Ok, I had to attack it from a different angle, but I think I figured it out (kind of).
If you already know how to do this stuff, don’t bother reading what I’m about to write. But if you’re still lacking a conceptual understanding of the pricing of a currency forward, this may or may not help. In fact, I’d probably just ignore it regardless. Anyway, here’s the answer (kind of) to my question:
The (1+rFC)^T term is actually missing something. The full denominator should be:
(units of foreign currency)[(1+rFC)^T]
The only reason that first term is omitted is because, when using direct quotes, the “units of foreign currency” will ALWAYS equal 1.
In other words, the CFA formula that I wrote in the first post can be better understood by doing the multiplication first, then dividing.
We can think of this as:
Compound S0 at the domestic interest rate, as you would with the price of any other forward. This amount should be equal to the amount that we would have if we exchanged DC for FC (at S0), invested at the foreign risk-free rate, and then converted back to DC. Here’s the key: We know every value in order to do this, except at time=0 we do not know what the final spot rate will be. We can call this E(ST) for sake of this post. So basically, we have to solve for this unkown spot rate that will make these two values equal. That “spot rate” is actually the price of the contract.
So, using numbers from the CFA text for sake of example, if S0 = .5987 /CHF, and compounded at the domestic interest rate for 180 days gives us .61472 dollars. Now, we should have an equal amount of dollars (.61472) at the end of the following transaction: we exchange .5987 dollars for 1 CHF, invest 1 CHF at the rFC (4.75%), and convert it back to dollars at T. So again, we know what the final dollar amount _should_ be (0.61472 dollars), and we know how many CHF we have after compounding that forward at the rFC (1.02315 CHF), but what we _don't_ know is the exchange rate that would make these two equal. Again, that unkown exchange rate (earlier I called it E(ST)) is basically the **price** of the forward contract. So to solve for that number, we divide 0.61472 dollars by 1.02315 CHF and that gives us 0.60081 /CHF, which is the correct price of the contract.
In order to tie the concepts back to my original question:
Basically, I was confused why we would compound the dollar amount forward, and then just discount it back again (albeit at the foreign interest rate). DON’T think of it like that. We are compounding the dollar amount forward, but the dividing by (1+rFC)^T is really just a way to “solve for x” to make sure that the price of the contract is equal whether we start with dollars and compound forward, or start with dollars, convert to CHF, invest, then convert back to dollars. So again, that bottom term is really (1)(1+rFC)^T, but obviously the one is not needed. If our exchange rate was somehow X amont of DC per 2 units of FC, then we would not simply be dividing by the foreign interest rate anymore, we’d be dividing by the foreign interest rate multiplied by the amount of foreign currency that we would have invested at that rate. Again, it just so happens that this number will always be 1 when using direct quotes.
Conclusions:
-
This is BY FAR the longest post I have ever written, and should probably be ignored unless you want to waste some time and then be more confused at the end.
-
It works best for me to think about compounding S0 forward at rDC, and then dividing by the amount of FC compounded at the rFC, rather than dividing first and then multiplying, like CFA has it.