Deriving Convexity From Bond Price Derivatives

Why Derive Convexity From Price?

A bond’s price changes when its yield changes. Duration describes the local first-order price sensitivity; convexity describes how that sensitivity itself changes as yield moves. To define convexity precisely, we need to specify which price function is being differentiated and which yield convention it uses.

This tutorial assumes a fixed schedule of annual cash flows and a single annual effective yield. The cash flows do not change when the yield changes. The derivative-based treatment is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Model and definition: Suppose a bond pays fixed amounts $C_t$ at times $t=1,\ldots,n$, with $C_t\geq 0$. Let $y>-1$ be the annual effective yield per year, and let $P(y)$ be the present value of those cash flows. Define the bond’s convexity at yield $y$ by $$ \mathcal{C}(y)=\frac{P''(y)}{P(y)}, $$ provided $P(y)>0$. Here $P'(y)$ and $P''(y)$ are the first and second derivatives of price with respect to yield.

The yield is measured as a decimal in differentiation. For instance, a change from 5% to 6% is a change of $0.01$, not a change of one unit. Under this annual-effective-yield convention, convexity has units of inverse yield squared. It is a local measure of curvature, not a price-change approximation by itself.

Differentiate the Present-Value Function

For fixed annual cash flows, the present value at yield $y$ is

P(y)=\sum_{t=1}^{n} C_t(1+y)^{-t}.

Each cash flow is discounted by the factor appropriate to its payment time. Because the cash flows and payment times are fixed, we can differentiate each term with respect to $y$.

Differentiate once.
For each term, use the power rule on $C_t(1+y)^{-t}$. The constant $C_t$ remains unchanged.

Differentiate again.
Apply the power rule to the resulting term. The exponent decreases by one again, and the derivative changes the sign from negative to positive.

Normalize by price.
Divide the second derivative by the price at the yield being considered. This gives convexity under the stated convention.

\begin{aligned} P'(y) &=\sum_{t=1}^{n} -tC_t(1+y)^{-(t+1)},\\ P''(y) &=\sum_{t=1}^{n} t(t+1)C_t(1+y)^{-(t+2)},\\ \mathcal{C}(y) &=\frac{P''(y)}{P(y)}. \end{aligned}

The second derivative follows term by term: differentiating $-tC_t(1+y)^{-(t+1)}$ multiplies by $-(t+1)$ and reduces the exponent by one. The two negative signs cancel, leaving $t(t+1)C_t(1+y)^{-(t+2)}$. When the cash flows are nonnegative and at least one is positive, every term in $P''(y)$ is nonnegative, $P(y)>0$, and therefore $\mathcal{C}(y)\geq 0$.

The normalization matters. $P''(y)$ is a dollar-valued derivative: it records the curvature of the dollar price. Dividing by $P(y)$ expresses that curvature relative to the current price. Convexity is not simply the second derivative, nor is it the first derivative with its sign changed.

Worked Example: A Two-Payment Bond at a 10% Yield

A hypothetical bond pays $110 at the end of year 1 and $121 at the end of year 2. Assume the payments are fixed, annual, and certain, and the annual effective yield is $y=0.10$. Find $P(y)$, $P'(y)$, $P''(y)$, and convexity. The method is to differentiate the present-value terms before substituting the yield; this avoids treating the discounted cash flows as constant while differentiating.

Price. Discount each payment to time 0:

P(0.10)=\frac{110}{1.10}+\frac{121}{(1.10)^2} =100+100 =200.

First derivative. There are terms at $t=1$ and $t=2$:

\begin{aligned} P'(0.10) &=-\frac{1(110)}{(1.10)^2} -\frac{2(121)}{(1.10)^3}\\ &=-90.9091-181.8182\\ &=-272.7273. \end{aligned}

Second derivative. Use $t(t+1)$ and the denominator power $t+2$:

\begin{aligned} P''(0.10) &=\frac{1(2)(110)}{(1.10)^3} +\frac{2(3)(121)}{(1.10)^4}\\ &=165.2893+495.8675\\ &=661.1568. \end{aligned}

Convexity. Divide the second derivative by the price:

\mathcal{C}(0.10)=\frac{661.1568}{200}=3.3058.

The check is that $P''(0.10)>0$, as expected for positive fixed cash flows, and the reported convexity is $P''/P$, not the dollar-valued $P''$. The figures above are rounded to four decimal places where applicable.

Reading the Derivative Results

The first derivative gives the instantaneous change in dollar price for a one-unit change in the decimal yield. It is usually negative for positive fixed cash flows: increasing the yield decreases the present value. The second derivative describes how that slope changes with yield. Under the assumptions above, it is nonnegative.

Convexity is a relative measure because it divides that second derivative by price. The definition uses derivatives evaluated at the current yield. For a finite yield change, the value of the derivative generally changes along the way, so convexity at one yield does not give the exact new price on its own.

The calculation depends on the model. If a cash flow itself changes with yield, it cannot be treated as a fixed $C_t$ when differentiating. For example, a payment determined by a yield-sensitive option exercise decision does not satisfy this fixed-cash-flow assumption without further modelling. Likewise, a yield quoted under another compounding convention requires a price function written in that convention before differentiating.

Worked Example: Three Payments With Equal Present Values

A hypothetical bond pays $105 at year 1, $110.25 at year 2, and $115.7625 at year 3. Assume fixed annual payments and an annual effective yield of $5\%$. Find its price, first and second price derivatives, and convexity. The method is to differentiate the general present-value terms, then use the fact that each payment has a present value of $100 at the stated yield.

Price. The discounted payments are:

\begin{aligned} \frac{105}{1.05}&=100,\\ \frac{110.25}{(1.05)^2}&=100,\\ \frac{115.7625}{(1.05)^3}&=100,\\ P(0.05)&=100+100+100=300. \end{aligned}

First derivative. For a payment with present value $100 at this yield, its term in $P'$ is $-t(100)/(1.05)$. Thus:

\begin{aligned} P'(0.05) &=-\frac{100}{1.05}(1+2+3)\\ &=-\frac{600}{1.05}\\ &=-571.4286. \end{aligned}

Second derivative. The corresponding term in $P''$ is $t(t+1)(100)/(1.05)^2$. Therefore:

\begin{aligned} P''(0.05) &=\frac{100}{(1.05)^2}\bigl(1(2)+2(3)+3(4)\bigr)\\ &=\frac{100}{1.1025}(2+6+12)\\ &=\frac{2000}{1.1025}\\ &=1814.0589. \end{aligned}

Convexity.

\mathcal{C}(0.05)=\frac{1814.0589}{300}=6.0469.

The price check is $300$, since all three discounted payments equal $100. The convexity is positive and equals the second derivative divided by that price. Values are rounded to four decimal places.

A Second Derivative With Unequal Present Values

The equal-present-value pattern in the previous example is a convenient shortcut, not a requirement. The derivative calculation works for any fixed cash flows under the stated annual-yield model. When payments have unequal present values, keep each term separate and apply its own time factor.

Worked Example: Unequal Payments at an 8% Yield

A hypothetical bond pays $216 at year 1, $349.92 at year 2, and $629.856 at year 3. Assume fixed annual payments and an annual effective yield of $8\%$. Compute price, both derivatives, and convexity. The method is to first verify the discounted cash-flow amounts, then use the time factors from the derivatives.

Price. Discount each payment:

\begin{aligned} \frac{216}{1.08}&=200,\\ \frac{349.92}{(1.08)^2}&=300,\\ \frac{629.856}{(1.08)^3}&=500,\\ P(0.08)&=200+300+500=1000. \end{aligned}

First derivative. Using the present values just calculated, each term in the first derivative is $-t$ times its present value, divided by $1.08$:

\begin{aligned} P'(0.08) &=-\frac{1(200)+2(300)+3(500)}{1.08}\\ &=-\frac{2300}{1.08}\\ &=-2129.6296. \end{aligned}

Second derivative. Each term is $t(t+1)$ times its present value, divided by $(1.08)^2$:

\begin{aligned} P''(0.08) &=\frac{1(2)(200)+2(3)(300)+3(4)(500)}{(1.08)^2}\\ &=\frac{400+1800+6000}{1.1664}\\ &=\frac{8200}{1.1664}\\ &=7030.1783. \end{aligned}

Convexity.

\mathcal{C}(0.08)=\frac{7030.1783}{1000}=7.0302.

The present values sum to $1,000, matching the price used in the normalization. Since the cash flows are positive, the positive second derivative is also consistent with the model. Values are rounded to four decimal places.

Common Derivative Trap

The most common algebra error is to use the wrong power in the denominator when differentiating a second time. After one differentiation, the term has denominator power $t+1$. Differentiating again raises that power to $t+2$, not $t+1$.

Common mistake: Writing $t(t+1)C_t/(1+y)^{t+1}$ for the second derivative. This retains the denominator power from the first derivative and omits the additional power produced by the second differentiation.

For the two-payment example at $y=0.10$, that incorrect expression would give:

\frac{220}{(1.10)^2}+\frac{726}{(1.10)^3} =181.8182+545.4545 =727.2727.

That is not $P''(0.10)$. Differentiating $(1+y)^{-(t+1)}$ contributes both the multiplier $-(t+1)$ and the new exponent $-(t+2)$. The correct calculation is:

P''(0.10)=\frac{220}{(1.10)^3}+\frac{726}{(1.10)^4}=661.1568, \qquad \mathcal{C}(0.10)=\frac{661.1568}{200}=3.3058.

A separate check is to distinguish the two quantities at the end: $P''(y)$ is the dollar price derivative; $\mathcal{C}(y)$ is that derivative divided by the current price.

Key takeaway: For fixed annual cash flows discounted at annual effective yield $y$, differentiate the present-value function twice to obtain $P''(y)$, then divide by $P(y)$. The resulting $\mathcal{C}(y)=P''(y)/P(y)$ is convexity under this stated yield convention.

Check Your Understanding

Assume fixed annual cash flows, an annual effective yield, and no yield-dependent payments in each question.

A bond pays $105 at year 1 and $110.25 at year 2. At a 5% yield, find its price and convexity. (Answer: $P=200$; convexity $=3.6281$.)
A bond pays $108 at year 1 and $233.28 at year 2. At an 8% yield, find $P''(y)$ and convexity. (Answer: $P''=1200.2743$; convexity $=4.0009$.)
At a 4% yield, the present values of a bond’s payments at years 1, 2, and 3 are respectively $50, $150, and $300. Find the price and convexity. (Answer: $P=500$; convexity $=8.5059$.)
A bond pays $100 at year 1 and $200 at year 3. At a 0% annual effective yield, compute $P''(0)$ and convexity. (Answer: $P''(0)=2600$; convexity $=8.6667$.)

What You'll Learn