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Yield-Based Bond Duration Measures and Properties

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Topic 11: Yield-Based Bond Duration Measures and Properties

Learning Objectives Coverage

LO1: Define, calculate, and interpret modified duration, money duration, and the price value of a basis point (PVBP)

Core Concept exam-focus

Modified duration measures a bond’s price sensitivity to yield changes as the first derivative of price with respect to yield. It builds directly on the Macaulay duration concept by dividing by (1 + r) to produce a metric expressed as a percentage price change per unit yield change. Money duration scales this into currency units, making it practical for position-level risk management, while PVBP (also called DV01, PV01, or BPV) measures the price change for a single basis point yield move — the workhorse metric for traders and hedgers. Together, these three measures form a hierarchy: MacDur to ModDur to MoneyDur to PVBP, each adding practical specificity. duration

Formulas & Calculations formula hp12c

  • Modified Duration:

    ModDur = MacDur/(1 + r)
%ΔPVFull ≈ -AnnModDur × ΔYield

  • Approximate Modified Duration (when analytical not available):

    AnnModDur ≈ (PV- - PV+) / [2 × (ΔYield) × (PV0)]

    Where: PV- = price if yield decreases, PV+ = price if yield increases

  • Money Duration:

    MoneyDur = AnnModDur × PVFull
ΔPVFull ≈ -MoneyDur × ΔYield

  • Price Value of a Basis Point:

    PVBP = MoneyDur × 0.0001
or
PVBP = (PV- - PV+) / 2  (using ±1bp change)

  • HP 12C Steps (Modified Duration):

    Example: MacDur = 10, YTM = 8%

10 [ENTER]                 (Macaulay duration)
1.08 [÷]                   (Divide by 1+r)
Result: 9.259              (Modified duration)

  • HP 12C Steps (PVBP calculation):

    Example: ModDur = 5, Price = $1,000

5 [ENTER] 1000 [×]         (Money duration = 5,000)
0.0001 [×]                 (Multiply by 0.0001)
Result: 0.50               (PVBP = $0.50)

Practical Examples

  • Traditional Finance Example: Corporate bond risk assessment

    $100M position in 5-year corporate bond
Modified duration: 4.43
Market value: $100.5M

Money duration = 4.43 × $100.5M = $445.2M
PVBP = $445.2M × 0.0001 = $44,520

If yields rise 25 bps:
Loss ≈ 25 × $44,520 = $1,113,000

  • Approximate Duration Calculation:

    Bond price at 5.00% yield: $100.00
Bond price at 4.95% yield: $100.23
Bond price at 5.05% yield: $99.77

AnnModDur ≈ (100.23 - 99.77)/(2 × 0.001 × 100)
AnnModDur ≈ 2.30

DeFi Application defi-application

In DeFi lending protocols like Aave, variable-rate supply positions see their modified duration change daily as rates adjust, because the effective time to next cash flow reset is continuously shifting. Money duration tracks the absolute USD exposure to rate changes, making it directly analogous to traditional portfolio risk management. The transparency of on-chain rate data enables real-time duration updates that TradFi systems achieve only through expensive infrastructure. However, DeFi rate volatility far exceeds traditional markets, requiring more frequent recalculation. A 20,000 and a PVBP of just $2 — illustrating the near-zero duration typical of floating-rate DeFi positions.

LO2: Explain how a bond’s maturity, coupon, and yield level affect its interest rate risk

Core Concept

  • Definition: Duration properties describe systematic relationships between bond characteristics and interest rate sensitivity, forming fundamental rules for bond portfolio management.
  • Why it matters: Understanding these relationships enables investors to select bonds matching their risk tolerance and construct portfolios with desired duration characteristics.
  • Key components:
    • Coupon effect (inverse relationship)
    • Yield effect (inverse relationship)
    • Maturity effect (direct relationship)
    • Interaction effects and exceptions

Formulas & Calculations

  • Duration Rules Summary:

    ↑ Coupon → ↓ Duration
↑ Yield → ↓ Duration
↑ Maturity → ↑ Duration (usually)

  • Perpetuity Duration (special case):

    MacDur = (1 + r)/r
ModDur = 1/r

  • Zero-Coupon Duration (special case):

    MacDur = Maturity
ModDur = Maturity/(1 + r)

  • Floating-Rate Duration:

    MacDur = (T - t)/T

    Where: T = coupon period, t = days since last reset

  • HP 12C Steps (Comparing durations):

    Bond A: 5% coupon, Bond B: 8% coupon
Same maturity and yield

Bond A duration > Bond B duration
(Lower coupon → Higher duration)

Practical Examples

  • Coupon Effect Example:

    10-year bonds at 6% yield:

2% coupon: ModDur = 8.12
6% coupon: ModDur = 7.44
10% coupon: ModDur = 6.89

Lower coupon → Higher duration

  • Yield Level Effect:

    10-year, 5% coupon bond:

At 3% yield: ModDur = 8.53
At 5% yield: ModDur = 7.99
At 7% yield: ModDur = 7.51

Lower yield → Higher duration

  • Maturity Effect with Exception:

    Deep discount bonds (1% coupon at 10% yield):

20-year: Duration = 10.50
30-year: Duration = 10.91
50-year: Duration = 10.96
100-year: Duration = 10.89 (decreases!)

Very long deep discount bonds can have decreasing duration

DeFi Application defi-application

Compound’s cToken system provides a compelling illustration of duration properties in DeFi. Because cTokens represent variable-rate positions with continuous rate resets, their effective duration is extremely short — cDAI typically has duration below 0.1 years, akin to an overnight floating-rate note. By contrast, fixed-rate protocols like Notional Finance and Element Finance create positions where duration precisely equals the remaining term, behaving like zero-coupon instruments. This composability allows DeFi users to construct portfolios with targeted duration profiles by blending variable-rate and fixed-rate positions across protocols.

Core Concepts Summary (80/20 Principle)

Critical 20% to Master:

  1. Modified Duration: MacDur/(1+r) - measures % price change per 1% yield change
  2. Money Duration: ModDur × Price - measures $ change per 1% yield change
  3. PVBP/DV01: MoneyDur × 0.0001 - measures $ change per 1bp yield change
  4. Duration Properties: Lower coupon/yield → Higher duration; Higher maturity → Higher duration (usually)

Key Relationships:

  • Duration Hierarchy: MacDur → ModDur → MoneyDur → PVBP
  • Approximation: Can calculate duration using small yield changes when analytical unavailable
  • Portfolio Duration: Weighted average of individual bond durations
  • Limitations: Linear approximation of non-linear relationship

Comprehensive Formula Sheet formula

Duration Measures

Modified Duration: ModDur = MacDur/(1 + r)
Annualized: AnnModDur = ModDur/k (k = periods per year)
Price Change: %ΔPVFull ≈ -AnnModDur × ΔYield

Money Duration & PVBP

Money Duration: MoneyDur = AnnModDur × PVFull
Price Change ($): ΔPVFull ≈ -MoneyDur × ΔYield
PVBP/DV01: PVBP = MoneyDur × 0.0001
Alternative: PVBP = (PV@yield-1bp - PV@yield+1bp)/2

Approximate Duration

AnnModDur ≈ (PV- - PV+) / [2 × (ΔYield) × (PV0)]
Where: Small yield changes (typically ±5bp)

Special Cases

Zero-Coupon: MacDur = Maturity, ModDur = Maturity/(1+r)
Perpetuity: MacDur = (1+r)/r, ModDur = 1/r
Floating-Rate: MacDur = Time to next reset

Duration Properties

∂Duration/∂Coupon < 0 (inverse)
∂Duration/∂Yield < 0 (inverse)
∂Duration/∂Maturity > 0 (usually direct)

HP 12C Calculator Sequences hp12c

Calculate Modified Duration

MacDur [ENTER]         Macaulay duration
1 [ENTER] r [+]       1 + yield
[÷]                   Modified duration

Calculate Money Duration

ModDur [ENTER]        Modified duration
Price [×]             Money duration

Calculate PVBP

MoneyDur [ENTER]      Money duration
0.0001 [×]           PVBP

Approximate Modified Duration

PV- [ENTER]          Price if yield falls
PV+ [-]              Difference
2 [÷]                Half the difference
ΔYield [÷]           Divide by yield change
PV0 [÷]              Divide by initial price
                     Result: Approximate ModDur

Compare Duration Effects

For coupon comparison:
Lower_Coupon_Dur [ENTER]
Higher_Coupon_Dur [-]
Result > 0 confirms lower coupon = higher duration

For yield comparison:
Lower_Yield_Dur [ENTER]
Higher_Yield_Dur [-]
Result > 0 confirms lower yield = higher duration

Practice Problems

Basic Level

  1. A bond has Macaulay duration of 7.5 years and YTM of 5%. Calculate its modified duration.

  2. A $10 million position has modified duration of 6. What is its PVBP?

  3. Which has higher duration: 10-year 3% coupon or 10-year 6% coupon bond?

Intermediate Level

  1. A bond trades at 99.42 and $97.59. Calculate approximate modified duration.

  2. A portfolio has 30M in 7-year bonds (ModDur=6.1). Calculate portfolio money duration.

  3. A perpetual bond yields 8%. Calculate its Macaulay and modified durations.

Advanced Level

  1. Compare durations of three 15-year bonds at 5% yield: 0% coupon, 5% coupon, 10% coupon. Explain the pattern.

  2. A trader holds 65,000. Design a hedge using futures with PVBP of $85 per contract.

  3. Explain why a 100-year deep discount bond might have lower duration than a 50-year bond.

Solutions Guide

  1. ModDur = 7.5/(1.05) = 7.14 years
  2. PVBP = 6 × 6,000
  3. 3% coupon has higher duration (lower coupon rule)
  4. AnnModDur ≈ (99.42-97.59)/(2×0.002×98.50) = 4.66
  5. MoneyDur = 2.8×30M = $323M
  6. MacDur = 1.08/0.08 = 13.5 years, ModDur = 12.5 years
  7. 0%: 14.29, 5%: 10.93, 10%: 9.14 (coupon effect)
  8. Need 765 contracts (65,000/85)
  9. Present value of distant cash flows approaches zero, limiting duration growth

DeFi Applications & Real-World Examples

Variable Rate Protocol Duration

Aave & Compound:

  • Supply positions: Duration ≈ 0 (rates adjust continuously)
  • Fixed-rate borrow: Duration = loan maturity
  • Variable borrow: Duration ≈ 0.25 years (quarterly adjustment assumption)

Example Calculation:

$1M USDC at 3% variable
Assumed rate volatility: 1% quarterly
Effective duration ≈ 0.25 years
Money duration = 0.25 × $1M = $250,000
PVBP = $25

Yield Tokenization Duration

Pendle Finance:

  • PT tokens: Duration = time to maturity
  • YT tokens: Complex duration (captures rate differentials)
  • Combined position: Replicates underlying duration

Element Finance:

  • Principal tokens: Zero-coupon profile
  • Duration precisely equals remaining term
  • Enables exact duration matching

Stablecoin Lending Duration

Traditional Stablecoins:

USDC in Compound: Duration ≈ 0.1 years
USDC in Aave: Duration ≈ 0.1 years
USDC in fixed protocols: Duration = term

Algorithmic Stablecoins:

DAI savings rate: Duration ≈ 0.25 years
sDAI (staked DAI): Duration ≈ 0.5 years
Fixed-rate DAI vaults: Duration = maturity

Real-World Applications

Portfolio Management:

  • Target duration: 5 years
  • Achieve through combination of:
    • 40% 2-year notes (ModDur = 1.9)
    • 60% 7-year notes (ModDur = 6.5)
    • Portfolio ModDur = 0.4×1.9 + 0.6×6.5 = 4.66

Risk Management:

Bank trading desk limits:
- Maximum PVBP: $100,000
- Maximum modified duration: 7.0
- Money duration limit: $500M

Common Pitfalls & Exam Tips

Calculation Errors to Avoid

  1. Forgetting to annualize: Semi-annual ModDur must be divided by 2
  2. Wrong PVBP formula: Use 0.0001, not 0.01
  3. Sign confusion: Duration is positive, price change is negative for yield increases
  4. Approximation accuracy: Only valid for small yield changes

Conceptual Traps

  1. Duration ≠ Time: Modified duration is not a time measure
  2. Money duration units: Currency units, not percentage
  3. PVBP alternatives: DV01, PV01, BPV all mean the same thing
  4. Portfolio duration: Must use market-value weights, not par weights

Exam Strategy Tips

  1. Quick checks: ModDur < MacDur always (for positive yields)
  2. Memory aids: “CMY” - Coupon↑ Maturity↑ Yield↑ affects duration ↓↑↓
  3. Approximation: If exact formula not given, use ±5bp approximation
  4. Units: Always specify - years for duration, $ for money duration

Red Flags in Problems

  • Modified duration > Macaulay duration (impossible for positive yields)
  • PVBP > Money duration (should be 10,000× smaller)
  • Duration increasing with coupon (violates inverse relationship)
  • Perpetuity with finite duration < (1+r)/r (too low)

Key Takeaways

Essential Concepts

  1. Modified duration measures percentage price sensitivity to yield changes
  2. Money duration converts this to dollar sensitivity for position sizing
  3. PVBP/DV01 provides per-basis-point sensitivity for trading
  4. Duration properties follow predictable patterns with coupon, yield, and maturity

Critical Formulas

  • ModDur = MacDur/(1+r)
  • MoneyDur = ModDur × Price
  • PVBP = MoneyDur × 0.0001
  • %ΔPrice ≈ -ModDur × ΔYield

DeFi Innovations defi-application

  • Variable rate protocols (Aave, Compound) have near-zero duration
  • Fixed-rate protocols (Notional Finance, Element) enable duration matching
  • Tokenized duration through Pendle Finance PT/YT splits
  • Composable duration strategies across protocols

Cross-References & Additional Resources

DeFi Protocol Documentation

Academic Resources

  • Fabozzi’s “Fixed Income Analysis” Ch. 5
  • Tuckman & Serrat “Fixed Income Securities” Ch. 4
  • Finance readings on duration and convexity

Practice Platforms

  • Bloomberg Terminal: PORT function for duration analysis
  • Python libraries: QuantLib for duration calculations
  • Excel templates: Duration and convexity calculators
  • DeFi simulators: Fork mainnet for testing

Review Checklist

Conceptual Understanding

  • Can explain difference between Macaulay and modified duration
  • Understand money duration vs PVBP distinction
  • Know how coupon, yield, maturity affect duration
  • Can identify when to use each duration measure

Calculation Proficiency

  • Calculate modified duration from Macaulay
  • Compute money duration and PVBP
  • Use approximate duration formula
  • Apply duration to estimate price changes

Application Skills

  • Analyze portfolio duration
  • Design duration hedges
  • Compare bonds using duration
  • Apply to DeFi protocols

Exam Readiness

  • Complete all practice problems
  • Master HP 12C sequences
  • Review common pitfalls
  • Understand duration limitations

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