# Forward rate agreement

### Below are links to the following topics:

**Forward Contract****Forward Rate Agreement (FRA)****Calculation of the Settlement Amount of a FRA****The Forward Rate Market****The Difference between FRAs and Futures****Forward Rate Agreement Pricing**

### Forward Contracts

A forward contract is an agreement negotiated between two parties for the delivery of a physical asset (e.g., oil or gas) at a certain time in the future for a certain price, fixed at the inception of the contract. The parties agreeing to the forward contract are known as counterparties. No actual transfer of ownership occurs in the underlying asset when the contract is negotiated. Instead, there is simply an agreement to transfer ownership of the underlying asset at some future delivery date.

The following example, illustrates a very simple, yet frequently occurring, type of forward contract. Having heard that a highly prized St. Bernard has just given birth to a litter of pups, the dog fancier rushes to the kennel to see the pups. After inspecting the pedigree of the parents, the dog fancier offers to buy a pup from the breeder. However, the exchange cannot be completed at this time, since the pup is too young to be weaned. The fancier and breeder thus agree that the dog will be delivered in six weeks and that the fancier will pay the $400 in six weeks upon delivery of the puppy. This contract is not a conditional contract; both parties are obligated to complete it as agreed. The puppy example represent a very basic type of forward contract. The example could have been made more complicated by the breeder requiring a deposit, but that would not change the essential character of the transaction. In this example, there is a buyer and a seller.

Many forward contracts are cash-settled forward contracts. At the maturity of such contracts, the long receives a cash payment if the spot price on the underlying prevailing at the contract's maturity date is above the purchase price specified in the contract. If the spot price on the underlying prevailing at the maturity date of the contract is below the purchase price specified in the contract, then the long makes a cash payment.

### Forward Rate Agreement (FRA)

### What is a Forward Rate Agreement (FRA)?

A forward rate agreement (FRA) is a cash-settled OTC contract between two counterparties, where the buyer is borrowing (and the seller is lending) a notional sum at a fixed interest rate (the FRA rate) and for a specified period of time starting at an agreed date in the future.

### Forward Rate Agreement

Buyer (the borrower) of an FRA is "short" price and "long" interest rates. Seller (the lender) of an FRA is "long" price and "short" interest rates.

### Futures

Buyer (the borrower) of an "futures contract" is "short" interest rates and "long" price. Seller (the lender) of an "futures contract" is "short" price and "long" interest rates.

An FRA is basically a forward-starting loan, but without the exchange of the principal. The notional amount is simply used to calculate interest payments. By enabling market participants to trade today at an interest rate that will be effective at some point in the future, FRAs allow them to hedge their interest rate exposure on future engagements.

Concretely, the buyer of the FRA, who locks in a borrowing rate, will be protected against a rise in interest rates and the seller, who obtains a fixed lending rate, will be protected against a fall in interest rates. If the interest rates neither fall nor rise, nobody will benefit.

FRA buyer is long INTEREST rates (rising interest rates)

FRA seller is short INTEREST rates (falling interest rates)

The life of an FRA is composed of two periods of time – the waiting period, or forward, and the contract period. The waiting period is the period up until the start of the notional loan and may last up to 12 months although durations of up to 6 months are most common. The contract period spans the duration of the notional loan and can also last up to 12 months.

### Key Dates of a Forward Rate Agreement (FRA)

The FRA however effectively ends with the settlement date, as there is no longer any contractual engagement between the two counterparties once the settlement amount has been paid. The contract period is merely one of the calculation parameters used to determine the settlement amount (FRAs are off-balance sheet instruments).

### FRA Terminology

Below is a short listing of the terms used for the different elements and events of an FRA:

**Contract rate (or FRA rate)**

- The interest rate the two contracting parties negotiate on trade date.
- This rate will be compared to the settlement rate when calculating the settlement amount.
- It starts on the settlement (d3) date and ends on maturity date (d4).

**Contract period**

- The time between the settlement date and maturity date of the notional loan. This period can go up to 12 months.

**Currency**

- The currency in which the FRA's notional amount is denominated.

**Fixing date**

- This is the date on which the reference rate is determined, that is, the rate to which the FRA rate is compared.

**FRA buyer**

- By convention, the buyer of an FRA is the contracting party that borrows at the FRA rate (contract rate).

**FRA seller**

- By convention, the seller of an FRA is the contracting party that lends at the FRA rate (contract rate).

**Master agreement**

- Usually, counterparties sign a master agreement between each other before entering into an OTC contract because doing so without a master agreement in place would mean huge amounts of paperwork having to be generated and processed for each single deal.

**Maturity date**

- The date on which the notional loan is deemed to expire.

**Notional amount**

- This is the notional sum for which the interest rate will be guaranteed and on which all interest calculations will be based.

**Reference rate**

- The interest rate index the FRA rate will be compared against in order to determine the settlement amount. This will generally be a LIBOR-type rate index with the same duration as the FRA's contract period (for example 6-month EURIBOR for an FRA in euros with a 6-month contract period).

**Settlement amount**

- The amount calculated as the difference between the FRA rate and the reference rate as a percentage of the notional sum, paid by one party to the other on the settlement date. The settlement amount is calculated after the fixing date, for payment on the settlement date.

**Settlement date**

- The date on which the notional loan period (the contract period) begins and on which the settlement amount is being paid.

**Spot date**

- The date on which the FRA. Usually two business days after the trade date.

**Trade date**

- The date on which the FRA is negotiated between the two counterparties.

**Waiting period**

- The period comprised between the value date (d1) and the settlement date (d3).

### Calculation of the Settlement Amount of an FRA

The amount to be exchanged on settlement date - the settlement amount - is calculated as described below. For the sake of clarity, the calculation has been split into two parts, but normally it is one single calculation.

**Step 1 - calculation of the interest differential**

The interest differential is the result of the comparison between the FRA rate and the settlement rate. It is calculated as follows:

Interest differential = | (Settlement rate - Contract rate) | × (Days in contract period/360) × Notional amount

**Step 2 - calculation of the settlement amount**

As stated above, the settlement amount is paid upfront (at the start of the contract period), whereas interbank rates like LIBOR or EURIBOR are for operations with interest payment in arrears (at the end of the loan period). To account for this, the interest differential needs to be discounted, using the settlement rate as a discount rate. The settlement amount is thus calculated as the present value of the interest differential:

Settlement amount = Interest differential / [1 + Settlement rate × (Days in contract period / 360)]

If the settlement rate is higher than the contract rate, then it is the FRA seller who has to pay the settlement amount to the buyer. If the contract rate is higher than the settlement rate, then it is the FRA buyer who has to pay the settlement amount to the seller. If the contract rate and the settlement rate are equal, then no payment is made.

The complete formula used to calculate the settlement amount is the following:

Settlement = ((r_{ref} - r_{FRA}) x M x n/B)/(1+(r_{ref} + n/B))

where r_{ref} is the reference interest fixing rate; r_{FRA} is the FRA rate or contract rate; M is the notional value; n is the number of days in the contract period; B is the day-count base (360 or 365).

To summarize:

If settlement rate > contract rate, the FRA buyer receives the settlement amount

If contract rate > settlement rate, the FRA seller receives the settlement amount

If settlement rate = contract rate, no settlement amount is being paid

### The Forward Rate (FRA) Market

FRAs are money market instruments, and are traded by both banks and corporations. The FRA market is liquid in all major currencies, also by the presence of market makers, and rates are also quoted by a number of banks and brokers.

### Notation and Quoting of FRAs

The format in which FRAs are noted is the term to settlement date and term to maturity date, both expressed in months and usually separated by the letter "x".

### Examples:

2x6 - An FRA having a 2-month waiting period (forward) and a 4 month contract period.

6x12 - An FRA having a 6-month waiting period (forward) and a 6 month contract period.

### Quotation of Forward Rate Agreements

FRA are quoted with the FRA rate. Thus, if an FRA 2x8 in US dollars quotes at 1.50%, and a future borrower anticipates the 6-month USD Libor rate in two months being higher than 1.50%, he should buy an FRA.

### Usages of an FRA

A FRA can be used for different purposes:

- As already mentioned above, it can be used by market participants to hedge future borrowing or lending engagements against adverse movements in interest rates by fixing an interest rate today.
- It can further be used for trading purposes in which a market participant wants to make profits based on his expectations on the future development of interest rates.
- Lastly, it can be employed in arbitrage strategies where a market participant tries to take advantage of price differences between an FRAs and other interest rate instruments.

### Example of an FRA

A corporation learns that it will need to borrow $1,000,000 in six months' time for a 6-month period. The interest rate at which it can borrow today is 6-month LIBOR plus 50 basis points. Let us further assume that the 6-month LIBOR currently is at 0.89465%, but the company’s treasurer thinks it might rise as high as 1.30% over the forthcoming months.

The treasurer chooses to buy a 6x12 FRA in order to cover the period of 6 months starting 6 months from now. He receives a quote of 0.95450% from his bank and buys the FRA for a notional of $1,000,000 on April 8th.

Characteristics of the FRA known on trade date: | |

Trade date | 4/8/2020 |

Spot date (t+2) | 4/12/2020 |

Fixing date | 10/10/2020 |

Settlement date | 10/12/2020 |

Maturity date | 4/12/2021 |

FRA rate | 0.95% |

Contract period: 182 days (= 4/12/2021 - 10/12/2021 = 182) |

On the fixing date (October 10, 2016), the 6-month LIBOR fixes at 1.26222, which is the settlement rate applicable for the company's FRA.

As anticipated by the treasurer, the 6-month LIBOR rose during the 6-month waiting period, hence the company will receive the settlement amount from the FRA seller. The settlement amount is calculated as follows:

Interest differential = (1.26222% - 0.95450%) × (182/360) × $1,000,000 = $1,555.50

Discounted at 1.26222% to the settlement date, the settlement amount the company will receive is: = 1,555.70 / [1 + 1.26222% × (182 / 360)] = 1,545.83

### The Difference Between FRAs and Futures

As a hedging vehicle, FRAs are similar to short-term interest rate futures (STIRs). There are however a couple of distinctions that set them apart.

- FRAs are not traded on an organized exchange but are over-the-counter instruments.
- Although FRAs have fairly standardized contract provisions, they are not fully standardized the way futures contracts are.
- When entering into an FRA, both parties to the contract entail credit risk exposure, because they face each other directly. Futures, on the other hand, do not bear credit risk exposure, because they are transacted through an exchange where each counterparty is facing the exchange.
- FRAs have the advantage that they can be traded for any maturity date. Futures, which are traded on exchanges, only mature on specific dates each year.
- The buyer of an FRA benefits from rising interest rates, whereas the buyer of a futures contract benefits from falling interest rates. This is due to the fact that both contracts have different underlyings. While an FRA’s underlying is an interest rate, the underlying for a futures contract is an interest rate instrument, and an interest rate instrument increases in value when interest rates fall. This is summarized in the table below:

Interest rates fall | Interest rates rise | |||

Party | FRA | Future | FRA | Future |

Buyer | Loses | Gains | Gains | Loses |

Seller | Gains | Loses | Loses | Gains |

### FRA Pricing

As their name implies, FRAs are forward rate instruments and are priced using forward rate principles. Consider an investor who has two alternatives, either a six-month investment at 5% or a one-year investment at 6%. If the investor wishes to invest for six months and then roll over the investment for a further six months, what rate is required for the rollover period such that the final return equals the 6% available from the one-year investment? If we view a FRA rate as the breakeven forward rate between the two periods, we simply solve for this forward rate and that is our approximate FRA rate. This rate is sometimes referred to as the interest rate "gap" in the money markets (not to be confused with an interbank desk’s gap risk, the interest rate exposure arising from the net maturity position of its assets and liabilities).

We can use the standard forward-rate breakeven formula to solve for the required FRA rate; we established this relationship earlier when discussing the calculation of forward rates that are arbitrage-free. The relationship given in equation 2.0 connects simple (bullet) interest rates for periods of time up to one year, where no compounding of interest is required. As FRAs are money market instruments we are not required to calculate rates for periods in excess of one year, where compounding would need to be built into the equation. This is given by equation 2.0.

Equation (2.0) (1+r_{2}t_{2}) = (1+r_{1}t_{1})*(1+r_{f}t_{f})

where

r_{2} is the cash market interest rate for the long period;

r_{1} is the cash market interest rate for the short period;

r_{f} is the forward rate for the gap period;

t_{2} is the time period from today to the end of the long period;

t_{1} is the time period from today to the end of the short period;

t_{f} is the forward gap time period, or the contract period for the FRA.

This is illustrated diagrammatically in Figure 2.

Figure 2: Rates used in FRA pricing

The time period t_{1} is the time from the dealing date to the FRA settlement date, while t_{2} is the time from the dealing date to the FRA maturity date. The time period for the FRA (contract period) is t_{2 minus t1. We can replace the symbol t for time period with n for the actual number of days in the time periods themselves. If we do this and then rearrange the equation to solve for rFRA the FRA rate, we obtain (3): }

r_{FRA} = (r_{2}n_{2} - r_{1}t_{1}) /( n_{FRA} (1+r_{1} x (n_{1}/365))

where

n_{1} is the number of days from the dealing date or spot date to the settlement date;

n_{2} is the number of days from the dealing date or spot date to the maturity date;

r_{1} is the spot rate to the settlement date;

r_{2} is the spot rate from the spot date in the FRA contract period;

n_{fra} is the number of days in the maturity date;

r_{FRA} is the FRA rate.

If the formula is applied to (say) the US money markets, the 365 in the equation is replaced by 360, the day count base for that market.

In practice FRAs are priced off the exchange-traded short-term interest rate future for that currency, so that sterling FRAs are priced off LIFFE short sterling futures. Traders normally use a spreadsheet pricing model that has futures prices directly fed into it. FRA positions are also usually hedged with other FRAs or short-term interest rate futures.

### Example 2: Hedging an FRA Position

An FRA market maker sells a US 100 million 3-v-6 FRA, that is, an agreement to make a notional deposit (without exchange of principal) for three months in three months' time, at a rate of 7.52%. He is exposed to the risk that interest rates will have risen by the FRA settlement date in three months' time.

Date 14 December

3-v-6 FRA rate 7.52%

March futures price 92.50%

Current spot rate 6.85%

### Action

The dealer first needs to calculate a precise hedge ratio. This is a three-stage process:

1. Calculate the nominal value of a basis point move (_{nomn}BPV) in LIBOR on the FRA settlement payment;

_{nomn}BPV = FRA_{nom} x 0.01% x n/360

Therefore: 100,000,000 x 0.01% x 90/360= 2500.

2. Find the present value of 1. By discounting it back to the transaction date using the FRA and spot rates;

Present value of a basis point move=

Nominal value of basis point / (1 + spot rate x (Days in hedge period/360)) x (1+ FRA rate x (Days in hedge period/360))

Therefore:

2,500 / ((1 + (6.85% x (90/360))) x (1 + (7.52% x (90/360)))) = $2,412.55

3. Determine the correct hedge ratio by dividing 25 by the futures tick value.

Hedge ratio = 2412/25 = 96.48.

Figure 3: FRA dates

The appropriate number of contracts for the hedge of a EUR 100,000,000 3-v-6 FRA would therefore be 96 or 97, as the fraction is under one-half, 96 is correct. To hedge the risk of an increase in interest rates, the trader sells 96 ECU three months' futures contracts at 92.50. Any increase in rates during the hedge period should be offset by a gain realized on the futures contracts through daily variation margin receipts.

Outcome

Date 15 March

Three month LIBOR 7.625%

March EDSP 92.38

The hedge is lifted upon expiry of the March futures contracts. Three-month LIBOR on the FRA settlement date has risen to 7.625% so the trader incurs a loss of EUR 25,759 on his FRA position (i.e., EUR 26,250 discounted back over the three month FRA period at current LIBOR rate), calculated as follows:

= 7.52-7.625*90/360*100,000,000 = $26,250

(LIBOR-FRA rate) × (days in FRAperiod/360) × Contract Nominal Amount1 + LIBOR rate × (days in FRA_{period}/360)

Discounted = 26,250* / 1 + 7.625% x (90/360) = 25,759

Therefore:

26,250* / 1 + 7.625% x (90/360) = 25,759

* i.e., 0.105% x 90/360 x 100,000,000 = 26,250

Futures P/L: 12 ticks (92.50-92.38)× €25 × 96 contracts = EUR 28,800.

The EUR 25,759 loss on the FRA position is more than offset by the EUR 28,800 profit on the futures position when the hedge is lifted. If the dealer has sold 100 contracts his futures profit would have been EUR 30,000, and, accordingly, a less accurate hedge. The excess profit in the hedge position can mostly be attributed to the arbitrage profit realized by the market maker (i.e., the market maker has sold the FRA for 7.52% and in effect bought it back in the futures market by selling futures at 92.50 or 7.50% for a 2 tick profit.)

Sources: Damodaran, Hull, Basu, Kolb, Brueggeman, Fisher, Bodie, Merton, White, Case, Pratt, Agee.

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