CHAPTER 4 Interest Rates

Practice Questions

Problem 4.1.

A bank quotes you an interest rate of 7% 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

?0.07?4ln?1???0.0694

4??

or 6.94% per annum.

(b) The rate with annual compounding is

?0.07??1???1?0.0719

4??

4or 7.19% per annum.

Problem 4.2.

Explain how LIBOR is determined

LIBOR is the London InterBank Offered Rate. It is calculated daily from borrowing rates estimated by a panel of banks

Problem 4.3.

The six-month and one-year zero rates are both 5% per annum. For a bond that has a life of 18 months and pays a coupon of 4% per annum (with semiannual payments and one having just been made), the yield is 5.2% 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 5.2%. The price is

22102???98.29 231.0261.0261.026

If the 18-month zero rate isR, we must have

22102???98.29 1.0251.0252(1?R/2)3

which gives R=5.204%.

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

1100?1?0?1

1000or 10% per annum.

(b) With semi-annual compounding the return is R where

?R?1000?1???1100

?2?i.e.,

R?1?1?1?0488 2so thatR?0?0976. The percentage return is therefore 9.76% per annum.

(c) With monthly compounding the return is R where

1?2

R??1000?1???1100

?12?i.e.

12R?12?1????1?1?1?00797 ?12?so that R?0?0957. The percentage return is therefore 9.57% per annum.

(d) With continuous compounding the return is R where:

1000eR?1100

i.e.,

eR?1?1

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 3.0 6 3.2 9 3.4 12 3.5 15 3.6 18 3.7

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 3.4% 3.8% 3.8% 4.0% 4.2% Problem 4.6.

Assuming that risk-free rates are as in Problem 4.5, what is the value of an FRA where the holder will pay LIBOR and receive 4.5% (quarterly compounded) for a three-month period starting in one year on a principal of $1,000,000. The forward LIBOR rate for the three month period is 5% quarterly compounded.

From equation (4.9), the value of the FRA is therefore

?1,000,000?0.25?(0.045?0.050)?e?0.036?1.25??1,195

or ?$1,195

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 8% per annum with monthly compounding?

The rate of interest is R where:

?0?08?e??1??

12??R12i.e.,

?0?08?R?12ln?1??

12??

?0?0797

The rate of interest is therefore 7.97% per annum.

Problem 4.10.

A deposit account pays 4% 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

e0.04?R???1?? ?4?4or

R?4(e0?01?1)?0?0402

The amount of interest paid each quarter is therefore:

0?040210?000??100.50

4or $100.50.

Problem 4.11.

Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are,

respectively, 4%, 4.2%, 4.4%, 4.6%, and 4.8% per annum with continuous compounding. Estimate the cash price of a bond with a face value of 100 that will mature in 30 months and pay 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?0?5y?4e?1?0y?4e?1?5y?4e?2?0y?4e?2?5y?104e?3?0y?104

Using the Solver or Goal Seek tool in Excel, y?0?06407 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