# How does interest parity work with settlement dates?

Interest rate parity is typically proven as follows.

Given one unit of a domestic currency, one can either convert it into $S$ units of the foreign currency and invest at the foreign risk free rate $r_f$ for time period $T$, to obtain $S e^{r_f T}$. Alternatively, one can invest at the domestic rate $r$, and then convert to foreign currency at the FX forward rate $F$ to obtain $Fe^{r T}$. To be arbitrage free, these quantities must be equal.

I'm interested in how to use this in practice, and how this can be complicated by settlement dates. For example, FX spot settlement does not occur immediately, but instead typically takes two days. So it seems like this would break down for short time spans.

So suppose the time period invested $T$ above is less than the number of days it takes to settle an FX spot transaction. Then the above argument does not work, since the time it would take to convert to foreign currency is non-trivial compared to time invested. How to prove that IR parity holds in this case?

Additionally, it would seem that given the settlement period, there should be some small adjustment to IR parity, and it would be something like

$F e^{rT} = S e^{r_f T'}$, where $T' = T -$ ($2$ days),

since we would not be able to invest the $S$ amounts of foreign currency until it is delivered after 2 days. Is this correct?

• All calculations in finance are of course done over the rights dates, so you are correct in assuming you have to price the actual timing of events. Note that you can orchestrate same day settlement if you trade early enough in the day, but then you wouldnt be trading at the spot FX rate since that is T+2, you would be trading a customised (and slightly adjusted FX rate) to account for the dynamic you cite. – Attack68 Jul 10 '18 at 6:34