A semiannual 4% bond on a 30/360 basis last paid on 15 March and settles on 20 June at a clean price of 98.50. Compute the accrued interest and the invoice amount on $2m of face, and say why the clean price does not jump on a coupon date.

A semiannual 4% bond on a 30/360 basis last paid on 15 March and settles on 20 June at a clean price of 98.50. Compute the accrued interest and the invoice amount on $2m of face, and say why the clean price does not jump on a coupon date.

Approach: Count the accrual days under the 30/360 convention, apply them to the semiannual coupon, add the accrued to the clean price and scale the result to face.

$1,991,111. Under 30/360 the accrual from 15 March to 20 June is 95 days, so accrued interest is 2.00 * 95/180, which is 1.0556 per 100. The dirty price is 98.50 + 1.0556, or 99.5556, and on $2m of face the invoice is 2,000,000 * 0.995556, which is $1,991,111. The clean price does not jump on a coupon date because accrued interest is stripped out of it by construction: the dirty price falls by the coupon and the accrued resets to zero on the same day, so the two moves cancel. Quoting clean is what makes a price series comparable through time, since a dirty series would show a sawtooth with the day count convention baked into it.

Follow-up: The same bond is quoted on an actual/actual basis instead. How does the accrued change and which convention favours the seller here?

Key concepts: accrued interest, clean price, dirty price, day count convention.