Modified Duration As Local Yield Sensitivity

Why Yield Sensitivity Matters

A bond’s price depends on the yield used to discount its future payments. When that yield changes, the price changes in the opposite direction: a higher yield reduces the present value of fixed future payments, while a lower yield increases it. Modified duration measures this response locally, near a specified current yield.

The word “locally” is important. Modified duration gives a first-order estimate for a small yield change, not an exact price for a substantially different yield. The estimate is useful because it turns a price derivative into an interpretable percentage: approximately how much the price changes for a given change in yield.

Definition: For a bond with fixed cash flows and price $P(y)$ at annual effective yield $y$, its modified duration at that yield is $$ D_{\mathrm{mod}}(y)=-\frac{1}{P(y)}\frac{dP(y)}{dy}. $$ It is the negative derivative of price with respect to yield, divided by the current price. Under this convention, yield is expressed as a decimal, so a change of one percentage point is $0.01$.

Deriving Modified Duration From the Price Function

Suppose a bond pays fixed amounts $C_t$ at times $t=1,2,\ldots,n$, measured in years. Assume one annual effective yield $y>-1$ discounts every payment, there are no embedded options that alter the payments, and the cash flows do not change as yield changes. The price is

P(y)=\sum_{t=1}^{n}\frac{C_t}{(1+y)^t}.

Differentiate each discounted payment with respect to $y$. The derivative of $(1+y)^{-t}$ is $-t(1+y)^{-t-1}$, so

\frac{dP(y)}{dy} =-\sum_{t=1}^{n}\frac{tC_t}{(1+y)^{t+1}}.

Dividing the negative of this derivative by the price gives the modified duration:

D_{\mathrm{mod}}(y) =\frac{\displaystyle\sum_{t=1}^{n}\frac{tC_t}{(1+y)^{t+1}}} {\displaystyle\sum_{t=1}^{n}\frac{C_t}{(1+y)^t}}.

This expression also explains why duration is a positive measure of sensitivity for ordinary bonds with positive payments: every term in the numerator and denominator is positive. The derivative itself is negative, and the minus sign in the definition makes modified duration positive.

Why the price-change estimate holds: For a small change $\Delta y$, the first-order change in a differentiable price function is $\Delta P\approx P'(y)\Delta y$. Since $P'(y)=-D_{\mathrm{mod}}(y)P(y)$, dividing by the current price gives $$ \frac{\Delta P}{P}\approx-D_{\mathrm{mod}}(y)\Delta y. $$ The approximation is local: as the yield change grows, the actual price change can differ more from this first-order estimate.

The negative sign captures the inverse price-yield relationship. If modified duration is $2.4$, for example, a yield increase of $0.001$ (0.10 percentage point) gives an estimated percentage price change of $-2.4(0.001)=-0.0024$, or approximately $-0.24\%$. The yield change must be entered as a decimal, not as the number of percentage points.

Using the Derivative in Bond Calculations

A practical calculation follows a short sequence: establish the price function and current yield, differentiate, evaluate the derivative at that yield, and normalize by the current price. Then apply the local estimate using the yield change in decimal form.

Set the yield convention.
Confirm that the stated yield is annual effective and that the payment times and discounting convention match it.

Write the current price.
Discount each fixed payment at the current yield and add the present values.

Differentiate and normalize.
Calculate $P'(y)$, then use $D_{\mathrm{mod}}=-P'(y)/P(y)$.

Estimate the change.
Use $\Delta P/P\approx-D_{\mathrm{mod}}\Delta y$; multiply by the current price if a dollar estimate is required.

Worked Example: A Two-Year Bond and a Small Yield Increase

A hypothetical bond pays \$40 at the end of year 1 and \$1,040 at the end of year 2. Its annual effective yield is 4%. Find its modified duration and estimate the price change if the yield rises by 0.10 percentage point. Use the local estimate, then check the calculation against the exact repriced value.

Method. Differentiate the fixed-cash-flow price function and divide the negative derivative by the current price. The payments are fixed, and all times are in years.

At $y=0.04$, the current price is

P(0.04)=\frac{40}{1.04}+\frac{1040}{1.04^2} =38.461538+961.538462 =\$1{,}000.00.

Differentiating the two terms gives

P'(0.04) =-\frac{1(40)}{1.04^2}-\frac{2(1040)}{1.04^3} =-(36.982249+1{,}849.112426) =-1{,}886.094675.

Therefore,

D_{\mathrm{mod}}=-\frac{-1{,}886.094675}{1{,}000} =1.886095.

A 0.10 percentage-point increase is $\Delta y=0.001$. The estimated price change is

\Delta P\approx-1{,}886.094675(0.001)=-\$1.886095.

To the nearest cent, the local estimate is a decrease of \$1.89. As a check, exact repricing at 4.10% gives

P(0.041)=\frac{40}{1.041}+\frac{1040}{1.041^2} \approx \$998.12, \qquad P(0.041)-P(0.04)\approx-\$1.88.

The exact change is close to the local estimate, as expected for this small shift. A common mistake is to omit the negative sign and predict a price increase when yield rises; the derivative is negative, so the estimate must be a decrease.

Percentage Sensitivity and Dollar Sensitivity

Modified duration is a percentage sensitivity: it is normalized by the bond’s price. The derivative $P'(y)$, by contrast, is a dollar change per unit change in decimal yield. Thus $P'(y)=-D_{\mathrm{mod}}P(y)$. A bond with a larger price can have a larger dollar response even when its modified duration is the same.

The phrase “per unit yield change” refers to a change of $1.00$ in the decimal yield, not a one-percentage-point change. For a more practical scale, multiply modified duration by the decimal yield movement. For instance, 25 basis points is $0.0025$, not $0.25$ or $25$.

Yield movement	Decimal $\Delta y$	Approximate percentage price change
Increase of 1 percentage point	0.01	$-D_{\mathrm{mod}}(0.01)$
Increase of 10 basis points	0.001	$-D_{\mathrm{mod}}(0.001)$
Decrease of 25 basis points	$-0.0025$	$+D_{\mathrm{mod}}(0.0025)$

Worked Example: Estimating a Price Change for a Coupon Bond

A hypothetical bond pays \$100 at the end of each of years 1 and 2, followed by \$1,100 at the end of year 3. Its annual effective yield is 10%. Estimate the dollar price change when yield falls to 9.5%, using modified duration.

Method. Calculate the current price and derivative from the three discounted payments, then apply the first-order estimate. The yield change is a decrease of 0.5 percentage point, or $\Delta y=-0.005$.

The current price is

P(0.10)=\frac{100}{1.1}+\frac{100}{1.1^2}+\frac{1100}{1.1^3} =90.909091+82.644628+826.446281 =\$1{,}000.00.

The derivative is

P'(0.10) =-\left( \frac{1(100)}{1.1^2} +\frac{2(100)}{1.1^3} +\frac{3(1100)}{1.1^4} \right) =-(82.644628+150.262960+2{,}253.944130) =-2{,}486.851718.

Thus the modified duration is

D_{\mathrm{mod}}=-\frac{P'(0.10)}{P(0.10)} =\frac{2{,}486.851718}{1{,}000} =2.486852.

Applying the yield change,

\Delta P\approx-D_{\mathrm{mod}}P(0.10)\Delta y =-(2.486851718)(1{,}000)(-0.005) =\$12.434259.

The estimated increase is \$12.43 to the nearest cent. The result has the expected positive sign because yield falls. Avoid treating a 0.5 percentage-point decrease as $-0.5$: the correct decimal change is $-0.005$, and using $-0.5$ would make the estimate 100 times too large.

Limits of the Local Estimate

The derivative describes the slope of the price function at one yield. It does not claim that price is a straight-line function of yield. Consequently, the estimate is most appropriate for small yield changes around the stated yield. Exact repricing is required when an exact new price is requested or when the yield movement is large.

The fixed-cash-flow model also has limits. If a bond is callable and the likelihood or timing of its payments changes with yield, its cash flows are not fixed under this model. The same issue arises when a contractual feature changes payment amounts or timing. In those settings, the simple derivative of a fixed-payment price function does not capture the full yield response.

Worked Example: A Zero-Coupon Bond

A hypothetical zero-coupon bond pays \$5,000 in five years. Its annual effective yield is 4%. Find its modified duration and estimate the price increase for a 0.20 percentage-point yield decrease. Check the estimate by repricing at the new yield.

Method. Treat the single maturity payment as the only cash flow, differentiate its present value, and then use the local estimate. The yield change is $\Delta y=-0.002$.

The current price is

P(0.04)=\frac{5{,}000}{1.04^5} =\frac{5{,}000}{1.2166529024} \approx \$4{,}109.63.

For one payment at time 5,

P'(0.04)=-\frac{5(5{,}000)}{1.04^6}, \qquad D_{\mathrm{mod}}=-\frac{P'(0.04)}{P(0.04)} =\frac{5}{1.04} \approx 4.807692.

The estimated price increase is

\Delta P\approx-(4.807692)(4{,}109.63)(-0.002) \approx \$39.52.

Exact repricing at 3.8% gives

P(0.038)=\frac{5{,}000}{1.038^5} \approx \$4{,}149.38, \qquad P(0.038)-P(0.04)\approx \$39.75.

The local estimate is close, but not exact. This comparison checks both the direction and the scale: the lower yield produces a higher price, and the exact increase is near the estimated \$39.52. The common mistake of multiplying the yield change by five alone would ignore the yield’s effect on price sensitivity; the derivative gives $5/1.04$, not 5.

Common Exam Traps

Dropping the minus sign. Since $P'(y)$ is negative for fixed positive cash flows, modified duration is defined with a leading minus sign. The estimated price change still uses a minus sign: $\Delta P/P\approx-D_{\mathrm{mod}}\Delta y$.
Using percentage points as decimals. A move from 6.0% to 6.2% is $\Delta y=0.002$, not $0.2$.
Confusing a derivative with a percentage sensitivity. $P'(y)$ is in dollars per unit decimal yield; $D_{\mathrm{mod}}$ is the negative derivative divided by price.
Calling an estimate an exact price. Modified duration gives a first-order approximation around the current yield. Substitute the new yield into the price function for exact repricing under the model.
Applying the fixed-cash-flow formula when payments change. The derivation assumes cash flows remain fixed as yield changes.

Key takeaway: Modified duration is $-P'(y)/P(y)$. For a small decimal yield change, the estimated percentage price change is $-D_{\mathrm{mod}}\Delta y$. Keep the negative sign, use the decimal yield movement, and distinguish a local estimate from an exact repricing.

Check Your Understanding

For each question, use annual effective yield and fixed cash flows unless stated otherwise. Round numerical answers to four decimal places where needed.

A zero-coupon bond pays \$2,000 in three years at a 4% annual effective yield. Find its modified duration and estimate the percentage price change for a 0.10 percentage-point yield increase. (Answer: $D_{\mathrm{mod}}=2.8846$; estimated price change $-0.2885\%$.)
A bond pays \$100 at the end of year 1 and \$1,100 at the end of year 2. At a 10% annual effective yield, find its current price and modified duration. (Answer: Price $=\$1{,}000.00$; $D_{\mathrm{mod}}=1.7355$.)
A fixed payment of \$1,000 is due in two years, and the current annual effective yield is 0%. Find the modified duration and estimate the price change for a 0.50 percentage-point yield increase. (Answer: $D_{\mathrm{mod}}=2.0000$; estimated price change $-1.00\%$, or $-\$10.00$.)
A bond has modified duration 3.2 and current price \$980. Estimate its dollar price change for a yield decrease of 25 basis points. (Answer: Increase of approximately $\$7.84$, using $\Delta y=-0.0025$.)
A bond’s annual effective yield rises from 5.00% to 5.30%. A candidate enters $\Delta y=0.30$ in the local price-change formula. Identify the error and give the correct decimal yield change. (Answer: The change must be entered as a decimal; $\Delta y=0.003$, not $0.30$.)

Yield movement	Decimal \(\Delta y\)	Approximate percentage price change
Increase of 1 percentage point	0.01	\(-D_{\mathrm{mod}}(0.01)\)
Increase of 10 basis points	0.001	\(-D_{\mathrm{mod}}(0.001)\)
Decrease of 25 basis points	\(-0.0025\)	\(+D_{\mathrm{mod}}(0.0025)\)

What You'll Learn