Estimating Bond Price Changes With Duration

From Duration to an Estimated Price Change

A bond’s modified duration turns a small change in yield into a practical estimate of its price response. The estimate can be expressed as a percentage or converted into dollars, which helps assess the impact of a rate movement on a position. It is a first-order approximation, not a replacement for valuing the bond again at the new yield.

The formula below uses an annual effective yield and fixed promised cash flows. This convention makes the yield change and modified duration directly compatible: enter the change in yield as a decimal, not as a number of percentage points or basis points. For bonds with a different yield-quoting convention, the duration must be defined consistently with that convention.

Definition: For a bond with price $P$ and modified duration $D_{\mathrm{mod}}$, a small change $\Delta y$ in its annual effective yield gives the approximate percentage price change $\Delta P/P \approx -D_{\mathrm{mod}}\Delta y$. The corresponding dollar change is $\Delta P \approx -P D_{\mathrm{mod}}\Delta y$, and the estimated new price is $P_{\mathrm{est}}=P+\Delta P$. A positive yield change therefore implies an estimated price decrease.

Modified duration is stated in years under this annual-yield convention, while $\Delta y$ is a decimal rate change. For example, 25 basis points is $0.25$ percentage points, or $0.0025$ as a decimal. The product $D_{\mathrm{mod}}\Delta y$ is a proportion: multiply it by 100 to express it as a percentage.

Why the Estimate Works

Suppose the bond has fixed cash flows $C_t$ at times $t=1,\ldots,n$, and an annual effective yield $y$. Its price is the present value of those cash flows:

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

Differentiate each discounted cash flow with respect to yield. The derivative is negative because a higher yield reduces the present value of every positive future payment. Dividing the resulting derivative by price gives the percentage sensitivity:

\frac{1}{P}\frac{dP}{dy} = -\frac{1}{P}\sum_{t=1}^{n}\frac{tC_t}{(1+y)^{t+1}} = -\frac{D_{\mathrm{Mac}}}{1+y} = -D_{\mathrm{mod}}.

Here $D_{\mathrm{Mac}}$ is Macaulay duration, the present-value-weighted average time of the cash flows, and under annual compounding $D_{\mathrm{mod}}=D_{\mathrm{Mac}}/(1+y)$. For a sufficiently small yield change, the derivative gives the approximate change:

\frac{\Delta P}{P}\approx \left(\frac{1}{P}\frac{dP}{dy}\right)\Delta y = -D_{\mathrm{mod}}\Delta y.

This derivation explains both the negative sign and the approximation. It assumes the cash flows are fixed and that the yield shift applies to the bond’s cash flows as a whole. It does not account for a change in promised payments, embedded option exercise, or a non-parallel shift in rates. Even when the assumptions hold, the price-yield relationship is curved, so the tangent-line estimate is most useful for small shocks.

Applying the Estimate and Checking It by Repricing

A reliable calculation keeps the percentage estimate, dollar estimate, and exact repricing distinct. First calculate the modified-duration estimate using the starting price and yield. Then, if the bond cash flows and new yield are given, discount those cash flows again at the new yield to obtain the exact price under the model. The difference between the two is the approximation error.

Put the yield change in decimal form.
Convert basis points or percentage points into a decimal rate change before multiplying by duration.

Estimate the percentage change.
Calculate $-D_{\mathrm{mod}}\Delta y$, retaining the sign.

Convert to dollars if needed.
Multiply the estimated percentage change, as a decimal, by the starting price.

Reprice at the new yield.
Discount every promised cash flow at the new yield and compare that price with the duration estimate.

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

A hypothetical bond has a face value of $ \$1{,}000$, pays a $5\%$ annual coupon, and matures in two years. Its annual effective yield is $5\%$, so its price is $ \$1{,}000$. Estimate the price change if the yield rises by 50 basis points, then compare the estimate with exact repricing.

Method and duration. The annual coupon is $0.05(1{,}000)=\$50$. At a $5\%$ yield, the present values of the two cash flows are $50/1.05=\$47.6190$ and $1{,}050/(1.05)^2=\$952.3810$. Their sum checks to the stated starting price of $ \$1{,}000$. Thus,

D_{\mathrm{Mac}} = \frac{1(47.6190)+2(952.3810)}{1{,}000} = 1.952381, \qquad D_{\mathrm{mod}} = \frac{1.952381}{1.05} = 1.859410.

Duration estimate. The 50-basis-point increase is $\Delta y=0.005$. Therefore, $\Delta P/P\approx -(1.859410)(0.005)=-0.00929705$, or an estimated decrease of $0.9297\%$. In dollars, $\Delta P\approx -\$1{,}000(0.00929705)=-\$9.30$. The estimated new price is $ \$1{,}000-\$9.30=\$990.70$.

Exact repricing. The new yield is $5.5\%$. Discount the same cash flows at that yield: $50/1.055=\$47.3934$ and $1{,}050/(1.055)^2=\$943.3740$. Thus the repriced bond is $\$47.3934+\$943.3740=\$990.7674$, or $ \$990.77$ to the nearest cent. The exact decrease is $ \$9.23$, equal to about $0.9233\%$ of the initial price. The estimate is close, but it predicts a slightly larger decrease. This comparison avoids treating the duration result as an exact price.

Worked Example: A Zero-Coupon Bond and a Small Rate Shock

A hypothetical five-year zero-coupon bond pays $ \$1{,}000$ at maturity and has an annual effective yield of $4\%$. Estimate its price change when the yield rises by 20 basis points, then reprice it exactly. Report dollar prices to the nearest cent and percentages to four decimal places.

Starting price and duration. The price is $P=1{,}000/(1.04)^5=\$821.9271$, or $ \$821.93$ to the nearest cent. All present value is attached to the year-five payment, so $D_{\mathrm{Mac}}=5$. Hence $D_{\mathrm{mod}}=5/1.04=4.807692$.

Duration estimate. A 20-basis-point increase is $\Delta y=0.0020$. The estimated percentage change is $-(4.807692)(0.0020)=-0.00961538$, or $-0.9615\%$. The estimated dollar change is $-\$821.9271(0.00961538)=-\$7.90$ to the nearest cent. The estimated new price is $ \$814.02$.

Exact repricing. At the new yield of $4.2\%$, the bond price is $1{,}000/(1.042)^5=\$814.07$ to the nearest cent. The exact decrease is about $ \$7.86$, or $0.9560\%$. The estimate again gives a close result, but its percentage decrease is slightly larger than the exact decrease.

For a zero-coupon bond, the single payment makes the timing especially clear: its Macaulay duration equals its maturity. The modified duration, not the maturity alone, is the quantity used in the price-change estimate.

Worked Example: Estimating the Effect of a Yield Decrease

A hypothetical two-year bond has a face value of $ \$1{,}000$, an annual coupon rate of $4\%$, and an annual effective yield of $4\%$. Its price is $ \$1{,}000$. Estimate the price change for a yield decrease of 40 basis points, then check against repricing.

Method and duration. The cash flows are $ \$40$ in year one and $ \$1{,}040$ in year two. Their present values at $4\%$ are $40/1.04=\$38.4615$ and $1{,}040/(1.04)^2=\$961.5385$, which sum to $ \$1{,}000$. Therefore,

D_{\mathrm{Mac}} = \frac{1(38.4615)+2(961.5385)}{1{,}000} = 1.961538, \qquad D_{\mathrm{mod}} = \frac{1.961538}{1.04} = 1.886095.

Duration estimate. The yield change is negative: $\Delta y=-0.0040$. The estimated percentage change is $-(1.886095)(-0.0040)=0.00754438$, or an increase of $0.7544\%$. The estimated dollar increase is $ \$1{,}000(0.00754438)=\$7.54$, giving an estimated price of $ \$1{,}007.54$.

Exact repricing. The new yield is $3.6\%$. The discounted cash flows are $40/1.036=\$38.6100$ and $1{,}040/(1.036)^2=\$968.9778$. Their sum is $ \$1{,}007.5878$, or $ \$1{,}007.59$ to the nearest cent. The exact increase is $ \$7.59$, approximately $0.7588\%$. The estimate is slightly smaller than the exact increase. The sign check is important: a yield decrease should produce an estimated price increase.

Reading the Estimate and Avoiding Common Errors

Modified duration is useful because it provides a quick, interpretable measure of first-order price sensitivity. For example, a modified duration of $4$ indicates an estimated price change of about $4\%$ in the opposite direction for a yield movement of one percentage point, provided the shock is small and the assumptions are appropriate. The estimate is local to the starting yield and price; it should not be treated as an exact forecast for a large movement.

To assess its accuracy for a particular yield shock, compare the estimate with exact repricing under the same fixed-cash-flow model. If the new yield is known, exact repricing is direct: discount each promised cash flow at that yield and sum the present values. This check also helps distinguish a calculation error from the approximation inherent in using duration.

Common exam trap: mixing up the sign or the units. Suppose $D_{\mathrm{mod}}=3.2$ and the yield rises by 30 basis points. The wrong calculation is $-3.2(30)=-96$, interpreted as a $96\%$ price decrease. It uses the number of basis points instead of a decimal yield change. The correction is $30\text{ basis points}=0.0030$, so the estimated change is $-3.2(0.0030)=-0.0096=-0.96\%$. For a $ \$50{,}000$ position, that corresponds to an estimated decrease of $ \$50{,}000(0.0096)=\$480$.

A second frequent error is to report $-D_{\mathrm{mod}}\Delta y$ as dollars. That expression is a proportion. Multiply it by the bond’s starting price to obtain a dollar change. Also retain the sign of $\Delta y$: replacing a negative yield change with its absolute value would incorrectly predict a price decrease when yields fall.

Key takeaway: Under annual compounding, estimate the relative price change as $-D_{\mathrm{mod}}\Delta y$, with the yield change expressed as a decimal. Multiply by the starting price for a dollar estimate, preserve the sign, and use exact repricing to check the approximation when the new yield is specified.

Check Your Understanding

For each problem, use the stated annual effective yield convention. Treat the bond’s promised cash flows as fixed.

A three-year zero-coupon bond with face value $ \$1{,}000$ has a yield of $5\%$. Use modified duration to estimate the percentage and dollar price change for a 10-basis-point yield increase. (Answer: approximately $-0.2857\%$, or $-\$2.47$, using a starting price of about $ \$863.84$.)
A one-year bond pays $ \$40$ in addition to $ \$1{,}000$ principal at maturity. Its yield is $4\%$ and its starting price is $ \$1{,}000$. Estimate the percentage and dollar price change for a 25-basis-point yield decrease. (Answer: approximately $+0.2404\%$, or $+\$2.40$.)
A bond has modified duration $4.2$ and a market value of $ \$250{,}000$. Estimate the dollar change for a 30-basis-point yield increase. State the direction of the change. (Answer: an estimated decrease of $ \$3{,}150$, or $-1.26\%$.)
A two-year zero-coupon bond has face value $ \$2{,}000$ and annual effective yield $4\%$. Calculate its starting price and modified duration, then estimate the dollar change for a 20-basis-point yield decrease. (Answer: starting price about $ \$1{,}849.11$, modified duration about $1.9231$, estimated increase about $ \$7.11$.)

What You'll Learn