Day count fraction

Six months is not always half a year. It might be 181 days out of 365, or 182 out of 366, or — if the market has agreed to pretend every month has 30 days — exactly 180 out of 360.

A day count fraction is the convention that settles the argument. The numerator says how the days in the payment period are counted; the denominator says how many days the year is deemed to contain. Nothing about it is derived from first principles. It is a market agreement, and different markets agreed differently.

One rule is universal: include the first day of the period, exclude the last. From 30 April to 5 May is one day in April and four in May — five days, not six.

India uses four conventions, each in its own corner of the market:

Actual/Actual counts the real days between the last coupon date and the next, which handles a leap year correctly.

30/360 European treats every month as 30 days and the year as 360. Where a date falls on the 31st, it is reset to the 30th — for both the start date and the end date. That one adjustment is where marks are lost.

Actual/365 and Actual/360 both count real days but divide by a fixed year, so Actual/360 produces a slightly larger fraction than Actual/365 over the same calendar period — which is exactly why swap markets adopted it.

For 30/360 European:

                  [360 × (Y2 − Y1)] + [30 × (M2 − M1)] + (D2 − D1)
Day count fraction = ──────────────────────────────────────────────────
                                        360

   D1 = start day, reset to 30 if it would be 31
   D2 = end day,   reset to 30 if it would be 31

And the use it is put to:

Accrued interest = Periodic coupon × Day count fraction of the elapsed period

The 31st rule, worked. An accrual period running from 31-Dec-2020 to 25-Jan-2021. The start day is 31, so it is shortened to 30; the end day, 25, needs no adjustment.

[360 × (2021 − 2020)] + [30 × (1 − 12)] + (25 − 30)
= 360 + (−330) + (−5)
= 25

Day count fraction = 25 ÷ 360

Twenty-five days, not twenty-six. The shortening of the 31st removed one.

What the choice costs in rupees. The same 6.90% semi-annual bond, previous coupon 13-Jan, next coupon 13-Jul, settling 5 March:

A gap of Rs 0.024568 per Rs 100 face. On a wholesale debt market lot of Rs 5 crore that is 5,00,000 units × 0.024568 = Rs 12,284; on a Rs 100 crore institutional block it is Rs 2,45,680, settling to the wrong side of the trade if the wrong convention is applied.

A money market example. A 91-day T-Bill bought at Rs 99.6898 with 34 days to run, on the Actual/365 basis Indian money markets use:

Bond equivalent yield = [(100 − 99.6898) ÷ 99.6898] × (365 ÷ 34) = 3.34%

Run the same trade on a 360-day discount basis instead and the number changes, because the year changed — which is precisely why 91-day T-Bill futures quote off a discount yield and not the bond equivalent yield.

Chapter 1, section 1.7.4 (Day count fraction), lists the four conventions, gives the 30/360 European formula and works the 31-December adjustment. Everything downstream in the paper inherits it: accrued interest in section 1.8, the money market yields in section 1.10.2.3, the futures financing cost in Chapter 3, and the Rs 411 basis point value of an Overnight MIBOR futures contract, which is a 30/365 accrual on Rs 5 crore.

Questions are direct recall — which convention applies to Indian G-Secs, to corporate bonds, to the money market, to swaps — and the day count arithmetic itself, usually with a 31st in it.

• Learn which market takes which convention. 30/360 European for G-Secs and Actual/Actual for corporate bonds is the pair most often swapped round in answers.
• The 31st is reset to 30 at both ends under 30/360 European, and the reset is what changes the answer.
• Include the first day, exclude the last. Counting both ends gives one day too many on every calculation you do.
• Actual/360 is not Actual/365. Same numerator, different denominator; the swap market and the money market genuinely disagree.
• The denominator can be the full coupon period, not a calendar year. For a semi-annual bond on Actual/Actual it is 181 or 184 days, not 365.
• One geography can run several conventions at once. India uses different bases for its money market and its bond market, and the workbook says so explicitly.