Deriving The Duration-Convexity Expansion

Why Expand Price Around the Current Yield?

A bond's price changes when its yield changes. Duration describes the first-order price response, while convexity captures the second-order curvature. The duration-convexity expansion follows directly from a Taylor expansion of price as a function of yield; it is not a separate pricing rule. Understanding that derivation makes clear which yield convention the measures use, why the convexity term has a positive coefficient, and when the approximation is appropriate.

Throughout this tutorial, cash flows are fixed, deterministic payments at known times. There is no default, option exercise, or mortality-contingent payment. The price is assumed to be a smooth function of the yield in a neighborhood of the current yield. These assumptions matter: if a yield change alters the expected cash flows, the derivatives of the original fixed-cash-flow price function may not describe the new price.

Definition: Let $P(y)$ be the price at yield $y$, with the yield convention held fixed. The modified duration with respect to $y$ is $D_{\mathrm{mod}}=-P'(y)/P(y)$, and the convexity with respect to $y$ is $C=P''(y)/P(y)$. These are normalized price derivatives: the first is the negative proportional first derivative, and the second is the proportional second derivative.

From Taylor Expansion to the Price-Change Formula

Write the new yield as $y+\Delta y$. A second-order Taylor expansion of the price function about the current yield gives the new price in terms of its value and derivatives at $y$. Subtracting the current price and dividing by that current price produces the proportional price change.

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

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

The minus sign on the duration term comes from the definition $D_{\mathrm{mod}}=-P'(y)/P(y)$. The convexity term has a plus sign because convexity is defined as $P''(y)/P(y)$, and the Taylor coefficient for the second derivative is positive. The omitted terms involve third and higher derivatives; for a smooth price function, the second-order approximation is most useful for a yield change small enough that those higher-order terms are not material.

This derivation also explains why both terms must use the same yield variable and units as the derivatives. For example, derivatives with respect to an annual effective yield are not interchangeable with derivatives with respect to a nominal rate convertible semiannually. The next examples show how the derivatives and formula are obtained under each convention.

Cash-Flow Formulas Under Annual Compounding

Suppose payments $C_k$ occur at times $t_k$, and the price is calculated using an annual effective yield $y$. Then each payment is discounted by $(1+y)^{-t_k}$. Differentiating the price term by term gives the first and second derivatives.

P(y)=\sum_k C_k(1+y)^{-t_k}

P'(y)=-\sum_k t_k C_k(1+y)^{-t_k-1}, \qquad P''(y)=\sum_k t_k(t_k+1)C_k(1+y)^{-t_k-2}

Dividing each derivative by price yields the measures. In the following expressions, the present-value weight of payment $k$ is $C_k(1+y)^{-t_k}/P(y)$. The weights sum to 1 when the price is the sum of the stated discounted payments.

D_{\mathrm{mod}} = \frac{\sum_k t_k C_k(1+y)^{-t_k}}{P(y)(1+y)}, \qquad C = \frac{\sum_k t_k(t_k+1)C_k(1+y)^{-t_k}}{P(y)(1+y)^2}

When payment times are measured in years and $y$ is an annual effective yield, the Macaulay duration is the present-value-weighted average payment time. Thus $D_{\mathrm{mod}}$ is Macaulay duration divided by $1+y$. The convexity expression above is the second derivative normalized by price under this same annual-effective-yield convention.

Worked Example: A Two-Year Zero-Coupon Payment

A hypothetical bond pays $100 at the end of 2 years and nothing earlier. Its current yield is an annual effective rate of 10%. There is no default or mortality risk, and the payment is fixed. Estimate the proportional price change if the annual effective yield rises by 0.01, from 10% to 11%.

Method. Differentiate the fixed-payment price function, then substitute its modified duration and convexity into the second-order expansion. This keeps the yield change in decimal-rate units.

At the current yield, the price is $P(0.10)=100(1.10)^{-2}=100/1.21=82.644628$. For a single payment at time $t=2$, the annual-compounding formulas reduce to $D_{\mathrm{mod}}=t/(1+y)$ and $C=t(t+1)/(1+y)^2$. Therefore, $D_{\mathrm{mod}}=2/1.10=1.8181818$, and $C=2(3)/(1.10)^2=6/1.21=4.9586777$.

The yield change is $\Delta y=0.01$. The duration contribution is $-1.8181818(0.01)=-0.0181818$. The convexity contribution is $\tfrac12(4.9586777)(0.01)^2=0.0002479$. Adding the terms gives \[ \frac{\Delta P}{P}\approx -0.0181818+0.0002479=-0.0179339. \] To four decimal places as a percentage, the estimated proportional price change is -1.7934%.

Check. A positive yield change gives a negative first-order price change, as expected for a fixed-payment bond. The positive convexity contribution makes the estimated decline smaller than the duration-only decline. A common mistake would be to enter 1, rather than 0.01, for the one-percentage-point yield change; the approximation requires the rate change in decimal form.

Why the Second Derivative Becomes Convexity

The second derivative measures how quickly the slope of the price function changes as yield moves. Normalizing it by the current price gives a scale-free measure of that curvature. In a second-order Taylor expansion, this derivative is multiplied by $(\Delta y)^2/2$; consequently the convexity term is positive when $P''(y)>0$, regardless of whether the yield change is upward or downward.

The derivation does not assume that the cash flows occur annually. It requires the price function to be differentiated with respect to the selected yield variable. With annual effective yield, a payment at time $t$ is discounted as $(1+y)^{-t}$. With a nominal annual rate convertible semiannually, the corresponding price function instead uses the half-year rate $j/2$ and the number of half-year periods. Differentiating that function produces measures with respect to $j$, not $y$.

Worked Example: Two Payments at a Zero Current Yield

A hypothetical bond pays $100 at the end of year 1 and $100 at the end of year 3. Its current yield is 0% effective annually, and the yield then rises by 0.02. Assume fixed payments and no default or mortality risk. Find the second-order proportional price-change estimate.

Method. Calculate the price and the two normalized derivatives from the annual cash-flow formulas. At $y=0$, each present value equals its payment: the price is $P=100+100=200$.

The numerator for modified duration is $1(100)+3(100)=400$, and the denominator is $P(1+y)=200(1)=200$. Thus $D_{\mathrm{mod}}=400/200=2$.

For convexity, the numerator is $1(2)(100)+3(4)(100)=200+1200=1400$. The denominator is $P(1+y)^2=200(1)^2=200$, so $C=1400/200=7$.

With $\Delta y=0.02$, the duration term is $-2(0.02)=-0.04$, and the convexity term is $\tfrac12(7)(0.02)^2=3.5(0.0004)=0.0014$. Hence \[ \frac{\Delta P}{P}\approx -0.04+0.0014=-0.0386. \] The second-order estimate is a proportional price change of -3.8600%.

Check. The price response is negative for this upward yield move, while the positive second-order contribution offsets part of the first-order decline. The calculation uses the yield change $0.02$, not the number 2 for two percentage points.

Using a Different Yield Variable

The same Taylor argument applies when the yield is quoted using a different compounding convention, but the derivative definitions must be taken with respect to that quoted variable. For a nominal annual rate $j$ convertible $m$ times per year, a payment at time $t$ years is discounted by $(1+j/m)^{-mt}$. The formulas below give the derivatives for a payment at the end of $n$ conversion periods; the second derivative is with respect to $j$.

P(j)=C(1+j/m)^{-n}, \qquad D_{\mathrm{mod},j}=-\frac{P'(j)}{P(j)} =\frac{n}{m(1+j/m)}, \qquad C_j=\frac{P''(j)}{P(j)} =\frac{n(n+1)}{m^2(1+j/m)^2}

The subscript $j$ emphasizes that these measures are derivatives with respect to the nominal rate. The proportional-change expansion is still Taylor's formula, but it uses $D_{\mathrm{mod},j}$, $C_j$, and $\Delta j$ together. Mixing these quantities with annual-effective-yield measures would be inconsistent.

Worked Example: A Two-Year Payment at a Semiannual Nominal Yield

A hypothetical bond pays $1,000 at the end of 2 years. Its current yield is a nominal annual rate of 8% convertible semiannually. The payment is fixed, with no default or mortality risk. Estimate the proportional price change if the nominal rate rises by 0.01, with the same semiannual conversion convention.

Method. Use $m=2$ and $n=4$, since two years contain four half-year periods. The half-year rate is $0.08/2=0.04$, so the current price is $P=1000(1.04)^{-4}=1000/1.16985856=854.8042$, or $854.80 to the nearest cent.

The modified duration with respect to $j$ is $4/[2(1.04)]=4/2.08=1.9230769$. The convexity with respect to $j$ is $4(5)/[2^2(1.04)^2]=20/[4(1.0816)]=20/4.3264=4.6227811$.

For $\Delta j=0.01$, the duration term is $-1.9230769(0.01)=-0.0192308$. The convexity term is $\tfrac12(4.6227811)(0.01)^2=0.0002311$. Therefore, \[ \frac{\Delta P}{P}\approx -0.0192308+0.0002311=-0.0189997. \] To four decimal places as a percentage, the estimated proportional price change is -1.9000%.

Check. Both derivative measures are with respect to the nominal rate $j$, so the shock used is $\Delta j=0.01$. Using a semiannual rate change of 0.01 instead would not match the stated one-percentage-point change in the nominal annual rate.

Common Mistake: Reversing the Convexity Sign

A frequent error is to write the second-order term as $-\tfrac12 C(\Delta y)^2$. That sign does not follow from the Taylor expansion: the coefficient of the second derivative is positive, and convexity is defined here as $P''(y)/P(y)$. For the two-year zero-coupon example, the incorrect calculation would be $-0.0181818-0.0002479=-0.0184297$, or -1.8430%. The correct calculation is $-0.0181818+0.0002479=-0.0179339$, or -1.7934%.

Another error is combining duration or convexity measured under one yield convention with a shock measured under another. Before applying the expansion, identify the variable in $P(\cdot)$, differentiate with respect to that variable, and express its change in matching decimal units. The formula is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Key takeaway: The second-order proportional price-change approximation is obtained by expanding price around the current yield, dividing by current price, and identifying the normalized derivatives. Use $-D_{\mathrm{mod}}\Delta y+\tfrac12 C(\Delta y)^2$ only when the measures and yield change use the same convention and the fixed-cash-flow price function is sufficiently smooth near the current yield.

Check Your Understanding

For each problem, use the stated yield variable and report the proportional price-change estimate to four decimal places as a percentage.

A hypothetical four-year zero-coupon payment is priced at a 0% annual effective yield. The yield rises by 0.01. Find the second-order estimate. (Answer: -3.9000%)
A hypothetical bond pays $50 at the end of year 1 and $150 at the end of year 2. At a 0% annual effective yield, the yield falls by 0.02. Find the second-order estimate. (Answer: +3.6000%)
A hypothetical three-year zero-coupon payment is priced at a nominal annual yield of 6% convertible semiannually. The nominal yield rises by 0.005. Using $n=6$ and $m=2$, find the second-order estimate. (Answer: -1.4439%)
A fixed payment occurs at the end of year 2, and the annual effective yield is 10%. A candidate calculates the duration term using $\Delta y=1$ for a one-percentage-point yield rise. What is the correct decimal yield change, and what second-order proportional estimate follows? (Answer: $\Delta y=0.01$; -1.7934%)

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