Question:

A $1,000 par, 4% coupon bond yields 8% and matures on 7/1/2010. It pays coupons on January and July 1st but SETTLES on June 1st, 2008.

(i) Compute the bond’s full (a.k.a., invoice, dirty) price on settlement. (two ways to compute this, bonus for doing it both ways!)
(ii) What is the accrued interest?
(iii) What is the bond’s clean price?

Answer:

Please click this link for the full set of calculations.

(i)
The “long way” is to discount the future cash flows. But we must be careful to discount based on fractional periods; e.g., the 5th and final cash flow is figured as follows:

PV [principal + final coupon] = $1020/[(1+8%/2)^(5-1+0.17),
where 0.17 = 30/180 since there are approximately 30 days to the next coupon.
Note: a different DAY COUNT CONVENTION will give a slighly different result.

The “short way” is to compute the value of the bond at the previous coupon and then compound that value forward to settlement.

Value on 1/1/2008 = $910.96 = PV (8%/2, 2.5 years * 2, $1000 * 4% / 2, $1000)
Value on 6/1/2008 = $910.96 * (1 + 8%/2)^(1-0.17) = $941.23

(ii)
Accrued interest = $20 coupon * (1 - 0.17) where 0.17 = 30 days/180 days

(iii)
Clean price = Full (Dirty) Price - Accrued Interest
$924.56 = $941.23 - $16.67

David,

How does this work? Using the short way, we are compounding forward using the YEM of the bond.  it does not take into account the coupon. I get the same numbers using the HP12c and using the bond function and entering in the dates but I can’t understand your calculation.  I used 4% and not 8% when I tried the short way and was wrong.

Frank

Hi Frank,

It impresses me, too. But if you take the XLS, it works under any scenario!

The YTM is an internal rate of return (IRR): it includes both sources of return, the price appreciation and the coupon. As such, you’ve got to PV to get the coupon implicitly included. David