Convexity Of A Coupon Bond

Why Coupon-Bond Convexity Matters

A bond’s price does not generally move in a straight line as its yield changes. Duration describes the first-order response: for a small yield change, it estimates the percentage price change. Convexity measures the curvature of the price-yield relationship and adds a second-order correction. This matters because coupon bonds have payments at several dates, so their price curvature reflects the timing and size of all those payments.

This tutorial uses a bond with fixed annual cash flows and an annual effective yield. The bond is assumed to be noncallable, or otherwise to have coupon and redemption payments that do not change when the yield changes. The result does not directly apply if a yield change alters the bond’s expected cash flows. For a different compounding convention, use yield-period units and the corresponding formula for that convention.

Definition: Suppose a bond pays fixed cash flows $C_t$ at the ends of years $t=1,\ldots,n$, and has annual effective yield $y>-1$. Its price is $P(y)=\sum_{t=1}^{n} C_t(1+y)^{-t}$. Its annual-compounding convexity is $K=P''(y)/P(y)$, where $P''(y)$ is the second derivative of price with respect to yield.

For a level-coupon bond with face value $F$, annual coupon $cF$, and redemption of $F$ at the end of year $n$, the cash flows are $cF$ in years $1$ through $n-1$, and $F+cF$ in year $n$. Here $c$ is the annual coupon rate. The price and convexity are calculated using those cash flows and the bond’s yield; a bond trading at par is not required.

Cash-Flow Formula and Justification

For each payment date $t$, differentiating its present value once with respect to yield gives a negative term, and differentiating a second time gives a positive term. Adding the terms across all payment dates gives the price derivatives.

P'(y)=-\sum_{t=1}^{n}\frac{tC_t}{(1+y)^{t+1}}, \qquad P''(y)=\sum_{t=1}^{n}\frac{t(t+1)C_t}{(1+y)^{t+2}}.

Dividing the second derivative by price gives the cash-flow formula for convexity. Equivalently, let $PV_t=C_t/(1+y)^t$ be the present value of the payment at time $t$. Then the same formula can be written using present-value weights.

K= \frac{\displaystyle\sum_{t=1}^{n}\frac{t(t+1)C_t}{(1+y)^{t+2}}}{P} = \frac{\displaystyle\sum_{t=1}^{n}t(t+1)PV_t}{P(1+y)^2}.

The factor $(1+y)^2$ in the second expression is essential. It appears because the second derivative discounts each cash flow for two additional powers of $1+y$, beyond the discounting already included in $PV_t$. In the fixed-cash-flow model, every cash flow is positive, so this convexity is positive.

The derivative formula also explains how convexity enters a price estimate. A second-order Taylor expansion around the current yield gives a percentage price change that includes both the first derivative and the curvature term. Modified duration under annual compounding is $D_{\mathrm{mod}}=-P'(y)/P(y)$.

\frac{\Delta P}{P} \approx -D_{\mathrm{mod}}\Delta y+\frac{1}{2}K(\Delta y)^2, \qquad \Delta P \approx P\left[-D_{\mathrm{mod}}\Delta y+\frac{1}{2}K(\Delta y)^2\right].

Use the yield change as a decimal: a rise of 50 basis points is $\Delta y=0.005$, not $50$ or $0.50$. The approximation is most useful for a sufficiently small yield shift; it is not an exact repricing formula.

A Calculation Process

List the fixed cash flows.
Include every coupon and add redemption to the final coupon payment.

Calculate the current price.
Discount each payment at the stated annual effective yield and add the present values.

Calculate duration and convexity.
Use the cash-flow formulas, keeping the time index $t$ in years and using the same yield convention throughout.

Apply the yield shift.
Convert basis points to a decimal, calculate the duration and convexity terms, and multiply the relative estimate by the current price for a dollar estimate.

Worked Example: A Two-Year Par Bond

A hypothetical bond has face value $\$1{,}000$, annual coupon rate $4\%$, and two years remaining. It pays coupons annually and redeems at face value. Its annual effective yield is $4\%$. Calculate its price, modified duration, and convexity, then estimate the price change if yield increases by 50 basis points. Round money to the nearest cent and other results to four decimal places.

Method. List and discount the fixed cash flows, then use the annual-compounding derivative formulas. This avoids treating the final coupon and redemption as separate payments at different dates.

The coupon is $0.04(\$1{,}000)=\$40$. The cash flows are $\$40$ at year 1 and $\$1{,}040$ at year 2. Their present values are

PV_1=\frac{40}{1.04}=\$38.4615, \qquad PV_2=\frac{1{,}040}{1.04^2} =\frac{1{,}040}{1.0816} =\$961.5385.

Thus $P=38.4615+961.5385=\$1{,}000.00$. The present-value weights are $0.0384615$ and $0.9615385$. The modified duration is

D_{\mathrm{mod}} = \frac{1(38.4615)+2(961.5385)}{1{,}000(1.04)} = \frac{1.9615385}{1.04} \approx 1.8861.

For convexity, the weighted sum in the numerator is $1(2)(0.0384615)+2(3)(0.9615385)=0.0769230+5.7692310=5.8461540$. The required yield adjustment is $(1.04)^2=1.0816$, so

K=\frac{5.8461540}{1.0816}\approx 5.4051.

A 50-basis-point increase is $\Delta y=0.005$. The duration contribution is $-1.8861(0.005)=-0.0094305$, and the convexity contribution is $\frac12(5.4051)(0.005)^2=0.0000676$. Therefore,

\Delta P \approx \$1{,}000\left(-0.0094305+0.0000676\right) \approx -\$9.36.

The estimate is a price decrease, as expected for a yield increase. The result has also been checked by keeping the yield shift in decimal form and applying the relative estimate to the $\$1{,}000$ price.

Worked Example: A Three-Year Par Bond

A hypothetical bond has face value $\$1{,}000$, annual coupon rate $7\%$, and three years remaining. It pays annually and redeems at face value. Its annual effective yield is $7\%$. Calculate its convexity and estimate the price change for a 25-basis-point yield increase. Round the price change to the nearest cent and other results to four decimal places.

Method. Use the present-value-weight form of convexity. Because the coupon rate equals the yield, the price is par, but the individual cash-flow weights still need to be calculated.

The cash flows are $\$70$, $\$70$, and $\$1{,}070$. Their present values are

PV_1=\frac{70}{1.07}=\$65.4206,\quad PV_2=\frac{70}{1.07^2}=\$61.1407,\quad PV_3=\frac{1{,}070}{1.07^3}=\$873.4387.

The present values sum to $\$1{,}000.00$. The corresponding weights are $0.0654206$, $0.0611407$, and $0.8734387$. Their convexity-weighted sum is $2(0.0654206)+6(0.0611407)+12(0.8734387)=10.9789502$.

K=\frac{10.9789502}{1.07^2} =\frac{10.9789502}{1.1449} \approx 9.5894.

For the price-change estimate, the modified duration is also needed. The time-weighted sum of present-value weights is $1(0.0654206)+2(0.0611407)+3(0.8734387)=2.8080181$, so $D_{\mathrm{mod}}=2.8080181/1.07\approx2.6243$. The yield change is $0.0025$. Its duration contribution is approximately $-2.6243(0.0025)=-0.0065608$, and its convexity contribution is $\frac12(9.5894)(0.0025)^2=0.0000300$.

\Delta P \approx \$1{,}000(-0.0065608+0.0000300) \approx -\$6.53.

The positive convexity term offsets a small part of the duration-based decrease. It does not reverse the sign here: the yield increase still produces an estimated price decline.

Worked Example: A One-Year Bond Below Par

A hypothetical one-year bond has face value $\$2{,}000$, an annual coupon rate of $7\%$, and annual redemption at face value. Its annual effective yield is $8\%$. Estimate the price change if yield decreases by 40 basis points. Round money to the nearest cent and other results to four decimal places.

Method. A one-year bond has a single payment date, so the cash-flow formulas simplify. This example highlights that convexity depends on payment timing as well as cash-flow size.

The year-end payment is the coupon of $\$140$ plus the $\$2{,}000$ redemption, or $\$2{,}140$. Its present value is

P=\frac{2{,}140}{1.08}=\$1{,}981.4815.

For a single payment at $t=1$, modified duration is $1/(1+y)$, and convexity is $1(2)/(1+y)^2$. Thus

D_{\mathrm{mod}}=\frac{1}{1.08}\approx0.9259, \qquad K=\frac{2}{1.08^2}\approx1.7147.

A decrease of 40 basis points means $\Delta y=-0.004$. The duration contribution is $-0.9259(-0.004)=0.0037037$, and the convexity contribution is $\frac12(1.7147)(0.004)^2=0.0000137$. Therefore,

\Delta P \approx \$1{,}981.4815(0.0037037+0.0000137) \approx +\$7.37.

The estimated increase is consistent with the direction of the yield shift. The calculations retain the unrounded price and coefficients until the final dollar result.

Common Mistakes and Interpretation

One frequent error is to use the present-value-weight expression but omit its yield adjustment. In the two-year example, the incorrect calculation would be $K=5.8462$, obtained by summing $t(t+1)PV_t/P$ and stopping there. The correction is to divide by $(1+y)^2$: $5.8462/1.0816=5.4051$. The adjustment is not optional; it follows directly from differentiating the discounted cash flows twice.

Other errors can change either the magnitude or the sign of an estimate. Check these points before reporting a result:

Combining final payments: include the last coupon and redemption together at maturity; both are discounted for the same number of years.
Mixing yield units: if the yield is annual effective and cash flows are annual, use annual time indices and a decimal annual yield change.
Misreading basis points: 25 basis points equals $0.0025$, not $0.25$.
Dropping the square: the convexity correction uses $(\Delta y)^2$, so it is positive for either direction of yield change when convexity is positive.
Confusing an estimate with a new price: the formula estimates $\Delta P$; add it to the current price only if the question asks for an estimated new price.

For fixed positive coupon and redemption cash flows, positive convexity makes the price-yield curve bend upward. The correction therefore raises the estimate relative to the duration-only result for either a yield rise or a yield fall. Duration and convexity together are useful for comparing price sensitivity and for making local estimates, but the calculation rests on the stated fixed-cash-flow and compounding assumptions. These methods are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

Use annual compounding and fixed cash flows in each question. Give money to the nearest cent and other numerical results to four decimal places.

A one-year bond pays $\$1{,}050$ at maturity and has annual effective yield $4\%$. Find its price and convexity. (Answer: $P=\$1{,}009.62$; $K\approx1.8491$.)
A two-year par bond has face value $\$1{,}000$, annual coupon rate $4\%$, and annual effective yield $4\%$. Find its convexity. (Answer: $K\approx5.4051$.)
A bond is priced at $\$1{,}200$, has modified duration $4.2$, and convexity $22$. Estimate its dollar price change for a yield increase of 30 basis points. (Answer: approximately $-\$15.00$.)
A one-year bond has face value $\$2{,}000$, annual coupon rate $7\%$, and annual effective yield $8\%$. Estimate its price change for a 40-basis-point yield decrease. (Answer: approximately $+\$7.37$.)

What You'll Learn