How to compute discount factor from yield curve when there are two daycounts in play?

Question

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 1

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.

$$(1+0.01)^{-1} = (1+0.0098630136*365/360)^{-1} = 100/101$$