Title: What Is Modified Duration — A Practical Guide for Bond Investors
Key takeaways
– Modified duration measures a bond’s price sensitivity to small changes in interest rates: roughly, the percent change in price for a 1% (100 bps) change in yield.
– Formula: Modified Duration = Macaulay Duration / (1 + YTM/n), where YTM is yield-to-maturity (as a decimal) and n is coupon periods per year.
– Macaulay duration = weighted average time (in years) until each cash flow is received, weighted by the present value of each cash flow.
– Use modified duration to estimate price changes (linear approximation) and to manage interest-rate risk (duration matching, immunization). For larger rate moves, include convexity for accuracy.
– Source: Investopedia / Ellen Lindner (https://www.investopedia.com/terms/m/modifiedduration.asp)
1) What modified duration measures
Modified duration expresses how much a bond’s price is expected to change (in percent) for a small parallel change in interest rates. Numerically:
– If modified duration = 5, a 1% increase in yield is expected to cause roughly a 5% price decline; a 1% decrease in yield would increase price by about 5%.
2) Key formulas
– Modified Duration (MD) = Macaulay Duration / (1 + YTM/n)
– YTM = yield to maturity (decimal)
– n = number of coupon periods per year (e.g., 2 for semiannual)
– Macaulay Duration (years) = [Sum over t: t * PV(CF_t)] / Price
– PV(CF_t) = present value of the cash flow at time t
– Price = sum of PV(CF_t) for all cash flows
– Linear price-change approximation:
– ΔP / P ≈ -MD * Δy
– Δy expressed in decimal (e.g., 0.01 for 1%)
– For better accuracy on larger yield moves, add convexity:
– ΔP / P ≈ -MD * Δy + 0.5 * Convexity * (Δy)^2
3) Worked example (from Investopedia)
Bond: face $1,000, 3-year maturity, 10% annual coupon ($100), YTM = 5% (annual).
a) Price = 100/1.05 + 100/1.05^2 + 1,100/1.05^3 = $1,136.16
b) Macaulay Duration = (95.24*1 + 90.70*2 + 950.22*3) / 1,136.16 = 2.753 years
c) Modified Duration = 2.753 / (1 + 0.05/1) = 2.753 / 1.05 = 2.623 (years)
Interpretation: For a 1% (0.01) parallel rise in yields, approximate price change = -2.623% → price falls by about $29.8 (1,136.16 * 0.02623).
4) Step-by-step: How to compute modified duration yourself
1. Gather bond inputs: coupon amount, payment frequency, maturity schedule, face value, current YTM (expressed consistent with payment frequency).
2. List cash flows at each payment date (coupons and principal at maturity).
3. Discount each cash flow to present value using the per-period YTM (YTM/n).
4. Compute the bond price = sum of PVs.
5. Compute weighted times: for each t, weight = PV(CF_t) / Price; then multiply weight * t (t in years).
6. Sum weights * t to get Macaulay Duration.
7. Compute Modified Duration = Macaulay Duration / (1 + YTM/n).
8. Estimate price sensitivity: ΔP ≈ -MD * P * Δy (for small Δy).
5) Practical ways investors and managers use modified duration
– Estimate interest-rate exposure: approximate percent change in bond prices for expected rate moves.
– Portfolio duration: compute market-value-weighted average of individual bond durations to get portfolio duration; use to match liabilities (immunization).
– Hedging: choose derivatives (futures, swaps) sized to offset portfolio duration.
– Comparing bonds: higher modified duration = greater interest-rate sensitivity, all else equal.
– Risk reporting: communicate interest-rate sensitivity in basis points or percent.
6) Important caveats & limitations
– Linear approximation: Modified duration is a first-order (linear) estimate. For large rate changes the estimate misstates price change; include convexity for second-order correction.
– Assumes parallel shifts of the yield curve; actual market moves may be non-parallel, affecting different maturities differently.
– Embedded options: callable/putable bonds change cash flows when rates move; use option-adjusted duration for these.
– Credit spread changes and liquidity/market factors can move prices independent of duration.
– Compounding convention: be consistent with how YTM is quoted (annual vs. semiannual) when computing MD.
7) Modified duration vs. Macaulay duration (quick difference)
– Macaulay duration measures the weighted average time (in years) to receive the bond’s cash flows.
– Modified duration converts Macaulay into a measure of price sensitivity to yield changes (percent change in price per unit yield change).
8) Does a zero-coupon bond pay interest?
– No. Zero-coupon bonds do not make periodic interest (coupon) payments. They sell at a discount to face value and are redeemed at par; the investor’s return is the difference between purchase price and redemption value. Zero-coupon bonds have duration equal to their maturity (Macaulay = modified when yield compounded annually).
9) Practical checklist for investors before using duration
– Verify YTM methodology and compounding frequency.
– Check whether the bond has embedded options — compute option-adjusted duration if needed.
– Consider convexity if you expect large rate moves.
– Use market values when aggregating to portfolio duration.
– Recompute duration after significant price, yield, or security changes.
10) The bottom line
Modified duration is a core tool for measuring and managing interest-rate risk. It provides a straightforward, interpretable estimate of how bond prices will move for small yield changes and forms the basis for portfolio immunization and hedging. Use it with an understanding of its linear nature and limitations (convexity, non-parallel yield curve moves, embedded options) and combine it with convexity and scenario analysis for more accurate risk assessment.
Source
– Investopedia — “Modified Duration,” Ellen Lindner. https://www.investopedia.com/terms/m/modifiedduration.asp
If you’d like, I can:
– Compute modified duration for a bond you specify (coupon, maturity, YTM, coupon frequency), or
– Show how to calculate portfolio duration from a list of bonds, or
– Demonstrate the convexity adjustment for a large yield change. Which would you prefer?