(10 of 18) Ch.21 - Interest rate parity: derivation of formula - YouTube

This one year forward rate, F1, let's review how it was derived.

This is how it was derived, if money is invested domestically, we take our $1 and multiply

it by 1 plus R U.S.A. if money's invested overseas, then we take our $1, first convert

it into the foreign currency, by multiplying it by the exchange rate, S0, and then we add

the interest rate to it.

So we multiply it by 1 plus, the interest rate in the foreign country, and then we divide

the whole thing by the one year forward rate, F1, to convert that overseas investment back

from the foreign currency into the U.S. dollar.

Let's simplify by, you know, getting rid of $1 on the left hand side and on the right

hand side.

So we have one plus R U.S.A. equals S0, times 1 plus R foreign country and then the whole

thing on the right hand side divided by the one year forward rate, F1.

Then we can rearrange it and it will look like this, F1 equals S0 times 1 plus R foreign

country, divided by 1 plus R U.S.A. or let's rewrite it the way that fits sort of the definition

of the, so called, interest rate parity formula where it says that the one year forward rate,

divide by the spot exchange rate today, equals 1 plus the interest rate in the foreign country,

for the risk free asset, divided by 1 plus the rate for the risk free asset in the United

States.

So this is kind of the definition of the, so called, interest rate parity formula that

kind of comes out of the example that we just looked at, which shows how the forward rate

should be determined in order for there to be, sort of, balanced investments across different

countries.

Let's do a little bit more rearrangement to this formula, let's take this formula, the

interest rate parity formula, and write an approximate version of it where the right

hand side of it is simplified to 1 plus and then we have the difference between the foreign

and the domestic risk free rates.

One more rearrangement so there's a whole bunch of them here, right?

And this will be the last one that actually fits what we had at the very beginning when

we started this whole topic on the interest rate parity, so this formula can be further

rearranged to F1 minus the S0, the difference is divided by S0, so it's the percentage difference

between the forward exchange rate and the spot exchange rate, and on the right hand

side, what we have remaining from this rearrangement is the difference in the interest rates, the

foreign interest rate for the risk free asset minus the domestic interest rate for the risk

free asset.

So this is kind of where, you know, this definition of the interest rate parity is coming from.

The difference in interest rates between two countries is equal to the percentage difference

between the forward exchange rate and the spot exchange rate.

Oh, sorry, I lied, we are rearranging it again.

Okay.

This is -- well, one -- I mean, there are so many different ways to write the same formula,

depending on what you leave on the left hand side, which, like, notion and what you decide

to keep on the right hand side.

So we're rearranging one last time as follows, one year forward rate, F1, that's all we have

on the left hand side, equals today's spot exchange rate as 0, multiplied by open parenthesis,

1 plus, the difference between the foreign countries risk free rate and the U.S. risk

free rate, closed parentheses this is the approximate interest rate parity and by the

way, I have an example coming up on the next slide on both the approximate and the exact

rate parity formulas.

So this formula calculates what the one year forward rate that you're locking into today

should be depending on what today's exchange rate is, S0, and how the risk free rates differ

between the foreign and the domestic countries, but we don't have to limit ourselves to the

calculation of the forward rate for one year, we can also use this formula and kind of generalize

it to find the forward rate one year from now, two years from now, three years from

now, so anytime T, in the future.

All we do is determine the parenthesis in this formula, we raise it to the power of

T, which shows that future time.

In real life, forward exchange rates don't typically go beyond one year foreign rates,

like, on one of the earlier slides where I was explaining how exchange rates are quoted

and showed you that table from the Wall Street Journal, you probably only remember seeing

the one month, the three month, and the six month forward rates, so anything beyond one

year forward rates, kind of, you know, does not really exist in real life and that's because

how would you be sure you're look locking in The right exchange rate today that would

be used in, let's say, two years, three years it's such a long amount of time that nobody

would want to bet on what the exchange rate might be at such distant future time.