Question

What is the clean price (invoice price) for a bond with face value of $1,000, annual coupon rate of 12%, semiannual payment, time to maturity of 17 years if the YTM is 14%? 12%? 10%? Respectively? What is the current yield for each of the three cases? If the last coupon is paid 60 days ago, what is the dirty price for each of three cases? (Assume 365 days a year)

Answer 1

Given Information:

FV = $1000

Coupon C = 12% semiannual payment = 0.12 *1000 = 120/2 = 60

Time to maturity = 17 years

Find the clean bond price if YTM = 14%, 12%, 10%

Bond price is PV of the future cash flows discounted by the discount rate. The discount rate to be considered is the YTM.

The cash flows are the semi-annual coupon and Face value at maturity. Since coupons are paid semi-annually, there will be a total of n = 17*2 = 34 payments. Since the coupons are paid 2 times a year, the ytm will have to be reduced to ytm/2 for calculation purposes.

Present Value = C/(1+ytm/2) + C/(1+ytm/2)2 + C/(1+ytm/2)³ + C/(1+ytm/2)⁴ + .... C/(1+ytm/2)³⁴ + FV/(1+ytm/2)³⁴  

By applying the geometric progression formula, we get

PV = C/(1+ytm/2) * [ 1/(1+ytm/2)ⁿ - 1 ] / [(1+ytm/2) - 1]

By simplifying we get,

PV = C*[ 1 - (1+ytm/2)^-34] / ytm/2 + FV/(1+ytm/2)³⁴

For YTM = 14%,

Clean price = PV = 60*[ 1 - (1+0.14/2)^-34] / 0.14/2 + 1000/(1+0.14/2)³⁴ = 871.4599

With YTM = 12%,

Clean price = PV = 60*[ 1 - (1+0.12/2)^-34] / 0.12/2 + 1000/(1+0.12/2)³⁴ = 1000

Since the coupon rate and YTM both are 12%, the FV and the clean price are exactly the same. This is also called as bond trading at par.

With YTM = 10%

Clean price = PV = 60*[ 1 - (1+0.10/2)^-34] / 0.10/2 + 1000/(1+0.10/2)³⁴ = 1161.929

What is the current yield of each of the three cases?

The yields are the same given as 14%, 12% and 10% respectively

If the last coupon is paid 60 days ago, what is the dirty price for each of three cases? (Assume 365 days a year)

We need to calculate accrued interest for each of the three cases and then add it to the clean price calculated above.

accrued interest = coupon * no of days passed/total number of days in period = 60*60/(365/2) = 19.726

Hence the dirty price with ytm = 14% is 871.4599 + 19.726 = 891.1859

the dirty price with ytm = 12% is 1000 + 19.726 = 1019.726

the dirty price with ytm = 10% is 1161.929 + 19.726 = 1181.655