Question

Q3. (1) From the following data obtain the discount factors for T=0.25, 0.5, 0.75, and 1:

• A zero coupon bond Pz(0, 0.5) = 99.20. • A coupon bond paying 3% quarterly P(0, 0.25) = 100.5485.

• A coupon bond paying 6% quarterly P(0, 0.75) = 103.1655.

• A coupon bond paying 5% semiannually P(0, 1) = 103.0325.

(2) Using the previous discount curve to price the following:

• A 6-month coupon bond paying 7% semiannually.

• A 1-year coupon bond paying 4% quarterly

Answer 1

The Discount Rate, i%, used in the discount factor formulas is the effective rate per period. It uses the same basis for the period (annual, monthly, etc.) as used for the number of periods, n. If only a nominal interest rate (rate per annum or rate per year) is known, you can calculate the discount rate using the following formula:

where
• r = nominal annual interest rate
• k = number of compounding periods per year
• p = number of periods per year corresponding to the basis for n

This formula for the effective rate per period is more general than the formula used in the Excel functions EFFECT and NOMINAL. The EFFECT and NOMINAL functions are only used for converting between the effective and nominal annual rates, where p=1.

A coupon bond paying 3% quarterly P (0, 0.25) = 100.5485.

I= [1+3/0.25) 0.25/4-1

I= 0.8885

A coupon bond paying 6% quarterly P(0, 0.75) = 103.1655..

I= [1+6/0.75) 0.75/4-1

I= 0.7921

• A coupon bond paying 5% semiannually P(0, 1) = 103.0325

I= [1+5/1) 1/2-1

I= 0.9070

2)

Considering that the bond price is higher than the par value the bond should be selling at a premium.

Assume the face value of a bond =100

Coupon amount = 100*7% =7

Year     Coupon            Discount rate

0          1                    1/0.8885

0.5          7                   1/0.8885

Bond price = $106.07

Assume the face value of a bond =100

Coupon amount = 100*4% =4

Year     Coupon            Discount rate

0          1                    1/0.8885

1             4                   1/0.8885

Bond price = $103.09