# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] FX: Covered Interest Parity

with one comment

We’ve already been exposed to forward rates when we talked about the term structure of interest rates. Our work-horse model for introducing forward exchange rates isn’t too different from the toy model we used when introducing forward rates in the interest rate world.

Forward on Foreign Exchange: Covered Interest Rate Parity

Let’s say there is an exporter who has receivables, say 1 million USD at the end of 1 year. If she does not have natural offsetting payable around the same time, she is exposed to the exchange rate fluctuations. And unless her business is dealing with exchange rates, she is faced with a business headache. It turns out that there is a way that she can get rid of her headache and lock-in a rate today itself.

And the way to do so is not too unlike the way we went about introducing forward rates in the interest rate world. There we said that a borrower who needed money at the end of 1 year for one year could do so by borrowing for two years and lending for one year. Here we do something similar.

If she had a natural import expense, she would not be bothered about exchange rates. USD comes in from export and goes out to import. But she needn’t be too bothered. If she does not have a natural payable, it does not mean she cannot artificially create one. Good that it is not too difficult either.

One way to create such a payable is to, well, simply borrow the ‘right’ amount of USD today itself for 1 year. This ensures that she has a payable to the bank at the same time she gets money from the business. So assuming there are no capital controls, if she can borrow PV of 1 million USD from a US bank, at the end of 1 year the money from the business could go straight to the bank.

In the process she is left with PV of 1 million USD today, which can be immediately converted into INR. Since our focus is end of 1 year, we know how much this is worth then. This is exactly the the future value of money of the INR equivalent today.

So graphically:

(Click to zoom; Note the color coding. The green arrow represents the final receivable in INR after paying back the 1 million USD  due to the US bank from the business.)

That is, by borrowing USD and investing in INR, the exporter has locked-in the receivables in the domestic currency, and the effective ‘forward exchange rate’ for the exporter is:

$\boxed{F_{FX} = \displaystyle \frac{S_0(1 + r_{IN})}{1 + r_{f}}}$

or using continuous compounding as:

$F_{FX} = \displaystyle S_0e^{r_{IN} - r_f}$

where, $r_f$ denotes the foreign currency interest rate. If the financial institution offered a ‘forward price’ different from this price there’ll be possible arbitrage opportunity.

In general, for a contract expiring at time $t$, using continuous compounding, the forward price of FX will be:

$\boxed{F_{FX} = \displaystyle S_0e^{(r_{IN} - r_f)t}}$

For completeness sake, this is how an importer will lock in her forward price (as should be clear all signs/direction of arrows have reversed, and borrowings have become lending and vice-versa):

(Click to zoom; Note the color coding. The green arrow represents the final payable in INR after getting ‘back’ the 1 million USD  from the US bank due to the business.)