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2.7.4. Forward rate agreements

General

A forward rate agreement (FRA) is an agreement that enables a user to hedge itself against unfavorable movements in interest rates by fixing a rate on a notional amount that is (usually) of the same size and term as its exposure that starts sometime in the future. It is akin to a foreign exchange forward contract in terms of which an exchange rate for F future date 3 determined upfront.

9: 3 x 6 FRA

Figure 9: 3 x 6 FRA

An example is a 3 x 6 FRA (3-month into 6-month): the 3 in the 3 x 6 refers to 3 months' time when settlement takes place, and the 6 to the expiry date of the FRA from deal date, i.e. the rate quoted for the FRA is a 3-month rate at the time of settlement. This may be depicted as in Figure 9.

This type of instrument is particularly useful for the company treasurer who is of the opinion that the central bank is about to increase the KIR and that the interest rates on commercial paper (his borrowing habitat) will rise sharply. He needs to borrow LCC20 million in three months' time for a period of three months. He approaches a dealing bank that he normally deals with on 4 March and obtains quotes on a series of FRAs as shown in Table 113.

Bid (% pa)

Offer (% pa)

explanation

3 X 6

10.00

10.10

3-month rate in 3 months' time

6 X 9

10.20

10.30

3-month rate in 6 months' time

9 X 12

10.40

10.50

3-month rate in 9 months' time

Table 1: fictional for quotes

The treasurer verifies these rates against the quoted FRA rates of another two banks (i.e. to ensure that he is getting a good deal), finds that they are fair and decides to deal at the 10.10% pa offer rate for the 3 x 6 FRA for an amount of LCC20 million, which matches the company's requirement perfectly. The applicable future dates are 4 June and 3 September (91 days).

The transaction means that the dealing bank undertakes to fix the 3-month borrowing rate in three months' time at 10.10% for the company. The transaction is based on a notional amount of LCC million. The notional amount is not exchanged; it merely acts as the amount upon which the calculation is made.

The rate fixed in the FRA is some benchmark (also called reference) rate, or a rate referenced on a benchmark rate, i.e. some rate that is readily accepted by market participants to represent the 3-month rate. We assume this is the 3-month JIBAR14 rate, which is a yield rate.

On settlement date, i.e. 4 June, the 3-month JIBAR rate is 10.50% pa. On this day the 3-month (91-day) commercial paper rate is also 10.50% pa (which it should be because the JIBAR rate is representative of the 3-month rate). The company borrows the LCC20 million required at 10.50% through the issue of commercial paper for 91 days. According to the FRA the dealing bank now owes the company an amount of money equal to the difference between the spot market rate (i.e. 3-month JIBAR = 10.50% pa) and the agreed FRA rate (i.e. 10.10% pa) times the notional amount. This is calculated as follows:

where

SA = settlement amount

NA = notional amount

ird = interest rate differential (10.50% pa - 10.10% pa = 0.40% pa)

t = term (forward period), expressed as number of days / 365

SA = LCC20 000 000 x 0.004 x (91 / 365) = LCC19 945.21.

example of FRA: bank settles difference

Figure 10: example of FRA: bank settles difference

Note that this formula applies in the case where settlement of this amount is made in arrears at month 6 (= 3 September). If the amount is settled at month 3 (= 4 June) it has to be discounted to present value (PV). The discount factor is:

where

rr = reference rate (= JIBAR rate)

t = term of agreement (number of days / 365)

df = 1 / [1 + (rr x t)] = 1 / [1 + (0.105 x 91 / 365)] = 0.97449.

Therefore (PVSA = present value of settlement amount):

PVSA = SA x df = LCC19 945.21 x 0.97449 = LCC19 436.41

This transaction may be illustrated as in Figure 10. It will be evident that the exchange of interest on LCC20 million does not takeplaœ;the0eflingbank only œttlestheaifferenœ.

Implied forward rate

money market yield curve

Figure 11: money market yield curve

The dealing bank would of course not have sucked the rates quoted out of thin air. It would have based its forward rates on the rates implicit in the spot market rates. An example is required (see Figure 11).

Shown here are the spot rates for various periods at a point in time15. This may also be called a money market yield curve (as opposed to a long-term yield curve which stretches for a number of years). This notional yield curve may also be depicted as in Figure 12 (this is an unrealistic yield curve, because the yield curve does not usually follow straight lines).

fabricated money market yield curve

Figure 12: fabricated money market yield curve

The rate now (spot rate) for three months is 9.0% pa and the rate now (spot rate) for six months is 10.5% pa, and we know that the latter rate covers the period of the first rate. The rate of interest for the three-month period beyond the three-month period can be calculated by knowing the two spot rates mentioned. This is the forward rate of interest, or the implied forward rate. This is done as follows (assumption 3-month period: 91 days; 6-month period: 182 days):

where

IFR = implied forward rate

irL = spot interest rate for the longer period (i.e. 6-month period)

irS = spot interest rate for shorter period (i.e. 3-month period)

tL = longer period, expressed in days / 365) (i.e. the 6-month period - 182 days)

tS = shorter period, expressed in days / 365) (i.e. 3-month period - 91 days)

IFR = {[1 + (0.105 x 182/365)] / [1 + (0.09 x 91/365)] -1} x 365/91

= [(1.0524 / 1.0224) -1] x 365/91

= (1.0293 - 1) x 365/91

= 0.1174

= 11.74% pa.

The bank, in the case of a 3 x 6 FRA, will quote a rate that is below the implied 3-month forward interest rate, i.e. below 11.74%.

 
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