Topic 7: Pricing and Valuation of Interest Rates and Other Swaps
Learning Objectives Coverage
LO1: Describe how swap contracts are similar to but different from a series of forward contracts
Core Concept
Swaps are derivatives that exchange cash flows over multiple periods using a single fixed rate, while a series of forward contracts would use different forward rates for each maturity. Both have zero initial value and symmetric payoff profiles, but they differ in rate structure and settlement timing. Understanding the swap-as-forward-strip relationship is crucial for grasping how the par swap rate is derived and for connecting swap pricing to the fixed income yield curve. exam-focus
Key Similarities and Differences
Similarities:
- Both are firm commitments (not options)
- Zero initial value (ignoring costs)
- Symmetric payoff profiles
- Counterparty credit exposure
- Can hedge or speculate on rates
Differences:
| Aspect | Swap | Series of Forwards |
|---|---|---|
| Rate Structure | Single fixed rate | Different forward rates |
| Settlement | End of period | Beginning (FRA) or end |
| Complexity | One contract | Multiple contracts |
| Liquidity | Standardized market | Less liquid |
| Documentation | Single ISDA agreement | Multiple agreements |
Practical Example
$10M 3-Year Quarterly Swap vs FRAs:
Swap Approach:
- Single contract at 3.5% fixed
- 12 identical fixed payments
- One counterparty relationship
FRA Approach:
- 12 separate FRA contracts
- Different rates: 3.2%, 3.4%, 3.6%…
- Multiple execution times
- Higher transaction costs
DeFi Application defi-application
Uniswap v3 concentrated liquidity positions act like swap books, with Automated Market Makers (AMMs) replacing traditional order books. ETH/USDC liquidity providers earn “swap fees” in a manner similar to fixed income coupon collection. Meanwhile, perpetual swaps on platforms like dYdX combine swap and futures concepts, creating a hybrid instrument unique to DeFi.
LO2: Contrast the value and price of swaps
Core Concept
The “price” of a swap is the fixed rate (par swap rate) that makes the initial value zero. The “value” is the mark-to-market worth of the swap position after inception, which changes with interest rate movements and time. As with forward contracts, the price-versus-value distinction is a frequent source of exam questions. exam-focus pricing
Key Formulas
Par Swap Rate (Price):
Σ(i=1 to N) IFRᵢ/(1 + zᵢ)ⁱ = Σ(i=1 to N) sₙ/(1 + zᵢ)ⁱ
Swap Value (After Inception):
Value = PV(Fixed Leg) - PV(Floating Leg)
Periodic Settlement:
Settlement = (MRR - sₙ) × Notional × Period
Practical Example
Interest Rate Swap Valuation:
- Initial: 5-year swap at 3% fixed
- Price = 3% (par swap rate)
- Value = $0 at inception
After 1 Year (rates rise to 4%):
- New 4-year swap price = 4%
- Existing swap value = PV of receiving 3% vs 4%
- MTM gain for fixed receiver ≈ 1% × 4 years × Notional
DeFi Application defi-application
In DeFi, Compound Finance cTokens represent floating rate positions, while fixed rate protocols like Element Finance create fixed yields. Users can effectively swap between variable supply APY and a fixed rate — for example, locking in 5% on USDC versus Compound’s variable rate. This mirrors the traditional interest rate swap, where a floating-rate payer converts to fixed.
Core Concepts Summary (80/20 Principle)
The 20% You Must Know:
- Swaps = Series of cash flows at a single fixed rate
- Par swap rate makes initial value zero
- Value changes with rates and time after inception
- Fixed receiver gains when rates fall
- Fixed payer gains when rates rise
The 80% That Matters Most:
- Swap pricing uses implied forward rates from the yield curve
- Settlement occurs at period end (unlike FRAs)
- Credit exposure exists throughout swap life
- Swaps are more efficient than multiple forwards
- Interest rate swaps dominate OTC derivatives market
Comprehensive Formula Sheet
Implied Forward Rates
Basic Formula:
(1 + zₐ)^A × (1 + IFRₐ,ᵦ₋ₐ)^(B-A) = (1 + zᵦ)^B
Solving for IFR:
IFRₐ,ᵦ₋ₐ = [(1 + zᵦ)^B / (1 + zₐ)^A]^(1/(B-A)) - 1
First Period:
IFR₀,₁ = z₁
Par Swap Rate Calculation
General Formula:
sₙ = [Σ(i=1 to N) IFRᵢ × DFᵢ] / [Σ(i=1 to N) DFᵢ]
Where:
DFᵢ = Discount Factor = 1/(1 + zᵢ)ⁱ
For Par Bonds:
100 = Σ(i=1 to N) sₙ/(1 + zᵢ)ⁱ + 100/(1 + zₙ)^N
Swap Valuation
Fixed Rate Receiver Value:
V = Σ(i=1 to N) (sₙ × Notional × Period)/(1 + zᵢ)ⁱ - Σ(i=1 to N) (IFRᵢ × Notional × Period)/(1 + zᵢ)ⁱ
Simplified:
V = Notional × Period × Σ(i=1 to N) (sₙ - IFRᵢ)/(1 + zᵢ)ⁱ
Settlement Payment:
Payment = (MRR - sₙ) × Notional × Period
(Negative = fixed payer pays)
Currency Swap Formulas
Initial Exchange:
Exchange notionals at spot rate
Periodic Payments:
Each party pays interest in their currency
Final Exchange:
Re-exchange notionals at original spot rate
Value = PV(Receive Currency) - PV(Pay Currency) × Spot Rate
HP 12C Calculator Sequences
Calculate Implied Forward Rate
Example: IFR₁,₁ from z₁=2.5%, z₂=3.0%
[f] [CLX]
1.03 [ENTER]
2 [y^x] // (1.03)²
1.025 [÷] // ÷1.025
1 [-] // Result: 3.502% = IFR₁,₁
Calculate Par Swap Rate
Given: z₁=2%, z₂=2.5%, z₃=3%
Find: 3-year par swap rate
Step 1: Calculate IFRs
IFR₀,₁ = 2.0%
IFR₁,₁ = 3.0%
IFR₂,₁ = 4.0%
Step 2: Calculate Discount Factors
DF₁ = 1/1.02 = 0.9804
DF₂ = 1/(1.025)² = 0.9518
DF₃ = 1/(1.03)³ = 0.9151
Step 3: Par Swap Rate
[f] [CLX]
0.02 [ENTER] 0.9804 [×] [STO] 1
0.03 [ENTER] 0.9518 [×] [STO] 2
0.04 [ENTER] 0.9151 [×] [STO] 3
[RCL] 1 [RCL] 2 [+] [RCL] 3 [+] // Numerator
0.9804 [ENTER] 0.9518 [+] 0.9151 [+] // Denominator
[÷] // Result: 2.98%
Swap Settlement Calculation
Notional: $50M, Period: 0.25
Fixed rate: 2.5%, MRR: 3.0%
[f] [CLX]
0.03 [ENTER]
0.025 [-]
50000000 [×]
0.25 [×] // Result: $62,500 (fixed payer receives)
Practice Problems
Basic Level
Problem 1: Calculate the 2-year par swap rate:
- z₁ = 2.0%
- z₂ = 2.5%
Solution:
IFR₀,₁ = 2.0%
IFR₁,₁ = (1.025)²/1.02 - 1 = 3.0%
DF₁ = 1/1.02 = 0.9804
DF₂ = 1/(1.025)² = 0.9518
s₂ = (0.02×0.9804 + 0.03×0.9518)/(0.9804 + 0.9518)
s₂ = 0.04816/1.9322 = 2.49%
Problem 2: Quarterly settlement on $100M swap:
- Fixed rate: 3.6%
- Current MRR: 4.0%
- Calculate payment
Solution:
Payment = (0.04 - 0.036) × $100M × 0.25
Payment = 0.004 × $100M × 0.25 = $100,000
Fixed receiver receives $100,000
Intermediate Level
Problem 3: Value a 3-year swap after 1 year:
- Original fixed rate: 3%
- Current 2-year swap rate: 4%
- Notional: $50M
Solution:
Fixed Receiver Position:
Annual advantage = 4% - 3% = 1%
Years remaining = 2
Approximate value = 1% × 2 × $50M = $1M gain
Precise calculation:
V = $50M × 0.01 × [1/1.04 + 1/(1.04)²]
V = $500,000 × 1.8861 = $943,050
Advanced Level
Problem 4: Construct a 5-year swap hedge for a floating rate loan:
- Loan amount: $25M
- Floating rate: 3-month LIBOR + 150bp
- Current 5-year swap rate: 3.5%
- Goal: Fix the all-in cost
Solution:
Strategy: Pay fixed, receive floating in swap
Swap cash flows:
Pay: 3.5% fixed
Receive: 3-month LIBOR
Combined position:
Pay on loan: LIBOR + 1.5%
Receive from swap: LIBOR
Pay on swap: 3.5%
Net cost: 3.5% + 1.5% = 5.0% fixed
Quarterly payment = 5% × $25M × 0.25 = $312,500
DeFi Applications & Real-World Examples
1. Interest Rate Swap Protocols
Pendle Finance: defi-application
- Separates yield-bearing tokens into PT (Principal) and YT (Yield)
- PT = Fixed rate position
- YT = Floating rate position
- Example: Split stETH into fixed 4% APY and variable rewards
Element Finance:
- Creates fixed rate positions from variable yield
- Uses “Principal Tokens” for fixed income
- Enables term structure trading
- Example: Lock in Compound yields for 6 months
2. Automated Market Maker Swaps
Curve Finance:
- Specialized for stablecoin swaps
- Low slippage for large trades
- Similar to FX swap mechanics
- Example: USDC/USDT with 0.04% fees
Balancer:
- Weighted pools act like currency swaps
- Automatic rebalancing
- Multi-asset exposure
- Example: 80/20 ETH/DAI pool
3. Cross-Chain Swaps
Thorchain:
- Native cross-chain swaps
- No wrapped tokens needed
- Similar to currency swaps
- Example: Native BTC to ETH swap
4. Real-World Corporate Examples
Apple Inc. Interest Rate Swap:
Situation: $1B floating rate debt at LIBOR + 50bp
Action: Enter pay-fixed swap at 2.5%
Result: Fixed rate of 3% (2.5% + 0.5%)
Benefit: Rate certainty for budgeting
Tesla Currency Swap:
EUR revenue hedging:
- Receive EUR fixed 1%
- Pay USD fixed 2.5%
- Notional: €500M
- Hedges FX and rate risk
Common Pitfalls & Exam Tips
Common Mistakes to Avoid
-
Confusing price vs value
- Price = par swap rate (fixed rate)
- Value = MTM of existing position
-
Wrong settlement timing
- Swaps settle at period end
- FRAs settle at period beginning
-
Forgetting compounding
- Use (1+z)^n for multi-period rates
- Don’t just multiply
-
Sign conventions
- Positive value = gain for that position
- Negative settlement = payer makes payment
Exam Strategy Tips
-
Quick identification:
- “Exchange cash flows” → Swap
- “Single rate” → Not forward strip
- “Periodic settlement” → Not bullet payment
-
Position shortcuts:
- Fixed receiver = Long bond position
- Fixed payer = Short bond position
- Rates ↓ → Fixed receiver gains
-
Calculation efficiency:
- Set up discount factor table first
- Use approximations for quick checks
- Remember: Par swap < Average forward rate
Key Takeaways
Must Remember:
- Swaps use one fixed rate for all periods
- Initial value = zero at par swap rate
- Fixed receiver gains when rates fall
- Settlement at period end unlike FRAs
- Swaps are more efficient than forward strips
Critical Insights:
- Par swap rate is weighted average of forward rates
- Credit exposure varies throughout swap life
- Swaps can transform asset/liability profiles
- Central clearing reduces counterparty risk
- DeFi enables programmable swap logic
Cross-References & Additional Resources
Related Topics:
- Topic 5: Forward contract pricing
- Topic 6: Futures vs forwards comparison
- Topic 8: Option pricing (different payoff structure)
- Fixed Income topics on duration and yield curves
Key Readings:
- ISDA Documentation: Master agreements
- Hull, J.: “Options, Futures, and Other Derivatives” Ch. 7
- BIS Quarterly Review: OTC derivatives statistics
Practice Resources:
- CME Group: Interest Rate Swap Futures
- ISDA: SwapsInfo.org for market data
- Bloomberg: SWPM function tutorials
DeFi Protocols to Study:
- Pendle: Yield tokenization and trading
- Element Finance: Fixed rate protocols
- Notional Finance: Fixed rate lending
- Voltz Protocol: Interest rate swap AMM
Review Checklist
Conceptual Understanding
- Can you explain swaps vs forward strips?
- Do you understand par swap rate derivation?
- Can you identify fixed receiver vs payer positions?
- Do you know when swap value changes?
Calculations
- Can you calculate implied forward rates?
- Can you determine par swap rates?
- Can you value existing swap positions?
- Can you compute settlement payments?
Applications
- Can you design hedging strategies with swaps?
- Do you understand asset-liability management uses?
- Can you explain DeFi yield splitting?
- Can you compare traditional vs DeFi swaps?
Exam Readiness
- Memorized key formulas
- Practiced discount factor calculations
- Reviewed settlement conventions
- Completed practice problems
DeFi Integration
- Understand AMM swap mechanics
- Know fixed yield protocols
- Can explain yield tokenization
- Familiar with cross-chain swaps