How to Solve CFA Level I Bond Duration and Convexity Questions Step by Step
Duration and convexity calculations are among the most challenging—and heavily tested—concepts in the CFA Level I Fixed Income curriculum. If you can calculate bond yields and discount cash flows but struggle with interest-rate sensitivity measures, this step-by-step guide will build your confidence in tackling these essential questions.
This tutorial focuses specifically on solving CFA Level I duration and convexity questions, covering the distinctions between duration types, worked calculation examples, and common exam traps that catch even well-prepared candidates.
Understanding the Three Types of Duration
Macaulay Duration: Weighted Average Time to Cash Flows
Macaulay duration measures the weighted average time (in years or periods) until a bond's cash flows are received. Each cash flow is weighted by its present value as a fraction of the bond's current price.
Formula: ``` Macaulay Duration = Σ [t × PV(CFt)] / P ```Where:
- t = time period
- PV(CFt) = present value of cash flow at time t
- P = current bond price
Key Point: Macaulay duration is expressed in time units (years or periods) and measures timing, not price sensitivity.
Modified Duration: Price Sensitivity to Yield Changes
Modified duration directly measures the approximate percentage price change for a small parallel shift in yield-to-maturity. It's derived from Macaulay duration.
Formula: ``` Modified Duration = Macaulay Duration / (1 + y) ```For semiannual bonds:
```
Modified Duration = Macaulay Duration / (1 + YTM/2)
```
Effective Duration: When Cash Flows Change
Effective duration measures price sensitivity to benchmark yield curve changes, accounting for changes in expected cash flows due to embedded options.
Formula: ``` Effective Duration = (P₋ - P₊) / (2 × P₀ × Δy) ```Where:
- P₋ = price if yield decreases by Δy
- P₊ = price if yield increases by Δy
- P₀ = current price
- Δy = yield change (in decimal form)
When to Use Each:
- Macaulay Duration: Pure timing questions about when cash flows occur
- Modified Duration: Price sensitivity for option-free bonds with fixed cash flows
- Effective Duration: Bonds with embedded options (callable, putable, convertible)
Worked Example: Modified Duration Price Change Calculation
Let's solve a typical CFA Level I problem step by step.
Problem: A 5-year corporate bond has a modified duration of 4.25. If yields increase by 75 basis points, estimate the percentage price change. Step 1: Convert Basis Points to Decimal 75 basis points = 75 ÷ 10,000 = 0.0075 Step 2: Apply the Modified Duration Formula %ΔP ≈ -Modified Duration × ΔYield %ΔP ≈ -4.25 × 0.0075 %ΔP ≈ -0.031875 or -3.19% Step 3: Interpret the Result The bond's price will decrease by approximately 3.19% when yields increase by 75 basis points. Note the negative relationship: higher yields lead to lower bond prices.Adding the Convexity Adjustment
Duration provides only a linear approximation of the bond price-yield relationship. Convexity captures the curvature, improving accuracy for larger yield changes.
The Convexity Formula
Price Change with Convexity Adjustment: ``` %ΔP ≈ -ModDur × Δy + 0.5 × Convexity × (Δy)² ``` Approximate Convexity Calculation: ``` Convexity = [P₋ + P₊ - 2P₀] / [P₀ × (Δy)²] ```Worked Example with Convexity
Problem: The same 5-year bond has modified duration of 4.25 and convexity of 18.5. Yields increase by 75 basis points. Calculate the price change using both duration and convexity. Step 1: Duration Effect Duration effect = -4.25 × 0.0075 = -0.031875 Step 2: Convexity Effect Convexity effect = 0.5 × 18.5 × (0.0075)² = 0.5 × 18.5 × 0.00005625 = 0.000520 Step 3: Combined Effect Total %ΔP = -0.031875 + 0.000520 = -0.031355 or -3.14% Step 4: Compare to Duration-Only Estimate- Duration only: -3.19%
- Duration + Convexity: -3.14%
- Convexity reduces the estimated price decline by 0.05 percentage points
When Convexity Matters Most
Convexity becomes increasingly important for:
- Larger yield changes (>50 basis points)
- Longer-maturity bonds
- Lower coupon rates
- Lower current yields
According to the CFA Institute 2026 curriculum, convexity is always positive for option-free bonds, meaning price gains from yield declines exceed duration estimates, while price losses from yield increases are smaller than duration estimates.
Common Calculation Traps and How to Avoid Them
Trap 1: Incorrect Yield Conversion
Wrong: Using basis points directly in formulas Right: Convert basis points to decimal (divide by 10,000) Example:- 50 basis points = 50 ÷ 10,000 = 0.005
- Not 50 or 0.50
Trap 2: Sign Confusion
Wrong: Forgetting the negative sign in the duration formula Right: %ΔP = -Modified Duration × ΔYieldRemember: Bond prices and yields move in opposite directions.
Trap 3: Annualization Errors
Wrong: Using periodic duration for annual yield changes Right: Match the compounding frequencyFor semiannual bonds:
- Modified Duration = Macaulay Duration ÷ (1 + YTM/2)
- Not ÷ (1 + YTM)
Trap 4: Using Modified Duration for Bonds with Options
Wrong: Applying modified duration to callable or putable bonds Right: Use effective duration when cash flows can change Why: Modified duration assumes fixed cash flows. Callable bonds have uncertain cash flows because the issuer may redeem early when rates fall.Trap 5: Ignoring Convexity for Large Yield Changes
Wrong: Using duration-only estimates for changes >50 basis points Right: Include convexity adjustment for better accuracyThe convexity adjustment becomes material as yield changes increase, especially beyond 100 basis points.
Practice Drill: Five Quick Questions
Test your understanding with these typical CFA Level I scenarios:
Question 1: Duration Type Selection
A municipal bond is callable at par after 3 years. Which duration measure is most appropriate for estimating price sensitivity? Answer: Effective duration. The callable feature means cash flows can change with interest rates, making modified duration inappropriate.Question 2: Yield Conversion
A bond has modified duration of 6.8. If yields decrease by 120 basis points, what is the estimated percentage price change? Solution:- Convert: 120 bps = 0.012
- Calculate: %ΔP = -6.8 × (-0.012) = +8.16%
- Answer: +8.16% (price increases when yields fall)
Question 3: Convexity Benefit
Two bonds have identical modified durations of 5.5. Bond A has convexity of 25, Bond B has convexity of 40. If yields fall by 100 basis points, which bond will outperform? Solution: Both have the same duration effect, but Bond B has higher convexity:- Bond B convexity effect = 0.5 × 40 × (0.01)² = 0.02 or +2%
- Bond A convexity effect = 0.5 × 25 × (0.01)² = 0.0125 or +1.25%
- Answer: Bond B outperforms by 0.75 percentage points
Question 4: Macaulay vs Modified
A bond has Macaulay duration of 4.5 years and YTM of 6% (annual compounding). What is its modified duration? Solution: Modified Duration = 4.5 ÷ (1 + 0.06) = 4.5 ÷ 1.06 = 4.25 yearsQuestion 5: Portfolio Duration
A portfolio contains two bonds:- Bond A: $500,000 market value, duration 3.2
- Bond B: $300,000 market value, duration 5.8
Step-by-Step Problem-Solving Framework
Follow this systematic approach for any duration or convexity question:
Step 1: Identify the Bond Type
- Option-free bond → Use modified duration
- Callable/putable bond → Use effective duration
- Timing question → Use Macaulay duration
Step 2: Check the Compounding Frequency
- Annual: ModDur = MacDur ÷ (1 + YTM)
- Semiannual: ModDur = MacDur ÷ (1 + YTM/2)
Step 3: Convert Units Properly
- Basis points to decimal: ÷ 10,000
- Percentage to decimal: ÷ 100
Step 4: Apply the Correct Formula
- Small yield changes: Duration only
- Large yield changes: Duration + Convexity
Step 5: Check Signs and Direction
- Yields up → Prices down (negative change)
- Yields down → Prices up (positive change)
Advanced Considerations for Portfolio Applications
Duration and convexity can be calculated for bond portfolios using weighted averages, but this approach has important limitations. According to CFA Institute guidance, portfolio measures assume parallel yield curve shifts, which rarely occur in practice.
Portfolio Duration Formula: ``` Portfolio Duration = Σ (wi × Durationi) ```Where wi is the market weight of bond i.
This calculation becomes less reliable when yield curve shapes change non-uniformly across maturities, highlighting why effective duration and key rate durations are used for more sophisticated interest rate risk management, as detailed in the CFA Institute's curve-based risk measures curriculum.
For students preparing for CFA Level I, focus on mastering the basic duration and convexity calculations before advancing to portfolio applications or curve risk measures covered in later levels.
Limitations of Duration and Convexity Measures
Understanding when these measures break down is crucial for the CFA exam:
1. Parallel Shift Assumption: Duration assumes all yields change by the same amount
2. Small Change Approximation: Accuracy decreases for very large yield changes (>200 bps)
3. Fixed Cash Flow Assumption: Modified duration fails for bonds with embedded options
4. Credit Risk Ignored: Duration only captures interest rate risk, not credit spread changes
These limitations explain why effective duration, key rate duration, and scenario analysis are used in professional portfolio management.
Integration with Other CFA Level I Topics
Duration and convexity calculations build directly on time value of money concepts and require strong calculator skills. Like the systematic approach needed for ethics case analysis, duration problems reward candidates who follow consistent computational steps and avoid common shortcuts that introduce errors.
The mathematical precision required for duration calculations also prepares you for the rigorous analytical thinking needed throughout the CFA curriculum, making these Fixed Income concepts excellent practice for developing exam-day accuracy.
Frequently Asked Questions
When should I use effective duration instead of modified duration?
Use effective duration whenever a bond has embedded options (callable, putable, convertible) or when cash flows depend on interest rates (mortgage-backed securities). Modified duration assumes fixed cash flows and will underestimate risk for bonds with options.
How do I know if a yield change is "large enough" to require convexity?
As a rule of thumb, include convexity for yield changes exceeding 50 basis points. The convexity adjustment becomes increasingly important for changes above 100 basis points, especially for long-maturity, low-coupon bonds.
What's the difference between annual and effective annual duration?
Annual modified duration uses the annual yield, while effective annual duration annualizes the periodic result. For semiannual bonds, divide Macaulay duration by (1 + YTM/2), not (1 + YTM), to get the correct modified duration.
Can convexity be negative?
Yes, for callable bonds when interest rates fall enough to make calling likely. The bond's price appreciation becomes limited (negative convexity), while putable bonds maintain positive convexity due to the put option's price floor.
How accurate is the duration-convexity approximation?
For most CFA Level I purposes, the approximation is quite accurate for yield changes up to 200 basis points. Beyond that, higher-order terms become significant, but they're beyond the Level I curriculum scope.
By mastering these duration and convexity concepts systematically, you'll build the foundation needed for the Fixed Income portion of the CFA Level I exam. Practice with various bond types and yield change scenarios to develop the speed and accuracy required for exam success.
Remember that Vidia's adaptive learning approach can help identify which specific duration and convexity calculation steps you need to practice most, personalizing your study plan around your actual knowledge gaps rather than reviewing concepts you already understand.
