Question

You are given a coupon-bond whose remaining term is 5 years with face value of 100$ and coupon rate of 5%, paid annually, with a first payment starting a year from now. Assume also that the annual yield is 6%.

a) Calculate the interest accrued as well as the dirty and clean bond prices
at times 0.5 and 1.6. Assume continuous compounding is used to model
the dirty price between coupon payments.
(b) Compute the dirty prices Bk?, Bk immediately prior and immediately after
the coupon payments in a table, and graph the dirty and clean price of
the bond over the remaining 5 year term.

Answer 1

For time = 0.5

Accrued interest = (Interest Rate*Face value)*(time till next coupon/Total time in coupon payment) = (0.05*100)*(0.5/1) = 2.50

Dirty Price = PV of all future cash Flows @ 6% annualized yield, compounded continuously:

Time Cash Flow PVF @ 6% PV
1              5.00         0.9704           4.85
2              5.00         0.9139           4.57
3              5.00         0.8607           4.30
4              5.00         0.8106           4.05
5          105.00         0.7634         80.15
Dirty Price         97.93

(For continuous compounding PVF, formula is PV = 1/e^(r*(T-t)), where:

r = rate per oeriod = 6% = 0.06
T= Time between two payments
t = time lapsed = 0.50 in the given case
Eg for year 1: PVF = 1/e^(0.06*(1-0.50)) = 0.9704

Clean Price = Dirty price - Accued Interest = 97.93-2.50 = $95.43

For t=1.6:

Accrued interest = (Interest Rate*Face value)*(time till next coupon/Total time in coupon payment) = (0.05*100)*(0.6/1) = 3.00

Dirty Price: Since, now the interest for year 1 would have already been received, the cash flows will incurr from year 2 onwards

Time Cash Flow PVF @ 6% PV
2              5.00         0.9763           4.88
3              5.00         0.9194           4.60
4              5.00         0.8659           4.33
5          105.00         0.8155         85.62
Dirty Price         99.43

(Eg for PVF: Year 2 = 1/e^(0.06*(2-1.60)) = 0.9763

Clean Price = Dirty price - Accued Interest = 99.43-3.00 = $96.43