Basis-Point Shifts And Duration-Based Price Estimates

Why Basis-Point Conversion Matters

Duration estimates are sensitive to the units used for a yield change. A shift of 25 basis points is a small change in yield, not a change of 25 percent or 0.25 in decimal form. Converting the shift correctly is essential: a unit error can make an otherwise correct duration calculation wrong by a factor of 100 or more.

This tutorial focuses on converting a stated basis-point shift into a decimal rate change and applying that change to modified duration or dollar duration. The estimates are first-order approximations: they describe local price sensitivity, not an exact repricing after a finite yield change. The methods are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: One basis point (one bp) is one hundredth of one percent, or $0.01\%$. As a decimal rate change, $1\text{ bp}=0.0001$. Therefore, a shift of $b$ basis points is $\Delta y=b/10{,}000$, where the sign is positive for a yield increase and negative for a yield decrease.

A percentage point is different from a basis point: one percentage point is 100 basis points, or $0.01$ as a decimal rate change. For example, a yield rising from $4.00\%$ to $4.25\%$ rises by $0.25$ percentage points, which is 25 basis points and a decimal change of $0.0025$. The new yield is $4.25\%$; the change in yield is $0.0025$.

Yield shift	Percentage equivalent	Decimal rate change
1 bp	0.01%	0.0001
10 bp	0.10%	0.0010
25 bp	0.25%	0.0025
100 bp	1.00%	0.0100

Using Modified Duration for a Price Estimate

For a bond with fixed contractual cash flows, let $P$ be its price at yield $y$. Modified duration, denoted $D_{\mathrm{mod}}$, measures the local percentage sensitivity of price to a change in the yield under a specified compounding convention. For a small yield change $\Delta y$, the duration estimate is

\frac{\Delta P}{P}\approx -D_{\mathrm{mod}}\Delta y, \qquad \Delta P\approx -P D_{\mathrm{mod}}\Delta y.

Here, $\Delta y$ is a decimal rate change in the yield convention to which the stated modified duration applies. The negative sign reflects the inverse price-yield relationship for a fixed-cash-flow bond: a yield increase generally lowers the present value of its cash flows, while a yield decrease generally raises it.

Definition: Dollar duration is $P D_{\mathrm{mod}}$, measured in currency units per unit change in yield. With a small yield change, the estimated price change is $\Delta P\approx -(\text{dollar duration})\Delta y$. Dollar duration is not the price change for a one-basis-point shift; for that, multiply dollar duration by $0.0001$.

The price estimate can be written as $P_{\mathrm{new}}\approx P+\Delta P$. For example, if a bond has a modified duration of 5 and its yield increases by $0.01$, its estimated percentage price change is $-5(0.01)=-0.05$, or $-5\%$. The yield change $0.01$ is 100 basis points, not 1 basis point.

Why the Duration Approximation Holds

Modified duration is based on the derivative of price with respect to yield. In a sufficiently small neighborhood of the current yield, a differentiable price function can be approximated by its tangent line. If $P(y)$ is the price as a function of the consistently quoted yield, the first-order change is

P(y+\Delta y)\approx P(y)+P'(y)\Delta y.

By definition, modified duration is $D_{\mathrm{mod}}=-P'(y)/P(y)$. Substituting $P'(y)=-P(y)D_{\mathrm{mod}}$ into the tangent-line approximation gives

P(y+\Delta y)-P(y)\approx -P(y)D_{\mathrm{mod}}\Delta y.

Dividing by $P(y)$ gives the percentage-change formula. This argument explains both the negative sign and why the estimate is local: it uses the slope at the starting yield and does not account for changes in slope as the yield moves. A larger shift can therefore produce a greater difference between the estimate and the price obtained by fully repricing the bond.

Worked Applications

Worked Example: An 18-Basis-Point Increase

A hypothetical fixed-cash-flow bond is valued at $ \$2{,}400{,}000$. Its modified duration is 5.75 with respect to the quoted annual yield. Estimate the price change and new price if that yield increases by 18 basis points.

Method. Convert the yield change to a decimal first, then use dollar duration. The stated duration is already modified duration with respect to the quoted annual yield, so no further compounding conversion is needed.

The yield change is

\Delta y=\frac{18}{10{,}000}=0.0018.

The estimated percentage price change is

\frac{\Delta P}{P}\approx -(5.75)(0.0018)=-0.01035.

Thus, the estimated percentage change is $-1.035\%$. Dollar duration is

P D_{\mathrm{mod}} =(\$2{,}400{,}000)(5.75) =\$13{,}800{,}000.

The estimated dollar change is

\Delta P\approx -(\$13{,}800{,}000)(0.0018) =-\$24{,}840.

The estimated new price is $\$2{,}400{,}000-\$24{,}840=\$2{,}375{,}160$. The percentage and dollar calculations agree because $(\$2{,}400{,}000)(-0.01035)=-\$24{,}840$. The common mistake avoided here is treating 18 basis points as $0.018$, which would overstate the yield shift by a factor of 10.

Worked Example: Annual Yield and a Zero-Coupon Bond

A hypothetical 5-year zero-coupon bond pays $\$1{,}000$ at maturity and has an effective annual yield of $4\%$. Estimate its price after the yield rises by 25 basis points. Assume annual compounding, a fixed payment, and no change in the payment date.

Method. First find the current price and modified duration under annual compounding. Then apply the converted yield change.

The current price is

P=\frac{\$1{,}000}{(1.04)^5} =\frac{\$1{,}000}{1.2166529024} \approx \$821.93.

For an annual-compounding zero-coupon bond, all of the present-value weight is at time 5, so its Macaulay duration is 5 years. Its modified duration is

D_{\mathrm{mod}}=\frac{5}{1.04}\approx 4.807692.

A rise of 25 basis points is a decimal change of

\Delta y=\frac{25}{10{,}000}=0.0025.

The estimated relative price change is

\frac{\Delta P}{P}\approx -(4.807692)(0.0025) \approx -0.01201923.

That is an estimated decrease of about $1.201923\%$. Dollar duration is approximately $(\$821.9271)(4.807692)=\$3{,}951.57$, so the estimated dollar change is

\Delta P\approx -(\$3{,}951.57)(0.0025) \approx -\$9.88.

The estimated new price is $\$821.93-\$9.88=\$812.05$. As a check, the estimated decrease is about $1.201923\%$ of the initial price: $(\$821.9271)(0.01201923)\approx \$9.88$. The common mistake avoided is applying a basis-point number without dividing by 10,000.

Worked Example: Semiannual Yield Convention

A hypothetical 2-year zero-coupon bond has a face value of $\$10{,}000$. Its quoted yield is $6\%$ nominal per year convertible semiannually. Estimate the price change when the quoted annual nominal yield increases by 20 basis points. Assume the bond’s cash flow and maturity do not change.

Method. Match the yield shift to the annual nominal yield convention. With $m=2$ conversion periods per year, a yield of $j$ gives a period rate $j/2$. The bond has four half-year periods to maturity, so its current price is

P=\frac{\$10{,}000}{(1+0.06/2)^4} =\frac{\$10{,}000}{(1.03)^4} =\frac{\$10{,}000}{1.12550881} \approx \$8{,}884.87.

The Macaulay duration is 2 years because the only payment is at the 2-year maturity date. For a nominal annual yield convertible twice a year, modified duration with respect to the quoted annual yield is

D_{\mathrm{mod}}=\frac{2}{1+0.06/2} =\frac{2}{1.03} \approx 1.941748.

The quoted annual yield shift is

\Delta j=\frac{20}{10{,}000}=0.0020.

The corresponding half-year rate change is $\Delta j/2=0.0010$. Using modified duration with respect to the annual quoted yield, the estimated relative price change is

\frac{\Delta P}{P}\approx -(1.941748)(0.0020) \approx -0.003883495.

The estimated decrease is about $0.388350\%$, or $(\$8{,}884.87)(0.003883495)\approx \$34.50$. The estimated new price is $\$8{,}884.87-\$34.50=\$8{,}850.37$. The common mistake avoided is using $0.0010$, the half-year rate change, with modified duration expressed with respect to the annual quoted yield; those quantities use different yield units. Alternatively, a derivative taken with respect to the half-year rate must use the half-year rate change.

Sign, Units, and Model Limits

Before calculating, identify what the given duration measures and what rate convention describes the yield shift. A quoted annual yield convertible $m$ times per year is not itself the rate for one payment period: the period rate is the quoted nominal annual rate divided by $m$. Modified duration must be expressed with respect to the same yield variable used for $\Delta y$. If the duration is given with respect to the annual quoted yield, use the change in that annual yield; if it is given with respect to a period yield, use the change in that period yield.

Duration-based price changes also depend on the pricing model. The approximation assumes that the bond’s promised cash flows and their timing remain fixed, and that the yield change is applied consistently to those cash flows. It does not include transaction costs, accrued-interest adjustments, or changes to contractual cash flows. It is a local estimate, not a guarantee of the price after a finite yield shift. In particular, it does not add a curvature adjustment.

A useful sign check is to multiply the signs before interpreting the result. If $\Delta y>0$ and $D_{\mathrm{mod}}>0$, then $\Delta P<0$. If $\Delta y<0$, then $\Delta P>0$. For a positive-price fixed-cash-flow bond, this direction is a basic check on the calculation, though the estimated size remains approximate.

Read the yield shift.
Record whether the yield rises or falls, and identify whether the quote is annual, nominal with a stated conversion frequency, or a period rate.

Convert basis points.
For a shift of $b$ basis points, calculate $b/10{,}000$ with the correct sign. Do not confuse the new yield with the change in yield.

Match duration units.
Use the modified duration defined with respect to the same yield variable as the converted change.

Estimate and check.
Calculate $-D_{\mathrm{mod}}\Delta y$, multiply by price if a dollar change is needed, and check the direction and scale.

Common Exam Traps

Using the basis-point count as a decimal rate change. A 25-basis-point increase is $0.0025$, not $25$, $0.25$, or $0.025$.
Confusing a yield level with a yield change. A move from $5.10\%$ to $5.35\%$ is a $0.25$-percentage-point increase, or $0.0025$ as a decimal change. The final yield level is $5.35\%$.
Dropping the negative sign. A positive yield change and positive modified duration imply a negative estimated price change.
Mixing annual and period units. With a nominal annual yield convertible semiannually, decide whether the supplied duration is with respect to the annual yield or the half-year rate before using the yield change.
Calling an estimate an exact repricing. The duration formula uses a local slope. It is an approximation, particularly when the yield shift is not small.

Key takeaway: Convert $b$ basis points to the signed decimal change $b/10{,}000$, match that change to the yield convention used by modified duration, and estimate $\Delta P\approx -P D_{\mathrm{mod}}\Delta y$. Check the sign and remember that the result is a first-order estimate.

Check Your Understanding

For each problem, convert the yield shift before applying duration. Treat each requested price change as a duration-based estimate.

A bond priced at $\$750{,}000$ has modified duration 4.2 with respect to its quoted annual yield. Estimate its new price after a 12-basis-point yield decrease. (Answer: $\$753{,}780$)
A 3-year zero-coupon bond with face value $\$5{,}000$ has an effective annual yield of $5\%$. Estimate its price after the yield decreases by 10 basis points. Round the estimated price to the nearest cent. (Answer: $\$4{,}331.53$)
A bond is priced at $\$1{,}200{,}000$, has Macaulay duration 6.3 years, and has a nominal annual yield of $4.8\%$ convertible semiannually. Estimate its new price after the quoted annual yield rises by 15 basis points. Round to the nearest cent. (Answer: $\$1{,}188{,}925.78$)
A bond has dollar duration of $\$4{,}800{,}000$ per unit yield and a current price of $\$960{,}000$. Estimate its new price after a 30-basis-point yield decrease. (Answer: $\$974{,}400$)
A fixed-cash-flow bond is priced at $\$12{,}500$ and has modified duration 3.6 with respect to the quoted yield. Estimate the new price after a 75-basis-point increase. (Answer: $\$12{,}162.50$)

What You'll Learn