but I’m completely missing the subtlety of why the S must be discounted (the second line of the formula derivation). When we’re at time = t, isn’t the spot at that time it’s present value?

This formula is too nutty and I keep forgetting it.

If you settle a currency forward early, you don’t settle it for the agreed amount of the base currency. You settle it for the present value of the agreed amount, discounted at the base currency risk-free rate.

You’re not discounting the spot rate, you’re discounting the spot _ amount _; the value you get in that formula is per unit of the agreed amount of the base currency.

Suppose that you have a currency forward to buy GBP10,000,000 in 60 days. If you settled it today, it would be for the present value of GBP10,000,000, discounted at the GBP risk-free rate. You can either divide GBP10,000,000 by 1 + r, then multiply that by the spot rate, or you can divide the spot rate by 1 + r, then multiply it by GBP10,000,000; the formula does the latter.

Your explanation makes sense, but it doesn’t seem to correspond to the example on this site. The example discounts the spot and forward rates. The difference between these discounted rates is then applied to the notional amount of the contract to arrive at the value of the forward.

I suppose this might make more sense tomorrow when I review some problems. it seemed to make sense at one point in time at least.

Thanks for replying. Your blog has been tremendously helpful to me.

In the example it shows that the spot rate is discounted. But note that that discounted number will be multiplied by the notional amount. What you’re really doing is discounting the notional amount and multiplying it by the (undiscounted) spot exchange rate. Mathematically, they’re equivalent ((a/b) × c = a × (c/b)), but it’s easier to have a formula where you multiply the whole thing by the notional amount, rather than having the first part (the spot rate) multiplied by the discounted notional amount, and the second part (the discounted forward rate) multiplied by the (undiscounted) notional amount.