Topic 13: Curve-Based and Empirical Fixed-Income Risk Measures

Learning Objectives Coverage

LO1: Explain why effective duration and effective convexity are the most appropriate measures of interest rate risk for bonds with embedded options

Core Concept exam-focus

Effective duration and convexity measure a bond’s price sensitivity to parallel shifts in the benchmark yield curve, accounting for how embedded options affect cash flows as interest rates change. Unlike yield-based duration measures that assume fixed cash flows, effective measures capture the path-dependent nature of callable, putable, and prepayable bonds — where the cash flow pattern itself changes with interest rates. This makes effective duration and effective convexity the required measures for any bond with embedded options, including mortgage-backed securities where prepayment behavior creates significant negative convexity. duration

Formulas & Calculations

Effective Duration:

EffDur = (PV⁻ − PV⁺) / [2 × (ΔCurve) × (PV₀)]

Where:
- PV⁻ = Price when curve shifts down
- PV⁺ = Price when curve shifts up
- ΔCurve = Parallel shift magnitude
- PV₀ = Current bond price

Effective Convexity:

EffCon = [(PV⁻) + (PV⁺) − 2(PV₀)] / [(ΔCurve)² × (PV₀)]

Price Change with Effective Measures:

%ΔPVFull ≈ (−EffDur × ΔCurve) + [½ × EffCon × (ΔCurve)²]

HP 12C Steps (Effective Duration):

Example: PV₀ = 101, PV⁺ = 99, PV⁻ = 103, ΔCurve = 0.25%

103 [ENTER] 99 [-]         (PV⁻ − PV⁺ = 4)
2 [ENTER] 0.0025 [×]       (2 × ΔCurve = 0.005)
101 [×]                    (× PV₀ = 0.505)
[÷]                        (4 ÷ 0.505)
Result: 7.92               (Effective duration)

Practical Examples

Callable Bond Example:

30-year 5% callable bond, callable at 100 in 5 years
Current price: 101.060

Benchmark shifts:
-25 bps: Price = 102.891 (limited upside due to call)
+25 bps: Price = 99.050

EffDur = (102.891 - 99.050)/(2 × 0.0025 × 101.060) = 7.60
EffCon = (102.891 + 99.050 - 202.120)/(0.0025² × 101.060) = -283

Note: Negative convexity from call option

Putable Bond Example:

10-year 4% putable bond, putable at 98
Current price: 97.50

Benchmark shifts:
-25 bps: Price = 99.20
+25 bps: Price = 96.80 (limited downside due to put)

EffDur = 4.92
EffCon = 125 (positive, put provides protection)

DeFi Application

Protocol example: Maple Finance callable loans
Implementation: Smart contracts with early redemption features require effective duration modeling
Advantages/Challenges:
- Advantages: Dynamic risk assessment, optimal refinancing triggers
- Challenges: On-chain option models complex, gas-intensive calculations
- Example: Maple’s institutional loans use effective duration to price prepayment options

LO2: Calculate the percentage price change of a bond for a specified change in benchmark yield, given the bond’s effective duration and convexity

Core Concept

Definition: The price change formula using effective measures accounts for both linear (duration) and non-linear (convexity) effects of benchmark curve shifts on bonds with embedded options.
Why it matters: Accurate price predictions for option-embedded bonds require effective measures; traditional duration/convexity will misestimate price changes significantly.
Key components:
- Asymmetric price responses
- Negative convexity regions
- Path dependency considerations
- Model calibration importance

Formulas & Calculations

Complete Price Change Formula:

%ΔPVFull ≈ (−EffDur × ΔCurve) + [½ × EffCon × (ΔCurve)²]

Components:
- Duration effect: −EffDur × ΔCurve
- Convexity adjustment: ½ × EffCon × (ΔCurve)²

Asymmetric Price Response (Callable bonds):

For rates down: Limited upside due to call option
For rates up: Full downside exposure

Example:
EffDur = 7.6, EffCon = -285

-100 bps: %ΔPrice = 7.6 - 1.43 = +6.17%
+100 bps: %ΔPrice = -7.6 - 1.43 = -9.03%

HP 12C Steps (Complete calculation):

Example: EffDur = 5.2, EffCon = -150, ΔCurve = +75 bps

5.2 [CHS] 0.0075 [×]       (Duration effect = -0.039)
0.5 [ENTER] 150 [CHS] [×]  (0.5 × EffCon = -75)
0.0075 [x²] [×]            (× (ΔCurve)² = -0.00422)
[+]                        (Total effect)
Result: -0.04322           (-4.32% price change)

Practical Examples

MBS Prepayment Example:

Mortgage-backed security
EffDur = 3.5 (prepayment-adjusted)
EffCon = -180 (negative due to prepayment option)

Scenario: Fed cuts rates 150 bps
Duration effect: +5.25%
Convexity effect: -2.03%
Net change: +3.22%

Without effective measures: Would overestimate at +7.5%

Convertible Bond Example:

Convertible with equity sensitivity
EffDur = 4.8, EffCon = 250

For various curve shifts:
-50 bps: +2.40% + 0.31% = +2.71%
+50 bps: -2.40% + 0.31% = -2.09%
+200 bps: -9.60% + 5.00% = -4.60%

DeFi Application

Protocol example: Ribbon Finance structured products
Implementation: Options vaults calculate effective duration for yield-bearing positions with embedded optionality
Advantages/Challenges:
- Advantages: Better hedge ratios, improved risk/reward optimization
- Challenges: Volatility surface modeling on-chain, oracle dependencies
- Example: DOV (DeFi Option Vaults) use effective convexity to optimize strike selection

LO3: Define key rate duration and describe its use to measure price sensitivity of fixed-income instruments to benchmark yield curve changes

Core Concept

Definition: Key rate duration (partial duration) measures a bond’s price sensitivity to a shift in the benchmark yield curve at a specific maturity point, holding all other rates constant.
Why it matters: Yield curves rarely shift in parallel; key rate durations capture sensitivity to twists, butterflies, and other non-parallel movements that dominate real markets.
Key components:
- Maturity buckets (0.5, 2, 5, 10, 20, 30 years)
- Shaping risk identification
- Portfolio construction applications
- Sum equals effective duration

Formulas & Calculations

Key Rate Duration Formula:

KeyRateDurₖ = −(1/PV) × (ΔPV/Δrₖ)

Or approximately:
KeyRateDurₖ = (PV⁻ₖ − PV⁺ₖ) / [2 × (Δrₖ) × (PV₀)]

Where k = specific maturity point

Sum Property:
```
Σ(KeyRateDurₖ) = EffDur
k=1 to n
```

Price Impact Formula:

Total %ΔPV = Σ(−KeyRateDurₖ × Δrₖ)

HP 12C Steps (Key rate calculation):

Example: 5-year key rate, PV₀ = 100
Shift 5-year rate ±10 bps: PV⁺ = 99.83, PV⁻ = 100.17

100.17 [ENTER] 99.83 [-]   (PV⁻ − PV⁺ = 0.34)
2 [ENTER] 0.001 [×]        (2 × Δr = 0.002)
100 [×] [÷]                (Divide by denominator)
Result: 1.70               (5-year key rate duration)

Practical Examples

Barbell Portfolio Analysis:

$100M portfolio: 50% 2-year, 50% 30-year bonds

Key Rate Durations:
2-year: 0.95
5-year: 0.20
10-year: 0.50
30-year: 14.35
Total (EffDur): 16.00

Curve Steepening Scenario:
2-year: -25 bps → Impact: +0.24%
30-year: +50 bps → Impact: -7.18%
Net impact: -6.94%

Ladder Portfolio Comparison:

Equal-weighted 2, 5, 10, 20, 30-year bonds

Key Rate Profile:
2-year: 0.40
5-year: 1.00
10-year: 2.00
20-year: 4.00
30-year: 6.00

More balanced exposure to curve reshaping

DeFi Application

Protocol example: Element Finance fixed-rate positions
Implementation: Principal/Yield token splits have different key rate exposures
Advantages/Challenges:
- Advantages: Precise yield curve positioning, optimized term structure trades
- Challenges: Limited maturity points in DeFi, liquidity concentration
- Example: Element’s convergence trades exploit key rate duration differentials

LO4: Describe the difference between empirical duration and analytical duration

Core Concept

Definition: Empirical duration is estimated from historical price/yield relationships using regression analysis, while analytical duration uses mathematical formulas assuming independence between benchmark yields and credit spreads.
Why it matters: During market stress, credit spreads often widen when government yields fall (negative correlation), making empirical duration more accurate for credit-risky bonds.
Key components:
- Statistical vs theoretical approach
- Credit spread correlation effects
- Flight-to-quality dynamics
- Model selection criteria

Formulas & Calculations

Analytical Duration (Formula-based):

Assumes: ΔPrice = f(ΔYield) + g(ΔSpread)
Where f and g are independent

ModDur or EffDur calculated from bond mathematics

Empirical Duration (Regression-based):

Regression model:
ΔPrice/Price = α + β₁(ΔBenchmark) + β₂(ΔSpread) + ε

Empirical Duration = −β₁

Accounts for correlation: Corr(ΔBenchmark, ΔSpread) ≠ 0

Flight-to-Quality Adjustment:

If Corr(ΔYield, ΔSpread) = -0.5:

Empirical Duration ≈ Analytical Duration × (1 + ρ × SpreadSensitivity)

Example:
Analytical Duration = 7.0
Spread Duration = 6.5
Correlation = -0.4

Empirical Duration ≈ 7.0 × (1 - 0.4 × 6.5/7.0) = 4.4

HP 12C Steps (Empirical adjustment):

Example: Analytical = 8, Spread sensitivity = 0.6, Correlation = -0.3

0.6 [ENTER] 0.3 [CHS] [×]  (Spread effect = -0.18)
1 [+]                      (1 + effect = 0.82)
8 [×]                      (Adjusted duration)
Result: 6.56               (Empirical duration)

Practical Examples

Investment Grade Corporate:

BBB-rated 10-year bond
Analytical Duration: 8.5

Normal markets (correlation ≈ 0):
Empirical Duration ≈ 8.5

Crisis period (correlation = -0.6):
Empirical Duration ≈ 5.1

Price predictions for -100 bps Treasury yield:
Analytical: +8.5%
Empirical: +5.1% (accounts for +60 bps spread widening)

High Yield Bond Example:

B-rated 5-year bond
Analytical Duration: 4.2
Spread Duration: 4.0

Historical regression (2008-2009 data):
ΔPrice = -0.02 - 2.8(ΔTreasury) - 3.9(ΔSpread)

Empirical Duration = 2.8 (vs 4.2 analytical)

Explains why high yield underperforms in flight-to-quality

DeFi Application

Protocol example: TrueFi uncollateralized lending
Implementation: Credit pools require empirical duration for accurate risk assessment
Advantages/Challenges:
- Advantages: Captures DeFi-TradFi correlations, better stress testing
- Challenges: Limited historical data, rapidly evolving correlations
- Example: TrueFi uses on-chain regression models to estimate empirical durations for loan portfolios

Core Concepts Summary (80/20 Principle)

Essential Knowledge (80% of value)

Effective measures: Required for bonds with embedded options
Key rate duration: Captures non-parallel yield curve shifts
Empirical vs Analytical: Credit correlation matters in stressed markets
Negative convexity: Callable bonds exhibit when rates fall

Advanced Concepts (20% remaining)

Shaping risk decomposition using key rates
Multi-factor regression for empirical duration
Option-adjusted spread (OAS) applications
Cross-curve key rate durations

Comprehensive Formula Sheet formula

Primary Formulas

1. Effective Duration:
   EffDur = (PV⁻ − PV⁺) / [2 × ΔCurve × PV₀]

2. Effective Convexity:
   EffCon = (PV⁻ + PV⁺ − 2×PV₀) / [(ΔCurve)² × PV₀]

3. Price Change (Effective):
   %ΔPV ≈ −EffDur × ΔCurve + ½ × EffCon × (ΔCurve)²

4. Key Rate Duration:
   KeyRateDurₖ = (PV⁻ₖ − PV⁺ₖ) / [2 × Δrₖ × PV₀]

5. Sum of Key Rates:
   Σ(KeyRateDurₖ) = EffDur

6. Empirical Duration (Regression):
   ΔPrice/Price = α + β₁(ΔBenchmark) + β₂(ΔSpread)
   EmpiricalDur = −β₁

Variable Definitions

PV⁺, PV⁻: Prices after positive/negative curve shifts
ΔCurve: Parallel benchmark curve shift
Δrₖ: Shift at maturity k only
β₁, β₂: Regression coefficients
ρ: Correlation between yields and spreads

HP 12C Calculator Sequences hp12c

Effective Duration Calculation

Given: PV₀ = 102, PV⁺ = 100.5, PV⁻ = 103.2, ΔCurve = 0.20%

103.2 [ENTER] 100.5 [-]    → 2.7
2 [ENTER] 0.002 [×]        → 0.004
102 [×]                    → 0.408
[÷]                        → 6.62 (Effective Duration)

Effective Convexity Calculation

Same data as above:

103.2 [ENTER] 100.5 [+]    → 203.7
102 [ENTER] 2 [×] [-]      → -0.3
0.002 [x²] [ENTER]         → 0.000004
102 [×] [÷]                → -735 (Effective Convexity)

Key Rate Duration Impact

Portfolio with three key rate exposures:
2-year: KeyDur = 0.5, Δr = -25 bps
10-year: KeyDur = 3.0, Δr = +50 bps
30-year: KeyDur = 8.0, Δr = +75 bps

0.5 [ENTER] 0.0025 [×]     → 0.00125
3.0 [ENTER] 0.005 [CHS] [×] → -0.015
8.0 [ENTER] 0.0075 [CHS] [×] → -0.06
[+] [+]                    → -0.07375 (-7.38% total impact)

Practice Problems

Basic Level

Effective Duration: Callable bond with PV₀ = 98, PV⁺ = 96.5 (curve up 50 bps), PV⁻ = 99.2 (curve down 50 bps). Calculate effective duration.
- Answer: 2.76 [(99.2-96.5)/(2×0.005×98)]
Price Change: Bond with EffDur = 4.5, EffCon = -120. Estimate price change for -75 bps curve shift.
- Answer: +3.72% [4.5×0.0075 + 0.5×(-120)×0.0075²]
Key Rate Sum: Bond has key rate durations: 2yr = 0.3, 5yr = 0.8, 10yr = 2.1, 30yr = 3.8. What is effective duration?
- Answer: 7.0 years (sum of all key rates)

Intermediate Level

Negative Convexity Impact:

Callable bond: EffDur = 6.2, EffCon = -250
Compare price changes for ±100 bps:

-100 bps: +6.2% - 1.25% = +4.95%
+100 bps: -6.2% - 1.25% = -7.45%

Asymmetry: 2.50% worse on downside

Key Rate Scenario:

Portfolio exposures:
5-year: KeyDur = 1.5
10-year: KeyDur = 4.0
30-year: KeyDur = 6.5

Curve flattening: 5yr +30bps, 10yr +10bps, 30yr -20bps

Impact: -1.5(0.003) - 4.0(0.001) + 6.5(0.002) = +0.45%

Empirical vs Analytical:

Corporate bond: Analytical Duration = 7.5
During crisis: Treasury yields -150 bps, Spreads +100 bps
Correlation = -0.6

Empirical Duration ≈ 4.5
Actual price change: +2.25% (not +11.25% from analytical)

Advanced Level

MBS Prepayment Modeling:

Current coupon MBS: Price = 102.5

Rate scenarios with prepayment model:
-100 bps: Price = 103.8 (high prepayments cap upside)
+100 bps: Price = 99.2

EffDur = 2.3 (vs 5.8 for comparable Treasury)
EffCon = -480

For -200 bps: Estimate vs actual price change

Multi-Factor Duration:

Convertible bond sensitive to:
- Interest rates (EffDur = 3.5)
- Credit spreads (SpreadDur = 3.0)
- Equity prices (Delta = 0.4)

Scenario: Rates -50bps, Spreads +30bps, Equity +5%

Total return: 1.75% - 0.90% + 2.00% = +2.85%

DeFi Protocol Risk:

Compound cDAI position: $10M supplied
Empirical analysis shows:
- Duration to USDC rates: 1.8
- Duration to ETH staking: 0.5
- Cross-effects: -0.3

If USDC rates +100bps, ETH staking -50bps:
Net impact on APY?

DeFi Applications & Real-World Examples

Perpetual Protocols

GMX and GLP Token Duration:

GLP exhibits negative empirical duration during market stress
Treasury yields down → Risk-off → GLP redemptions
Empirical duration captures this better than analytical
Key rate exposure mainly to short-term DeFi rates

Fixed-Rate Protocol Design

Pendle Finance YT Tokens:

Yield Token characteristics:
- High effective duration (leveraged rate exposure)
- Negative convexity near maturity
- Key rate duration concentrated at maturity date

Example: 1-year YT-aUSDC
EffDur = 0.95 × (1 + Yield/Price)
If trading at 5% discount: EffDur ≈ 20

Options Protocols

Dopex Interest Rate Options:

Atlantic options on rates require effective convexity modeling
Key rate durations determine optimal strike placement
Empirical correlations between DeFi/TradFi rates critical
Delta-hedging uses effective duration calculations

Lending Protocol Risk

Aave V3 Rate Sensitivity:

Variable rate positions:
- Analytical duration ≈ 0.25 (quarterly reset assumption)
- Empirical duration ≈ 0.10 (actual daily adjustments)

Stable rate positions:
- Effective duration depends on rebalancing threshold
- Key rate exposure to both supply/borrow curves

Common Pitfalls & Exam Tips

Calculation Errors to Avoid

Wrong curve shift: Using YTM change instead of benchmark curve
Sign confusion: Negative convexity still uses same formula
Key rate sum: Forgetting sum must equal effective duration
Regression interpretation: Empirical duration is negative of coefficient

Conceptual Traps

Effective ≠ Modified: Even for option-free bonds if curve not flat
Negative convexity: Only for callable bonds when rates fall
Empirical always lower: Only true in flight-to-quality scenarios
Key rates independent: Actually have some correlation in practice

Exam Strategy

Identify bond type first: Embedded options → Use effective measures
Check for correlations: Credit risk → Consider empirical duration
Curve shape matters: Non-parallel → Use key rate durations
Time allocation: These problems typically require 3-4 minutes

Quick Recognition Patterns

“Callable/putable” → Effective duration/convexity
“Twist/butterfly” → Key rate duration
“Flight-to-quality” → Empirical duration
“Benchmark curve” → Not YTM-based measures

Key Takeaways

Must-Know Concepts

✅ Effective measures required for embedded options
✅ Key rate durations sum to effective duration
✅ Empirical duration < Analytical during flight-to-quality
✅ Callable bonds can have negative effective convexity
✅ Curve-based measures use benchmark shifts, not YTM

Critical Formulas

✅ EffDur = (PV⁻ − PV⁺)/(2 × ΔCurve × PV₀)
✅ %ΔPV ≈ −EffDur × ΔCurve + ½ × EffCon × (ΔCurve)²
✅ Σ(KeyRateDurₖ) = EffDur

Practical Applications

✅ Use effective for MBS, callable, putable bonds
✅ Apply key rates for curve positioning strategies
✅ Consider empirical for credit-sensitive portfolios
✅ DeFi protocols need all three approaches

Cross-References & Additional Resources

Topic 11: Yield-Based Duration (foundation) duration
Topic 12: Convexity (complementary)
Topic 14: Credit Risk (empirical duration applications) credit-analysis
Topics 17-19: Securitization (MBS effective duration)
Topic 9: Term Structure (benchmark curves) yield-curve

DeFi Protocol Documentation

Pendle Docs - Fixed yield tokenization
Element Finance - Principal/Yield splits
Maple Finance - Institutional lending with options
Notional V3 - Fixed-rate lending

Advanced Reading

Fabozzi: “Fixed Income Analysis” Ch. 7-8
Tuckman & Serrat: “Fixed Income Securities” Ch. 7
Research: “Key Rate Durations: Measures of Interest Rate Risks” (J.P. Morgan)
DeFi Paper: “Interest Rate Risk in DeFi” (Gauntlet)

Online Tools & Calculators

Bloomberg Terminal - Professional key rate analysis
QuantLib Python - Open-source duration calculations
DeFi Pulse - DeFi rate tracking
Dune Analytics - On-chain empirical analysis

Review Checklist

Conceptual Understanding

Explain why effective measures needed for embedded options
Describe negative convexity in callable bonds
Understand key rate duration applications
Differentiate empirical vs analytical duration

Calculation Proficiency

Calculate effective duration and convexity from prices
Estimate price changes using effective measures
Apply key rate durations to curve scenarios
Adjust for empirical correlations

Application Skills

Identify when to use each duration type
Analyze MBS and callable bond risks
Design curve positioning strategies
Apply concepts to DeFi protocols

Exam Readiness

Complete practice problems without errors
Recognize problem types quickly
Remember all formula variations
Understand all Learning Objectives

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Curve-Based and Empirical Fixed-Income Risk Measures

Topic 13: Curve-Based and Empirical Fixed-Income Risk Measures

Learning Objectives Coverage

LO1: Explain why effective duration and effective convexity are the most appropriate measures of interest rate risk for bonds with embedded options

Core Concept exam-focus

Formulas & Calculations

Practical Examples

DeFi Application

LO2: Calculate the percentage price change of a bond for a specified change in benchmark yield, given the bond’s effective duration and convexity

Core Concept

Formulas & Calculations

Practical Examples

DeFi Application

LO3: Define key rate duration and describe its use to measure price sensitivity of fixed-income instruments to benchmark yield curve changes

Core Concept

Formulas & Calculations

Practical Examples

DeFi Application

LO4: Describe the difference between empirical duration and analytical duration

Core Concept

Formulas & Calculations

Practical Examples

DeFi Application

Core Concepts Summary (80/20 Principle)

Essential Knowledge (80% of value)

Advanced Concepts (20% remaining)

Comprehensive Formula Sheet formula

Primary Formulas

Variable Definitions

HP 12C Calculator Sequences hp12c

Effective Duration Calculation

Effective Convexity Calculation

Key Rate Duration Impact

Practice Problems

Basic Level

Intermediate Level

Advanced Level

DeFi Applications & Real-World Examples

Perpetual Protocols

Fixed-Rate Protocol Design

Options Protocols

Lending Protocol Risk

Common Pitfalls & Exam Tips

Calculation Errors to Avoid

Conceptual Traps

Exam Strategy

Quick Recognition Patterns

Key Takeaways

Must-Know Concepts

Critical Formulas

Practical Applications

Cross-References & Additional Resources

Related Finance Topics

DeFi Protocol Documentation

Advanced Reading

Online Tools & Calculators

Review Checklist

Conceptual Understanding

Calculation Proficiency

Application Skills

Exam Readiness

Graph View

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