What Is Modified Duration? Formula and How to Use It

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Modified duration is the approximate percentage change in a bond’s price for a 1 percentage point (100 basis point) change in its yield to maturity. A bond with a modified duration of 6 would be expected to lose about 6% of its price if its yield rose by one point, and gain about 6% if the yield fell by one point. Bond prices and yields move in opposite directions, so the measure is applied with a negative sign. It answers a practical question: how much interest rate risk does this bond or portfolio carry?

The Modified Duration Formula

Modified duration is built on Macaulay duration, so the calculation has two steps.

Step 1: Macaulay duration

Macaulay duration is the weighted average time, in years, until a bondholder receives the bond’s cash flows. Each cash flow’s weight is its present value as a share of the bond’s price:

Macaulay duration = Σ (t × PV of cash flow at t) / bond price

Here t is the time to each cash flow in years. You discount each coupon and the final principal at the bond’s yield to maturity, multiply each present value by its timing, sum the results and divide by the market price.

Step 2: Adjust for the yield

Modified duration = Macaulay duration / (1 + YTM / n)

YTM is the annual yield to maturity and n is the number of coupon periods per year (1 for annual, 2 for semiannual). The adjustment uses the per-period yield, so a semiannual bond divides by 1 + YTM/2, not 1 + YTM.

Example of Modified Duration

Hypothetical example. A bond has a $1,000 face value, a 10% annual coupon ($100 per year), three years to maturity, and a yield to maturity of 5%.

Price. Discount each cash flow at 5%:

• Year 1: $100 / 1.05 = $95.24
• Year 2: $100 / 1.05² = $90.70
• Year 3: $1,100 / 1.05³ = $950.22

Price = $1,136.16

Macaulay duration. Weight each period by its present value:

(1 × 95.24 + 2 × 90.70 + 3 × 950.22) / 1,136.16 = 3,127.31 / 1,136.16 ≈ 2.753 years

Modified duration. With annual coupons, n = 1:

2.753 / 1.05 ≈ 2.62

Interpretation. If the yield rises from 5% to 6%, the estimated price change is −2.62 × 1% ≈ −2.62%, or about −$29.78, taking the price to roughly $1,106.38.

Repricing the bond exactly at 6% gives $1,106.93, a drop of 2.57%. At 4%, the exact price is $1,166.50, a gain of 2.67%. The duration estimate (2.62% either way) slightly overstates the loss and understates the gain. That gap is convexity, covered below.

Interpreting Modified Duration

Percentage and dollar price change

The core relationship is:

% change in price ≈ −modified duration × Δyield

To express the change in dollars, multiply modified duration by the bond price. That product is called dollar duration:

Δprice ≈ −modified duration × price × Δyield

A common version measures the dollar change for a one basis point move, known as DV01 or the price value of a basis point: modified duration × price × 0.0001. For the example bond, that is 2.62 × $1,136.16 × 0.0001 ≈ $0.30 per $1,000 face value. Traders use this figure to size hedges, because it states risk in money rather than percent.

What raises or lowers modified duration

Holding other factors constant:

• Longer maturity generally means higher duration, because more of the value arrives later.
• Higher coupon means lower duration, because more value is paid out early.
• Higher yield means lower duration. Distant cash flows are discounted more heavily, so the bond becomes less sensitive to further rate increases.

A zero-coupon bond pays nothing until maturity, so its Macaulay duration equals its maturity. Hypothetical: a 10-year zero yielding 4% with semiannual compounding has a Macaulay duration of 10 and a modified duration of 10 / 1.02 ≈ 9.80. That is why long zero-coupon bonds such as STRIPS are among the most rate-sensitive bonds.

Effective Duration: Estimating It From Price Changes

When you can price a bond at different yields but do not want to work through the cash flow formula, you can approximate duration from the difference in present values:

Approximate duration = (P₋ − P₊) / (2 × P₀ × Δy)

P₀ is the current price, P₋ and P₊ are the prices after yields fall and rise by Δy, and Δy is in decimal form. Using the example bond with a one-point shift: (1,166.50 − 1,106.93) / (2 × 1,136.16 × 0.01) ≈ 2.62, matching the formula result.

This repricing approach is called effective duration when the prices come from a model that lets cash flows change with rates. That matters for callable bonds, putable bonds and mortgage-backed securities, where a rate move can change when or whether principal is repaid. Modified duration assumes fixed cash flows, so for an option-free corporate bond it estimates the price sensitivity well, while for bonds with embedded options effective duration is the appropriate measure.

Why Modified Duration Is Important

Uses include:

• Comparing bonds. Two bonds with similar yields but different durations carry different price risk.
• Scenario analysis. Multiply duration by an assumed rate move to estimate the potential loss.
• Portfolio management. A portfolio’s duration is approximately the market-value-weighted average of its holdings’ durations. Managers lengthen duration when they expect rates to fall and shorten it when they expect rates to rise.
• Funds. Bond mutual funds and bond ETFs commonly report duration, which indicates how their NAV may react to rate changes.
• Hedging and matching. Pension funds and insurers match the duration of assets to liabilities so that rate moves affect both sides similarly.

Limitations of Modified Duration

It is a linear approximation. The true price/yield relationship is curved. Modified duration is the slope of that curve at the current yield, so it is accurate for small moves and loses accuracy for large ones. In the example, the error is visible even at one point. Adding convexity, the second-order term, improves the estimate: % change ≈ −ModD × Δy + ½ × convexity × (Δy)².

The first-order estimate uses the current yield. Duration is computed at the starting rate, not the new rate. Once yields move, duration itself changes and must be recalculated.

It assumes a parallel shift. A single duration number treats all yields as moving by the same amount. When the curve steepens or flattens, a portfolio’s actual price change can differ from the estimate. Key rate durations address this by measuring sensitivity to moves at specific maturities (such as 2, 5, 10 and 30 years); their sum approximately equals the effective duration.

It ignores embedded options, as described above, and it measures only rate risk, not credit or liquidity risk.

Macaulay vs. Modified vs. Effective Duration

Modified duration is always slightly smaller than Macaulay duration whenever the yield is positive, because of the division by 1 + YTM/n.

The Bottom Line

Bond investors, fund holders and portfolio managers can use modified duration to compare rate risk across option-free bonds and to estimate losses from a rate move. It works best for small, parallel yield shifts. For large moves, add convexity, and for bonds with embedded options or uneven curve changes, rely on effective or key rate durations.