When Floating-Leg Day-Count Fractions Do Not Cancel
Summary
The document explains why a floating swap leg does not always become independent of day-count conventions, even when the accrual fraction appears to cancel between the coupon amount and the forward rate. That cancellation requires the day-count fraction used to calculate the index rate to match the fraction used to accrue the swap coupon.
It gives several cases where the fractions differ: an IMM swap period referencing three-month Euribor, a floating rate with a simply compounded spread, and a customized leg whose accrual convention differs from the index convention. In each case, the coupon’s accrual fraction remains relevant to valuation. The explanation is conceptual and uses an illustrative calendar example; it does not cover every floating-rate convention or provide a full pricing model.
Key ideas
- The apparent cancellation depends on matching the index rate’s day-count fraction with the coupon accrual fraction.
- A mismatch between the swap period dates and index dates leaves the coupon accrual factor relevant.
- A simply compounded spread prevents the day-count factor from cancelling completely.
- Customized legs can also retain day-count dependence when their convention differs from the underlying index.
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Full text
# When are daycounts needed for the floating leg of a swap?
# When are daycounts needed for the floating leg of a swap?
For the fixed leg, the daycount is needed since the PV is $\sum_t N\delta_{t}r DF_t$ where $\delta_t$ is the daycount accrual factor at time $t$.
However for the floating leg it looks like $\sum_t N \delta_t F_t DF_t$ where $F_t = (P(t-1)/P(t) - 1)/\delta_t$. Here the daycounts cancel out, so when is the daycount accrual factor $\delta_t$ needed for the floating leg?
## Answer by Attack68 (score 3)
https://quant.stackexchange.com/a/78319
You are reliant on the formula for the rate of the floating period being as stated:
$$ F_t = \left ( \frac{P(t-1)}{P(t)} - 1 \right ) \frac{1}{\delta_t} $$ and the period DCF $\delta_t$ being equal. This is a special case and is only true sometimes.
Examples of when it is not true:
- Take an IMM swap period between Wed 20th March 2024 and Wed 19th June 2024, versus 3M Euribor. The DCF for the accrual period on the swap, $\delta_p$ is between those dates. The DCF for 3M Euribor rate, $\delta_t$, runs between 20th March and 20th June. $\delta_p \ne \delta_t$
- Suppose that the floating rate has a spread applied with a simple compounding method then $$ F_t = \left ( \frac{P(t-1)}{P(t)} - 1 \right ) \frac{1}{\delta_t} + z $$ In this case one observes that the $\delta_t$ do not cancel due the presence of the spread.
- Suppose you defined a rare kind of customised leg where the day count applied to the period did not align with the convention for the underlying index. Then the two '$\delta_t$' would again not be the same.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.