Keeping Discount Factors Consistent Across Day-Count Conventions
Summary
The document addresses how to discount a payment when the yield curve uses one day-count convention and the instrument uses another. Its central principle is that the discount factor for a given payment date must be the same regardless of how the rate is expressed. If different conventions implied different present values for the same future cash flow, the discrepancy would create an arbitrage opportunity.
The example converts a rate quoted on an ACT/365 basis into its equivalent on ACT/360 while preserving the present and future values. Thus, the appropriate approach is not to apply a rate from one convention with the time fraction from another indiscriminately: the rate and accrual fraction must be paired consistently, yielding the same discount factor. The example assumes a particular simple compounding setup and illustrates the equivalence numerically; it does not cover interpolation, curve construction, or other compounding conventions.
Key ideas
- A payment date should have one consistent discount factor, regardless of the rate’s day-count expression.
- Rates expressed under different day-count conventions can differ while representing the same present value.
- Use a rate and time fraction that follow compatible conventions when computing discount factors.
- Inconsistent discount factors for the same cash flow could imply an arbitrage opportunity.
- The example’s conversion relies on its stated compounding setup and does not explain broader curve construction.
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Full text
# How to compute discount factor from yield curve when there are two daycounts in play?
# How to compute discount factor from yield curve when there are two daycounts in play?
Let's say I have a yield curve, i.e. a series of times $t_1, ..., t_n$ and associated rates $r_{t_1}, ..., r_{t_n}$, such that my discount factors are $DF_{t_i} = (1+r_{t_i})^{(-t_i)}$. The curve has been computed using an implicit ACT/365 daycount convention.
I have a payment occuring exactly 365 days from now, from a bond with a ACT/360 daycount convention.
The curve tells me that $r_{1} = r_{365 / 365} = 1\%$ and $r_{1.0319}=r_{365 / 360} = 1.01 \%$.
What then is my discount factor? $(1+1/100)^{-1}$ or $(1 + 1.01/100)^{-1.0319}$?
On one hand, I think it's the former, because after all somebody computed that rate for that exact date in question, but just happened to convert it to $t=1$ using the ACT/365 convention. On the other hand, it could be the latter, since that's what one would do if one did not know the yield curve's implicit daycount convention.
## Answer by AlRacoon (score 4)
https://quant.stackexchange.com/a/78303
The discount factors have to be identical for the same date. If not, you would have 2 different present values, which would be closed by arbitrageurs. the rates using different day count conventions would be different but the PV must be the same.
In your example, if you had $101 due a year from now:
Using 1%, Act/365:
$$PV = FV * DF$$ $$PV = 101 * (1+ 0.01)^{-1}$$ $$PV = 100$$
The present value and future values would be the same but the rate using Act/360 basis would be different.
$$FV=PV*(1+r*Act/360)$$
$$101=100*(1+r*365/360)$$
Solving for r, 1% Act/365 would be equivalent to earning 0.98630136% on and Act/360 basis.
The discount factors would be the same.
$$(1+0.01)^{-1} = (1+0.0098630136*365/360)^{-1} = 100/101$$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.