Forwards futures and swaps valuation - FRA valuation and app...

Learning Outcomes

After studying this article, you will be able to calculate the value of a forward rate agreement (FRA) both at inception and during its life, interpret FRA payoff formulas, explain how FRAs are used to hedge interest rate risk, and distinguish FRAs from other forward and swap contracts. You will also identify the critical exam pitfalls associated with FRA applications.

CFA Level 2 Syllabus

For CFA Level 2, you are required to understand how forward rate agreements (FRAs), forwards, and swaps are valued and applied in practice. This article focuses on:

• Pricing and valuing FRAs at and after contract initiation
• Calculating the mark-to-market value of FRAs and identifying settlement payments
• Comparing FRA, forward, and swap structures and their mechanics
• Applying FRAs for interest rate risk management and hedging strategies
• Recognizing potential valuation and calculation errors in exam scenarios

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

1. What is the settlement value of a 3x9 FRA contract for the long position if the current 6-month LIBOR set in 3 months exceeds the FRA rate?
2. Why is the payoff on a FRA discounted to the settlement date, not the loan's end date?
3. True or false? A rise in the reference rate above the FRA rate benefits the FRA long position at settlement.
4. What key element distinguishes a swap from a series of FRAs?

Introduction

Forward rate agreements (FRAs), forwards, and swaps are fundamental derivatives for managing and transferring interest rate risk. Understanding FRA valuation—both at contract initiation and at settlement—is essential for the Level 2 CFA exam, as is recognizing how they interconnect with swaps and other forward commitments. This article outlines the FRA pricing logic, walks through step-by-step calculations, and illustrates common pitfalls in marking-to-market and exam scenarios.

Key Term: forward rate agreement (FRA)
A contract where parties agree today on an interest rate to apply to a notional loan over a future period. At settlement, a cash payment reflects the difference between the agreed FRA rate and the actual reference rate.

Key Term: mark-to-market value (of an FRA)
The current value of a FRA after its initiation, reflecting market interest rate moves and calculated as the present value of expected future settlement cash flows.

Key Term: settlement date (FRA)
The date when the net cash payment on the FRA is exchanged, based on the difference between market and contract rates and discounted to present value.

Key Term: notional principal
The face amount on which FRA interest differences are calculated, but which itself is never exchanged.

FRA Pricing and Payoff Logic

A FRA specifies an interest rate for a future loan/deposit over a predetermined period for a notional amount. At settlement, instead of exchanging principal or the actual loan, the counterparties transfer a cash amount reflecting the difference between the contract rate and the newly observed reference rate.

The FRA's value at initiation is set so that it is zero to both parties. If market rates change before the settlement date, the value of the FRA moves, favoring one side and disadvantaging the other.

The payout on settlement is:

$\text{Payoff} = \frac{\text{Notional} \times [\text{Reference Rate} - \text{FRA Rate}] \times \frac{\text{Days}}{360}}{1+ \text{Reference Rate} \times \frac{\text{Days}}{360}}$

The payoff is discounted to the settlement date because the interest difference would be realized at the maturity of the relevant loan, not at settlement. However, the actual cash is paid up front.

Key Term: reference rate (FRA)
The market interest rate observed at the FRA's settlement; determines the size of any settlement payment.

Worked Example 1.1

A company enters a $10 million 2x8 FRA at 4.2% (meaning a 6-month loan starting in 2 months). At settlement, the 6-month LIBOR set for the loan is 5.0%. Calculate the settlement value to the FRA long. Assume 180 days in the accrual period.

Answer:

• Reference Rate = 5.0%
• FRA Rate = 4.2%
• Notional = $10,000,000
• Days = 180

\text{Payoff} = \frac{10,000,000 \times (0.050 - 0.042) \times \frac{180}{360}}{1 + 0.050 \times \frac{180}{360}} = \frac{10,000,000 \times 0.008 \times 0.5}{1 + 0.025} = \frac{40,000}{1.025} = \39,024 $

The long (borrower) receives $39,024 at settlement.

Exam Warning

Many candidates forget to discount the payoff to the settlement date. FRA settlement is based on present value, not the maturity value. Always apply the denominator in the formula above.

Mark-to-Market Valuation of FRAs

After initiation but before settlement, the value of a FRA may move away from zero due to changes in expected interest rates. The new value is determined by calculating the expected payoff using the current forward rates and discounting back to today.

Mark-to-market value:

1. Calculate expected future settlement cash flow: use current forward rate minus FRA rate, times notional and day-count fraction.
2. Discount the expected payment to present using the current market rate for the settlement period.

This reflects the value if you were to close out or offset your FRA early.

Key Term: forward rate (FRA context)
The market-implied rate for the future loan/deposit period, used to value the FRA.

Worked Example 1.2

Suppose 30 days after entering the FRA in Example 1.1 (with 60 days to settlement remaining), the new 6-month forward rate is 4.6%. What is the current value of the FRA to the long?

Answer:

• Notional = $10,000,000

• New Forward Rate = 4.6%

• FRA Rate = 4.2%

• Days in accrual period = 180

• Days to settlement = 60

1. Expected settlement payoff (using 4.6%):
   $10,000,000 \times (0.046 - 0.042) \times 0.5 = 20,000$
2. Discount this payoff from settlement back 60 days using the present 6-month forward rate (use 4.6%, 60/360 = 0.1667):
   \text{PV} = \frac{20,000}{1 + 0.046 \times 0.1667} = \frac{20,000}{1.00767} = \19,847 $

The FRA long position is currently worth $19,847.

Revision Tip

On exam day, always start with the formula in the question and follow the inputs exactly—double-check the dates, rates, and period lengths.

FRAs, Forwards, and Swaps—Comparison

While FRAs and standard forwards are both over-the-counter contracts setting future rates or prices, FRAs settle only on the interest differential and do not involve funding or exchanging principal. Interest rate swaps, meanwhile, can be viewed as a series of FRAs, each covering successive periods, netting cash flows periodically.

Key Term: interest rate swap
A derivative in which two parties exchange fixed and floating interest payments on a notional amount over multiple periods.

Worked Example 1.3

If you enter a 3-year quarterly interest rate swap, each periodic payment's value at inception is the same as a corresponding FRA for that period. If rates move after one year, how is the next payment valued?

Answer:
Each quarter, payment is based on the difference between fixed and newly set floating rates on the notional. The present value of expected future differences is calculated like a FRA, using current forward or market rates.

Common Applications of FRAs

FRAs are used to hedge interest rate risk (e.g., protecting a future loan or deposit from adverse rate moves), to speculate on rate direction, or to synthetically lock in borrowing or lending costs.

Exam Warning:

Confusing the directions: FRA long (buyer) wins if rates rise; short (seller) wins if rates fall. Interpret the question carefully.

Summary

• The value of an FRA at initiation is always zero.
• At settlement, value is based on the present value of the interest rate differential.
• Mark-to-market value between initiation and settlement reflects changes in forward rates.
• The FRA long benefits if the observed reference rate at settlement exceeds the contract rate, and vice versa for the short.
• FRAs can be replicated or hedged using forwards or series of swap payments.

Key Point Checklist

This article has covered the following key knowledge points:

• The settlement and mark-to-market valuation of FRAs
• Settlement cash flows depend on discounted interest differential for the notional period
• The difference between FRAs, forwards, and swaps in practice
• Appropriate uses of FRAs for managing interest rate risk
• Correct calculation and common pitfalls on exam-style FRA questions

Key Terms and Concepts

• forward rate agreement (FRA)
• mark-to-market value (of an FRA)
• settlement date (FRA)
• notional principal
• reference rate (FRA)
• forward rate (FRA context)
• interest rate swap