Convexity as the Second-Order NPV Sensitivity

Why Convexity Adds to Duration

Duration describes the local, first-order response of present value to a small yield change. It gives a linear estimate: as yield rises, the present value of fixed positive payments falls. But a present-value function is generally curved, not a straight line. Convexity measures that curvature and supplies the second-order term in a local approximation.

This tutorial considers a single annual effective yield applied to fixed cash flows at integer future times. It derives convexity directly from the second derivative of present value. The result is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Suppose fixed cash flows $C_k$ are paid at the end of year $k$, for $k=1,\ldots,n$, and are discounted at annual effective yield $i>-1$. Their present value is $P(i)=\sum_{k=1}^{n} C_k(1+i)^{-k}$. When $P(i)\ne 0$, define the normalized convexity by $\mathcal{C}(i)=P''(i)/P(i)$. If $P(i)>0$, this is the second derivative expressed relative to present value.

The cash flows must remain fixed as $i$ changes. The yield is one annual effective rate for all the cash-flow dates, and the cash-flow timing is measured in years. A cash flow at time 0, if present, adds a constant to $P(i)$ and contributes zero to its derivatives. If cash flows themselves depend on yield, this fixed-cash-flow formula does not capture their full sensitivity. The formulas for derivatives still hold for signed cash flows, but convexity need not be positive if some payments are negative.

Deriving the Curvature Measure

For each payment at time $k$, differentiate its discounted value with respect to $i$. The payment amount and date are fixed, so only the discount factor changes.

P(i)=\sum_{k=1}^{n} C_k(1+i)^{-k}

P'(i)=-\sum_{k=1}^{n} kC_k(1+i)^{-k-1}, \qquad P''(i)=\sum_{k=1}^{n} k(k+1)C_k(1+i)^{-k-2}.

The factor $k$ in the first derivative comes from differentiating the power $-(k)$. Differentiating again multiplies by the next exponent factor $k+1$, while the two negative signs produce a positive term. Thus a positive payment contributes positively to the second derivative.

D_{\mathrm{mod}}(i)=-\frac{P'(i)}{P(i)} =\frac{\sum_{k=1}^{n} kC_k(1+i)^{-k-1}}{P(i)}, \qquad \mathcal{C}(i)=\frac{\sum_{k=1}^{n} k(k+1)C_k(1+i)^{-k-2}}{P(i)}.

Here $D_{\mathrm{mod}}$ is modified duration: the negative proportional first derivative of present value with respect to yield. For positive cash flows and positive present value, duration is positive and convexity is positive. The latter means the present-value curve bends upward as yield changes. It does not mean that present value rises when yield rises: its first derivative remains negative in this setting.

From the Second Derivative to a Price Estimate

A second-order Taylor expansion around the current yield $i$ explains how convexity is used. For a small change $\Delta i$, the present value at the new yield is approximated by the current value plus the first- and second-derivative terms.

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

Divide by $P(i)$, use $P'(i)/P(i)=-D_{\mathrm{mod}}(i)$, and use $P''(i)/P(i)=\mathcal{C}(i)$. This gives the proportional-change approximation:

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

The duration term is linear in the yield change and changes sign when the direction of the yield change reverses. The convexity term is quadratic, so it is positive for either direction when convexity is positive. It adjusts duration’s straight-line estimate to account for curvature. This is a local approximation, not an identity; its quality depends on the size of the yield change and on higher derivatives omitted from the expansion.

Worked Example: Deriving Duration and Convexity at a 10% Yield

A hypothetical contract pays $100 at the end of year 1 and $100 at the end of year 2. Using an annual effective yield of $i=0.10$, calculate present value, modified duration, and normalized convexity. Then estimate the present-value change for a yield increase of 0.01.

Method. Discount each payment at the stated yield, differentiate the fixed-cash-flow present value, and divide the derivatives by present value. This avoids treating the second derivative itself as normalized convexity.

Present value. The two discounted payments are:

\frac{100}{1.1}=90.909091,\qquad \frac{100}{1.1^2}=82.644628, \qquad P(0.10)=173.553719.

First derivative. Apply the derivative formula with $k=1$ and $k=2$:

P'(0.10) =-\frac{100}{1.1^2}-\frac{2(100)}{1.1^3} =-82.644628-150.262960 =-232.907588.

Therefore:

D_{\mathrm{mod}}(0.10) =-\frac{-232.907588}{173.553719} =1.341991.

Second derivative. The $k=1$ term has multiplier $1(2)=2$, and the $k=2$ term has multiplier $2(3)=6$:

P''(0.10) =\frac{2(100)}{1.1^3}+\frac{6(100)}{1.1^4} =150.262960+409.808073 =560.071033.

Normalize by present value:

\mathcal{C}(0.10)=\frac{560.071033}{173.553719}=3.227076.

Yield-change estimate. A 0.01 increase means a one-percentage-point increase, expressed as a decimal change in the rate. The second-order estimate is:

\Delta P\approx (-232.907588)(0.01) +\frac{1}{2}(560.071033)(0.01)^2 =-2.329076+0.028004 =-2.301072.

The estimated present value is $173.553719-2.301072=171.252647$. The positive convexity term offsets part of the first-order decrease. As a check, both cash flows are positive, so the calculated positive convexity is consistent with the model.

Interpreting Convexity Across Cash-Flow Patterns

Each payment contributes to convexity according to its discounted value and its time. The factor $k(k+1)$ grows with payment time, and the payment is discounted by $(1+i)^{k+2}$. Consequently, later cash flows often have substantial curvature exposure, although the actual contribution depends on their size, discounting, and the other cash flows in the present value.

Convexity can also distinguish cash-flow patterns that have the same present value and duration. Duration fixes the local slope relative to present value; it does not fix the curvature. This distinction matters when comparing how two fixed-payment positions respond to a nonzero yield movement.

Worked Example: Same Duration, Different Convexity

At an annual effective yield of 0%, compare two hypothetical portfolios. Portfolio A pays $100 at the end of year 2. Portfolio B pays $50 at the end of year 1 and $50 at the end of year 3. For each, calculate present value, modified duration, and normalized convexity. Use the measures to estimate the change in value for a yield increase of 0.05.

Method. At a zero yield, every payment has discount factor 1. Calculate $P$, $P'$, and $P''$ from the derivative formulas for each portfolio, then apply the second-order approximation. This avoids assuming that equal duration implies equal curvature.

Portfolio A. Its present value is $P_A=100$. Its derivatives are:

P_A'(0)=-2(100)=-200,\qquad P_A''(0)=2(3)(100)=600.

Thus $D_{\mathrm{mod},A}=200/100=2$ and $\mathcal{C}_A=600/100=6$. The estimated change for $\Delta i=0.05$ is:

\Delta P_A\approx(-200)(0.05)+\frac{1}{2}(600)(0.05)^2 =-10+0.75=-9.25.

Portfolio B. Its present value is $P_B=50+50=100$. Its derivatives are:

P_B'(0)=-1(50)-3(50)=-200, \qquad P_B''(0)=1(2)(50)+3(4)(50)=100+600=700.

Thus $D_{\mathrm{mod},B}=200/100=2$, while $\mathcal{C}_B=700/100=7$. Its estimated change is:

\Delta P_B\approx(-200)(0.05)+\frac{1}{2}(700)(0.05)^2 =-10+0.875=-9.125.

Both portfolios have the same present value and first-order sensitivity, but B has greater convexity and therefore a less negative second-order estimate for this yield increase. The calculations check: both durations equal 2, while their convexities differ. These are Taylor estimates, not exact repricings.

Common Exam Trap: Rate Units and Signs

The yield change in the Taylor formula is a change in the rate as a decimal. If yield rises from 4% to 5%, then $\Delta i=0.01$, not 1. Also, the first-order term uses a negative sign because modified duration is defined as $-P'/P$.

For example, suppose $P=500$, $D_{\mathrm{mod}}=4$, $\mathcal{C}=20$, and yield rises by one percentage point. The correct proportional estimate is:

\frac{\Delta P}{P}\approx -(4)(0.01)+\frac{1}{2}(20)(0.01)^2 =-0.04+0.001=-0.039.

The estimated dollar change is $(500)(-0.039)=-\$19.50$. A common incorrect calculation substitutes 1 for the rate change and obtains $-(4)(1)+\frac12(20)(1)^2=6$, or a positive change of 600%. That error both misreads one percentage point as a 100-percentage-point change and ignores that the approximation is intended for a small change. The correction is to convert percentage points to decimal rate units before using either term.

Key takeaway: For fixed cash flows discounted at one annual effective yield, normalized convexity is $P''(i)/P(i)$. It captures the second-order curvature omitted by duration’s local linear estimate. Use the yield change as a decimal, retain the negative duration term, and interpret the result as an approximation.

Check Your Understanding

For each problem, use the fixed-cash-flow model and state the requested measure or estimate.

A hypothetical asset pays $300 at year 1 and $200 at year 2, with an annual effective yield of 0%. Find its present value, modified duration, and normalized convexity. (Answer: $P=500$, $D_{\mathrm{mod}}=1.4$, $\mathcal{C}=3.6$.)
A hypothetical zero-coupon payment of $1,210 is due at the end of year 2. The annual effective yield is 10%. Find the present value, modified duration, and normalized convexity. (Answer: $P=\$1{,}000$, $D_{\mathrm{mod}}=2/1.1\approx1.8182$, $\mathcal{C}=6/1.1^2\approx4.9587$.)
A fixed-payment position pays $100 at year 1 and $200 at year 3. At a 0% annual effective yield, calculate its present value, modified duration, and normalized convexity. (Answer: $P=\$300$, $D_{\mathrm{mod}}=7/3\approx2.3333$, $\mathcal{C}=26/3\approx8.6667$.)
At a 0% yield, Portfolio A pays $200 at year 2. Portfolio B pays $100 at year 1 and $100 at year 3. Compare their duration and convexity, then estimate each value change for a 2-percentage-point yield increase. (Answer: Both have $P=\$200$ and $D_{\mathrm{mod}}=2$; A has $\mathcal{C}=6$ and estimated $\Delta P=-\$7.76$; B has $\mathcal{C}=7$ and estimated $\Delta P=-\$7.72$.)
A position has present value $800, modified duration 3, and normalized convexity 14. Estimate the dollar change and new value for a yield decrease of 0.01. (Answer: $\Delta P/P\approx0.0307$, $\Delta P\approx\$24.56$, and estimated new value $\$824.56$.)

What You'll Learn