Bond Payment Date Adjustments, Accrued Interest, and Yield Conventions

# 76150


# Does the rolling of bond payments from non-business days to the next or previous business day affect the calculation of accrued interest and YTM?












(1) Does the rolling of bond payment from non-business days to the next or previous business day affect the coupon payment and the accrued interests within the coupon period? In other words, are the `adjusted` (bumped) or the `unadjusted` (unbumped) payment days used in the calculations?

(2) Is there a correspondence between the use of `adjusted` or `unadjusted` payment days and the different day-count conventions?

(3) Which conventions tend to be used for fixed-rate bonds and which for floaters?

(4) Are `unadjusted` or `adjusted` payment dates used in standard publicized YTMs when the day-count convention for their calculation is actual/365?

Tentative Answer (using the information from @Dimitri Vulis's answer, comments by @LuigiBallabio to another question, and other sources)

(1) We compute coupon payments and the accrued interests within the coupon periods using the same convention for payment dates, either `adjusted` or `unadjusted`. HKD and RMB bonds mentioned below are exceptions.

(2) There is some correspondence.

We use `unadjusted` payment dates in conjunction with

```
(a) the 30/360 convention**s** and

(b) actual/actual (ICMA) convention.
```

By definition, the value of a regular coupon with convention (b) is

```
Principal × Rate / Frequency,
```

Therefore, all regular coupon payments of a bullet bond are equal. The accrued interest within each coupon period is computed with the Actual_accrued_days/Actual_days_in_period rule.

Under the 30/360 conventions, regular coupons are also constants satisfying the previous formula. There might be rare exceptions like the following example: end-of-month rule; the payment date of the previous coupon is the last day of February; the current coupon's payment date is August 31st; the day count convention is 30/360 European or 30-360 Bond Basis, which do not adjust the last day of February to a notional 30th. Issuers can easily avoid this oddity by setting a fixed payment day that is available in all payment months, e.g., days 1 to 28 if February is one of the payment months.

Using unadjusted dates maintains or reinforces the simplicity pursued by the 30/360 convention. Using adjusted dates would lead to differences between regular coupon payments, thus breaking the simplicity intended by the convention.

Using unadjusted dates reinforces the simplicity of the Actual/Actual (ICMA). Furthermore, using unadjusted dates tends to decrease the variability of daily accrued interest across coupon periods. The following is a quote from the "ICMA Primary Market Handbook, Appendix A5", Day count fraction: ICMA Actual/Actual (March 2022):

> With a few notable exceptions (such as HKD and RMB bonds), a fixed-rate bond interest period (and so the related accrued interest amount) is not generally adjusted due to a scheduled interest payment day falling on a non-business day. (The interest amount is just paid on the next following business day without adjustment.) The ICMA Actual/Actual day count fraction has thus not been specifically envisaged to operate in the context of adjusted interest periods.

We use `adjusted` payment dates in conjunction with

- actual/fixed_base conventions (Actual/360, Actual/365, etc.)

except in a few isolated cases among fixed-rate bonds.

I do not have information about the prevalence of the Adjusted or the Unadjusted convention with non-ICMA actual/actual conventions. Actually, I do not know if there are still bonds using these conventions nowadays. The actual/actual ISDA convention is still used in the SWAP market but is losing ground to the ICMA version.

(3) (i) Floaters use an actual/fixed_base convention with `adjusted` dates. (ii) Most fixed-rate bonds use `unadjusted` dates with actual/actual ICMA or 30/360 conventions. (iii) A few fixed-rate bonds use an actual/fixed_base convention, mostly with `adjusted` dates.

(4) The use of the Adjusted convention for YTM (actual/365) is the natural choice whenever the accrued interests follow the actual/fixed_base convention with Adjusted dates. Otherwise, the YTM would be distorted: the rolling forward of a payment date could increase the coupon value without having it discounted for the extra day(s) or vice-versa.

If interest payments are computed with Unadjusted dates, then either Adjusted or Unadjusted dates could be used to compute YTM. On the one hand, `unadjusted` dates for YTM would match the convention for coupons and avoid the computation of adjusted dates. On the other hand, `adjusted` dates for YTM would be more precise and consistent with the case of the previous paragraph.

The so-called "true yield" uses the actual/365 convention with adjusted dates. (See Donald J. Smith's Bond Math, 2nd Ed., Wiley (2014), pp. 50-51, also available at https://ebrary.net/14260/economics/yield_statistics)

The "street convention" for calculating YTM uses unadjusted payment dates and the ICMA actual/actual convention or 30/360 conventions, according to which is used for interest accrual.

While the "true yield" is more precise, the "street convention" is more efficient because it allows the use of annuity formulas with all regular coupons and does not require the computation of adjusted days.

A variation of the "street convention" for bonds using the 30/360 convention for interest accrual is the "government equivalent" yield, which uses the ICMA actual/actual convention for discounting (while keeping the 30/360 convention to compute accrued interest so as to get the correct full/dirty price). This convention is used to compute spreads between 30/360 corporate bonds to ICMA actual/actual government bonds.

The annual effective rate (AER) equivalent to the "street convention" yield and, even more, the AER equivalent to the "government equivalent" yield are very close to the "true yield". The "true yield" is slightly lower than the AER corresponding to the "government equivalent" yield. To see why, consider the time to a payment in the computation of the "true yield" as the fraction actual/365. On average, the numerator will be bigger because the "following business day" adjustment applies to the "true yield" but not the "street convention". On average, 365 is lower than the denominator under the government convention because of leap years. Because of these two effects, the fraction used to measure the years to any payment is bigger in the "true yield" calculation. To compensate so as to obtain the same price, the yield has to be lower.

Please complete and correct the above. Thanks!

## Answer by Dimitri Vulis (score 7)

https://quant.stackexchange.com/a/76151

It all depends on the individual bond/loan. Read the bond prospectus.

Fixed $C\%$ per year, coupon frequency $n$, usually means exactly $C/n$ regular coupons. The daycounting convention is used if the bond is traded in the middle of a coupon period, so the accrued partial coupon is daycounted. If the first or last coupon periods are long / short / stub, they are pro-rated or daycounted. However, for some products, all coupons are daycounted, so e.g. with Actual/360, they pay a few days' more than $C$ per year. And in some markets, e.g. Brazil BNTNs, the coupons are something like $\sqrt[n]{1+C}-1$, so the bond only pays the quoted $C$ after compounding.

Usually, for fixed-coupon bonds, if the payment date falls on a weekend or local holiday, then the date is bumped using the "modified following" rule, but the coupon amount is not daycounted. But you should not assume this behavior, because there are exceptions. For example, fixed-coupon Mexican government bonds (MBONOs) - the date is bumped backwards, from Thursday to Wednesday, and the coupon amounts are daycounted.

The coupons are usually daycounted for swap legs, including fixed-coupon, and for loans and loan products, such as participation notes, again including fixed-coupon.

EU bureaucrats have done a good job regularizing bond market conventions in their markets. In the rest of the world, there's much illogical legacy / traditional complexity that benefits no one.

## Answer by DavidJN (score 0)

https://quant.stackexchange.com/a/85171

For Treasury bonds, coupon payments are not adjusted if they fall on non-business days, and the payment date will use the Following or "next good business day" roll. In conventional bond yield-to-maturity based math, the day count impacts the treatment of the current coupon period only, and subsequent coupon periods are counted as whole integer values. The Bloomberg app offers a "true yield" calculation that adjusts all bond cash flow dates to good business dates. The US government Treasury website (is it Treasury Direct?) has downloadable pdfs describing the terms and conditions associate with its liabilities.