Yield-Based Bond Convexity and Portfolio Properties

Topic 12: Yield-Based Bond Convexity and Portfolio Properties

Learning Objectives Coverage

LO1: Calculate and interpret convexity and describe the convexity adjustment

Core Concept exam-focus

Convexity is a complementary risk metric that measures the second-order (non-linear) effect of yield changes on price for an option-free fixed-rate bond. Where modified duration provides only a linear approximation of price changes, convexity captures the curvature in the price-yield relationship — the mathematical second derivative. For larger yield movements, the convexity adjustment becomes crucial for accurate price estimation, especially for long-duration bonds or volatile markets. Convexity is always positive for option-free bonds (negative convexity arises only with embedded options, as explored in Topic 13), which means that bondholders always benefit from convexity: gaining more than duration predicts when yields fall, and losing less when yields rise. duration

Formulas & Calculations formula hp12c

• Convexity Adjustment Formula:

  %ΔPVFull ≈ (−AnnModDur × ΔYield) + [1/2 × AnnConvexity × (ΔYield)²]
  

• Approximate Convexity:

  ApproxCon = (PV⁻) + (PV⁺) − [2 × (PV₀)] / [(ΔYield)² × (PV₀)]
  

  Where: PV⁻ = price if yield decreases, PV⁺ = price if yield increases, PV₀ = current price

• Spreadsheet Convexity (exact calculation):

  For each cash flow:
  Convexity_CF = (Time) × (Time + 1) × (Weight) × (1 + r)^(-n)
  
  AnnConvexity = Σ(Convexity_CF) / (Periods per year)²
  

• Money Convexity:

  MoneyCon = AnnConvexity × PVFull
  ΔPVFull ≈ −(MoneyDur × ΔYield) + [1/2 × MoneyCon × (ΔYield)²]
  

• HP 12C Steps (Convexity Adjustment):

  Example: ModDur = 8, Convexity = 50, ΔYield = 1%
  
  8 [CHS] 0.01 [×]           (Duration effect = -0.08)
  0.5 [ENTER] 50 [×]         (0.5 × Convexity)
  0.01 [ENTER] 0.01 [×] [×]  (× ΔYield²)
  [+]                        (Add to duration effect)
  Result: -0.0775            (-7.75% price change)
  

• HP 12C Steps (Approximate Convexity):

  Example: PV₀ = 100, PV⁺ = 99.77, PV⁻ = 100.23, ΔYield = 0.05%
  
  100.23 [ENTER] 99.77 [+]   (PV⁻ + PV⁺ = 200)
  100 [ENTER] 2 [×] [-]      (- 2×PV₀)
  0.0005 [ENTER] [x²]        (ΔYield²)
  100 [×] [÷]                (Divide by ΔYield²×PV₀)
  Result: 24.24              (Approximate convexity)
  

Practical Examples

• Traditional Finance Example: 30-Year Treasury Bond

  30-year Treasury, 4.625% coupon, YTM = 4.75%
  Price: 98.022448
  Modified Duration: 15.91
  Convexity: 369.64
  
  For +100 bps yield change:
  Duration effect: -15.91%
  Convexity adjustment: +1.85% (0.5 × 369.64 × 0.01²)
  Total estimated change: -14.06%
  Actual change: -14.04%
  
  Error without convexity: 1.87%
  Error with convexity: 0.02%
  

• Money Convexity Application:

  $100M position in 5-year bond
  Modified Duration: 4.59
  Convexity: 24.24
  
  Money Duration: $459M
  Money Convexity: $2,424M
  
  For +100 bps:
  Duration loss: -$4,590,000
  Convexity gain: +$121,200
  Net change: -$4,468,800
  

DeFi Application defi-application

Curve Finance’s StableSwap invariant provides a direct application of convexity optimization in DeFi. Traditional AMMs like Uniswap use the constant-product formula (x*y=k), whose high convexity creates large impermanent loss for stablecoin swaps. Curve’s innovation was to flatten the bonding curve near equilibrium, reducing convexity and delivering roughly 80% lower impermanent loss for stable pairs. This is mathematically analogous to choosing between high-convexity and low-convexity bond portfolios: the shape of the price function matters enormously for expected outcomes under volatility.

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

Core Concept

• Definition: The complete price estimation formula combines linear (duration) and quadratic (convexity) components to accurately predict bond price changes for any yield movement.
• Why it matters: Duration alone underestimates gains when yields fall and overestimates losses when yields rise. The convexity adjustment corrects this asymmetry, crucial for risk management and portfolio optimization.
• Key components:
  • Duration component (first-order, linear)
  • Convexity component (second-order, quadratic)
  • Error analysis and approximation limits
  • Practical application thresholds

Formulas & Calculations

• Complete Price Change Formula:

  %ΔPVFull = −(ModDur × ΔYield) + [0.5 × Convexity × (ΔYield)²]
  
  Components:
  - Linear term: −ModDur × ΔYield (always negative for yield increases)
  - Quadratic term: 0.5 × Convexity × (ΔYield)² (always positive)
  

• Error Analysis:

  Error without convexity = Actual − Duration estimate
  Error with convexity = Actual − (Duration + Convexity estimate)
  
  Relative improvement = 1 − (Error with convexity / Error without convexity)
  

• Decision Rules:

  Use convexity adjustment when:
  - |ΔYield| > 50 bps
  - Duration > 10 years
  - Convexity > 100
  - Required accuracy < 0.1%
  

• HP 12C Steps (Complete calculation):

  Example: ModDur = 6, Convexity = 40, ΔYield = -0.5%
  
  6 [ENTER] 0.005 [×]        (Duration effect = 0.03)
  0.5 [ENTER] 40 [×]         (0.5 × Convexity = 20)
  0.005 [x²] [×]             (× (ΔYield)² = 0.0005)
  [+]                        (Total effect)
  Result: 0.0305             (+3.05% price change)
  

Practical Examples

• Comparison: Small vs Large Yield Changes:

  Bond: ModDur = 10, Convexity = 150
  
  For ±25 bps:
  Duration only: ±2.50%
  With convexity: -2.49% / +2.51%
  Convexity contribution: 0.01%
  
  For ±200 bps:
  Duration only: ±20.00%
  With convexity: -17.00% / +23.00%
  Convexity contribution: 3.00%
  

• Portfolio Impact Analysis:

  $500M portfolio, Weighted ModDur = 7.5, Weighted Convexity = 85
  
  Scenario: Fed raises rates 75 bps
  Duration loss: -5.625% = -$28.125M
  Convexity gain: +0.239% = +$1.195M
  Net loss: -5.386% = -$26.930M
  
  Saved by convexity: $1.195M (4.2% reduction in loss)
  

DeFi Application

• Protocol example: Yearn Finance yield optimization
• Implementation: Convexity calculations optimize vault rebalancing thresholds
• Advantages/Challenges:
  • Advantages: Better timing for strategy switches, reduced transaction costs
  • Challenges: On-chain convexity computation expensive, oracle dependencies
  • Example: yvUSDC vault uses convexity triggers to switch between Compound/Aave when rate differential exceeds convexity-adjusted threshold

LO3: Calculate portfolio duration and convexity and explain the limitations of these measures

Core Concept

• Definition: Portfolio duration and convexity are weighted averages of individual bond statistics, providing aggregate measures of interest rate sensitivity for the entire portfolio.
• Why it matters: Portfolio-level metrics enable risk management at scale, facilitate hedging decisions, and allow comparison across different portfolio strategies (barbell vs bullet).
• Key components:
  • Market value weighting methodology
  • Aggregation assumptions and limitations
  • Parallel shift requirement
  • Yield curve considerations

Formulas & Calculations

• Portfolio Duration (Weighted Average Method):

  Portfolio Duration = Σ(wi × Durationi)
  where: wi = MVi / Σ(MV) = market weight of bond i
  

• Portfolio Convexity (Weighted Average Method):

  Portfolio Convexity = Σ(wi × Convexityi)
  

• Alternative Method (Aggregate Cash Flows):

  1. Aggregate all portfolio cash flows by time period
  2. Calculate duration/convexity using aggregate flows
  3. More theoretically correct but computationally intensive
  

• Portfolio Price Change:

  %ΔPortfolio ≈ −(Portfolio Duration × ΔYield) + 
                [0.5 × Portfolio Convexity × (ΔYield)²]
  

• HP 12C Steps (Portfolio metrics):

  Example: Two bonds, 60%/40% weights
  Bond A: Duration = 5, Convexity = 30
  Bond B: Duration = 10, Convexity = 80
  
  Portfolio Duration:
  0.6 [ENTER] 5 [×]          (Bond A contribution = 3)
  0.4 [ENTER] 10 [×] [+]     (Bond B contribution = 4)
  Result: 7                   (Portfolio duration)
  
  Portfolio Convexity:
  0.6 [ENTER] 30 [×]         (Bond A contribution = 18)
  0.4 [ENTER] 80 [×] [+]     (Bond B contribution = 32)
  Result: 50                  (Portfolio convexity)
  

Practical Examples

• Barbell vs Bullet Portfolio Comparison:

  Both portfolios: $100M, Duration = 10 years
  
  Bullet Portfolio (100% in 10-year bonds):
  - Convexity: 120
  - For ±100 bps: -8.80% / +11.20%
  
  Barbell Portfolio (50% 5-year, 50% 30-year):
  - Convexity: 200
  - For ±100 bps: -8.50% / +11.50%
  
  Barbell advantage: 30 bps in both directions
  Annual value in volatile markets: ~$300,000
  

• Real Portfolio Example:

  Investment Grade Corporate Portfolio:
  
  Holdings:
  - 30% in 2-5 year bonds (Duration = 3.5, Convexity = 15)
  - 50% in 5-10 year bonds (Duration = 7.2, Convexity = 65)
  - 20% in 10-30 year bonds (Duration = 15.8, Convexity = 280)
  
  Portfolio metrics:
  Duration = 0.3(3.5) + 0.5(7.2) + 0.2(15.8) = 7.81
  Convexity = 0.3(15) + 0.5(65) + 0.2(280) = 93
  
  For +50 bps shock:
  Estimated change: -3.68%
  Actual (non-parallel): -3.95%
  Error from assumptions: 27 bps
  

DeFi Application

• Protocol example: Balancer v2 weighted pools
• Implementation: Portfolio duration/convexity metrics for multi-asset pools with different rate sensitivities
• Advantages/Challenges:
  • Advantages: Dynamic rebalancing based on aggregate convexity, automated risk parity
  • Challenges: Cross-asset correlations, different yield conventions, oracle complexity
  • Example: 80/20 WETH/USDC pool adjusts weights based on implied duration differentials to maintain target portfolio duration

Core Concepts Summary (80/20 Principle)

Essential Knowledge (80% of value)

1. Convexity Basics: Second-order price sensitivity, always positive for option-free bonds
2. Price Change Formula: %ΔPrice ≈ −Duration × ΔYield + 0.5 × Convexity × (ΔYield)²
3. Portfolio Metrics: Weighted averages of individual bond statistics
4. Key Relationships: Higher convexity from longer maturity, lower coupon, lower yield

Advanced Concepts (20% remaining)

1. Money convexity for position-level risk
2. Barbell vs bullet portfolio construction
3. Limitations of parallel shift assumption
4. Effective convexity for bonds with options

Comprehensive Formula Sheet formula

Primary Formulas

1. Convexity Adjustment:
   %ΔPVFull ≈ (−ModDur × ΔYield) + [0.5 × Convexity × (ΔYield)²]

2. Approximate Convexity:
   ApproxCon = [(PV⁻) + (PV⁺) − 2(PV₀)] / [(ΔYield)² × (PV₀)]

3. Money Convexity:
   MoneyCon = AnnConvexity × PVFull

4. Portfolio Duration:
   Portfolio Duration = Σ(wi × Durationi)

5. Portfolio Convexity:
   Portfolio Convexity = Σ(wi × Convexityi)

6. Exact Convexity (Spreadsheet):
   ConvexityCF = t × (t+1) × WeightCF × (1+r)^(-periods)
   AnnConvexity = Σ(ConvexityCF) / (periods per year)²

Variable Definitions

• ModDur: Modified duration
• ΔYield: Change in yield-to-maturity
• PV⁺, PV⁻, PV₀: Prices at higher yield, lower yield, current yield
• wi: Market value weight of bond i
• t: Time to cash flow receipt
• r: Periodic yield

HP 12C Calculator Sequences hp12c

Convexity Adjustment Calculation

Given: ModDur = 8, Convexity = 50, ΔYield = +1%

Step 1: Duration effect
8 [CHS] [ENTER]
0.01 [×]
Store: -0.08

Step 2: Convexity effect
0.5 [ENTER]
50 [×]
0.01 [ENTER] 0.01 [×] [×]
Store: 0.0025

Step 3: Total effect
[RCL] [+]
Result: -0.0775 (-7.75%)

Portfolio Convexity Calculation

Bond A: Weight = 40%, Convexity = 25
Bond B: Weight = 60%, Convexity = 75

0.40 [ENTER] 25 [×]        → 10
0.60 [ENTER] 75 [×]        → 45
[+]                        → 55 (Portfolio Convexity)

Approximate Convexity from Prices

PV₀ = 100, PV⁺ = 99.5, PV⁻ = 100.51, ΔYield = 0.1%

99.5 [ENTER] 100.51 [+]    → 200.01
100 [ENTER] 2 [×] [-]      → 0.01
0.001 [x²] [ENTER]         → 0.000001
100 [×] [÷]                → 100 (Approximate Convexity)

Practice Problems

Basic Level

1. Simple Convexity Adjustment: A bond has modified duration of 5 and convexity of 25. Calculate the percentage price change for a 50 bp increase in yield.

   • Answer: -2.44% [−5(0.005) + 0.5(25)(0.0025)]

2. Portfolio Duration: Portfolio has 70% in Bond A (Duration = 4) and 30% in Bond B (Duration = 8). Find portfolio duration.

   • Answer: 5.2 years [0.7(4) + 0.3(8)]

3. Money Convexity: Bond with convexity of 40 and price of $950. Calculate money convexity.

   • Answer: $38,000 [40 × 950]

Intermediate Level

4. Complete Price Estimation: Bond with ModDur = 12, Convexity = 180. Estimate price changes for: a) -75 bps: +9.51% [12(0.0075) + 0.5(180)(0.005625)] b) +150 bps: -15.97% [−12(0.015) + 0.5(180)(0.0225)]

5. Portfolio Metrics: Three-bond portfolio:

   • Bond X: 25%, Duration = 3, Convexity = 12
   • Bond Y: 45%, Duration = 6, Convexity = 45
   • Bond Z: 30%, Duration = 10, Convexity = 125

   Portfolio Duration: 6.45 years Portfolio Convexity: 60.75

6. Barbell Construction: Create a barbell with same duration as 7-year bullet but higher convexity using 2-year and 15-year bonds.

   • Solution: 62% in 2-year, 38% in 15-year
   • Duration matches at 7 years
   • Convexity: 95 vs 60 for bullet

Advanced Level

7. Error Analysis: 20-year bond, ModDur = 14.5, Convexity = 310 For +200 bps:

   • Duration only: -29.00%
   • With convexity: -22.80%
   • If actual = -22.65%, calculate approximation errors
   • Answer: Duration error = 6.35%, Combined error = 0.15%

8. Optimal Portfolio: Given yield volatility of 100 bps annually, compare:

   • Portfolio A: Duration = 8, Convexity = 100
   • Portfolio B: Duration = 8, Convexity = 150

   Expected outperformance of B: 25 bps annually On $100M:$250,000 advantage

9. DeFi Application: Aave position with $10M USDC supplied:

   • Effective duration: 2.5 years
   • Convexity: 8
   • If DeFi rates drop 200 bps, estimate position value change
   • Answer: +5.08% = +$508,000

DeFi Applications & Real-World Examples

AMM Convexity Optimization

Curve Finance StableSwap:

• Traditional AMM (Uniswap): High convexity causes large IL
• StableSwap: Flattened curve reduces convexity near equilibrium
• Result: 100x better pricing for stablecoin swaps
• Mathematical basis: Convexity minimization subject to arbitrage constraints

Leveraged Yield Farming

Convexity in Leveraged Positions:

Base position: $1M at 5% yield, Duration = 3
3x leverage at 2% borrow cost

Effective duration: 9 years
Effective convexity: 81
Net yield: 11%

For -100 bps rate move:
Position gain: +9.41%
Demonstrates convexity amplification with leverage

Fixed-Rate Lending Protocols

Notional Finance Example:

• Issues fixed-rate loans with embedded convexity
• fCash tokens have bond-like convexity properties
• Liquidity pools must manage aggregate convexity risk
• Uses convexity limits to prevent pool imbalances

Yield Aggregator Strategies

Yearn v3 Vaults:

• Dynamically adjusts between protocols based on convexity-adjusted yields
• Factors in rate volatility and convexity benefits
• Rebalancing triggers incorporate convexity thresholds
• Optimizes for total return including convexity gains

Common Pitfalls & Exam Tips

Calculation Errors to Avoid

1. Forgetting the 0.5 factor in convexity adjustment
2. Using wrong units (annual vs semi-annual convexity)
3. Sign confusion (convexity always adds to price, whether rates rise or fall)
4. Decimal/percentage mixing (0.01 vs 1% for yield changes)

Conceptual Traps

1. Parallel shift assumption: Remember this is the key limitation
2. Negative convexity: Only occurs with embedded options (callable bonds)
3. Convexity ≠ Duration²: These are independent risk measures
4. Portfolio convexity: Cannot simply average without weights

Exam Strategy

1. Check if convexity needed: For <50 bps, duration often sufficient
2. Units consistency: Match yield change format to formula
3. Approximate methods: Acceptable if analytical not feasible
4. Time management: Convexity problems typically 2-3 minutes

Quick Recognition Patterns

• “Curvature” or “second-order” → Convexity topic
• “Large yield change” → Must include convexity
• “Barbell vs bullet” → Compare convexities
• “More accurate estimate” → Add convexity adjustment

Key Takeaways

Must-Know Concepts

1. ✅ Convexity measures price-yield curvature (second derivative)
2. ✅ Always positive for option-free bonds
3. ✅ Price change = Duration effect + Convexity adjustment
4. ✅ Portfolio metrics use market-value weighted averages
5. ✅ Higher convexity is always beneficial to bondholders

Critical Formulas

1. ✅ %ΔPrice ≈ −ModDur × ΔYield + 0.5 × Convexity × (ΔYield)²
2. ✅ Portfolio Duration = Σ(wi × Durationi)
3. ✅ Portfolio Convexity = Σ(wi × Convexityi)

Practical Applications

1. ✅ Use convexity for yield changes >50 bps
2. ✅ Barbell portfolios have higher convexity than bullets
3. ✅ Money convexity for position-level risk management
4. ✅ DeFi protocols embed convexity in bonding curves

Cross-References & Additional Resources

Related Finance Topics

• Topic 10: Interest Rate Risk and Return (duration foundation) duration
• Topic 11: Duration Measures (first-order sensitivity)
• Topic 13: Curve-Based Risk Measures (key rate durations)
• Topic 6: Bond Valuation (price-yield relationship)

DeFi Protocol Documentation

• Curve StableSwap Whitepaper - Convexity optimization
• Balancer V2 Docs - Weighted pool duration
• Notional Finance - Fixed-rate convexity
• Yearn Strategies - Convexity in yield optimization

Advanced Reading

1. Fabozzi: “Bond Markets, Analysis, and Strategies” Ch. 4
2. Tuckman: “Fixed Income Securities” Ch. 5-6
3. Research: “Convexity Bias in Eurodollar Futures” (CME Group)
4. DeFi Research: “Automated Market Makers and Convexity” (Paradigm)

Online Tools & Calculators

• Bond Calculator Pro - Duration/convexity
• Desmos Convexity Visualizer - Interactive graphs
• DeFi Rate - Real-time DeFi yields and durations

Review Checklist

Conceptual Understanding

• Can explain why convexity is always positive for option-free bonds
• Understand the relationship between convexity and bond characteristics
• Know when convexity adjustment is necessary vs optional
• Can describe limitations of portfolio duration/convexity measures

Calculation Proficiency

• Calculate convexity adjustment for any yield change
• Compute portfolio duration and convexity from components
• Use approximate convexity formula with price changes
• Apply money convexity for position sizing

Application Skills

• Compare barbell vs bullet portfolio convexities
• Identify convexity in DeFi protocols
• Recognize convexity’s role in risk management
• Evaluate convexity trade-offs in portfolio construction

Exam Readiness

• Complete all practice problems without calculator errors
• Can solve convexity problems in <3 minutes
• Memorized key formulas and relationships
• Understand all Learning Objectives thoroughly