Estimating A Yield Change From A Price Change

From an Observed Price Change to a Yield Change

Duration and convexity are often used to estimate how a bond price responds when its yield changes. In some problems, the price change is observed and the yield change is unknown. The same duration-convexity approximation can be rearranged to estimate that unknown change, but the rearrangement produces a quadratic equation. Solving it requires attention to the price-change units, the yield units, and which root is appropriate.

The estimate is local: it uses duration and convexity measured at the starting yield and approximates the bond’s price response near that yield. It is not an exact repricing. The method is most useful for a reasonably small yield shift when the bond’s cash flows and the yield convention used to calculate the measures remain consistent.

Definition: Let $P$ be the bond’s initial price, $D$ its modified duration, and $C$ its convexity, all evaluated at the initial yield. Let $\Delta y$ be the yield change expressed in the same yield units used to calculate $D$ and $C$. Define the proportional price change as $r=\Delta P/P$, where $\Delta P=P_{\text{new}}-P$. The duration-convexity approximation is $r\approx-D\Delta y+\frac{1}{2}C(\Delta y)^2$.

For example, if a bond priced at $1,000 falls to $970, then $\Delta P=-\$30$ and $r=-30/1000=-0.03$, a 3% price decline. Use $r$, not the dollar change $-30$, in the duration-convexity equation. Yield changes also need consistent units: a change of 1 percentage point is $0.01$, not $1$.

Why the Quadratic Appears

The approximation comes from a second-order Taylor expansion of the price function around the initial yield. With the usual modified-duration and convexity definitions, $D=-P'(y)/P(y)$ and $C=P''(y)/P(y)$. The first derivative captures the local price response to yield; the second derivative captures curvature. Keeping terms through the second derivative gives the approximation below.

\frac{P(y+\Delta y)-P(y)}{P(y)} \approx \frac{P'(y)}{P(y)}\Delta y +\frac{1}{2}\frac{P''(y)}{P(y)}(\Delta y)^2 = -D\Delta y+\frac{1}{2}C(\Delta y)^2.

This also explains why the estimate can differ from an exact repricing: the expansion omits third- and higher-order terms. The approximation assumes a yield change small enough for those omitted terms not to dominate. Use duration and convexity calculated under the same compounding and yield-unit convention as $\Delta y$; otherwise the coefficients do not match the variable being solved for.

Rearranging and Selecting a Root

Set the observed proportional change $r$ equal to the approximate price change. Writing $x=\Delta y$ makes the quadratic form explicit. For $C\ne0$, the quadratic formula gives two mathematical roots, when the discriminant is nonnegative.

r\approx-Dx+\frac{1}{2}Cx^2 \quad\Longrightarrow\quad \frac{1}{2}Cx^2-Dx-r=0

x=\frac{D\pm\sqrt{D^2+2Cr}}{C}.

For the usual local estimate with positive duration and positive convexity, choose the root near zero:

\Delta y\approx\frac{D-\sqrt{D^2+2Cr}}{C} = \frac{-2r}{D+\sqrt{D^2+2Cr}}.

The second form follows by multiplying the first fraction by its conjugate and simplifying. It can be numerically convenient when the two terms in the numerator are close. When $r=0$, the local root is $0$, as expected; the other quadratic root is $2D/C$, which is generally not a local yield change. If $C=0$, the quadratic reduces to the duration-only estimate $\Delta y\approx-r/D$, provided $D\ne0$.

Condition: A real quadratic estimate requires $D^2+2Cr\ge0$. A nonnegative discriminant only means the quadratic has real roots; it does not establish that the approximation is accurate or that either root is a plausible local change. If there are two roots, choose the one consistent with a small change near the starting yield, and check its meaning in context.

Convert the price movement to a proportion.
Calculate $r=(P_{\text{new}}-P)/P$, retaining its sign.

Check units and inputs.
Use duration and convexity at the initial yield, and express the yield change in matching units.

Solve for the local root.
For the usual positive-duration, positive-convexity case, use the root with the minus sign before the square root.

Interpret and check.
State whether the yield rises or falls, then substitute the estimate into the approximation to see whether it reproduces the observed $r$.

Worked Applications

Worked Example: Estimating a Yield Rise After a Price Decline

A bond has modified duration $D=5$ and convexity $C=25$, measured per unit change in annual yield. Its observed price decline is 4.875% of its initial price. Estimate the yield change using the duration-convexity approximation, and explain which root is relevant.

Method. Use the proportional price change $r=-0.04875$, solve the quadratic, and select the root near zero. The negative sign is essential because the price declined.

The equation is

-0.04875\approx-5x+12.5x^2, \qquad 12.5x^2-5x+0.04875=0.

The discriminant in the duration-convexity formula is

D^2+2Cr=5^2+2(25)(-0.04875) =25-2.4375=22.5625, \qquad \sqrt{22.5625}=4.75.

Thus the two roots are

x=\frac{5\pm4.75}{25} \quad\Longrightarrow\quad x=0.01\ \text{or}\ x=0.39.

The local root is $x=0.01$: the estimated yield rises by 0.01, or 1.00 percentage point. The other root, a 39-percentage-point rise, is not a plausible local change for this approximation.

Check. Substituting the local root gives

$$ -5(0.01)+12.5(0.01)^2 =-0.05+0.00125 =-0.04875. $$

This matches the observed proportional decline. The common mistake is to treat the 4.875% decline as a positive number; doing so would reverse the direction of the estimated yield change.

Worked Example: Estimating a Yield Fall After a Price Gain

A fixed-cash-flow bond has $D=4$ and $C=20$ at its current yield. Its price rises by 8.4% of its initial price. Estimate the yield change and verify the result.

Method. Record the price gain as $r=+0.084$, then solve for the nearby root. A price gain ordinarily corresponds to a yield decrease for a bond with positive duration.

The discriminant is

D^2+2Cr=4^2+2(20)(0.084) =16+3.36=19.36, \qquad \sqrt{19.36}=4.4.

Therefore,

x=\frac{4\pm4.4}{20} \quad\Longrightarrow\quad x=-0.02\ \text{or}\ x=0.42.

The nearby root is $\Delta y=-0.02$, so the estimated yield falls by 2 percentage points. The other root, a 42-percentage-point increase, is not the local solution.

Check. Using the selected root,

-4(-0.02)+\frac{1}{2}(20)(-0.02)^2 =0.08+10(0.0004) =0.08+0.004 =0.084.

The approximation reproduces the 8.4% gain. The common mistake is to report $0.02$ as the yield change without its negative sign: the estimated change is a fall, not a rise.

Worked Example: A Smaller Price Gain and the Percentage-Point Answer

A bond has $D=5$, $C=25$, and an observed price gain of 3.00% of its initial price. Estimate the yield change. Report the yield change as a decimal and as a change in percentage points, rounding only at the end.

Method. Set $r=0.03$ and use the local root of the quadratic.

D^2+2Cr=25+2(25)(0.03)=26.5, \qquad \sqrt{26.5}=5.1478150705\ldots

The two roots are

x=\frac{5\pm\sqrt{26.5}}{25}.

The local root is

x=\frac{5-\sqrt{26.5}}{25} =-0.0059126028\ldots.

To express this as percentage points, multiply the decimal yield change by 100: $100x=-0.59126028\ldots$ percentage points. Rounded to four decimal places, the yield change is $-0.0059$, or a decrease of 0.5913 percentage points.

Check. The quadratic equation used to obtain the root is $12.5x^2-5x-0.03=0$, so the selected root satisfies $12.5x^2-5x=0.03$, the required approximate price gain. The common mistake is to round the percentage-point decrease too early; rounding the unrounded value to four decimal places gives 0.5913, not 0.5912.

Worked Example: When the Approximation Has No Real Root

Suppose a bond has $D=5$, $C=25$, and an observed price decline of 60% of its initial price. Determine whether the duration-convexity quadratic produces a real yield-change estimate.

Method. Test the discriminant before attempting to interpret a root. Here $r=-0.60$, so

$$ D^2+2Cr=25+2(25)(-0.60)=25-30=-5. $$

The discriminant is negative, so there is no real root. Equivalently, the quadratic price-change expression is $-5x+12.5x^2$. Its minimum occurs at $x=5/25=0.20$, where its value is $-0.50$. Thus this quadratic model cannot produce a decline greater than 50%, and it cannot match the stated 60% decline.

This does not prove that the bond’s actual price cannot fall by 60%; it says only that this particular second-order approximation cannot explain that observed change. The common mistake is to take the square root of the negative discriminant as though a real yield estimate had been found. In this situation, the appropriate conclusion is that the approximation does not give a real solution.

Interpretation and Exam Traps

The quadratic is useful because it incorporates curvature while estimating an unknown yield movement, but a root is not automatically a reliable estimate. The approximation was built around the initial yield. A distant root may solve the polynomial while lying far outside the range in which the local expansion is useful. Even the nearby root should be treated cautiously when the observed price move is large.

Use a proportion, not a dollar change. Divide the dollar price movement by the initial price before setting $r$.
Preserve the sign. A price decline has $r<0$; a price gain has $r>0$.
Match yield units. If $\Delta y$ is a decimal change, enter one percentage point as $0.01$.
Choose the local root. A second algebraic root can be mathematically valid but economically remote from the starting yield.
Check the discriminant and the approximation. A negative discriminant means no real root; a nonnegative one does not guarantee accuracy.

Key takeaway: Convert the observed price move to $r=\Delta P/P$, solve $\frac{1}{2}C(\Delta y)^2-D\Delta y-r=0$, and interpret the root near zero using consistent yield units. Then substitute it back into the approximation and assess whether a local estimate is reasonable.

Check Your Understanding

For each problem, use the duration-convexity approximation and identify the local yield-change estimate.

A bond has $D=3.5$, $C=14$, and a price decline of 3.43% of its initial price. Estimate the yield change. (Answer: $+0.0100$, a rise of 1.00 percentage point.)
A bond has $D=6$, $C=40$, and its price rises by 6.2%. Estimate the local yield change. (Answer: $-0.0100$, a fall of 1.00 percentage point.)
A bond has $D=4$, $C=16$, and its price falls by 4.875%. Find the local yield change. (Answer: $+0.0125$, a rise of 1.25 percentage points.)
A bond has $D=4$, $C=10$, and its price rises by 2.00%. Estimate the yield change, rounded to six decimals as a decimal and four decimals in percentage points. (Answer: approximately $-0.004969$, a decrease of 0.4969 percentage points.)
A bond has $D=3$, $C=12$, and its price falls by 40%. Does the quadratic produce a real yield-change estimate? (Answer: No. The discriminant is $9+24(-0.40)=-0.6$, so there is no real root.)

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