Interest Rate Risk and Return

Topic 10: Interest Rate Risk and Return

Learning Objectives Coverage

LO1: Calculate and interpret the sources of return from investing in a fixed-rate bond

Core Concept

• Definition: Fixed-rate bonds generate returns from three distinct sources: (1) scheduled coupon and principal payments, (2) reinvestment of coupon payments at prevailing rates, and (3) capital gains/losses if sold before maturity.
• Why it matters: Understanding return sources enables investors to assess total return potential and identify which components are most sensitive to interest rate changes, crucial for portfolio management and risk assessment.
• Key components:
  • Coupon and principal payments (certain if held to maturity)
  • Interest-on-interest from reinvestment (variable with rates)
  • Capital gain/loss component (realized only if sold early)
  • Holding period return calculation methodology

Formulas & Calculations

• Holding Period Return:

  r = (FV/PV)^(1/T) - 1
  

  Where: FV = Future value (reinvested coupons + sale price), PV = Purchase price, T = Holding period

• Future Value of Reinvested Coupons:

  FV = PMT × [(1+r)^n - 1]/r
  

  Excel/Calculator: = -FV(Rate, Nper, Pmt, Pv, Type)

• Total Future Value:

  Total FV = Future value of coupons + Sale price (or par at maturity)
  

• HP 12C Steps (Reinvestment calculation):

  Example: 10-year, 6.2% bond, reinvest at 6.2%
  
  [f] [FIN]                   (Clear financial registers)
  10 [n]                      (10 periods)
  6.2 [PMT]                   (Annual coupon)
  6.2 [i]                     (Reinvestment rate)
  0 [PV]                      (No initial value)
  [FV]                        (Result: -82.493)
  [CHS]                       (82.493 = FV of coupons)
  

Practical Examples

• Traditional Finance Example: Corporate bond total return analysis

  10-year, 6.2% bond purchased at par (100)
  
  Scenario 1: Hold to maturity, rates unchanged
  - Coupons: 62.00
  - Interest-on-interest: 20.49
  - Principal: 100.00
  - Total return: 82.49% (6.20% annualized)
  
  Scenario 2: Rates rise to 7.2%
  - Coupons: 62.00
  - Interest-on-interest: 24.48
  - Principal: 100.00
  - Total return: 86.48% (6.43% annualized)
  

• Early Sale Example: Impact of rate changes

  5-year bond, 4% coupon, sold after 3 years
  Initial rate: 4%, New rate: 5%
  
  - FV of reinvested coupons: 12.49
  - Sale price: 98.14 (2-year bond at 5%)
  - Total FV: 110.63
  - HPR = (110.63/100)^(1/3) - 1 = 3.42%
  

DeFi Application

• Protocol example: Notional Finance fixed-rate lending
• Implementation: fCash tokens represent fixed-rate claims, automatically calculating reinvestment at maturity rates
• Advantages/Challenges:
  • Advantages: Transparent return sources, no reinvestment uncertainty during fixed period
  • Challenges: Limited secondary market liquidity, gas costs for compounding
  • Example: Lending 10,000 DAI at 5% fixed for 1 year guarantees 500 DAI return, eliminating reinvestment risk

LO2: Describe the relationships among a bond’s holding period return, its Macaulay duration, and the investment horizon

Core Concept

• Definition: The duration gap (Macaulay duration minus investment horizon) determines whether price risk or reinvestment risk dominates, directly impacting holding period returns under different rate scenarios.
• Why it matters: When investment horizon equals Macaulay duration, price and reinvestment risks offset perfectly, immunizing the portfolio against parallel interest rate shifts.
• Key components:
  • Duration gap analysis
  • Risk dominance determination
  • Immunization conditions
  • Return sensitivity to rate changes

Formulas & Calculations

• Duration Gap:

  Duration Gap = Macaulay Duration - Investment Horizon
  

• Risk Relationships:

  Negative gap (Horizon > MacDur): Reinvestment risk dominates
  Zero gap (Horizon = MacDur): Risks offset (immunized)
  Positive gap (Horizon < MacDur): Price risk dominates
  

• Return Impact Analysis:

  If rates ↑:
    - Reinvestment return ↑
    - Price return ↓
  Net effect depends on duration gap
  

• HP 12C Steps (Duration gap analysis):

  Example: MacDur = 7.74, Horizon = 5 years
  
  7.74 [ENTER]               (Macaulay duration)
  5 [-]                      (Investment horizon)
  Result: 2.74               (Positive gap → price risk dominates)
  

Practical Examples

• Immunization Example:

  Pension fund liability in 7.74 years
  Buy bond with MacDur = 7.74 years
  
  Rates rise 1%: Loss on price = Gain on reinvestment
  Rates fall 1%: Gain on price = Loss on reinvestment
  Net position protected
  

• Risk Dominance Example:

  3-year horizon, 5-year duration bond
  Gap = 5 - 3 = 2 (positive)
  
  If rates rise 100 bps:
  - Price loss: -4.8%
  - Reinvestment gain: +0.9%
  - Net impact: -3.9% (price risk dominated)
  

DeFi Application

• Protocol example: Pendle Finance maturity matching
• Implementation: Match YT token maturity with investment horizon to minimize duration gap
• Advantages/Challenges:
  • Advantages: Precise horizon matching, tradeable duration exposure
  • Challenges: Limited maturity options, requires active management
  • Example: 2-year DeFi treasury using 2-year PT tokens achieves near-zero duration gap

LO3: Define, calculate, and interpret Macaulay duration

Core Concept

• Definition: Macaulay duration is the weighted-average time to receipt of a bond’s cash flows, where weights are the present value of each cash flow divided by the bond’s price.
• Why it matters: It measures interest rate sensitivity, indicates the holding period for immunization, and serves as the foundation for all duration-based risk metrics.
• Key components:
  • Time-weighted cash flow calculation
  • Present value weighting system
  • Relationship to maturity and coupon
  • Special cases (zeros, perpetuities)

Formulas & Calculations

• General Macaulay Duration Formula:

  MacDur = Σ[(t × PV(CFt))/Price]
  
  Where: t = time to cash flow, PV(CFt) = present value of cash flow at time t
  

• Closed-Form Formula (for bonds on coupon date):

  MacDur = {(1+r)/r - (1+r+[N×(c-r)])/(c×[(1+r)^N-1]+r)}
  

  Where: r = YTM per period, N = periods to maturity, c = coupon rate

• Between Coupon Dates:

  MacDur = MacDur(on coupon date) - (t/T)
  

  Where: t/T = fraction of period elapsed

• HP 12C Steps (Weighted average calculation):

  Example: 3-year, 5% bond at 5% yield
  
  Year 1: 5/(1.05) = 4.762, weight = 0.04762, time-weight = 0.04762
  Year 2: 5/(1.05)² = 4.535, weight = 0.04535, time-weight = 0.09070
  Year 3: 105/(1.05)³ = 90.703, weight = 0.90703, time-weight = 2.72109
  
  Sum of time-weights = 2.859 years
  

Practical Examples

• Zero-Coupon Bond:

  10-year zero, yield 4%
  MacDur = 10 years (equals maturity)
  All cash flow at maturity
  

• High vs Low Coupon Comparison:

  Both 10-year bonds, 5% yield
  
  2% coupon: MacDur = 8.98 years
  8% coupon: MacDur = 7.56 years
  Higher coupon → Lower duration
  

• Yield Level Impact:

  10-year, 6% coupon bond
  
  At 3% yield: MacDur = 8.53 years
  At 6% yield: MacDur = 7.80 years
  At 9% yield: MacDur = 7.19 years
  Higher yield → Lower duration
  

DeFi Application

• Protocol example: Element Finance’s Principal Tokens
• Implementation: PT tokens have duration equal to time-to-maturity, simplifying duration calculations
• Advantages/Challenges:
  • Advantages: Exact duration matching, no coupon complexity
  • Challenges: Requires combining with yield tokens for full exposure
  • Example: 6-month PT-DAI has exactly 0.5-year Macaulay duration

Core Concepts Summary (80/20 Principle)

Critical 20% to Master:

1. Three Return Sources: Coupons, reinvestment, and price changes combine for total return
2. Duration Gap: MacDur - Horizon determines dominant risk type
3. Immunization Point: When horizon = MacDur, interest rate risks offset perfectly
4. Duration Properties: Always ≤ maturity (except zeros), decreases with higher coupons/yields

Key Relationships:

• Rising rates: Help reinvestment, hurt prices
• Falling rates: Hurt reinvestment, help prices
• Duration gap sign: Determines which effect dominates
• Zero duration gap: Creates hedged position

Comprehensive Formula Sheet formula

Return Calculations

Holding Period Return: r = (FV/PV)^(1/T) - 1
Future Value: FV = Σ[Coupons × (1+r)^(n-t)] + Sale Price
Horizon Yield = IRR of all cash flows

Duration Formulas

Macaulay Duration = Σ[(t × PV(CFt))/Price]
Duration Gap = MacDur - Investment Horizon
Closed-form: MacDur = {(1+r)/r - (1+r+[N×(c-r)])/(c×[(1+r)^N-1]+r)}

Risk Relationships

If Duration Gap > 0: Price risk dominates
If Duration Gap = 0: Immunized position
If Duration Gap < 0: Reinvestment risk dominates

HP 12C Calculator Sequences hp12c

Calculate Reinvestment Value

[f] [FIN]              Clear registers
n [n]                  Number of periods
PMT [PMT]              Coupon payment
r [i]                  Reinvestment rate
0 [PV]                 No initial value
[FV] [CHS]            Future value of coupons

Calculate Macaulay Duration (Manual)

For each cash flow:
CF [ENTER]            Cash flow amount
(1+r) [ENTER] t [y^x] [÷]    Present value
Price [÷]             Weight
t [×]                 Time-weighted value
[Σ+]                  Add to memory

[RCL] [Σ]             Total = Macaulay Duration

Duration Gap Analysis

MacDur [ENTER]        Macaulay duration
Horizon [-]           Subtract investment horizon
                      Result: Duration gap
[x>0?]                Test if positive (price risk)
[x=0?]                Test if zero (immunized)
[x<0?]                Test if negative (reinvestment risk)

Practice Problems

Basic Level

1. A 5-year, 4% annual coupon bond is purchased at par. Calculate the future value of reinvested coupons if rates remain at 4%.

2. A bond has a Macaulay duration of 6 years. An investor with a 4-year horizon buys this bond. Which risk dominates?

3. Calculate the Macaulay duration of a 2-year, 6% annual coupon bond priced at par.

Intermediate Level

4. A 10-year, 5% bond is purchased at 95. If held to maturity with coupons reinvested at 6%, calculate the holding period return.

5. An investor needs $1 million in 8 years. She buys bonds with 8-year Macaulay duration. If rates rise 2%, estimate the impact on her terminal wealth.

6. Compare the Macaulay duration of two 5-year bonds: one with 2% coupon, one with 8% coupon, both yielding 5%.

Advanced Level

7. A bond has MacDur = 7.5 years. An investor with a 5-year horizon experiences rates rising from 4% to 5%. Estimate the net impact on holding period return.

8. Design an immunization strategy for a $10 million liability due in 6.3 years using available bonds.

9. A DeFi protocol offers 1-year fixed rate lending at 8% or variable rate starting at 7%. Under what interest rate scenarios would fixed rate provide higher returns?

Solutions Guide

1. FV = 4 × [(1.04)^5 - 1]/0.04 = 21.67
2. Duration gap = 6 - 4 = 2 > 0, price risk dominates
3. MacDur = (1×4.762 + 2×90.238)/100 = 1.857 years
4. FV coupons = 67.16, Total FV = 167.16, HPR = (167.16/95)^0.1 - 1 = 5.82%
5. With perfect immunization, minimal impact (< 0.1%)
6. 2% coupon: 4.85 years, 8% coupon: 4.55 years
7. Price loss ≈ -3.75%, reinvestment gain ≈ +2.5%, net ≈ -1.25%
8. Buy bonds totaling $10M with aggregate MacDur = 6.3 years
9. Fixed better if variable rates average > 8% over the year

DeFi Applications & Real-World Examples

Fixed-Rate Lending Protocols

Notional Finance:

• Creates fixed-rate markets using fCash system
• Macaulay duration equals loan maturity
• Example: 3-month fDAI loan has 0.25-year duration

Yield Protocol:

• fyTokens are zero-coupon tokens
• Duration precisely equals time-to-maturity
• Enables exact duration matching strategies

Duration Management in DeFi

Pendle Finance:

• Splits yield-bearing tokens into PT and YT
• PT duration = time to maturity
• YT captures duration gap returns
• Example: stETH split into PT-stETH (fixed duration) and YT-stETH (variable yield)

Element Finance:

• Principal tokens for fixed-rate exposure
• Duration ladders using multiple maturities
• Convergence trades based on duration gaps

Real-World Applications

Traditional Finance:

• Pension fund immunization strategies
• Insurance company ALM matching
• Bank duration gap management
• Structured product design

DeFi Treasury Management:

• DAO treasury duration matching
• Protocol revenue immunization
• Yield farming optimization
• Cross-protocol arbitrage strategies

Common Pitfalls & Exam Tips

Calculation Errors to Avoid

1. Wrong time units: Ensure periods match (annual vs semi-annual)
2. Missing reinvestment: Don’t forget interest-on-interest component
3. Sign conventions: FV calculations often need sign changes
4. Between coupon dates: Adjust duration for elapsed time

Conceptual Traps

1. Duration ≠ Maturity: Except for zero-coupon bonds
2. Immunization limits: Only works for parallel shifts
3. Gap interpretation: Positive gap means price risk, not reinvestment risk
4. Yield changes: Higher yields decrease duration, not increase

Exam Strategy Tips

1. Quick checks: Zero-coupon duration always equals maturity
2. Relationships: Higher coupon/yield → lower duration
3. Gap analysis: Draw timeline to visualize horizon vs duration
4. Calculator efficiency: Store intermediate values in memory

Red Flags in Problems

• Duration > Maturity (impossible except for some exotic bonds)
• Negative Macaulay duration (impossible for normal bonds)
• Perfect immunization with non-parallel shifts (doesn’t work)
• Ignoring convexity for large rate changes (introduces error)

Key Takeaways

Essential Concepts

1. Three return sources create total bond returns, each with different risk characteristics
2. Duration gap determines risk exposure: positive = price risk, negative = reinvestment risk
3. Macaulay duration is the weighted-average time to cash flows and immunization horizon
4. Perfect immunization occurs when investment horizon equals Macaulay duration

Critical Formulas

• Holding period return: r = (FV/PV)^(1/T) - 1
• Duration gap = MacDur - Horizon
• Immunization condition: Duration gap = 0

DeFi Innovations

• Fixed-rate protocols eliminate reinvestment uncertainty
• Tokenized duration through PT/YT splits
• On-chain duration matching capabilities
• Composable duration strategies across protocols

Cross-References & Additional Resources

Related Finance Topics

• Modified Duration and Money Duration (extends duration concepts) duration
• Convexity (second-order effects)
• Empirical Duration (practical applications)
• Bond Valuation and Yield Measures (pricing foundations)

DeFi Protocol Documentation

• Notional V3 Docs: Fixed-rate lending
• Pendle Docs: Yield tokenization
• Element Docs: Principal token mechanics
• Yield Protocol: fyToken system

Academic Resources

• Fabozzi’s “Bond Markets, Analysis, and Strategies”
• Tuckman’s “Fixed Income Securities”
• Hull’s “Options, Futures, and Other Derivatives” (Ch. 4)

Practice Platforms

• Finance Learning Ecosystem
• Kaplan Schweser QBank
• Bloomberg BMC courses
• DeFi simulation environments

Review Checklist

Conceptual Understanding

• Can explain three sources of bond returns
• Understand price risk vs reinvestment risk trade-off
• Know when each risk type dominates
• Can interpret duration gap meanings

Calculation Proficiency

• Calculate future value of reinvested coupons
• Compute holding period returns
• Determine Macaulay duration manually
• Analyze duration gaps correctly

Application Skills

• Design immunization strategies
• Compare bonds based on duration
• Assess interest rate risk exposure
• Apply concepts to DeFi protocols

Exam Readiness

• Complete all practice problems
• Review common pitfalls
• Master HP 12C sequences
• Understand DeFi applications