Estimated study time: 60 minutes
Content:
An interest rate swap is an agreement between two counterparties to exchange a series of fixed cash flows for a series of floating cash flows (or vice versa) based on a notional principal amount. The notional is not exchanged — only the net interest differential is paid. In a plain vanilla interest rate swap, one party pays a fixed rate (the swap rate) and receives a floating rate (such as SOFR), while the counterparty does the opposite. The fixed rate that makes the swap have zero net present value at initiation (fair market value = 0) is called the par swap rate. At Level 2, candidates must price swaps by equating the present value of the fixed payments to the present value of the floating payments using the current spot rate curve.
Swap pricing uses the no-arbitrage principle. The floating leg of a swap has a known present value — the first floating payment is set in advance (the current reference rate), and because the notional is returned (hypothetically) at the end, the floating leg's value always equals par at each reset date. Therefore: PV(fixed leg) = PV(floating leg) = par (at initiation). The fixed swap rate (c) is determined by: Sum [c / (1+z_t)^t] + [1/(1+z_T)^T] = 1, where z_t are the spot rates for each payment period. Solving: c = (1 - 1/(1+z_T)^T) / Sum[1/(1+z_t)^t]. This is analogous to setting a bond's coupon so it prices at par, where the coupon rate equals the par yield (which equals the swap rate for equivalent maturities).
After initiation, a swap's value changes as interest rates change. The value of the fixed-rate payer (who pays fixed and receives floating) increases when rates rise (the new floating payments received are worth more, while the fixed payments remain unchanged). The value of the fixed-rate payer at any time during the swap's life equals: Value = PV(floating leg) - PV(fixed leg). If rates rise significantly above the swap rate, the fixed-rate payer's swap has positive value (it's receiving above-market floating while paying below-market fixed). Conversely, the fixed-rate receiver's value falls when rates rise. These mark-to-market values are important for collateral management and for computing the economic effect of entering or unwinding swaps.
Currency swaps involve exchanging cash flows in two different currencies, including the exchange of notional principal at initiation and maturity (unlike interest rate swaps). A standard cross-currency swap involves: exchanging notional in two currencies at initiation, paying fixed or floating interest in one currency while receiving fixed or floating in the other, and re-exchanging notional at maturity. Currency swaps are used by multinationals to hedge long-term foreign currency cash flow exposures (e.g., a US company issuing EUR bonds and swapping EUR fixed payments for USD floating) and by foreign entities seeking funding in local currency markets. Cross-currency basis swaps (floating-for-floating) capture the cross-currency basis spread — the premium or discount that emerges in the swap market relative to covered interest rate parity due to dollar funding demand.
A swaption is an option to enter into a swap at a future date. A payer swaption gives the holder the right to enter into a swap as the fixed-rate payer (and floating-rate receiver). A receiver swaption gives the holder the right to enter as the fixed-rate receiver. Payer swaptions increase in value when interest rates rise (the right to pay a below-market fixed rate becomes more valuable). Receiver swaptions increase in value when rates fall. At Level 2, swaptions are analyzed using the same conceptual framework as interest rate options: payer swaption ≈ call option on interest rates (benefits from rate rises); receiver swaption ≈ put option on interest rates (benefits from rate declines). Swaptions are used by corporations to cap the fixed rate on anticipated debt issuance, by mortgage servicers to hedge prepayment-driven interest rate risk, and by bond portfolio managers for strategic interest rate positioning.
Key Terms:
Quiz Questions:
Q1. The following spot rates are given: 1-year = 3%, 2-year = 3.5%, 3-year = 4%. Calculate the par 3-year annual-pay swap rate.
A) c = (1 - 1/(1.04)^3) / [1/(1.03) + 1/(1.035)^2 + 1/(1.04)^3] = (1 - 0.8890) / [0.9709 + 0.9335 + 0.8890] = 0.1110 / 2.7934 = 3.97%. B) c = (3% + 3.5% + 4%) / 3 = 3.5%. C) c = 4% (the longest spot rate). D) c = (1 - discount factor) / sum of discount factors = (1 - 0.9709) / 2.7934 = 0.0291 / 2.7934 = 1.04%.
Answer: A — Discount factors: DF1 = 1/1.03 = 0.9709; DF2 = 1/(1.035)^2 = 0.9335; DF3 = 1/(1.04)^3 = 0.8890. Sum of DFs = 2.7934. Par swap rate = (1 - DF3) / Sum(DFs) = (1 - 0.8890) / 2.7934 = 0.1110 / 2.7934 = 3.97%. This is the fixed rate that makes the swap's fixed leg equal in present value to its floating leg (which always equals par at initiation). The swap rate (3.97%) is between the 3-year spot rate (4%) and the 1-year rate (3%), as expected for an averaging of spot rates.
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Q2. A company entered a 5-year pay-fixed interest rate swap 2 years ago at a fixed rate of 3.5%. Today, the remaining 3-year par swap rate is 5.0%. The notional principal is $10M. Has the fixed-rate payer benefited or been harmed by the rate move, and approximately what is the swap's current value to the fixed-rate payer?
A) The fixed-rate payer benefited — rates rose, so the floating received is higher than the fixed paid; the swap has positive value to the fixed-rate payer. B) The fixed-rate payer was harmed — rising rates increase the value of the fixed leg, which the payer owes. C) The value of the swap to the fixed-rate payer = PV(floating leg) - PV(fixed leg at 3.5%) = par - (3.5% annuity at 5% + par at 5%) = $10M - PV of 3.5% coupon bond at 5% yield for 3 years. D) Both A and C are correct — the fixed-rate payer has benefited (positive value) and C describes the calculation correctly.
Answer: D — When market swap rates rose from 3.5% to 5.0%, the fixed-rate payer benefits: the fixed payments remain at 3.5% but the floating received is now based on the higher market rate. Value to fixed-rate payer = PV(floating leg at par) - PV(old fixed leg at 3.5%). The fixed leg is like a 3.5% coupon bond; at 5% discount rate: PV = $350K/1.05 + $350K/1.05^2 + $10,350K/1.05^3 (using $10M notional) = $333.3K + $317.4K + $8,941.5K ≈ $9,592K. Value to fixed-rate payer = $10,000K - $9,592K = +$408K. The positive value confirms the fixed-rate payer has benefited from rising rates.
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Q3. A US corporation plans to issue EUR 50M in bonds in 6 months to fund European operations. The company is concerned that EUR interest rates will rise before the issuance, increasing its borrowing cost. Which swaption position best hedges this risk?
A) Buy a receiver swaption in EUR to lock in a cap on the EUR fixed rate paid. B) Buy a payer swaption in EUR — the right to pay fixed in EUR at a predetermined rate. If EUR rates rise, the payer swaption allows the company to lock in the lower predetermined rate, offsetting the higher rates it would otherwise pay on the new bonds. C) Sell a payer swaption to generate premium income that offsets higher borrowing costs. D) Enter a receiver swaption to benefit from falling rates; the company need not hedge rising rates.
Answer: B — The company will issue EUR fixed-rate bonds and pay EUR fixed interest. It wants to cap the EUR fixed rate it will pay. A payer swaption (right to pay fixed/receive floating) gives the company the right to enter a pay-fixed swap at a predetermined rate. If EUR rates rise above the swaption strike, the company exercises the swaption, pays the below-market fixed rate on the swap, and offsets the higher coupon on the newly issued bonds. The net effect: the company's effective fixed borrowing cost is capped at the swaption strike rate (plus the premium cost).
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Q4. An investor is constructing a position equivalent to a receiver swaption without using swaptions. Which combination of instruments replicates a receiver swaption (right to receive fixed, pay floating)?
A) Long a payer swaption + short a bond. B) Long a bond (with the same maturity and coupon as the underlying swap) + sell an interest rate cap (or equivalently, buy an interest rate floor), within the put-call parity framework for swaptions. C) Long a payer swaption and short a receiver swaption at the same terms creates a forward swap, not a receiver swaption. D) Receiver swaption = Long cap - Short floor is incorrect; it is Long floor only.
Answer: C — This question tests put-call parity for swaptions: Payer Swaption - Receiver Swaption = Forward Swap (pay fixed, receive floating). Equivalently, Receiver Swaption = Payer Swaption - Forward Swap. The most direct replication: long payer swaption + long the receiver side of a forward swap (which can itself be replicated with bond positions). Option C correctly identifies that long payer + short receiver = forward swap — rearranging gives receiver swaption = payer swaption - forward swap. This is the put-call parity equivalent for interest rate derivatives.
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Q5. A corporation has $100M in floating-rate debt (SOFR + 150 bps). The CFO is concerned that SOFR will rise significantly and wants to convert the floating-rate exposure to fixed-rate. She enters a pay-fixed, receive-floating swap with a notional of $100M at a fixed rate of 4.5%, receiving SOFR. If SOFR is currently 3.0%, what is the company's effective annual cost of borrowing after the swap?
A) Effective rate = 4.5% + 150 bps = 6.0% fixed. B) Effective rate = 3.0% + 150 bps = 4.5% (current floating rate, unswapped). C) Effective rate = (SOFR + 150 bps) - SOFR + 4.5% = 4.5% + 150 bps = 6.0% fixed, regardless of where SOFR moves. D) Effective rate = 4.5% - SOFR + SOFR + 150 bps = 4.5% + 1.5% = 6.0% — same as A and C.
Answer: A — The swap converts floating to fixed. The company pays: (SOFR + 150 bps) on the bond; receives SOFR from the swap; pays 4.5% fixed on the swap. Net cost = (SOFR + 1.50%) - SOFR + 4.50% = 4.50% + 1.50% = 6.0%. The SOFR components cancel, leaving a fixed effective rate of 6.0% regardless of where SOFR moves. This demonstrates the swap's hedging function — the company has locked in a 6.0% total borrowing cost, eliminating exposure to SOFR increases. Options A, C, and D all give the same answer through different algebraic paths.
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