Library/CFA (Chartered Financial Analyst)/Derivatives (CFA Program Curriculum 2026 • Level I • Volume 7)/Learning Module 7 Pricing and Valuation of Interest Rates and Other Swaps

Learning Module 7 Pricing and Valuation of Interest Rates and Other Swaps

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Building blocks: discount factors and implied forward rates5 min

This chapter develops the pricing and valuation framework for multi-period interest-rate derivatives built from single-period building blocks: discount factors, implied forward rates, forward rate agreements (FRAs), interest-rate futures, and interest-rate swaps. Key definitions: discount factor DF(T) = 1/(1 + z_T)^T is the present value of one currency unit at time T; implied forward rate IFR_{A,B-A} solves (1 + z_A)^A (1 + IFR_{A,B-A})^{B-A} = (1 + z_B)^B and represents the no-arbitrage breakeven reinvestment rate for the period from A to B. Forward rate agreements are contracts that fix a future floating market reference rate (MRR) for a single interest period; settlement typically occurs at the start of the interest period with the net payment discounted back to the settlement date using the observed MRR. FRAs are predominantly used by financial intermediaries to hedge rate-sensitive asset or liability exposures and are priced at the implied forward rate.

Key Points

Discount factor DF(T) = 1/(1 + z_T)^T is PV of one unit at T
Implied forward rate IFR_{A,B-A} links z_A and z_B via no-arbitrage
FRAs fix a future MRR; settlement discounted to start of the period
FRAs mainly used by financial intermediaries to hedge balance-sheet items

Futures pricing and daily mark-to-market5 min

Interest-rate futures are standardized, exchange-traded instruments with futures price convention f = 100 - 100M, where M is the implied yield (MRR); at initiation f0(T)=S0(1 + r)^T adjusted for PV of costs and benefits of ownership. The principal practical difference between interest-rate futures and FRAs is daily margining and mark-to-market which changes cash-flow timing relative to a forward contract; this produces a pricing difference that depends on the correlation between futures prices and interest rates and on interest-rate volatility. For short-term interest-rate contracts, the linear price/yield relationship produces a BPV (basis point value) equal to Notional 0.0001 * Period. Convexity bias arises because FRA settlement is discounted at the realized MRR whereas futures settlement is linear; the FRA payoff shows nonlinearity (convexity) relative to futures, especially for longer discounting periods.

Key Points

Futures price convention for rates: f = 100 - 100 * MRR
At initiation f0(T) equals S0 compounded at risk-free rate adjusted for PV of costs/benefits
Daily mark-to-market creates different cash flows vs forwards; affects pricing
BPV for interest futures: Notional * 0.0001 * Period
Convexity bias: FRA discounting vs futures linear payoff

Central clearing and effect on forward/futures pricing4 min

The emergence of central clearing for OTC derivatives has reduced the practical cash-flow difference between exchange-traded futures and cleared OTC forwards by imposing futures-like margining on cleared bilateral trades, reducing the futures-versus-forward price gap. Dealers that must post margin to a central counterparty typically pass margin/collateral requirements to end users, reducing the relative pricing difference driven by margin timing. The sign and magnitude of the futures-minus-forward price differential depend on the correlation of futures prices with interest rates and interest-rate volatility.

Key Points

Central clearing imposes margining on OTC forwards, making cash flows more like futures
This reduces futures vs forward price differences
Correlation between futures prices and rates and volatility determine sign/magnitude of difference

Swaps as series of FRAs and par swap rate6 min

Swaps are series of periodic fixed-for-floating exchanges: an interest-rate swap with fixed rate s_N is priced so that the PV of fixed payments equals the PV of expected floating payments (derived from implied forward rates), yielding the par swap rate. A swap at inception has zero value; its MTM evolves as time passes (some forward-periods drop out) and as forward curve shifts. From the fixed-rate payer perspective, an increase in expected forward rates raises the PV of floating receipts and yields an MTM gain; a decrease yields an MTM loss. Swaps may be replicated or decomposed into a series of FRAs; swaps are commonly used by issuers/investors to transform balance-sheet cash flows since a single constant fixed swap rate replaces a sequence of differing FRA fixed rates. Practical valuation uses zero rates (discount factors) to compute the par swap rate: solve sum(PV of IFRs) = s_N * sum(discount factors) for s_N.

Key Points

Swap fixed rate (par swap rate) equates PV(floating) and PV(fixed)
Swap equals series of FRAs but with constant fixed rate across periods
Swap MTM depends on time passage and forward curve shifts
Use zero curve and discount factors to compute par swap rate

Swap applications and analogies to bonds5 min

Swaps are often preferred by issuers/investors to match periodic cash flows to liabilities or assets. Paying fixed is economically like shorting a fixed-rate bond and going long a floating-rate note; receiving fixed is like holding a fixed-coupon bond. Using swaps is operationally simpler than entering multiple FRAs and allows portfolio managers to change duration efficiently. The chapter provides worked examples of Esterr and Fyleton demonstrating swap cash flows, valuation, and MTM behavior.

Key Points

Pay-fixed swap ~ short fixed coupon bond + long floating-rate note
Receive-fixed swap ~ long fixed coupon bond
Swaps used to adjust portfolio duration and hedge reinvestment risk
Examples show fixed vs floating net payments and MTM effects

Practical conventions and settlement mechanics4 min

Throughout valuation examples the chapter emphasizes correct day-count/convention handling for period fractions, the use of PV discounting for FRA cash settlement (discounting final net payment back to settlement using observed MRR), and futures margining mechanics (initial and maintenance margins, variation margin). BPV computations and notional conventions are given for interest-rate futures. Swap settlement conventions (periodic netting, payment at period end or start depending on market) are highlighted as important for accurate calculations.

Key Points

FRA settlement is discounted to start of period using MRR for PV
Futures require initial and maintenance margins and daily variation margin
BPV formula: Notional * 0.0001 * Period
Day-count and period fraction conventions affect PV and BPV

Questions

Question 1

Given zero rates z3 = 2.4485% and z4 = 2.6690% (annual compounding), calculate the implied one-year forward rate starting in three years, IFR3,1. Use the formula (1 + z3)^3 * (1 + IFR3,1) = (1 + z4)^4.

Learning Module 7 Pricing and Valuation of Interest Rates and Other Swaps

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