CHAPTER 4 Interest Rates
Practice Questions
Problem 4.1.
A bank quotes you an interest rate of 14% per annum with quarterly compounding. What is the equivalent rate with (a) continuous compounding and (b) annual compounding?
(a) The rate with continuous compounding is
0?14??4ln?1???0?1376
4??or 13.76% per annum.
(b) The rate with annual compounding is
0?14??1????1?0?1475
4??4or 14.75% per annum.
Problem 4.2.
What is meant by LIBOR and LIBID. Which is higher?
LIBOR is the London InterBank Offered Rate. It is calculated daily by the British Bankers Association and is the rate a AA-rated bank requires on deposits it places with other banks. LIBID is the London InterBank Bid rate. It is the rate a bank is prepared to pay on deposits from other AA-rated banks. LIBOR is greater than LIBID.
Problem 4.3.
The six-month and one-year zero rates are both 10% per annum. For a bond that has a life of 18 months and pays a coupon of 8% per annum (with semiannual payments and one having just been made), the yield is 10.4% per annum. What is the bond’s price? What is the 18-month zero rate? All rates are quoted with semiannual compounding.
Suppose the bond has a face value of $100. Its price is obtained by discounting the cash flows at 10.4%. The price is
41?052?41?0522?1041?0523?96?74
If the 18-month zero rate isR, we must have
41?05?41?052?104(1?R?2)3?96?74
which gives R?10?42%.
Problem 4.4.
An investor receives $1,100 in one year in return for an investment of $1,000 now. Calculate the percentage return per annum with a) annual compounding, b) semiannual compounding, c) monthly compounding and d) continuous compounding.
(a) With annual compounding the return is
11001000?1?0?1
or 10% per annum.
(b) With semi-annual compounding the return is R where
R??1000?1???1100
2??2i.e.,
1?R2?1?1?1?0488
so thatR?0?0976. The percentage return is therefore 9.76% per annum.
(c) With monthly compounding the return is R where
R??1000?1??12??12?1100
i.e.
R??1????12??121?1?1?00797
so that R?0?0957. The percentage return is therefore 9.57% per annum.
(d) With continuous compounding the return is R where:
1000e?1100
R i.e.,
e?1?1
R so thatR?ln1?1?0?0953. The percentage return is therefore 9.53% per annum.
Problem 4.5.
Suppose that zero interest rates with continuous compounding are as follows: Maturity (months) Rate (% per annum) 3 8.0 6 8.2 9 8.4 12 8.5 15 8.6 18 8.7
Calculate forward interest rates for the second, third, fourth, fifth, and sixth quarters.
The forward rates with continuous compounding are as follows to
Qtr 2 Qtr 3 Qtr 4 Qtr 5 Qtr 6 8.4% 8.8% 8.8% 9.0% 9.2% Problem 4.6.
Assuming that zero rates are as in Problem 4.5, what is the value of an FRA that enables the holder to earn 9.5% for a three-month period starting in one year on a principal of $1,000,000? The interest rate is expressed with quarterly compounding.
The forward rate is 9.0% with continuous compounding or 9.102% with quarterly compounding. From equation (4.9), the value of the FRA is therefore
[1?000?000?0?25?(0?095?0?09102)]e?0?086?1?25?893?56
or $893.56.
Problem 4.7.
The term structure of interest rates is upward sloping. Put the following in order of magnitude:
(a) The five-year zero rate
(b) The yield on a five-year coupon-bearing bond
(c) The forward rate corresponding to the period between 4.75 and 5 years in
the future
What is the answer to this question when the term structure of interest rates is downward sloping?
When the term structure is upward sloping, c?a?b. When it is downward sloping, b?a?c.
Problem 4.8.
What does duration tell you about the sensitivity of a bond portfolio to interest rates? What are the limitations of the duration measure?
Duration provides information about the effect of a small parallel shift in the yield curve on the value of a bond portfolio. The percentage decrease in the value of the portfolio equals the duration of the portfolio multiplied by the amount by which interest rates are increased in the small parallel shift. The duration measure has the following limitation. It applies only to parallel shifts in the yield curve that are small.
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:
0?15??Re??1??
12??12i.e.,
0?15??R?12ln?1??
12???0?1491
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
e0?12R????1??
4??4or
R?4(e0?03?1)?0?1218
The amount of interest paid each quarter is therefore:
10?000?0?12184?304?55
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
2e?0?04?0?5?2e?0?042?1?0?2e?0?044?1?5?2e?0?046?2?102e?0?048?2?5?98?04
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
?4e?4eUsing the Solver or Goal Seek tool in Excel y?0?06407
4e?0?5y?4e?1?0y?4e?1?5y?2?0y?2?5y?104e?3?0y?104
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, m?2, d?e?0?07?2?0?8694. Also
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)?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