Forward Rate Agreement (FRA) Valuation

Forward rate agreement is an instrument by using which a party can eliminate the interest rate risk. If you are a lender of money and you feel that interest rate can decrease in future, then you can enter into a forward rate agreement and short a FRA contract to fix your interest at the current rates. Similarly, for a borrower, if you think that interest rate can rise in future, then you can take a long position in FRA contract to fix your interest payments at the current interest rates. This is very important to know the rationale behind the FRA positions.  In CFA curriculum, level I deals with only payments of FRA contracts at contract expiry and does not deal with valuation of FRA. Valuation of FRA is a part of CFA level II curriculum and it is important topics as well.

CFA Level I:

Let us understand the notation of FRA first. A 2X5 FRA means that contract will expire in 2 months and loan is for a period of 3 (=5-2) months. If you enter into 2X5 FRA contract through a long position, then you are eliminating your risk that interest rate may rise in future. If you enter into the contract at a rate of 5% and at the contract expiry i.e. after 2 months, the 3-month interest rates (usually LIBOR or LIBOR+ some spread). You will get a loan for 3 months after 2 months at an interest of 5% whatever be the market rate at the end of 2 months. The profit or loss due to a position in a contract can be calculated by comparing the FRA rate with the market interest rate at the end of contract expiry. If the market rate is more than the FRA rate i.e. 5%, then the long position will gain as he will get the loan at an interest rate of 5% rather than the current higher market interest rate. Similarly, if the market interest rate is lower than the FRA rate, then the long position will lose as he could have got the loan at a lower market interest rate.

To calculate the gain at the FRA contract expiration, we need to know the notional principal on which there was an agreement. Suppose the principal amount is $1 million.  Now if the market interest rate for 3 month period is 6% at the expiration of the contract, then you will gain as you will get the loan at 5% while the market rate is 6%. Now, you need to pay that interest at the end of 3 months (loan period). So, you will gain total $2,500 [$1million*(0.06-0.05)*(3/12)] at the end of the loan period. At the contract expiry, you will gain the present value of that amount and that have to be discounted at the current market rates. So, it would be $2,500/ (1+0.06*90/360) = $2,463.05.

CFA Level II:

In CFA level II, we need to calculate the FRA rate as well depending on the current market rates. Then we have to calculate the value of FRA at different times: at the start of the contract (which will be zero), at the expiration of the contract, in between the expiration and start of the contract.

Let us try to understand the FRA valuation. For a 1X4 FRA contract i.e. 3 months loan after one month. If the current market interest rate is 5% for 30-days and 6% for 120-days. Then, the FRA rate will be based on arbitrage pricing i.e. if you invest for a 1-month period and 1X4 FRA, then your investment’s total worth will be same if you invest directly for 4 months. So, the FRA rate should be such that if you invest $100 in 1-month security, it will become 100*(1+R1) in one month and then 100*(1+R1)*(1+1X4 FRA) after the end of 4 months. And, by directly investing in 4-month instrument we will get 100*(1+R4). So, both of these values should be equal.

100*(1+R1)*(1+1X4 FRA) = 100*(1+R4)

=> 1X4 FRA = [(1+R4)/ (1+R1)] – 1

Similarly, we can calculate any FRA rate at any time. Now, the value of FRA contract will be zero for both the parties (long and short) at the contract initiation. At the expiration, its value will depend on the interest rate for the loan period and that can be calculated simply as I had explained earlier.

To calculate the FRA contract value in between contract initiation and contract expiry, our approach will be the same. Let us discuss it with an example. Suppose in 1X4 FRA contract, we initiated the contract at 6.5% and after 10 days the interest rates for 20 days and 110 days are 5.5% and 6.5% respectively. After 10 days, the new FRA rate will be equal to [(1+R110)/ (1+R20) – 1]. (Do note that here we have used notation as R110 which is using days while earlier we used R4 where 4 was the number of months. It can be anything as long as we use the proper rates, notations do not matter.) The new FRA rate will be (1+0.065*110/360)/ (1+0.055*20/360) – 1 = 0.016754 which is equivalent to 0.016754*360/90 = 0.067017 per annum i.e. 6.70%. As this rate is more than the FRA rate at the contract initiation, the FRA contract will have positive value for the long position. The long position holder will gain the interest rate differential multiplied by the notional principal as interest savings. But the long will gain that interest saving at the end of the loan period i.e. 110 days after today. So, we need to discount that value by 110-days current market rate and for 110 days period. For a notional principal of $1 million, the interest rate savings will be $1million*(0.067-0.065)*(90/360) = $500. And the present value of those savings (i.e. the FRA contract value) will be $500/ (1+0.065*110/360) = $490.27.

That is all about FRA valuation. I hope that you have got the concept and won’t get it wrong ever. Please do try the problems given below to check the concepts learned. All the best.

Please answer the question numbering from 1 to 6 using the data given below:

Vaibhav Aggarwal, a CFA level I candidate has attempted his level I exam today and he is sure about his passing and wants a loan of $700 after 2 months (when results will be out) for a period of 4 months. But he is afraid that the interest rates might rise in the coming future and he might get a loan at a higher interest rate. So, he decides to enter into a long position for a 2X6 FRA contract. Konvexity Institute agrees to enter from the short position. The loan rates in the market are decided as LIBOR+100 basis points. The LIBOR rates are given below:

60-day LIBOR = 4.0%

120-day LIBOR = 4.5%

180-day LIBOR = 5.0%

After 50 days from today, the LIBOR rates are:

10-day LIBOR = 4.2%

70-day LIBOR = 4.7%

130-day LIBOR = 5.2%

At the expiration of contract, the LIBOR rates are:

60-day LIBOR = 4.7%

90-day LIBOR = 4.9%

120-day LIBOR = 5.2%

Mrinal Gaurav, a friend of Vaibhav, advises him to square off the position after the end of 50 days. Mrinal thinks that interest rate may fall in next few days. Vaibhav listens to his advice but doesn’t act on that and keep the FRA contract till expiry.

Konvexity Institute lost its time because of getting into this contract and also some money in getting the paperwork done. The total transaction cost for it was $2.

1.         What is the forward rate agreement contract interest rate at the initiation of the contract?

(a)       5.46%

(b)       6.46%

(c)       7.28%

2.         What is the value of the forward contract 50 days after the contract initiation?

(a)       $0.3926

(b)       $0.4254

(c)       $0.4349

3.         What is the value of the forward contract at the contract expiry?

(a)       $0.5630

(b)       $0.5747

(c)       $0.6045

4.         Vaibhav did not act on the advice of his friend. What is the impact of this action on him?

(a)       Positive

(b)       No Impact

(c)       Negative

5.         Which party has the credit risk at the end of 50 days?

(a)       Konvexity Institute

(b)       Vaibhav Aggarwal

(c)       Mrinal Gaurav

6.         What should have been the minimum notional principal of the contract to cover the costs for Konvexity Institute?

(a)       $2,315.84

(b)       $2,486.59

(c)       $2,507.31

Answers: (1) c (2) b (3) b (4) a (5) b (6) a

The above answers are wrong. To get the correct answers consider c as b, b as a, and a as c.


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A blog of the NYU Colloquium on Market Institutions and the Leipzig Colloquium on the Market Order

Fundoo Professor

Thoughts of a teacher & practitioner of value investing and behavioral economics

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