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Thursday, December 21, 2006

How to calculate accrued interest

Looking at the logs from this website, I see that every day at least a couple people land here looking for how to calculate accrued interest on some interest bearing instrument. Since it’s a slow holiday week, I thought why not actually post the calculation? So the following is a little bond 101.

What is accrued interest?

Bonds generally pay interest every six-months. The interest payment is called a coupon. Whoever is on record as holding the bond on the coupon paying date receives the entire coupon payment regardless of whether they've held the bond for 1 day or for the whole six months.

If that were the end of the story, bonds would gyrate in price based on how close to a coupon date you are, creating all kinds of distortions in the market. To prevent this, bonds trade with accrued interest. Any time a bond trades, the buyer pays the seller a fraction of the upcoming coupon payment, with the fraction being equal to the fraction of the coupon period which has already passed.

How to calculate accrued interest

In order to calculate accrued interest, you must first know what day count fraction (DCF) is to be used. The most common is 30/360, which means that each month is assumed to be 30 days long, and the year is assumed to be 360 days. So if 15 days have passed since the most recent coupon paying day, the accrued interest on a

5% coupon, semi-annual paying bond would be…

15/360 * 5%

That’s 15 days out of a 360 day year, so the fraction of coupon earned is about 4.17% of 5%.

Note that because we've assumed a 30 day month, any time there is a 31st, no interest accrues. Also at the end of February, its possible to have more than 1 day accrued between the 28th or 29th and March 1.

Treasury bonds are done on an actual/actual (sometimes noted as ACT/ACT) basis, which simply means you take the actual number of days that have past and the actual number of days in the year when calculating the fraction.

There are also some conventions where the divisor is 365, which works just like the ACT calculation except leap years are ignored.

Why does the 30/360 convention exist? I've heard different stories, but one reason is that it makes various couponing periods easy to calculate. You can do monthly, quarterly, or semi-annual couponing easily, because 360 divides evenly into 12, 4, or 2. You’d never run into a problem where one period is actually longer than another, resulting in more accrued interest being paid than the coupon! The actual/actual DCF doesn't have that advantage. The 30/360 convention is also easy to calculate by hand, which before the days of Bloomberg was probably helpful.

I’m thinking of running these Bond 101 posts from time to time, so if anyone reading this is interested in the definition of something bond related, please post a comment.

Category: Forex

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