Question

Assume a bond matures for $1000 ten years and two months from today. It also pays semiannual coupons with an annual coupon rate of 8%. Calculate the dirty (cash) and clean prices of the bond if the bond’s yield to maturity equals 9%. (Show Work)

Answer 1

Calculating Dirty price of the bond

Face Value = $1000, Annual coupon rate = 8%, Yield to maturity of bond = YTM = 9%

Since the bond pays coupon semi-annually

Semi annual coupon = (Coupon rate x par value) / 2 = (8% x 1000) / 2 = 80 / 2 = 40

Semi annual yield to maturity = r = Yield to maturity / 2 = 9% / 2 = 4.5%

Time to maturity = 10 years and two months

Using 30/360 day convention for bonds, there are 30 days in a month and 360 days in a year

Since the bond pays coupon semi annually, therefore

No of months between two coupon payments = 6

No of months since last coupon payment = 4

Days since last coupon payment = t = no of months since last coupon payment x 30 = 4 x 30 = 120 days

Days between two coupon payments = T = no of months between two coupon payment x 30 = 6 x 30 = 180 days

We know that Dirty Price = A (1+r)^t/T

where A = Present value at last coupon late of all future cash flows of bond

No of half years between last coupon date and maturity of bond = 21 half years

A = 40/(1+4.5%) + 40/(1+4.5%)² + 40/(1+4.5%)³ + ...................................+40/(1+4.5%)²⁰ + (40+1000)/(1+4.5%)²¹

We can find A by using PV function in excel

Formula to be used in excel: =PV(rate,nper,-pmt,-fv)

Using PV function in excel, we get present value at last coupon date of all future cash flows = A = 932.976380

Dirty Price = A (1+r)^t/T

Dirty price = 932.976380 x (1+4.5%)^120/180 = 932.976380 x (1.045)^120/180 = 932.976380 x 1.0297793 = 960.7597635 = 960.7598 (Rounded to four decimal places)

Hence Dirty price = 960.7598

Calculating Clean Price of Bond

Accrued interest = Semi Annual coupon payment x (t/T) = 40 x (120/180) = 26.6666666

Clean price = Dirty price - Accrued interest = 960.7597635 - 26.6666666 = 934.0930969 = 934.0931 (rounded to four decimal places)

Hence Clean price = 934.0931