Improving Duration Estimates With Convexity

Why Add Convexity to Duration?

Modified duration provides a useful first estimate of how a bond’s price responds to a small change in yield. But a bond’s price does not move along a straight line as yield changes: for a conventional fixed-cash-flow bond, its price-yield curve bends. Duration captures the local slope; convexity captures the curvature. Adding a convexity adjustment generally makes the duration estimate more accurate, especially when the yield change is not very small.

This tutorial uses an annual effective yield $i$ and annual cash-flow periods unless stated otherwise. Assume the bond has fixed, known cash flows, positive cash flows, and no default or embedded options. A yield change $\Delta i$ is expressed as a decimal: a change of 0.50 percentage points is $0.005$. The estimate is a local approximation, not an exact repricing.

Definition: For a bond with cash flow $CF_t$ at the end of year $t$, its price at annual effective yield $i$ is $P(i)=\sum_t CF_t(1+i)^{-t}$. Its modified duration is $D_{\text{mod}}=-P'(i)/P(i)$, and its convexity is $C=P''(i)/P(i)$. For fixed positive cash flows, modified duration is positive and convexity is positive.

The first-order duration estimate is $\Delta P/P\approx-D_{\text{mod}}\Delta i$. It predicts the direction of the price change correctly for an ordinary fixed-cash-flow bond: a yield increase produces an estimated price decrease, and a yield decrease produces an estimated price increase. The convexity adjustment adds a second-order term:

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

Here $P$ is the bond’s current price, $\Delta P$ is its estimated dollar price change, and $\Delta P/P$ is the estimated percentage change. Once the percentage change is calculated, multiply it by the current price to obtain the dollar change.

Why the Convexity Term Appears

The formula follows by approximating the price function near the current yield. The second-order Taylor expansion of $P(i+\Delta i)$ is

P(i+\Delta i)\approx P(i)+P'(i)\Delta i+\frac{1}{2}P''(i)(\Delta i)^2.

Subtract $P(i)$ from both sides and divide by $P(i)$. By definition, $P'(i)/P(i)=-D_{\text{mod}}$ and $P''(i)/P(i)=C$. Therefore,

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

For fixed positive cash flows, differentiating each discounted cash flow twice gives

P'(i)=-\sum_t\frac{t\,CF_t}{(1+i)^{t+1}}, \qquad P''(i)=\sum_t\frac{t(t+1)CF_t}{(1+i)^{t+2}}.

The second derivative is positive, so the price-yield curve bends upward. The convexity term is positive whether $\Delta i$ is positive or negative because it contains $(\Delta i)^2$. Thus it offsets some of the duration-only decline when yields rise, and adds to the duration-only increase when yields fall. This does not mean the price rises for a yield increase: the negative duration term ordinarily remains larger for a modest change.

Worked Example: Convexity of a Zero-Coupon Bond

A hypothetical three-year zero-coupon bond pays $1,000 at maturity. Its annual effective yield is 5%. Estimate its price change if the yield rises by 1 percentage point, using duration alone and then duration plus convexity. Compare the adjusted estimate with exact repricing. Use the fixed-cash-flow annual-compounding model.

Method and current price. The only cash flow is $1,000 at year 3, so $P=1{,}000/(1.05)^3=\$863.84$, to the nearest cent.

Calculate the sensitivity measures. For a zero-coupon bond, all present value is at time 3. The derivative formulas give $D_{\text{mod}}=3/(1.05)=2.8571$ and $C=3(4)/(1.05)^2=10.8844$.

Duration estimate. The yield change is $\Delta i=0.01$, not 1. Substitution gives $-2.8571(0.01)=-0.028571$, or an estimated decrease of 2.8571%.

Add convexity. The adjustment is $\frac12(10.8844)(0.01)^2=0.0005442$, or 0.0544 percentage points. The combined estimate is $-0.028571+0.000544=-0.028027$, a decrease of about 2.8027%.

Convert to dollars and check. The adjusted dollar change is $\$863.84(-0.028027)\approx-\$24.21$, so the estimated new price is about $839.63. Exact repricing at 6% gives $\$1{,}000/(1.06)^3=\$839.62$. The exact decrease is about 2.8036%, close to the convexity-adjusted estimate. The method avoids the common mistake of treating a 1-percentage-point yield change as $1$ rather than $0.01$.

Calculating Convexity From Cash Flows

For a bond with annual cash flows, divide the weighted second-derivative sum by the current price. Each cash flow contributes according to its amount, its payment time, and its discounted value. In this convention, the factor $t(t+1)$ and the two additional powers of $1+i$ come from differentiating the discounted cash flow twice.

C=\frac{1}{P}\sum_t\frac{t(t+1)CF_t}{(1+i)^{t+2}}

This formula’s units correspond to changes in an annual effective yield expressed as a decimal. If a bond is quoted at a nominal annual yield $i^{(2)}$ convertible semiannually, its period yield is $i^{(2)}/2$. When measuring sensitivity to a change in that quoted annual rate, the derivative formulas must include the conversion factor $1/2$ for each differentiation. Do not apply the annual-period formula directly to semiannual cash flows while treating each half-year as a full year.

Worked Example: A Coupon Bond When Yields Fall

A hypothetical two-year bond with face value $1,000 pays a 6% annual coupon. Its annual effective yield is 5%. Estimate the percentage price change when the yield falls by 2 percentage points. Use duration plus convexity, then compare with the new price from exact repricing.

Method and current price. The cash flows are $60 at year 1 and $1,060 at year 2. Their present values are $60/1.05=\$57.142857$ and $1{,}060/(1.05)^2=\$961.451247$. Thus $P=\$57.142857+\$961.451247=\$1{,}018.594104$, or $1,018.59 to the nearest cent.

Calculate modified duration. Using the first derivative,

D_{\text{mod}} =\frac{1}{P}\left( \frac{1(60)}{(1.05)^2} +\frac{2(1{,}060)}{(1.05)^3} \right) =\frac{1{,}885.7575}{1{,}018.5941} \approx1.8513.

Calculate convexity. Using the second derivative,

C=\frac{1}{P}\left( \frac{1(2)(60)}{(1.05)^3} +\frac{2(3)(1{,}060)}{(1.05)^4} \right) =\frac{5{,}336.0483}{1{,}018.5941} \approx5.2386.

Estimate the change. Here $\Delta i=-0.02$. The duration contribution is $-1.8513(-0.02)=0.037026$, or an increase of 3.7026%. The convexity contribution is $\frac12(5.2386)(0.02)^2=0.0010477$, or an additional 0.1048 percentage points. The estimated total increase is $0.037026+0.0010477=0.038074$, or about 3.8074%.

Check by repricing. At a 3% yield, the price is $60/1.03+1{,}060/(1.03)^2=\$1{,}057.40$, to the nearest cent. Relative to the original price, that is an increase of approximately $(1{,}057.40-1{,}018.59)/1{,}018.59=3.81\%$, close to the convexity estimate. The correction matters here because the yield move is large enough for curvature to contribute noticeably. The calculation avoids the mistake of subtracting the convexity term when yields fall; the squared yield change makes the adjustment positive.

When the Correction Matters

The size of the convexity contribution is $\frac12C(\Delta i)^2$. Holding convexity fixed, doubling the size of the yield change makes the correction four times as large. Holding the yield change fixed, a bond with greater convexity receives a larger correction. These comparisons concern the approximation term, not a guarantee about the total error against exact repricing.

A useful practical interpretation is to compare the duration-only estimate with the convexity adjustment in percentage points. If the adjustment is tiny relative to the duration contribution, reporting many decimals in the adjusted estimate would be misleading. If it is material, omitting it can understate the estimated price increase when yields fall or overstate the estimated price decline when yields rise.

Convexity does not make a second-order approximation exact. Taylor expansion omits third- and higher-order terms, so a large yield move can leave a meaningful residual error. Also, the fixed-cash-flow formulas assume the cash flows do not change when yields change. For a bond with an embedded option, expected cash flows may respond to rates; a model-based effective duration and effective convexity, obtained by repricing under yield shifts, are more appropriate than the fixed-cash-flow formulas.

The formula also assumes the yield change applies consistently to the cash flows being valued. If yields at different maturities change by different amounts, one parallel shift is not a complete description of the price response. In that case, the single duration-plus-convexity estimate may not represent the effect of the actual yield-curve movement.

Worked Example: From Sensitivity Measures to Dollars

A hypothetical fixed-cash-flow bond portfolio is currently worth $500,000. Its modified duration is 4.2 and its convexity is 28, calculated for an annual effective yield. Estimate the dollar price change if yield rises by 1.5 percentage points. Assume a parallel yield shift, unchanged cash flows, and no default. No exact repricing is requested.

Method. Convert the change to $\Delta i=0.015$. The duration contribution is $-4.2(0.015)=-0.063$, or a decrease of 6.3%. The convexity contribution is $\frac12(28)(0.015)^2=14(0.000225)=0.00315$, or an increase of 0.315 percentage points.

Combine and convert. The estimated percentage change is $-0.063+0.00315=-0.05985$, a decrease of 5.985%. The estimated dollar change is $\$500{,}000(-0.05985)=-\$29{,}925$. The estimated new value is $\$500{,}000-\$29{,}925=\$470{,}075$.

Check and common mistake. Since the yield rises, the final change should be negative; the positive convexity adjustment reduces the estimated loss but does not reverse its sign. Using $1.5$ instead of $0.015$ would produce $-4.2(1.5)+\frac12(28)(1.5)^2=25.2$, an implausible estimated increase of 2,520%. Yield changes stated in percentage points must first be written as decimal changes.

Estimating Convexity by Repricing

When a suitable formula is unavailable, convexity can be estimated by repricing at yields a small distance above and below the current yield. Let $P_0$ be the current price, $P_+$ the price after a yield increase of $h$, and $P_-$ the price after a decrease of $h$. For a small, equal shock $h$, the central-difference estimate is

C\approx\frac{P_-+P_+-2P_0}{P_0h^2}.

The numerator compares the sum of the two shocked prices with twice the current price. The first-order price effects in opposite directions largely cancel, leaving information about curvature. The same yield units must be used throughout: if $h$ is 0.01, the denominator contains $0.01^2$, not $1^2$. This is a numerical estimate of local convexity; its quality depends on the repricing model and the size of the shocks.

Key takeaway: For an ordinary bond with fixed positive cash flows, estimate the percentage price change using $-D_{\text{mod}}\Delta i+\frac12C(\Delta i)^2$, with the yield change expressed as a decimal. Convexity improves the local duration estimate, but large moves, nonparallel yield changes, and cash flows that vary with rates require additional care. This treatment is aligned with the SOA VEE Accounting and Finance syllabus.

Check Your Understanding

Use decimal yield changes. Unless a question says otherwise, assume an ordinary fixed-cash-flow bond and a parallel yield shift.

A two-year zero-coupon bond has an annual effective yield of 4%. Find its modified duration and convexity, then estimate the percentage price change for a 0.50-percentage-point yield increase. (Answer: $D_{\text{mod}}=1.9231$, $C=5.5473$, estimated change $-0.9546\%$.)
A bond portfolio is worth $800,000, with modified duration 3.6 and convexity 18. Estimate its dollar and percentage price change for a 1-percentage-point yield decrease. (Answer: estimated increase $3.6900\%$, or $\$29{,}520$; estimated value $\$829{,}520$.)
A fixed-cash-flow bond has modified duration 2.4 and convexity 8.5. Estimate its percentage price change for a 1.25-percentage-point yield increase. State the separate duration and convexity contributions. (Answer: duration contribution $-3.0000\%$, convexity contribution $+0.0664$ percentage points, total estimated change $-2.9336\%$.)
A bond has current price $P_0=\$102.40$. Repricing gives $P_-=\$104.35$ after a yield decrease of 0.01 and $P_+=\$100.55$ after an increase of 0.01. Estimate its convexity using the central-difference formula. (Answer: $C\approx9.7656$.)
Two bonds have the same modified duration. Bond A has convexity 12 and Bond B has convexity 30. For a parallel yield change of either $+0.02$ or $-0.02$, by how many percentage points does Bond B’s convexity adjustment exceed Bond A’s? (Answer: $0.3600$ percentage points, for either direction of the yield change.)

What You'll Learn