Modified Duration
Modified duration is a measure of a fixed income instrument's price sensitivity to changes in interest rates, expressed as the percentage change in the bond's price for a 1% (100 basis point) change in yield. It is derived from Macaulay duration (the weighted-average time to receipt of a bond's cash flows) by dividing by (1 + yield/n), where n is the number of compounding periods per year.
Key takeaways
- A bond with modified duration of 5 will decrease approximately 5% in price for each 1% increase in yield, and increase approximately 5% for each 1% decrease in yield.
- Modified duration is a linear approximation; for large yield changes, convexity correction is needed to improve accuracy: ΔP/P ≈ −D_mod × Δy + (1/2) × Convexity × (Δy)².
- Duration increases with time to maturity and decreases with coupon rate, yield to maturity, and frequency of coupon payments.
- Portfolio duration is the dollar-weighted average of individual bond durations, enabling aggregate interest rate risk management for multi-security portfolios.
- DV01 (dollar value of a basis point) = Modified Duration × Price × 0.0001, and is the preferred measure for expressing interest rate risk in dollar terms.
Explanation
Modified duration is the workhorse interest rate risk metric of the fixed income world, providing an intuitive and computationally tractable measure of how much a bond's price will change when interest rates change. Its development from Macaulay duration — which measures the weighted-average time to receive a bond's cash flows — transforms a time-based measure into a price-sensitivity metric by accounting for the mathematical relationship between yields and bond prices.
The derivation of modified duration proceeds from the basic bond pricing formula. The price of a bond equals the present value of all future cash flows discounted at the yield to maturity: P = Σ [CF_t / (1+y)^t]. Taking the derivative of price with respect to yield and dividing by price yields the negative of modified duration: dP/P ≈ −D_mod × dy. This relationship implies that duration measures the elasticity of price with respect to (1+y) — a proportional change in discount factor.
Modified duration has important properties that practitioners must understand. First, it is an approximation that is most accurate for small yield changes. For larger changes, the convexity of the price-yield relationship means that modified duration will underestimate the actual price increase from a yield decline and overestimate the price decrease from a yield increase (because the price-yield curve is convex). This asymmetry — where bonds gain more from rate declines than they lose from equivalent rate increases — is itself a valuable property that investors pay for in the form of lower yields on higher-convexity bonds.
For portfolio managers, modified duration is the primary tool for interest rate risk management. By calculating the portfolio's aggregate duration (the dollar-weighted average of individual bond durations) and comparing it to a benchmark duration, managers can identify their interest rate positioning relative to the benchmark. Duration management using Treasury futures — buying futures to increase duration, selling to decrease it — allows managers to adjust overall interest rate sensitivity without buying or selling individual bonds. Duration mismatches between assets and liabilities (as in pension fund asset-liability management) create interest rate risk that must be carefully monitored and hedged.
Formula
D_mod = D_mac / (1 + y/n); ΔP/P ≈ −D_mod × Δy
Example
A 10-year Treasury note with a 4% semiannual coupon, priced at par ($100) to yield 4%, has a Macaulay duration of approximately 8.11 years. Modified duration = 8.11 / (1 + 0.04/2) = 8.11 / 1.02 = 7.95 years. If yields rise by 50 basis points (0.50%), the approximate price change is: ΔP/P ≈ −7.95 × 0.005 = −3.975%, implying a price decline from $100 to approximately $96.03. The DV01 = $100 × 7.95 × 0.0001 = $0.0795 per $100 face value, meaning each basis point of yield change moves the price by approximately $0.0795.
Related terms
Basis Bond Convexity Duration Dv01 Eurodollar Face Value Federal Funds Rate Implied Repo Rate Interest Rate Macaulay Duration Par Value