Bond Duration: Understanding Interest Rate Risk
Duration measures how long (in years) it takes, on average, to be repaid the bond’s price by cash flows. It is central to interest-rate risk management: longer duration means greater sensitivity to interest rate changes. There are multiple duration measures — Macaulay and Modified are most common. This article explains the math, intuition, and practical uses.
1. Cash flows and time value
Coupon bonds pay periodic interest and repay principal at maturity. Each payment's present value contributes to the bond price; duration weights payment times by their PV share. If earlier payments make up most PV, duration is shorter; if later principal dominates PV, duration is longer.
2. Macaulay duration formula
Macaulay duration = Σ (t × PV(CF_t)) / Price, where t is time in years (or fractions) and PV(CF_t) is present value of cash flow t. For multi-period coupon bonds, compute PVs using yield per period.
3. From Macaulay to Modified duration
Modified duration converts Macaulay into a measure of price sensitivity: Modified = Macaulay / (1 + y/m), where y is nominal annual yield, m is compounding frequency. The approximated percentage price change for a small yield change Δy (in decimal) is ≈ −Modified × Δy.
4. Example (start)
Example setup: 1,000 par, 5% coupon, 10-year maturity, semi-annual coupons, yield 4%. Part 2 will compute cash flows, PVs, Macaulay and Modified durations, and show approximate vs exact price change for a 50 bps yield move.
5. Calculating step-by-step (continued)
To calculate durations we follow these steps:
- List all cash flows (coupon payments each period and final principal repayment).
- Discount each cash flow to present value using yield per period.
- Sum PVs to get the bond price.
- Compute time-weighted PVs (t × PV(CF_t)), where t is expressed in years (period / frequency).
- Macaulay duration = Σ (t × PV(CF_t)) / Price.
- Modified duration = Macaulay / (1 + y/m), where y is annual yield (decimal) and m is periods per year.
6. Intuition: What duration tells you
If a bond has a Macaulay duration of 7 years, that means the weighted-average time to receive the bond's cash flows is 7 years. If its modified duration is 6.5, an approximate 1% (0.01) rise in yield would lead to a −6.5% change in price. Duration is linear only for small yield changes — for larger yield moves convexity matters (explained below).
7. Convexity — the curvature correction
Duration provides a first-order (linear) approximation. Convexity measures curvature and provides a second-order correction, improving the accuracy of price-change estimates for larger yield moves. The convexity-adjusted approximation is:
ΔP/P ≈ −ModifiedDuration × Δy + 0.5 × Convexity × (Δy)^2
Where convexity is the weighted average of squared time periods weighted by PV, scaled appropriately. We do not compute convexity in this simple tool, but many fixed-income desks include it when hedging interest-rate exposure.
8. Use cases
- Portfolio immunization: Match asset and liability durations to reduce interest-rate risk.
- Risk monitoring: Track portfolio duration to control sensitivity to rate shifts.
- Pricing & trading: Estimate how a small change in market yields affects bond prices for trade sizing and stress tests.
9. Limitations and caution
Duration assumes parallel shifts in the yield curve and a stable yield for each bond. In reality, yield curve movements can be non-parallel, and credit spreads may change independently. Duration also assumes coupons are reinvested at the same yield — a simplifying assumption.
10. Example (worked)
Using the example from Part 1 (par 1,000, coupon 5% annual, 10 years, semi-annual, yield 4%):
- Coupon per period = 1000 × 0.05 / 2 = 25.
- Yield per period = 0.04 / 2 = 0.02.
- Compute PVs of 20 coupon payments and final 1,000 principal; sum gives price.
- Compute weighted times and derive Macaulay and Modified durations.
The calculator below will compute these precisely and show both the approximate and exact price change for a sample yield move.
11. Practical tips
- For zero-coupon bonds, Macaulay duration equals maturity (in years) because all cash flow occurs at maturity.
- Higher coupon bonds have shorter durations than lower coupon bonds with the same maturity and yield (because more cash flow arrives earlier).
- When yields fall, duration typically increases modestly (duration is yield-dependent).
12. Final thoughts
Duration is a powerful yet intuitive tool to measure interest-rate risk. Use it alongside convexity and scenario analysis for robust fixed-income portfolio management. This calculator gives you fast, accurate Macaulay and Modified durations and a practical sensitivity check for small yield changes.