What Is Modified Duration?

Translating duration into a price-change estimate — the formula, the worked numbers, and where the linear approximation breaks down.

Modified duration is the percentage change in a bond's price for a one percentage-point change in yield. It is the single most practical rate-sensitivity number in fixed income: give it a yield move and it hands back an estimated price move. If you want to know how much a portfolio loses when rates rise 50 basis points, modified duration is where you start. Run the numbers yourself with the bond pricing calculator, then come back to understand exactly what the output is telling you.

From Macaulay duration to modified duration

Macaulay duration measures the weighted-average time to receive a bond's cash flows, expressed in years. It is useful on its own, but it cannot be applied directly to a yield change. The conversion is one step:

ModDur = MacDur / (1 + y/k)

where y is the bond's yield to maturity expressed as a decimal and k is the number of compounding periods per year. For a semiannual-pay bond with a YTM of 6% (y = 0.06, k = 2), the denominator is 1 + 0.06/2 = 1.03. A Macaulay duration of 4.326 years becomes a modified duration of 4.326 / 1.03 = 4.20. That number is now dimensionless — or more precisely, its unit is "percent per unit change in yield," which is what makes it directly useful.

The working approximation

Modified duration linearizes the price-yield relationship around a single point:

%ΔPrice ≈ −ModDur × Δy

The negative sign reflects the inverse relationship between bond prices and yields. If modified duration is 4.20 and yields rise by 50 basis points (Δy = +0.005), the estimated price change is:

%ΔPrice ≈ −4.20 × 0.005 = −0.021 = −2.1%

A bond priced at $980 would fall to roughly $959.42. The approximation is fast and intuitive. It is the calculation a trader does mentally when a central bank moves rates.

Dollar duration and DV01

Percentage changes are useful for comparing securities, but risk management usually works in dollar terms. Two derived measures handle that:

• Dollar duration scales the percentage sensitivity by the bond's full price: Dollar Duration = ModDur × Price. For the bond above, that is 4.20 × $980 = $4,116 per unit change in yield. A 1% yield move costs roughly $4,116 per $980 face.
• DV01 (dollar value of a basis point, also called PVBP) is the price change for a one-basis-point move in yield: DV01 = Dollar Duration / 10,000. Here, $4,116 / 10,000 = $0.41 per basis point per $980 face. DV01 is the standard unit for hedging — you size an offsetting position by matching DV01s.

A quick reference for the three related measures:

Measure	Formula	Example (ModDur 4.20, Price $980)
Modified duration	MacDur / (1 + y/k)	4.20
Dollar duration	ModDur × Price	$4,116 per unit Δy
DV01	Dollar Duration / 10,000	$0.41 per bp

The linear-approximation limitation

The price-yield curve is not a straight line — it is convex. Modified duration approximates it with a tangent line at the current yield. For small moves (say, ±25 bp) the error is negligible. For larger moves the approximation systematically understates the actual price. When yields fall, the bond rises more than modified duration predicts; when yields rise, the bond falls less. Both deviations work in the holder's favor — that asymmetry is convexity, and it has value.

The full second-order approximation is:

%ΔPrice ≈ −ModDur × Δy + ½ × Convexity × (Δy)²

For the same 50 bp rise, if convexity is 28, the convexity adjustment adds ½ × 28 × (0.005)² = +0.035%, partially offsetting the −2.1% duration estimate. For a 200 bp shock the convexity correction becomes material. The interaction between the two is covered in detail in the discussion of duration and convexity.

Hedging and portfolio rate risk

Modified duration's main operational use is constructing rate hedges. The logic is straightforward: if you hold a portfolio with a DV01 of $5,000 per basis point and want to neutralize that exposure, you need an offsetting position with DV01 of −$5,000. In practice that means selling Treasury futures or entering a pay-fixed interest-rate swap, sized so the instrument's DV01 matches the portfolio's.

Portfolio-level modified duration is the market-value-weighted average of the individual durations:

ModDur_portfolio = Σ (w_i × ModDur_i)

where w_i is each position's share of total market value. A $10 million portfolio with 60% in a 4.20-duration bond and 40% in a 1.80-duration floating note has portfolio duration of 0.60 × 4.20 + 0.40 × 1.80 = 2.52 + 0.72 = 3.24. A 100 bp parallel shift in the yield curve costs roughly 3.24% of the portfolio, or about $324,000. That number drives decisions on whether to add or reduce duration — through outright bond trades, futures, or swaps — before a rate-sensitive event.

Modified duration is a first-order tool. It is fast, additive across positions, and gives you a number you can act on immediately. Its limitations — the linear approximation, the assumption of a parallel yield-curve shift, the exclusion of embedded options — are well understood and corrected for when precision matters. Used correctly, it turns an abstract yield forecast into a concrete dollar estimate.