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.

39 comments:

  1. That does not seem right. Shouldn't it be [(5%/360)*15] or put your way [(15/180)*(5%/2)]?

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  2. Wow. You are right. I should resign from blogger immediately. I can't believe I did that!

    ReplyDelete
  3. don't resign !!!

    thanks for this great blog , you do fantastic work and it's appreciated

    Happy Holidays !!!

    JJ

    ReplyDelete
  4. How about for a foreign bond? Would you simply multiply the accrued interest in the local currency times the exchange rate? If so, if the exchange rate went against you for a period of time, you could actually have a negative accrual, which is funky.

    ReplyDelete
  5. I think the order of operations would be as follows:

    1) You get accrued interest in the foreign currency (yen, euros, whatever).

    2) You convert the accrued into USD.

    3) You record the net P/L on the whole trade.

    Because #1 is distinct from #2, you don't actually have a negative accrued even though you might have a net loss. Its not unlike the fact that interest rates could rise on a USD bond and you wind up with a loss. Doesn't mean you have negative accrued.

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  6. I think you post your gain or loss into foreign currency translation adjustments (FCTA) and leave your liabilities, etc. on the books at nominal values. Let's ask an accountant.

    ReplyDelete
  7. Thanks for the Bonds 101 post - just found your great blog.

    What I would love to hear you talk about

    o Bond laddering techniques
    o Muni/corporate/agency/treasury market basics
    o Retail buying - where, how ...
    o Hedging for retail investors

    Thanks!

    ReplyDelete
  8. I can post about some of that stuff, sure. What do you mean by "retail buying?"

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  9. If it is Actual/Actual and semi-annual coupon paid in May and Nov.
    Sould I use 366 for both 5/2008 and 11/2008 paymnets? Or just the May?

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  10. If something paid interest on 5/15 and 11/15, On 5/14 there would be 181 days of accrued. On 11/14, there would be 183. The coupon is the same for each period. In the second period, each day of accrued is worth slightly less.

    ReplyDelete
  11. i need help with a problem; probably so easy but i want to make sure it is correct

    200,000 par val, 3% fixed coupon rate paying semi annualy and a 30/360 accrual; what is 1 day's interest? single coupong payment?

    ReplyDelete
  12. 200,000 * 0.03 = 6,000

    6,000 / 360 = 16.67

    So that's 1-days interest on almost every day. The exceptions are any 31st of a month there is no accrued and between 2/28 and 3/1 there is 3 days.

    ReplyDelete
  13. That does not seem right. Shouldn't it be [(5%/360)*15] or put your way [(15/180)*(5%/2)]?

    9:11 PM

    Why not??? (15/180)*(5%/2) is the same as 15/360*5% - it is just the way you are looking at it. One is annualized and the other is semi-annual - since the coupon rates are annual rates then by dividing by 2 u r converting it to semi......

    ReplyDelete
  14. This is a phenomenal blog! Your post on how a CDS works was the clearest and most complete explanation of structured finance that I've come across so far. I just finished a mid-career MBA, and I can tell you that most of the finance instruction is not as good as yours.

    Particularly important to my understanding of the issues, was how, after you laid out the simple example, you explained how the rocket scientists had complicated the securities so that no one could tell who was ging to get paid when conditions changed.

    This is great stuff. Please keep up the good work. I'm definitely bookmarking your site.

    ReplyDelete
  15. Thanks and welcome to the blog.

    ReplyDelete
  16. Hi. I was wondering if you could help me out. There seems to be a slightly different convention when computing for first interest payment for US Treasuries and I was hoping you could possibly give me an explanation. When is there a "notional coupon period" and does it apply for all first interest payments?

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  17. Often the first couponing period is longer or shorter than 6-months. Is that what you're talking about?

    ReplyDelete
  18. hi. if a treasury note is issued and has a coupon of 4.625%, first accrual date of march 31, 2006 and a maturity of march 31, 2008 and it pays coupons every march 31 and september 30. how would you compute for the first interest payment value? there are 183 days in between march 31 and sept. 30. would the computation be (4.625%/2)*(183/183) in which case it would be 2.3125% or would it be (4.625%/2)*(183/184)?

    ReplyDelete
  19. I find that The Accrued interest is negative at 5th March 2008 for a bond dated 7th March 2012
    I downloaded data from UK DMO website
    The data is dead right as the site is a goverment one

    Could you please explain the negative Accrued int rate.

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  20. Gilts have negative Accrued Interst from ex-date until the next payment date.

    ReplyDelete
  21. What if you have a regular, plain-vanilla, bond that has a last coupon, on the maturity date, that has a different amount (call it A) than a regular coupon?

    If the method is ACT/ACT I guess it's more or less straight-forward, you could do A * (days since last coupon) / (days in period between last coupon and final special coupon).

    But what if you're using ACT/360?

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  22. Not sure what you mean? Like where the last couponing period is less than 6 months?

    ReplyDelete
  23. What I really want to know is if there is a correct (standardized) way to calculate accrued interest when there is no concept of "annual coupon" and you only have a series of payments with arbitrary dates and amounts?

    For example, given the following payments (incomes) on a security:

    2009-04-31: $10
    2009-10-02: $5
    2010-01-26: $15
    2010-04-01: $100, maturity date

    what is the accrued interest on 2009-11-26 for example, given a certain day count convention? It should be some fraction of $15, but what is that fraction? If using ACT/360 for example you cannot divide the number of days between 2009-10-02 and 2010-01-26 with 360 since that would result in a too small amount.

    ReplyDelete
  24. *****
    If something paid interest on 5/15 and 11/15, On 5/14 there would be 181 days of accrued. On 11/14, there would be 183. The coupon is the same for each period. In the second period, each day of accrued is worth slightly less.

    What in the answer for this? and is there an excel model to calcualte the accrued interest at anytime?

    ReplyDelete
  25. Joseph:

    Every 31st day of a month doesn't earn any accrued on a 30/360 bond.

    ReplyDelete
  26. There are several industry-standard methods. Even English and European methods are different, and sometimes one company would issue different bonds with different methods: see
    http://hubpages.com/hub/CalculatingDayCountforAccruedInterestandMarketValues

    ReplyDelete
  27. can anyone tell me the accounting or IFRS standards for calculation of accrued interest on bonds for valuation purposes. Should it be till the valuation date or till the settlement date as per market?

    ReplyDelete
  28. Great blog! Just found it today. Are you aware of any standards for calculating yield for a callable bond when the call date falls on a non-business date? Should you calculate the yield to the actual day (despite the non-business day status) or should you calculate to the next business day?

    ReplyDelete
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  32. Hi

    Please let me know the difference in the interest calculation method 30/360 for US and Europe.

    Thnks :))

    ReplyDelete
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    ReplyDelete
  34. Check out this online accrued interest calculator:

    http://knowpapa.com/acint/

    Calculates for many popular date conventions in the bond market

    ReplyDelete
  35. I work with an UK based fund administration major , where in we do Bond Interest Reconciliations for an European Hedge Fund. I found this blog extremely helpful, as we do calculate Accrued Interests on bonds while investigating breaks. Thanks to the blogger. Great post.

    ReplyDelete

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