Question

You enter into a five-to-eight-month forward rate agreement with a firm. You agree to lend the firm a 3-month loan of $5 million starting 5 months from now, with a quarterly compounded forward interest rate of 2.5% per annum. Currently, the continuously compounded 5-month and 8-month interest rates are 3% per annum and 3.5% per annum, respectively.

1) What is the implied forward rate for the 3-month period starting 5 months from now?

2) What is the present value of this forward rate agreement to you now?

Answer 1

Answer to question 1.

So the implied forward rate for 3 months starting 5 months from now can be calculated like below:

Step 1;

To calculate the implied forward rate for 3 month first we need to compute the conitinuos compunded rate for 3 months, which can be calculated by the given formula below:

Now this 4.33% is continuosly compunded rate per annum.

Now to get the continuous compunded rate for 3 months or a quarter we need to divide this rate by 4 because there are 4 quarters in a year.

Hence, by doing so we get :  %

This 1.083% is for 3 months continuous rate.

Step 2:

Now we per annum effective rate or the rate compounded every year.

To reach there we need to this rate to the power of exponential. Hence,

%

Now this is the effective rate for 3 months and we are gonna multiply it by 4 to get per annum quarterly compunded rate.

%

Hence, the implied rate for 3 months is 4.35%

Answer to question 2.

So, as we can see the implied rate for 3 months is higher than the rate i.e, 2.5% per annum compounded quarterly in which we entered the contract.

so the value of the FRA now is,

rate prevailing now vs rate in which we entered  

So the value is:

Hence, the value is $23,125