Act/365 Canadian Bond Accrual Rules and Their Edge Cases

# Act/365 (Canadian Bond) day count convention


# Act/365 (Canadian Bond) day count convention












Seemingly the only reference for the Act/365 (Canadian Bond) day count convention, that I've come across online, is the document from which I take the name of the convention: Canadian Conventions in Fixed Income Markets, which I'll refer to as CCFIM.

This day count convention is used with Canada Government Bonds and some Canadian corporate bonds. I guess it is an attempt to address this feature of government bonds and the corporate bonds that use the same structure:

> Interest The Bonds shall accrue interest from the issuance date (“Issue”) to the date immediately prior to the maturity date (“Maturity”), as specified in the Specific Terms, inclusively. In accordance with the Specific Terms, interest shall be paid on the specific dates (the “Coupon Payment Dates”) and at the rate per annum (the “Coupon Rate”) determined upon Issue, semi-annually, in arrears and in two equal payments in each year until Maturity. Where interest is payable for a period of less than six months, it shall be calculated on the basis of a 365-day year. Interest will cease to accrue on the Bonds on Maturity.

(From Legal Terms and Conditions for Government of Canada Domestic Nominal Bonds. Highlighting is my own.)

Those two highlighted conditions together are tricky to handle.

For regular coupon accrued interest, we get these formulas from section 2.2.2 of CCFIM for "fraction of a coupon period":

> Denoting the annual payment frequency (or number of coupon periods per year) as $f$, Act/365 (Canadian Bond) measures the fraction of a coupon period represented by a given number of days as follows: (i) If the number of days of interest accrual is less than the actual number of days in the coupon period: $$\text{Fraction of coupon period} = \frac{Days\cdot f}{365}$$ (ii) If the number of days of interest accrual exceeds $365/f$, or 182.5 days for a semi-annual pay bond: $$\text{Fraction of coupon period} = 1 - \left[\frac{DaysRemainingInPeriod\cdot f}{365}\right]$$ Where $DaysRemainingInPeriod$ is the actual number of days from Valuation Date to Next Coupon Date.

For settlement accrued interest, these formulas are applied in section 6.1 of CCFIM. Essentially you divide both formulas by $f$ to get the year fraction that will be applied to the annual coupon rate. (Dividing by $f$ converts the "fraction of coupon period" into a year fraction.)

(Irregular front and back stubs are dealt with separately but in a manner consistent with the regular coupon periods.)

There are several quirks with this definition, which I'll outline.

## Question

I am wondering whether this definition is "canonical"? Does anyone have any other sources that describe it? How have you handled this day count convention or seen it handled?

I'll provide an answer for what I do now given the CCFIM definition but I'm looking for others.

## Quirks

The way CCFIM defines Act/365 (Canadian Bond) there are some problems or, in my judgement, undesirable behaviours.

- In section 2.2.2 of CCFIM, conditions (i) and (ii) are not mutually exclusive. We could be on day 183 of a 184 day coupon period, with $f = 2$, so that $365 / f = 182.5$. Day 183 satisfies both conditions, and the two formulas do not give the same result. However section 6.1 perhaps corrects this probable error/oversight by distinguishing the two cases according to accrued days being less than or greater than $365 / f$. Since $f$ is meant to take on only values that are among positive integer divisors of 12, equality (number of accrued days is equal to $365 / f$) is only possible when $f = 1$. When $f = 1$ we have either a 365 day or a 366 day coupon period. If we're on day 365 of a 365 day period both formulas agree, so no issue. But if we're on day 365 of a 366 day coupon period, then formulas (i) and (ii) disagree, and using formula (ii) seems to me more consistent with the convention (in contrast to assigning zero accrued interest for the last day).

- Accrued interest is frequently discussed in a trade settlement context. The convention is that interest accrual for the settlement day itself "belongs to the buyer". Accrued interest from coupon period accrual start day to settlement day is typically calculated from beginning of day to beginning of day, that is, it includes the accrual start day and does not include the settlement day. When calculating accrued interest to beginning of day we won't need the case when the number of days of accrual is equal to the number of days in the period, since the number of days of accrual measured beginning of day to beginning of day is equal to the number of days in the period on the coupon pay date, and on that day accrued interest resets to zero. (Beginning of day accrued interest is reported as zero on that date.) Bloomberg's `INT_ACC` field shows accrued interest to beginning of day. But, sometimes we want end of day accrued interest instead. The way to handle this is to equate end of day accrued interest with beginning of day accrued interest the next calendar day, so add one day to the right endpoint and use the usual beginning of day to beginning of day formulas. (The approach chosen can make a noticeable difference in day count conventions for which the dates themselves are important, not just the number of days elapsed, such as 30/360 conventions.) For example, if we want to calculate yield to maturity using Bloomberg's end of day `PX_MID` for a vanilla fixed coupon bond, we'll need to add end of day accrued interest to get the dirty price, and then solve for the yield with that. (I've matched Bloomberg yields to 3 percentage decimal places - as far as Bloomberg shows them - with this approach, whereas using beginning of day accrued interest for that day will be off slightly.) Or we may need earned accrued interest to end of day for accounting purposes for a held position, not for trading purposes. For example on the first day of a new coupon period, there is 1 day of interest accrued at the end of the day, whereas standard beginning of day accrued interest reports zero. On the last day of the accrual period, at end of day, the entire coupon value should have accrued, and pays out on the next (business) day. A problem with the day count convention as described in CCFIM is that at end of day on the last day of a 181 day coupon period, the formulas don't support producing a value equal to the whole coupon. Here $f = 2$. At the beginning of day on the last accrual period day we've accrued 180 days, and the formula gives us a year fraction of $180 / 365$. At the end of the day we should have accrued the whole coupon, but neither of the conditions in formula (i) and formula (ii) of section 2.2.2 apply to 181 days of accrual in a 181 day period. If instead we use the conditions in section 6.1, formula (i) applies because $Days < 365/f = 182.5$, and we get a year fraction of $181 / 365$, which is too small. (It should be $182.5 / 365 = 1/2$.) (The same problem holds at the end of a 182 day coupon period.)

- Most days in a coupon period have an interest weight (i.e. year fraction) of $1/365$, but at the end of a coupon period days can take on quite different year fractions. (30/360 day count conventions do similar things, with the unusual year fractions occurring at Feb. and 31 day month ends, and the rest having $1/360$ year fractions.) In the example of a 181 day coupon period above, the last day must have a year fraction of $2.5 / 365$ in order to accrue the entire coupon by the end of the period. In a 184 day coupon period, accrued interest decreases from 182 to 183 days. At 182 days we have a year fraction from the period accrual start of $182 / 365 \approx 0.49863$ (formula (i) from 6.1), but at 183 days we have $1/2 - 1/365 \approx 0.49726$ (formula (ii) from 6.1). So day 183 has a negative year fraction. An example of this can be observed on Bloomberg: the Canadian corporate (Rogers) bond with ISIN CA775109CT61. The coupon period of interest is 2023-05-09 to 2023-11-09, a period of 184 days. Bloomberg reports (on a face value of 1000) beginning of day accrued interest of 33.47 for 2023-11-06; 33.66 for 2023-11-07; 33.57 for 2023-11-08; and 0 for 2023-11-09. The coupon is 33.75.

## How I'm handling it now

In order to address the quirks 1 and 2, (3 is unavoidable with the given formulas,) I think the answer is to combine the condition on formula (i) from section 2.2.2 with the condition on formula (i) from section 6.1 and to include equality in formula (ii), to arrive at these conditions (the formulas stay the same):

i. If the number of days of interest accrual is less than the actual number of days in the coupon period, and less than $365 / f$, then use the year fraction formula $$\frac{Days}{365}.$$

ii. If the number of days of interest accrual equals or exceeds $365/f$, or is equal to the actual number of days in the coupon period, then use the year fraction formula $$\frac{1}{f} - \frac{DaysRemainingInPeriod}{365}.$$

Then there is no problem calculating end of day accrued interest in the way I normally approach it. When the number of days of accrual is equal to the length of the coupon period, the year fraction is just $1/f$ (i.e. $DaysRemainingInPeriod = 0$). When $f = 1$ and the coupon period has 366 days, the accrued interest for 365 days uses a consistent approach. The two conditions are mutually exclusive.

The possibility of an accrued interest decrease corresponding to an accrual days increase seems generally undesirable to me. I believe it can happen only with $f = 2$, but semi-annual coupons are very common.

I wish we could all agree on a single day count convention for a security type. There is way too much effort that goes into understanding and implementing all of these details.