# Misunderstanding of 'day counts' and accrued interest

I'm totally new to the fixed income world.

My goal with this question is to gain an understanding how interest is accrued day-by-day for a particular instrument. This will obviously be done by an app in production, not by hand; This question is purely to increase my own knowledge on the topic.

I have access to a Bloomberg terminal if you'd like to point me to any helpful screens that I may have missed.

The Question

Consider this fixed income instrument:

3130A3GE8
Federal Home Loan Bank
Maturity: 2024-12-13
Coupon:  2.75 % (CPN)
Previous Coupon Date: 2019-06-13
Next Coupon Date: 2019-12-13


My goal is to figure out how much to accrue each day on this product. My original approach was:

(1)Determine the coupon frequency (CPN_FREQ). It's 2.

(2)Divide the coupon by the coupon frequency to obtain the interest rate for the current coupon period. In our case we end up with 1.375 % (2.75 / 2). This is confirmed by consulting the Bloomberg mnemonic ACCRUED_CALC_INT_RT.

(3) Determine the number of days in the period. There are 183 days in the period (2019-06-13 => 2019-12-13).

(4)Divide the coupon period's interest rate by the number of days in the period to obtain the daily accrual rate. This is 0.00751366% (1.375/183).

This process seems pretty straightforward. What I don't understand is where the "day count" (DAY_CNT) convention is used. The day count for this item is "30/360." But where does this come into play? I've Googled it and it appears that the day count is used when computing accrued interest, that is, the amount of interest that's added on to your clean purchase price (or sale price) when you make a trade.

So presumably if you submit a trade to me and provide me with a dirty price, I should use the day count method (which is well documented) to compute the amount of interest to subtract so as to obtain the clean price for use in computing the capital gain/loss portion of the trade. But that presents a problem. I don't understand how we could have two differing methods for calculating accrued interest, both of which I found online. What happens if the two methods generate different values? How does this affect the dirty=>clean price calculation? Presumably my confusion is due to context; i.e. both methods are correct but are used in different situations? Please enlighten me.

Thanks,

Yippie McSmashmouth

## 2 Answers

What you've calculated is essentially ACT/ACT day count basis, since you use the actual number of days between the dates and the actual number of dates in the coupon period. With a 30/360 DCC, you treat each month as if it has 30 days, and that there are 360 days in a year (which means that there are 12 even interest periods).

So the calculation is done in two steps:

• Calculate the number of full months between the two dates, and multiply that by the interest rate divided by 12
• Calculate the number of remaining days after subtracting the months, divide it by 30 and multiply that times the interest rate divided by 12.

So the accrued interest from the previous coupon (2019-06-13) to today (2019-10-03), you'd take the 3 full months that have passed times the interest rate over 12, plus the 20 remaining days, divided by 30, times the interest rate/12. So the accrued interest for one $1,000 bond would be: AccrInt = 1,000 * 2.75%/12 * (3 + (20/30))$8.40


or

AccrInt = 1,000 * 2.75% * (30*3 + 20)/360 = $8.40  So you'd pay the "clean" price of the bond plus$8.40 in accrued interest.

• Got it. What would happen in a case where the 30/360 day count convention produced an accrued interest value LESS THAN what would be obtained using actual/actual? Wouldn't the purchaser of the instrument be paying slightly less for the accrued interest than he should, thereby getting some interest "for free"? Oct 3, 2019 at 19:43
• The price should reflect this difference as bond yields are calculated with a dirty price. Two otherwise equivalent bonds with different daycounts should have the same yield but different clean prices to account for the difference. Oct 3, 2019 at 21:33
• @Bond Wiz: Makes sense. Thank you! Oct 3, 2019 at 21:35

This is due to day count convention, among them 30/360, Actual/360, and Actual/Actual. Convention varies by market and product, so you need to be clear on conventions used for the instrument you're interested in to do the calculation by hand.