Actual/Actual ISMA Stub Coupons in Fixed-Rate Bonds

# Calculation of Cashflows Using ISMA Day Count in Fixed-Rate Bond


# Calculation of Cashflows Using ISMA Day Count in Fixed-Rate Bond












I'm working with a fixed-rate bond in QuantLib, and I have set the day count convention to ISMA, but I would like to understand how this specific day count convention is used in the calculation of the bond's first cashflow I have gone through the link https://www.isda.org/a/pIJEE/The-Actual-Actual-Day-Count-Fraction-1999.pdf still not able to understand how quantlib is calculating the first short period or long period cashflow. Here are the relevant code and cashflows:

```
issue_date = ql.Date(1, 1, 2021)
maturity_date = ql.Date(15, 1, 2022)
stubdate = ql.Date(28, 2, 2021)
coupon_rate = 8.40 / 100
face_value = 1000000
calendar = ql.NullCalendar()
day_count = ql.ActualActual(ql.ActualActual.ISMA)
compounding = ql.Simple
payfreq = ql.Monthly
payment_schedule = ql.Schedule(issue_date, maturity_date, ql.Period(payfreq), calendar, ql.Unadjusted, ql.Unadjusted, ql.DateGeneration.Forward, True, stubdate)
lst_pysch = list(payment_schedule)
lst_pysch.pop(0)
lst_pysch.insert(0, issue_date)
new_paysch = ql.Schedule(lst_pysch)

fixedrate_leg = ql.FixedRateLeg(
            schedule=new_paysch, dayCount=day_count, nominals=[face_value], couponRates=[coupon_rate])

bond = ql.Bond(0, calendar, 100.0, maturity_date, issue_date, fixedrate_leg)

[(a.date(), a.amount()) for a in fixedrate_leg]
```

Output:

```
[(Date(28, 2, 2021), 14000.000000000013),
 (Date(31, 3, 2021), 6999.999999999895),
 ...
 (Date(15, 1, 2022), 3452.054794520487)]
```

My specific question is, how is the first cashflow value of 14,000 calculated? I would like to understand the logic and calculations behind this particular cashflow.

Thank you for your help!

## Answer by Luigi Ballabio (score 4, accepted)

https://quant.stackexchange.com/a/76775

Calculate the schedule first, and then create the day counter as:

```
day_count = ql.ActualActual(ql.ActualActual.ISMA, payment_schedule)
```

this allows the act/act instance to have all the information it needs to perform the calculations so well detailed by @Attack68. After this change, the output of your code is:

```
[(Date(28,2,2021), 13774.193548387047),
 (Date(31,3,2021), 6999.999999999895),
 (Date(30,4,2021), 6999.999999999895),
 (Date(31,5,2021), 6999.999999999895),
 (Date(30,6,2021), 6999.999999999895),
 (Date(31,7,2021), 6999.999999999895),
 (Date(31,8,2021), 6999.999999999895),
 (Date(30,9,2021), 6999.999999999895),
 (Date(31,10,2021), 6999.999999999895),
 (Date(30,11,2021), 6999.999999999895),
 (Date(31,12,2021), 6999.999999999895),
 (Date(15,1,2022), 3387.0967741935765)]
```

## Answer by Attack68 (score 5)

https://quant.stackexchange.com/a/76774

I cannot speak for `Quantlib`, I do not believe that calculation is correct actually.

But I can tell you how this calculation is performed in `Rateslib`.

```
from rateslib import *

bond = FixedRateBond(
    effective=dt(2021, 1, 1),
    termination=dt(2022, 1, 15),
    front_stub=dt(2021, 2, 28),
    stub="LongFrontShortBack",
    roll=31,
    convention="ActActICMA",
    modifier="none",
    frequency="M",
    fixed_rate=8.4
)
curve = Curve({dt(2021, 1, 1): 1.0, dt(2022, 12, 31): 0.96})
bond.cashflows(curve)
```

In this case the first period is identified as a `Stub` under `ActActICMA` mode and the calculation proceeds as follows:

i) Subtract a regular period from 28-feb-21 to get to 31-Jan-21.

ii) The start is less than 31-Jan-21 so it must be a LongStub. Set `fraction=1.0`.

iii) Subtract another regular period from 31-Jan-21 to get to 31-Dec-20.

iv) 1-Jan-21 is after this date so we add the following amount to `fraction`:

$$ \frac{31/Jan/21 - 1/Jan/21}{31/Jan/21 - 31/Dec/21} = 0.967741935483871 $$

v) `fraction = 1.967741935483871` and `coupon=8.4% / 12 * fraction * 1mm = 13,774.19 `

### Other Factors

#### Roll day

I observed from your bond schedule that the roll was end-of-month (31) so this has been added to generate the schedule. If the roll was `28` this would change the above calculations. Nothing else about the bond needs to change except the `roll` and this will produce different cashflows in the front and back stubs.

```
other_bond = FixedRateBond(
    effective=dt(2021, 1, 1),
    termination=dt(2022, 1, 15),
    front_stub=dt(2021, 2, 28),
    stub="LONGFRONTSHORTBACK",
    roll=28,
    fixed_rate=8.4,
    frequency="M",
    convention="ActActICMA",
)
other_bond.cashflows(curve)
```

#### Calculation Mode Convention

Some conventions, such as Canadian Government Bonds work differently with ActActICMA. Under this calculation any identified Stub period adopts an Act365F calculation in the fractional part. `Rateslib` has a specifically named day count convention for these bonds: 'ActActICMA_stub365f':

```
another_bond = FixedRateBond(
    effective=dt(2021, 1, 1),
    termination=dt(2022, 1, 15),
    front_stub=dt(2021, 2, 28),
    stub="LONGFRONTSHORTBACK",
    roll=31,
    fixed_rate=8.4,
    frequency="M",
    convention="ActActICMA_stub365f",
)
another_bond.cashflows(curve)
```

In this latter case the fraction is assessed as: $$ fraction = 1.0 + \frac{31/Jan/21 - 1/Jan/21}{365} * 12 = 1.986301369.. $$ And the cashflow is calculated as: $$ 1mm * 8.4\% / 12 * fraction = 13,904.11 $$

### QuantLib Speculation

If you assert that `quantlib` is using something similar to the "ActActICMA_stub365f" calculation but is using an 'Act360' convention instead of 'ACT365f' convention then the first stub cashflow is actually very close to \$14,000.