.?.-.?-.?.-.?+++=. The present value of $1 received in two years, d , is 00493852090595e -.?=.. From the formula in Section 4.4 the par yield is
(100100090595)2
498337748
-?.?=.. or 4.983%.
Problem 4.32.
The following table gives the prices of bonds
*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(1298)40405%+/=.. The zero rate for a maturity of one year, expressed with continuous compounding is ln(1595)51293+/=.. The 1.5-year rate is R where
004040505005129311531311031101R e e e -.?.-.?-?..+.+.= The solution to this equation is 0054429R =.. The 2.0-year rate is R where
004040505005129310054429152444104104R e e e e -.?.-.?-.?.-?+++= The solution to this equation is 0058085R =.. These results are shown in the table below
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
0040405050051293100544291500580852353535103510213e e e e -.?.-.?-.?.-.?.+.+.+.=. e) The yield on the bond, y satisfies
05101520353535103510213y y y y e e e e -?.-?.-?.-?..+.+.+.=. f) The solution to this equation is 0057723y =.. The bond yield is therefore 5.7723%.