Intuition for the forward FX equation

Every quant knows the expression that defines a forward FX rate on date t with maturity T:

where B_f is the foreign discount factor and B_d is the domestic discount factor. But what is the best way to explain this intuitively? Here is my suggestion.

Let’s pick an example pair, say EUR and CHF, and see how the forward FX rate compares with spot, and use the rates differential to explain the differences.

Quick side note on how to read an FX rate

There are lots of different ways to read an FX rate, and I find most of them don’t help remove confusion. For example, EURCHF is about 1.22 at the moment of writing, so people might say:

“you can exchange 1 EUR for 1.22 CHF”, or

“1 unit of EUR is equal to 1.22 units of CHF”.

In my experience it is better to think of 1.22 as the price of something:

Learn this instead: “1.22 is the price of EUR (in CHF obviously)”.

The advantage of this approach is that you make FX look just like buying and selling something. (See my note here on trader jargon for a longer discussion of why the world of trading is easier to deal with if you treat everything you ever do as either buying or selling).

It’s much nicer this way, here are some examples to help you get convinced:

EURGBP ~ 0.87: “the EUR costs about 87p”,

EURUSD ~ 1.29: “the EUR costs about 1 dollar 29″,

USDJPY ~ 77.7: “the dollar costs less than 78 yen”.

A second advantage to this approach is that now you have a direct interpretation of the terms weakening and stengthening:

Weakening = the currency is getting cheaper.

Strengthening = the currency is getting richer.

Of course, whether this means the ‘FX number’ is going up or down depends on the quotation convention. For example, if GBP is strenghening then EURGBP is going down, but if USD is strengthening then USDJPY is going up.

So the steps to understanding FX quotations are:

  1. think of currencies as a commodity with a price,
  2. work out whether it is getting more or less valuable,
  3. then think about what that means for the FX quotation.

And this is really the key to it all: separate the concept of value from the convention of the FX quotation.

Back to the main topic

Now that we see we are just buying and selling something, the forward FX intuition becomes a simple corollary of:

The no free lunch principle: the values of all the bits of the trade must add up to zero.

Let’s write the problem like:

  1. Today you can buy EUR for 1.22.
  2. You are going to agree a price today at which you will sell the EUR back at time T in the future.
  3. Will this price be higher or lower than 1.22?

Here is the intuition which gives the answer:

  1. EUR has higher interest rates than CHF.
  2. So you are today buying the more valuable currency, in the sense that its yield is better if you put it in a deposit account.
  3. The no-free-lunch principle then implies that you must pay back this good value in the other part of the trade, the selling at maturity.
  4. In other words, the price you agree to sell at must be bad.
  5. A bad sell price means ‘sell low’.

All this adds up to say:

Forward EURCHF < 1.22.

In general this means:

Forward FX intuition: if you buy the more valuable currency then you must sell at a discount.

If you check with the FX equation you’ll see that the intuitive understanding agrees with the ratio of the discount factors.

End note: how to remember the forward FX equation

All of this intuition is lovely, but oftentimes you just need to remember the equation.

If you also want a visual way to remember the forward FX equation then try this: I rely on having noticed that ‘forward FX’ often gets shortened to ‘FFX'; the first term on the right is ‘FX’ and it is the foreign discount factor on the top, so the equation is a ‘visual gathering of Fs at the top’.

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