Yield Shocks In Basis Points

Why Basis-Point Conversion Matters

Yield changes are often quoted in basis points, while duration and convexity calculations use a yield change expressed as a decimal. Confusing these units can make a price estimate wrong by a factor of 100 or 10,000. A reliable calculation therefore begins by converting the shock, then applies the duration-convexity approximation, and only afterward converts the estimated percentage change into dollars.

This tutorial assumes a bond or portfolio has an initial price $P$, modified duration $D_{\text{mod}}$, and convexity $C$, all evaluated at the initial yield. The yield changes by $\Delta y$, expressed as a decimal in the same yield convention used to calculate duration and convexity. The estimate is local: it is intended for a sufficiently small yield change and assumes that the relevant cash flows and valuation model do not change during the shock.

Definition: One basis point (bp) is $0.01\%$, or $0.0001$ when expressed as a decimal. Thus $b$ basis points correspond to the decimal yield change $\Delta y=b/10{,}000$. A rise in yield has $\Delta y>0$; a fall has $\Delta y<0$.

For example, a rise of 25 bp is $25/10{,}000=0.0025$, which is also $0.25\%$. A fall of 25 bp is $-0.0025$. Do not enter 0.25 or 25 into the approximation: those are not the decimal yield change.

The Duration-Convexity Estimate

Modified duration measures the first-order sensitivity of price to yield, while convexity accounts for the second-order curvature. With $D_{\text{mod}}$ and $C$ measured under a consistent yield convention, the estimated relative price change is

\frac{\Delta P}{P}\approx -D_{\text{mod}}\Delta y+\frac{1}{2}C(\Delta y)^2.

Here $\Delta P$ is the estimated change in price, not the new price. Once the relative change is found, the estimated dollar change is $P(\Delta P/P)$, and the estimated new price is $P+\Delta P$. The duration term changes sign when the yield shock changes sign. The convexity term contains a squared shock, so it is nonnegative when convexity is positive, whether yields rise or fall.

Unit check: Use the decimal yield change in both terms. If the shock is quoted as $b$ basis points, substitute $\Delta y=b/10{,}000$, including a negative sign for a yield decrease. Do not use the basis-point count itself.

The reason the two terms have these forms follows from a second-order Taylor expansion of price as a function of yield. At the initial yield, the local expansion is $P(y+\Delta y)\approx P(y)+P'(y)\Delta y+\tfrac12P''(y)(\Delta y)^2$. Modified duration is $D_{\text{mod}}=-P'(y)/P(y)$, and convexity is $C=P''(y)/P(y)$. Subtracting $P(y)$ from the expansion and dividing by $P(y)$ gives the stated approximation. The factor $1/2$ comes from the second-order Taylor term; omitting it doubles the convexity adjustment.

This relation is an estimate, not an exact repricing formula. It is appropriate when the price-yield relationship is locally smooth and the shock is small enough for higher-order terms to be neglected. Use duration and convexity calculated for the same compounding convention and yield measure as the shock. If a problem supplies them directly, use those supplied values rather than trying to convert them using an unrelated convention.

Worked Example: A Yield Increase Quoted in Basis Points

A hypothetical bond portfolio is valued at $160,000. Its modified duration is 6.4 and its convexity is 52, both measured using the same annual yield convention. Estimate the percentage and dollar price changes when yield rises by 35 bp. State the estimated new value to the nearest cent.

Method. Convert the shock to a decimal, calculate the duration and convexity contributions separately, and then apply the resulting relative change to the starting value. This avoids treating the basis-point number as a decimal yield.

Step 1: Convert the shock. Because the yield rises, the change is positive:

\Delta y=\frac{35}{10{,}000}=0.0035.

Step 2: Calculate the duration contribution.

-D_{\text{mod}}\Delta y=-6.4(0.0035)=-0.0224.

Step 3: Calculate the convexity contribution. First, $(0.0035)^2=0.00001225$. Therefore,

\frac{1}{2}C(\Delta y)^2 =\frac{1}{2}(52)(0.00001225) =26(0.00001225) =0.0003185.

Step 4: Combine the contributions.

\frac{\Delta P}{P}\approx -0.0224+0.0003185=-0.0220815.

The estimated relative change is $-0.0220815$, or a decrease of $2.20815\%$. The estimated dollar change is

\Delta P\approx 160{,}000(-0.0220815)=-3{,}533.04.

Thus the estimated new value is $160{,}000-3{,}533.04=\$156{,}466.96$. The estimate has the expected direction: a yield increase produces a price decrease, and positive convexity offsets part of the duration-only decrease.

Check. The relative change is negative, and the dollar decrease is less than the duration-only decrease of $160{,}000(0.0224)=\$3{,}584$, because the convexity contribution is positive.

From Relative Change to Dollar Price

Duration and convexity produce a relative change, such as $-0.0220815$, not a dollar amount. To estimate dollars, multiply that unrounded relative change by the initial price. Keep the full value through the multiplication and round the final price or dollar change only at the requested precision. Rounding an intermediate amount can alter the last cent of the final result.

The sign convention is particularly useful as a check. For positive duration, a yield rise makes the duration term negative and a yield fall makes it positive. For positive convexity, the convexity term is positive for either direction of shock. It adjusts the duration-only estimate upward; it does not reverse the usual price direction for a modest shock.

Worked Example: A Yield Decrease and a Dollar Gain

A hypothetical fixed-cash-flow bond position has a current market value of $87,500, modified duration 3.75, and convexity 28. The yield decreases by 18 bp. Estimate the percentage gain and new market value, rounding the final market value to cents.

Method. Preserve the negative sign of the yield change in the duration term, but square the signed change in the convexity term. This avoids incorrectly making the convexity adjustment negative when yields fall.

Step 1: Convert the shock.

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

Step 2: Find the duration contribution.

-D_{\text{mod}}\Delta y=-3.75(-0.0018)=0.00675.

Step 3: Find the convexity contribution. The squared yield change is $(-0.0018)^2=0.00000324$. Hence,

\frac{1}{2}C(\Delta y)^2 =\frac{1}{2}(28)(0.00000324) =14(0.00000324) =0.00004536.

Step 4: Combine and convert to dollars.

\frac{\Delta P}{P}\approx0.00675+0.00004536=0.00679536.

This is a gain of $0.679536\%$. The estimated dollar gain is

\Delta P\approx87{,}500(0.00679536)=594.594.

The estimated new value is $87{,}500+594.594=88{,}094.594$, which rounds to $88,094.59 to the nearest cent.

Check. Both contributions are positive for this yield decrease and positive convexity, so the estimate is a gain. The duration-only gain would be $87{,}500(0.00675)=\$590.625$, which is smaller than the convexity-adjusted gain.

Common Unit and Rounding Traps

Using the basis-point count directly. A 20 bp shock is $20/10{,}000=0.002$, not 20 or 0.20.
Confusing percent with decimal form. A 0.35% change is $0.0035$ as a decimal. Entering 0.35 instead makes the shock 100 times too large.
Dropping the sign. A yield decrease of 18 bp is $-0.0018$, not $+0.0018$. The sign determines the direction of the duration contribution.
Applying the sign to the squared term. The convexity term uses $(\Delta y)^2$. A negative shock still has a positive square.
Forgetting the one-half. The convexity contribution is $\tfrac12 C(\Delta y)^2$, not $C(\Delta y)^2$.
Calling a relative change a dollar change. A result of $0.006$ means a 0.6% estimated increase. Multiply by the initial price to obtain the estimated dollar increase.
Rounding too early. Carry the unrounded relative change through the dollar calculation, then round the final monetary amount as requested.

Worked Example: Detecting a Basis-Point Scaling Error

A hypothetical bond portfolio is worth $310,000. Its modified duration is 4.8 and its convexity is 36 under the yield convention being used. Yields rise by 45 bp. Estimate the new value to the nearest cent, and identify the result of a common incorrect conversion.

Method. Convert the shock before using either sensitivity measure. The likely trap is reading 45 bp as a decimal change of 0.45 rather than converting it to a decimal yield.

Correct conversion.

\Delta y=\frac{45}{10{,}000}=0.0045.

Duration term.

$$ -4.8(0.0045)=-0.0216. $$

Convexity term. Since $(0.0045)^2=0.00002025$,

\frac{1}{2}(36)(0.00002025) =18(0.00002025) =0.0003645.

Estimated relative change.

\frac{\Delta P}{P}\approx-0.0216+0.0003645=-0.0212355.

The estimated decrease is $2.12355\%$. The dollar change and new value are

\Delta P\approx310{,}000(-0.0212355)=-6{,}583.005, \qquad P+\Delta P=310{,}000-6{,}583.005=303{,}416.995.

Rounded to the nearest cent using ordinary half-up rounding, the estimated new value is $303,417.00.

Incorrect calculation and correction. If the basis-point number is mistaken for a decimal yield change, the duration term would be $-4.8(45)=-216$, an implausible relative change of $-21{,}600\%$. The correct input is $0.0045$, not 45. A separate but also incorrect conversion would use $0.45$, confusing 45 bp with a 45% yield change; the correct decimal change is 0.0045.

Check. A positive yield shock gives a negative estimated price change. Positive convexity partially offsets that decline: the duration-only estimate is $-2.16\%$, while the combined estimate is $-2.12355\%$.

A Repeatable Calculation Process

When a problem gives a shock in basis points, use a fixed sequence. It makes units visible and makes sign errors easier to catch.

Record the starting price and sensitivities.
Identify $P$, $D_{\text{mod}}$, and $C$, and confirm that the supplied sensitivities use the stated yield convention.

Convert the shock.
For a signed shock of $b$ bp, calculate $\Delta y=b/10{,}000$. A rise is positive; a fall is negative.

Calculate both approximation terms.
Evaluate $-D_{\text{mod}}\Delta y$ and $\tfrac12C(\Delta y)^2$ separately, retaining the full precision.

Combine and interpret.
Add the terms to obtain the estimated relative change. Check that its sign and size make sense for the shock.

Convert to money and round once.
Multiply the relative change by $P$ for the estimated dollar change, add it to $P$ for the new value, and round only the requested final amount.

This process is aligned with the SOA Exam FM (Financial Mathematics) syllabus because it combines yield-unit conversion with the standard duration-convexity price estimate. In a timed solution, writing the converted $\Delta y$ explicitly is a simple safeguard: it exposes the most consequential unit error before it affects every later step.

Check Your Understanding

For each problem, use the duration-convexity approximation and keep the yield shock in decimal form. Round monetary answers to cents where needed.

A hypothetical bond is worth $50,000, with modified duration 5 and convexity 30. Estimate its percentage price change and new price after a yield increase of 10 bp. (Answer: estimated change $-0.4985\%$; new price $49,750.75.)
A fixed-cash-flow position has value $120,000, modified duration 4.2, and convexity 25. Estimate its dollar gain and new value if yield falls by 24 bp. (Answer: estimated gain $1,218.24; new value $121,218.24.)
A portfolio worth $900,000 has modified duration 6 and convexity 48. Estimate its dollar change after yields rise by 25 bp. (Answer: estimated decrease $13,365; new value $886,635.)
A hypothetical bond position has value $200,000, modified duration 3.6, and convexity 20. Estimate the relative change and new value when yields fall by 40 bp. (Answer: estimated change $+1.456\%$; new value $202,912.)

What You'll Learn