Annual-Compounding Convexity Calculation

Why the Compounding Convention Matters

Convexity calculations depend on how the yield is compounded. This tutorial treats a bond with fixed cash flows at integer-year times and an effective annual yield. Under that convention, a cash flow at time $t$ is discounted by $(1+i)^t$. The resulting derivatives give the duration and convexity used to estimate a price change when the annual yield changes.

Assume the bond is valued immediately after a coupon date, so its remaining payments occur at the ends of years $1,2,\ldots,n$. Assume its cash flows are fixed, and it has no embedded option that changes those cash flows when yields move. The annual-compounding formulas below do not apply unchanged to a yield quoted with semiannual compounding.

Definition: Let $C_t$ be the fixed payment at the end of year $t$, and let $i>-1$ be the effective annual yield. The bond price is $P(i)=\sum_{t=1}^{n} C_t(1+i)^{-t}$. Its annual-compounding modified duration is $D_{\mathrm{mod}}=-P'(i)/P(i)$, and its annual-compounding convexity is $C_{\mathrm{conv}}=P''(i)/P(i)$.

For positive cash flows, the price generally falls as the yield rises, so $P'(i)<0$ and modified duration is positive. The second derivative is also positive, so convexity is positive. These measures are defined relative to changes in the annual yield $i$, not to changes in a nominal rate with another compounding frequency.

Deriving the Annual-Compounding Formulas

Differentiate each discounted cash flow with respect to the annual yield. This gives the cash-flow formulas directly and explains the factors of $1+i$ in their denominators.

\frac{d}{di}\left[C_t(1+i)^{-t}\right] =-tC_t(1+i)^{-t-1}, \qquad \frac{d^2}{di^2}\left[C_t(1+i)^{-t}\right] =t(t+1)C_t(1+i)^{-t-2}.

Summing over the payments gives $P'(i)$ and $P''(i)$. Substituting these derivatives into the definitions of modified duration and convexity yields

D_{\mathrm{mod}} =\frac{\displaystyle\sum_{t=1}^{n}tC_t(1+i)^{-t-1}}{P(i)} =\frac{\displaystyle\sum_{t=1}^{n}t\,PV_t}{(1+i)P(i)}, \qquad C_{\mathrm{conv}} =\frac{\displaystyle\sum_{t=1}^{n}t(t+1)C_t(1+i)^{-t-2}}{P(i)} =\frac{\displaystyle\sum_{t=1}^{n}t(t+1)\,PV_t}{(1+i)^2P(i)},

where $PV_t=C_t(1+i)^{-t}$ is the present value of the year-$t$ payment at the valuation yield. The factors $t$ and $t(t+1)$ come from differentiating the discount factor; they are not optional adjustments.

For a small yield change $\Delta i$, the second-order Taylor approximation to the price change is

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

The first term estimates the change from duration; the second is the convexity adjustment. A basis-point shock must first be converted to a decimal change in annual yield: for example, 50 basis points is $0.005$, not $0.50$. This approximation is local: for a large shock, higher-order terms can make it differ materially from exact repricing.

Price the cash flows.
Discount each fixed payment at the stated effective annual yield and add the present values.

Calculate the measures.
Use the present values with weights $t$ for modified duration and $t(t+1)$ for convexity, including their respective yield denominators.

Apply the shock.
Write the yield change as a decimal, calculate both approximation terms, and multiply the percentage estimate by the original price for the dollar estimate.

Check the result.
For positive fixed cash flows, a yield increase should give a negative estimated price change, while a yield decrease should give a positive one.

Worked Example: Uneven Payments and a Yield Decrease

A hypothetical bond pays $20$ at the end of year 1 and $121$ at the end of year 3, with no payment at the end of year 2. Its effective annual yield is $10\%$. Calculate its price, modified duration, and convexity, then estimate the dollar price change for a 50-basis-point yield decrease. Use the second-order duration-convexity approximation.

Method. First calculate each payment’s present value. Then apply the annual-compounding formulas to those present values. The yield shock is $\Delta i=-0.005$.

The discount factors are $1.10$ for year 1 and $1.10^3=1.331$ for year 3. Thus

PV_1=\frac{20}{1.10}=18.181818, \qquad PV_3=\frac{121}{1.331}=90.909091, \qquad P=18.181818+90.909091=109.090909.

For duration, the weighted present-value sum is $1(18.181818)+3(90.909091)=290.909091$. The denominator is $1.10P=120.000000$. Therefore

D_{\mathrm{mod}}=\frac{290.909091}{120.000000}=2.424242.

For convexity, the weights are $1(1+1)=2$ in year 1 and $3(3+1)=12$ in year 3. The weighted sum is $2(18.181818)+12(90.909091)=1{,}127.272727$. The denominator is $1.10^2P=132.000000$. Hence

C_{\mathrm{conv}}=\frac{1{,}127.272727}{132.000000}=8.539945.

Using the negative yield change, the estimated percentage price change is

-2.424242(-0.005) +\frac{1}{2}(8.539945)(-0.005)^2 =0.012121210+0.000106749 =0.012227959.

The dollar estimate is $109.090909(0.012227959)=1.33396$, or an increase of approximately $\$1.33$, rounded to the nearest cent. As a check, both terms are positive for this yield decrease, as expected for a bond with positive fixed payments. The method also avoids a common error: $121/1.331$ is $90.909091$, not $100$; using an incorrect present value would distort the price and both risk measures.

Interpreting the Calculation

Modified duration and convexity are calculated from the same present values, but the cash-flow weights differ. Duration uses the time $t$; convexity uses $t(t+1)$. A later payment therefore receives a larger weight in the convexity numerator. The final denominators also differ: duration divides by $(1+i)P$, while convexity divides by $(1+i)^2P$. Keeping those factors visible is a useful safeguard when working from a table of present values.

The approximation estimates a relative change first. Its dollar version must use the original price $P$, not an estimated new price. The yield change is squared in the convexity term, so that term is positive for either direction of shock when convexity is positive. It therefore partly offsets the duration-only estimate for a yield increase, and adds to the estimated gain for a yield decrease.

The formulas use derivatives with respect to the effective annual yield itself. They are not formulas for a nominal annual yield convertible semiannually. A quoted yield’s compounding convention determines its discount factors and derivatives; substituting an annual rate into a formula derived for a different convention can produce an internally inconsistent answer.

Worked Example: A Coupon Bond With Two Payments

A hypothetical bond pays $30$ at the end of year 1 and $80$ at the end of year 2. Its effective annual yield is $10\%$. Calculate its price, modified duration, and convexity, then estimate its dollar price change for a 100-basis-point yield increase. Round money to the nearest cent and the risk measures to four decimal places.

Method. Discount both payments at the annual yield, form the weighted sums, and apply the second-order approximation with $\Delta i=0.010$.

The present values and price are

PV_1=\frac{30}{1.10}=27.272727, \qquad PV_2=\frac{80}{1.10^2}=\frac{80}{1.21}=66.115702, \qquad P=27.272727+66.115702=93.388430.

For modified duration, the weighted sum is $1(27.272727)+2(66.115702)=159.504131$, and the denominator is $1.10(93.388430)=102.727273$. Thus

D_{\mathrm{mod}}=\frac{159.504131}{102.727273}=1.552695.

For convexity, the year-1 weight is $2$, and the year-2 weight is $6$. The weighted sum is $2(27.272727)+6(66.115702)=451.239669$. The denominator is $1.21(93.388430)=113.000000$, so

C_{\mathrm{conv}}=\frac{451.239669}{113.000000}=3.993271.

The relative price-change estimate is

-(1.552695)(0.010) +\frac{1}{2}(3.993271)(0.010)^2 =-0.015526950+0.000199664 =-0.015327286.

Multiplying by the original price gives $93.388430(-0.015327286)=-1.43139$, so the estimated dollar change is a decrease of $\$1.43$, rounded to the nearest cent. The negative estimate has the expected sign because the yield rises. The positive convexity term partially offsets the negative duration term; it does not reverse the price-decrease direction here.

A common mistake is to use $t^2$ instead of $t(t+1)$ in the convexity numerator. That would incorrectly use weights $1$ and $4$, rather than $2$ and $6$. The second derivative of $(1+i)^{-t}$ establishes the correct weights.

Worked Example: Zero-Coupon Bond and an Exact Check

A hypothetical three-year zero-coupon bond has a price of $\$200.00$ at an effective annual yield of $5\%$. Estimate its dollar price change for a 50-basis-point yield increase using duration and convexity. Then compare with exact repricing. Round dollar amounts to the nearest cent.

Method. A zero-coupon bond has one cash flow, at year 3. Its present value is the entire price, so the time weights can be applied directly.

For duration, the weighted present value is $3(200)=600$, and the denominator is $1.05(200)=210$. For convexity, the weighted present value is $3(4)(200)=2{,}400$, and the denominator is $1.05^2(200)=220.5$. Therefore

D_{\mathrm{mod}}=\frac{600}{210}=2.857143, \qquad C_{\mathrm{conv}}=\frac{2{,}400}{220.5}=10.884354.

The yield change is $\Delta i=0.005$. The relative estimate is

-(2.857143)(0.005) +\frac{1}{2}(10.884354)(0.005)^2 =-0.014285715+0.000136054 =-0.014149661.

Thus the estimated dollar change is $200(-0.014149661)=-2.829932$, or a decrease of $\$2.83$, rounded to the nearest cent.

For an exact check, the maturity payment is $200(1.05)^3=\$231.525$. At the new yield of $5.5\%$, exact repricing gives $\$231.525/(1.055)^3$, approximately $\$197.17$. The exact decrease is approximately $\$2.83$, consistent with the second-order estimate at this small shock. Exact repricing checks the approximation; it is not a replacement for calculating the requested duration-convexity estimate.

Common Exam Traps

Using a shock in percentage points as if it were a decimal. A 50-basis-point increase is $\Delta i=0.005$. Substituting $0.50$ makes both approximation terms much too large.
Omitting a present value or discounting it incorrectly. Compute each $C_t/(1+i)^t$ before applying its time weight. Check divisions and powers independently.
Using the wrong convexity weight or denominator. Under annual compounding, the weight is $t(t+1)$, and the denominator is $(1+i)^2P$.
Reporting a relative change as a dollar amount. The expression in brackets is a proportion. Multiply it by the original bond price to estimate dollars.
Using the annual formula for a differently compounded yield. Confirm the yield convention and payment timing before calculating. This tutorial’s formulas assume an effective annual yield and annual payment dates.
Treating the approximation as an exact new price. Duration and convexity provide a local estimate. Exact repricing discounts the cash flows at the new yield.

Key takeaway: For annual-compounding fixed cash flows, calculate price from the annual discount factors, duration with weights $t$, and convexity with weights $t(t+1)$. Convert the yield shock to a decimal, include the one-half in the convexity term, and multiply the resulting proportion by the original price. This calculation is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, assume fixed cash flows, annual payment dates, and an effective annual yield. Use the second-order duration-convexity approximation where a yield shock is given.

A two-year zero-coupon bond is priced at $\$300.00$ at a $5\%$ annual yield. Calculate its convexity and estimate the dollar price change for a 40-basis-point yield increase. (Answer: convexity $=5.4422$; estimated change $=-\$2.27$.)
A three-year zero-coupon bond is priced at $\$500.00$ at a $6\%$ annual yield. Calculate its modified duration and convexity, then estimate the dollar price change for a 25-basis-point yield decrease. (Answer: modified duration $=2.8302$, convexity $=10.6805$; estimated change $=+\$3.56$.)
A bond pays $\$22$ at the end of year 1 and $\$133.10$ at the end of year 3. The annual yield is $10\%$. Calculate its price, modified duration, and convexity, then estimate the dollar price change for a 50-basis-point yield decrease. (Answer: price $=\$120.00$, modified duration $=2.4242$, convexity $=8.5399$; estimated change $=+\$1.47$.)
A two-year bond pays $\$40$ at the end of year 1 and $\$140$ at the end of year 2. Its annual yield is $5\%$. Calculate its price, modified duration, and convexity, then estimate the dollar price change for a 75-basis-point yield increase. (Answer: price $=\$167.80$, modified duration $=1.6580$, convexity $=3.2754$; estimated change $=-\$2.06$.)
A three-year zero-coupon bond is priced at $\$250.00$ at a $4\%$ annual yield. Calculate its convexity and estimate the dollar price change for a 60-basis-point yield increase. (Answer: convexity $=11.0947$; estimated change $=-\$4.17$.)

What You'll Learn