Pricing and Valuation of Forward Rate Agreements
What you'll learn
- LOS 28.cDescribe how interest rate forwards and futures are priced, and calculate and interpret their no arbitrage value.
Eurodollar deposit is the term for deposits in large banks outside the United States denominated in U.S. dollars. The lending rate on dollar denominated loans between banks is based on the secured overnight funding rate (SOFR), hereafter referred to generically as the MRR (“market reference rate”). It is quoted as an annualized rate based on a 360 day year.
In contrast to T bill discount yields, MRR is an add on rate, like a yield quote on a short term certificate of deposit.
Compute the amount that must be repaid on a $1 million loan for 30 days if 30 day MRR is quoted at 6%.
Solution
The add on interest is calculated as $1 million × 0.06 × (30 / 360) = $5,000. The borrower would repay $1,000,000 + $5,000 = $1,005,000 at the end of 30 days.
The long position in a forward rate agreement (FRA) is the party that is effectively borrowing money (long the loan, with the contract price being the interest rate on the loan). If the floating rate at contract expiration is above the rate specified in the forward agreement, the long position in the contract can be viewed as the right to borrow at below market rates, and the long will receive a payment. If the floating rate at the expiration date is below the rate specified in the forward agreement, the short will receive a cash payment from the long.
(The right to lend at above market rates has a positive value.)
The notation for FRAs is unique. There are two numbers associated with an FRA: the number of months until the contract expires, and the number of months until the underlying loan is settled. The difference between these two is the maturity of the underlying loan.
For example, a 2 × 3 FRA is a contract that expires in two months (60 days), and the underlying loan is settled in three months (90 days). The underlying rate is 1 month (30 day) MRR on a 30 day loan in 60 days. See Figure 28.3.
Figure 28.3: Illustration of a 2 × 3 FRA
Pricing FRAs
There are three important things to remember about FRAs when we’re pricing and valuing them:
1. MRR rates in the Eurodollar market are add on rates and are always quoted on a 30/360 day basis in annual terms. For example, if the MRR quote on a 30 day loan is 6%, the actual unannualized monthly rate is 6% × (30/360) = 0.5%.
2. The long position in an FRA is, in effect, long the rate and benefits when the rate increases.
3. Although the interest on the underlying loan won’t be paid until the end of the loan (e.g., in three months in Figure 28.3), the payoff on the FRA occurs at the expiration of the FRA (e.g., in two months). Therefore, the payoff on the FRA is the present value of the interest savings on the loan (e.g., discounted one month in Figure 28.3).
The forward “price” in an FRA is actually a forward interest rate. The calculation of a forward interest rate is presented in Level I as the computation of forward rates from spot rates. We will illustrate this calculation with an example.
Calculate the price of a 1 × 4 FRA (i.e., a 90 day loan, 30 days from now). The current 30 day MRR is 4% and 120 day MRR is 5%.
Solution
The actual (unannualized) rate on the 30 day loan is:
The actual (unannualized) rate on the 120 day loan is:
We wish to calculate the actual rate on a 90 day loan from day 30 to day 120:
We can annualize this rate as:
This is the no arbitrage forward rate, the forward rate that will make the values of the long and the short positions in the FRA both zero at the initiation of the contract.
The time line is shown in the following figure.
Pricing a 1 × 4 FRA
Valuing an FRA at Maturity
To understand the calculation of the value of the FRA after the initiation of the contract, recall that in the previous example the long in the FRA has the “right” to borrow money 30 days from inception for a period of 90 days at the forward rate. If interest rates increase (specifically the 90 day forward contract rate), the long will profit as the contract has fixed a borrowing rate below the now current market rate. These “savings” will come at the end of the loan term, so to value the FRA we need to take the present value of these savings.
An example incorporating this fact will illustrate the cash settlement value of an FRA at expiration.
Continuing the prior example for a 1 × 4 FRA, assume a notional principal of $1 million. Suppose that, at contract expiration, the 90 day rate has increased to 6%, which is above the contract rate of 5.32%. Calculate the value of the FRA at maturity, which is equal to the cash payment at settlement.
Solution
The interest savings at the end of the loan term (compared to the market rate of 6%) will be:
The present value of this amount at the FRA settlement date (90 days prior to the end of the loan term) discounted at the current rate of 6% is:
This will be the cash settlement payment from the short to the long at the expiration of the contract. Note that we have discounted the savings in interest at the end of the loan term by the market rate of 6% that prevails at the contract settlement date for a 90 day term, as shown in the following figure.
Valuing a 1 × 4 FRA at Maturity
Valuing an FRA Before Maturity
To value an FRA before the settlement date, we need to know the number of days that have passed since the initiation of the contract. For example, let’s suppose we want to value the same 1 × 4 FRA 10 days after initiation. Originally, it was a 1 × 4 FRA, which means the FRA now expires in 20 days.
The calculation of the “savings” on the loan will be the same as in our previous example, except that we need to use the “new” FRA price that would be quoted on a contract covering the same period as the original “loan.” In this case the “new” FRA price is the now current market forward rate for a 90 day loan made at the settlement date (20 days in the future). Also, we need to discount the interest savings implicit in the FRA back 110 days (i.e., an extra 20 days) instead of 90 days as we did for the value at the settlement date.
Value a 5.32% 1 × 4 FRA with a principal amount of $1 million 10 days after initiation if 110 day MRR is 5.9% and 20 day MRR is 5.7%.
Solution
Step 1: Find the “new” FRA price on a 90 day loan 20 days from today. This is the current 90 day forward rate at the settlement date, 20 days from now.
Step 2: Calculate the interest difference on a $1 million, 90 day loan made 20 days from now at the forward rate calculated previously compared to the FRA rate of 5.32%.
Step 3: Discount this amount at the current 110 day rate.
Valuing a 1 × 4 FRA Before Settlement
Futures Contracts
Futures contracts are very much like the forward contracts except that they trade on organized exchanges. Each exchange has a clearinghouse. The clearinghouse guarantees that traders in the futures market will honor their obligations.
The clearinghouse does this by splitting each trade once it is made and acting as the opposite side of each position. To safeguard the clearinghouse, the exchange requires both sides of the trade to post margin and settle their accounts on a daily basis. Thus, the margin in the futures markets is a performance guarantee.
Marking to market is the process of adjusting the margin balance in a futures account each day for the change in the value of the contract from the previous trading day, based on the settlement price.
The pricing relationship discussed earlier also generally applies to futures contracts as well (see for example, our earlier discussion on Treasury bond futures contract). Like forward contracts, futures contracts have no value at contract initiation. Unlike forward contracts, futures contracts do not accumulate value changes over the term of the contract.
Since futures accounts are marked to market daily, the value after the margin deposit has been adjusted for the day’s gains and losses in contract value is always zero. The futures price at any point in time is the price that makes the value of a new contract equal to zero. The value of a futures contract strays from zero only during the trading periods between the times at which the account is marked to market:
If the futures price increases, the value of the long position increases. The value is set back to zero by the mark to market at the end of the mark to market period.