:: Collaborations ::

:: Forward rate of agreement ::

In this article we introduce the concept of forward rate agreement (FRA) which is very important in investment decision that are sensitive to interest rates, for example, fowards and futures.

Consider the two investment opportunities A and B.  In investment A you are offered by a bank to deposit £10000 for a period of at least one year at a rate of 4% per annum (p/a).  Investment B offers you a more generous rate of 6% p/a with a minimum depositing period of two years.  In both investments the interest is calculated at the continuous compound rate, however, for practice purposes it is received only at the end of each financial year.  If the money is withdrawn before the minimum depositing period the accumulated interest in the that period is forfeited.   [It is normal for banks to offer higher interest rates for money deposited over longer periods.  Usually higher interest rates also apply the larger the amount of money deposited].

From our the article on 'How to Calculate Future and Present Values' we know that Investment A is worth

if extended over two years.

Likewise, investment B is worth

over the same period.  Clearly you will earn more money in investment B than taking investment A over two years.  However, the advantage investment A has is that you are free to take your money and invest it in other ways that may be more attractive at the end of the first year.  For example, interest rates might shoot up between years 1 and 2 such that if you invest the money plus interest after year 1 in another deposit account it will be higher than the total earned by investment B.  Investment B does not have this flexibility, therefore, it carries a higher opportunity cost than investment A.

Now you are presented with a third investment options - investment C.  Investment C is an agreement (similar to a foward) to deposit the initial money in investment A plus the interest earned between years 1 and 2 at a rate of 7% p/a -this is known as the forward rate of agreement (FRA).  Therefore, investment C (with the interest from A) is worth

over the two years.  Therefore, investment C is still worth less than investment B and the opportunity cost is the same despite the 7% interest p/a offered between years 1 and 2.  Thus, we would choose investment B over C in a two year commitment.  Whilst the interest rate offered between periods 1 and 2 is higher in C than in B, the money on which this interest is calculated is greater in investment B at then end of year 1.  If there was no commitment between years 1 and 2 in investment C (it was an option) then the flexibility at the end of year 1 would lower the opportunity cost of investment C, and perhaps investment C will be chosen over investment B.

So what FRA does one need to equal the return on investment B when combined with A?  We require

Thus,

FRA = 0.06 x 2  - 0.04 x 1  =  0.08 ,

or 8%.

So we need a FRA between years 1 and 2 of 8% to make investments B and C equivalent in terms of financial gain.

Consider the case where we are told a bank offers for the first years and between years and .   What is the forward rate of agreement between years and ?

Generally, this can be written as

and rearrange to get

If one wishes to calculate the interest return expected on an investment till we can take it to be till and interpolate the FRA from   till .

Forward rates of agreement are needed when one wishes to calculate the interest earned between two periods and direct quotes for in between periods may not exist.  It is also important when pricing interest rate future / foward contacts and using the bootstrap method to calculate Treasury Zero Rates for pricing bonds.

Written by Alessio Farhadi

Back for more