Convexity For Unevenly Dated Cash Flows

Why Uneven Payment Dates Require Care

A bond or other fixed-income asset need not pay only at regular annual or semiannual dates. A project may produce scheduled payments at dates such as 0.5, 1.4, and 2.8 years after valuation. Convexity still measures the curvature of price as yield changes, but the calculation must use each payment’s actual time from valuation. Replacing those times with payment numbers or rounded periods changes the result.

This tutorial uses an annual effective yield $y$. A cash flow $C_j$ paid $t_j$ years from the valuation date has present value $C_j(1+y)^{-t_j}$. The times $t_j$ may be fractional and need not be equally spaced. The cash flows are fixed: their amounts and payment dates do not change when the yield changes.

Definition: For fixed cash flows $C_j$ at nonnegative times $t_j$, the price at annual effective yield $y>-1$ is $P(y)=\sum_j C_j(1+y)^{-t_j}$. The yield-based convexity is $\mathcal{C}(y)=P''(y)/P(y)$, provided $P(y)\ne 0$. For nonnegative cash flows with positive price, this is the second derivative of price divided by price.

The definition uses the actual elapsed time for every payment. A payment at 1.4 years is discounted by $(1+y)^{-1.4}$, not by the factor for the second payment, nor by a factor for two complete years. The stated yield convention matters as well: the formulas below are for an annual effective yield, not a nominal rate with a different compounding frequency.

Deriving the Formula for Unevenly Dated Cash Flows

For one cash flow $C_j$ paid at time $t_j$, its present value is $C_j(1+y)^{-t_j}$. Differentiating with respect to $y$ gives

\frac{d}{dy}\left[C_j(1+y)^{-t_j}\right] =-t_jC_j(1+y)^{-t_j-1}

Differentiating a second time gives

\frac{d^2}{dy^2}\left[C_j(1+y)^{-t_j}\right] =t_j(t_j+1)C_j(1+y)^{-t_j-2}.

Price is the sum of the present values, so its derivatives are the sums of the corresponding derivatives. Dividing the second derivative by price produces the cash-flow formula. The justification does not require equal spacing: each term is differentiated using its own $t_j$.

Theorem: For fixed cash flows at times $t_j\ge 0$, discounted at an annual effective yield $y>-1$, convexity is $$ \mathcal{C}(y) =\frac{\displaystyle\sum_j t_j(t_j+1)C_j(1+y)^{-t_j-2}} {\displaystyle\sum_j C_j(1+y)^{-t_j}}. $$ Equivalently, if $PV_j=C_j(1+y)^{-t_j}$, then $$ \mathcal{C}(y)=\frac{\displaystyle\sum_j PV_j\,t_j(t_j+1)} {P(y)(1+y)^2}. $$

The second form is often convenient in a hand calculation. First find each payment’s present value at the yield, then multiply that present value by $t_j(t_j+1)$. Divide the sum by total price times $(1+y)^2$. The $(1+y)^2$ denominator is essential: the second derivative contains two additional powers of $1+y$ beyond the present-value discounting.

Record the valuation yield and time unit.
Use the yield convention specified by the problem and measure every $t_j$ in years from the valuation date.

Discount each cash flow separately.
Calculate $PV_j=C_j(1+y)^{-t_j}$, using the actual time for that payment.

Form the curvature-weighted sum.
For each payment, multiply its present value by $t_j(t_j+1)$, and add the results.

Normalize by price and yield.
Divide the weighted sum by $P(y)(1+y)^2$, where price is the sum of all present values.

Worked Calculations

Worked Example: Fractional Payment Dates at a 10% Yield

A hypothetical contract pays three fixed amounts at 0.5, 1.5, and 2.5 years. The annual effective yield is 10%. To keep the present values transparent, the payments are specified as $40(1.10)^{0.5}$, $60(1.10)^{1.5}$, and $100(1.10)^{2.5}$, respectively. Find the price and convexity.

Method. Discount each payment at its own actual time. The specified payment amounts are approximately $41.95, $69.22, and $126.91; calculations retain the exact expressions rather than these rounded displays.

The yield discount factors are $ (1.10)^{-0.5}\approx0.953463$, $ (1.10)^{-1.5}\approx0.866784$, and $ (1.10)^{-2.5}\approx0.787986$. Multiplying each exact payment by its corresponding factor gives present values of $40, $60, and $100. Thus $P=40+60+100=\$200$.

The curvature-weighted present values are $40(0.5)(1.5)=30$, $60(1.5)(2.5)=225$, and $100(2.5)(3.5)=875$. Their sum is $30+225+875=1{,}130$. The denominator is $200(1.10)^2=200(1.21)=242$. Therefore $\mathcal{C}=1{,}130/242\approx4.6694$, to four decimal places.

Check. The present values sum to the stated price of $200, and the calculation uses 0.5, 1.5, and 2.5—not payment counts 1, 2, and 3. The common mistake avoided is treating irregular dates as equally spaced periods.

Worked Example: Including a Payment at the Valuation Date

A fixed-cash-flow arrangement has an annual effective yield of 8%. It pays $25 immediately, then $75(1.08)^{0.75}$ at 0.75 years, and $100(1.08)^{2.25}$ at 2.25 years. Find its price and convexity.

Method. Calculate the present value of each payment at 8%. The immediate payment has $t=0$, so its discount factor is $1$ and its present value is $25. The other payments have present values exactly $75 and $100 because their stated amounts include the corresponding yield accumulation factors. Hence $P=25+75+100=\$200$.

For convexity, the time-zero payment contributes $25(0)(1)=0$ to the weighted sum. The other contributions are $75(0.75)(1.75)=75(1.3125)=98.4375$ and $100(2.25)(3.25)=100(7.3125)=731.25$. The numerator is $98.4375+731.25=829.6875$. The denominator is $200(1.08)^2=200(1.1664)=233.28$. Thus $\mathcal{C}=829.6875/233.28\approx3.5566$, to four decimal places.

Check. The time-zero payment increases price but contributes nothing to the second derivative, because its present value is independent of yield. The common mistake avoided is assigning an arbitrary positive period to a payment made at valuation.

Worked Example: Recovering Price and Convexity at 21%

A hypothetical security pays $50(1.21)^{0.25}$ at 0.25 years, $100(1.21)^{1.25}$ at 1.25 years, and $150(1.21)^{2.5}$ at 2.5 years. The annual effective yield is 21%. Find its price and convexity.

Method. Discount at the 21% yield, keeping the fractional exponents. By construction, the three present values are $50, $100, and $150. Therefore the price is $P=50+100+150=\$300$.

The weighted terms are $50(0.25)(1.25)=50(0.3125)=15.625$, $100(1.25)(2.25)=100(2.8125)=281.25$, and $150(2.5)(3.5)=150(8.75)=1{,}312.5$. Their sum is $15.625+281.25+1{,}312.5=1{,}609.375$. The denominator is $300(1.21)^2=300(1.4641)=439.23$. Thus $\mathcal{C}=1{,}609.375/439.23\approx3.6641$, to four decimal places.

Check. The weighted numerator and denominator use the unrounded yield factor $1.4641$. The common mistake avoided is using the number of payments or an incorrect squared-yield denominator in place of the actual time factors and $P(1+y)^2$.

Properties and Practical Checks

For nonnegative cash flows at nonnegative times, every present value is nonnegative, as is every factor $t_j(t_j+1)$. If price is positive, the formula therefore gives nonnegative convexity. It is strictly positive if at least one positive cash flow occurs at a time greater than zero. If all positive cash flows occur at time zero, price is positive but convexity is zero.

A useful audit is to make a table with one row per payment and columns for time, cash flow, discount factor, present value, and $PV_jt_j(t_j+1)$. This makes it easier to identify whether a date was measured from the right valuation point and whether a yield factor was applied consistently. Keep full precision during the calculation and round only the final convexity.

Check	What to verify
Time	Each $t_j$ is elapsed time in years from valuation, including fractional years.
Discounting	Each present value uses its own factor $(1+y)^{-t_j}$.
Price	The denominator uses the sum of all present values at the stated yield.
Normalization	The convexity denominator is $P(y)(1+y)^2$.
Time zero	A payment at $t=0$ contributes to price but has zero curvature contribution.

This formula assumes fixed cash flows and a single annual effective yield used to discount every payment. It is not a substitute for repricing when cash flows themselves depend on yield, or when separate spot rates are specified for different dates. Those situations require a model consistent with their own cash-flow or term-structure assumptions.

Key takeaway: For unevenly dated fixed cash flows, convexity is calculated payment by payment. Use each actual time $t_j$, discount by $(1+y)^{-t_j}$, and normalize the time-weighted present-value sum by price times $(1+y)^2$.

Check Your Understanding

For each problem, use the annual effective yield stated and the actual payment times. In the first four problems, the payment amounts are expressed so that their present values at the stated yield are directly visible.

A contract pays $80(1.05)^{0.5}$ at 0.5 years and $120(1.05)^2$ at 2 years. At a 5% annual effective yield, find convexity. (Answer: 3.5374)
An asset pays $40 at time zero, $60(1.10)^{1.25}$ at 1.25 years, and $100(1.10)^{2.5}$ at 2.5 years. At a 10% annual effective yield, find price and convexity. (Answer: price $200; convexity 4.3130)
At a 4% annual effective yield, payments have present values $30 at 0.25 years, $70 at 1.75 years, and $100 at 3.5 years. Find convexity. (Answer: 8.8815)
At a 6% annual effective yield, payments have present values $50 at time zero, $80 at 0.5 years, and $120 at 2.5 years. Find price and convexity. (Answer: price $250; convexity 3.9516)
Explain why a positive payment at time zero contributes zero to convexity but still changes the convexity of a cash-flow stream that also has future payments. (Answer: Its own present value is constant in yield, so its second-derivative contribution is zero; it increases total price, the denominator of $P''/P$, and can therefore reduce the stream’s normalized convexity.)

Check	What to verify
Time	Each \(t_j\) is elapsed time in years from valuation, including fractional years.
Discounting	Each present value uses its own factor \((1+y)^{-t_j}\).
Price	The denominator uses the sum of all present values at the stated yield.
Normalization	The convexity denominator is \(P(y)(1+y)^2\).
Time zero	A payment at \(t=0\) contributes to price but has zero curvature contribution.

What You'll Learn