Why Payment Timing Matters
Two securities can have the same present value and the same average payment time, yet have different convexities. The reason is that convexity does not weight time in the same way as duration: it places more weight on cash flows at later dates. To make that comparison correctly, use the discounted cash-flow weights, not just the nominal payment amounts or the final maturity.
This tutorial considers fixed, positive cash flows discounted at one annual effective yield. Cash-flow times are measured in years, and a payment at time \(t\) is discounted by \((1+y)^{-t}\), where \(y>-1\). The results apply to this fixed-cash-flow model; they do not account for cash flows that change when yields change, such as payments affected by an embedded option. The timing and weighting ideas are aligned with the SOA Exam FM (Financial Mathematics) syllabus.
A cash-flow weight is the share of the security’s price contributed by that payment. For example, a payment with a large nominal amount far in the future may have a smaller weight than a more modest payment sooner, because the future payment is discounted for longer. Convexity comparisons use these present-value shares.
Deriving the Weighted-Time Formula
The formula follows by differentiating the price of each fixed cash flow with respect to the yield. For one payment, \(C_t(1+y)^{-t}\), the first derivative is \(-tC_t(1+y)^{-t-1}\). Differentiating again gives \(t(t+1)C_t(1+y)^{-t-2}\). Summing over the payments and dividing by price yields the convexity formula.
This is a weighted-average representation: each time contributes \(t(t+1)/(1+y)^2\), and its weight is its share of the price. For \(t\geq 0\), \(t(t+1)\) increases as \(t\) increases. Thus a payment at a later time has a larger convexity contribution per unit of discounted cash flow, at the same yield. This is the precise sense in which later timing tends to raise convexity.
For comparison, Macaulay duration in this fixed-cash-flow setting is the weighted average of payment times, \(\sum_t w_t t\). Convexity depends on a higher-order function of time, not just that average. Consequently, knowing a security’s duration alone does not determine its convexity.
Worked Example: Comparing Two Single-Payment Securities
Two hypothetical securities each pay $1,000 once. Security A pays in 2 years and Security B pays in 5 years. The annual effective yield for both is 5%, and there are no other cash flows. Calculate each price and convexity, then compare them.
Method. For a single payment at time \(t\), its discounted cash-flow weight is 1. The price is \(1{,}000(1.05)^{-t}\), and convexity is \(t(t+1)/(1.05)^2\).
Security A, payment at year 2. The discount factor is \(1/(1.05)^2=1/1.1025\). Therefore,
Its convexity is
Security B, payment at year 5. The discount factor is \(1/(1.05)^5=1/1.2762815625\). Therefore,
Its convexity is
Rounded to four decimal places, the prices are $907.0295 and $783.5262, and the convexities are 5.4422 and 27.2109, respectively. Security B has the greater convexity because its only discounted cash flow occurs later. The calculation avoids a common mistake: comparing the prices and assuming the lower-priced security must also have lower convexity. Price level and normalized convexity are different quantities.
Same Price and Duration, Different Convexity
The single-payment comparison changes both payment time and price. A more informative timing comparison can hold price and duration constant. The following example uses cash flows chosen so that their discounted weights are easy to verify. This isolates how the distribution of those weights across time affects convexity.
Worked Example: Equal Price and Duration, Unequal Convexity
At an annual effective yield of 10%, Security A pays $1,210 at year 2. Security B pays $550 at year 1 and $665.50 at year 3. Assume all payments are fixed and positive. Calculate the price, duration, and convexity of each security.
Method. Discount each payment to find its weight. Then use the weighted average of payment times for duration and the weighted average of \(t(t+1)/(1.10)^2\) for convexity.
For Security A, the year-2 discount factor is \(1/(1.10)^2=1/1.21\). Its price is
There is one payment, so its weight at year 2 is \(1{,}000/1{,}000=1\). Its duration is \(1(2)=2\) years, and its convexity is
For Security B, the present values are
The price is \(500+500=1{,}000\), and both payments have weight \(500/1{,}000=0.5\). Its duration is
Its convexity is
Rounded to four decimal places, both prices are $1,000 and both durations are 2 years, but the convexities are 4.9587 for A and 5.7851 for B. Security B spreads its discounted value across an earlier and a later payment instead of concentrating it at year 2. The later payment’s larger \(t(t+1)\) contribution is not offset by the earlier payment, so the weighted convexity is higher. The common mistake is to infer equal convexity from equal duration; duration fixes a weighted average of time, not a weighted average of \(t(t+1)\).
Shifting an Entire Payment Schedule
A useful special comparison shifts every payment date by the same number of years while leaving every nominal payment unchanged. If all dates move later by one year, each present value is multiplied by the same factor \(1/(1+y)\). The price changes, but the normalized discounted weights remain the same. Since every payment time increases, each \(t(t+1)\) contribution also increases.
For a one-year shift from \(t\) to \(t+1\), the increase in the time factor is
Therefore, under this specific whole-schedule shift, convexity must increase for positive cash flows at nonnegative times and a fixed yield \(y>-1\). This conclusion relies on the weights staying unchanged. Moving only one payment does not leave all the other weights unchanged and needs a fresh calculation.
Worked Example: Shifting Every Payment One Year Later
At an annual effective yield of 10%, a hypothetical security pays $220 at year 1, $363 at year 2, and $665.50 at year 3. Compare it with a version that pays the same amounts at years 2, 3, and 4, respectively. Assume fixed, positive cash flows and calculate both prices and convexities.
Method. First find the original present values and weights. Then observe how a one-year shift changes each present value and payment-time factor.
For the original schedule, the present values are
The price is \(200+300+500=1{,}000\), so the weights at years 1, 2, and 3 are 0.2, 0.3, and 0.5. Its convexity is
After shifting each payment one year later, the present values are
Their sum is \(909.090909\ldots\). Each shifted present value is the corresponding original present value divided by 1.10, so the weights remain 0.2, 0.3, and 0.5. The shifted payments occur at years 2, 3, and 4. Thus,
Rounded to four decimal places, the original price is $1,000.00 and convexity is 6.7769; the shifted price is $909.09 and convexity is 12.2314. The calculation confirms the whole-schedule result: the price falls because the same payments are further away, while convexity rises because the discounted weights are unchanged and the time factors are larger. The mistake to avoid is assuming that unchanged nominal payments imply unchanged price or unchanged convexity.
What “Later Payments Increase Convexity” Does and Does Not Mean
A precise comparison should say how the discounted weights change. If one security’s discounted weights are shifted later as a distribution—for example, each weighted payment time is increased by the same amount—then its weighted average of the increasing function \(t(t+1)\) rises. More generally, a distribution of weights that is later in the sense of first-order stochastic dominance has a larger average for every increasing function of time, including \(t(t+1)\).
But a change in nominal payment dates can also change the weights. If only one payment is moved, its present value changes relative to the other payments, so its share of price changes too. It is not safe to conclude that every isolated delay must raise the normalized convexity without checking the resulting weights. Calculate the new price, recompute each weight, and then apply the weighted formula.
- Use payment times and weights together. Maturity alone does not describe a coupon security’s timing profile.
- Keep the yield convention consistent. The displayed formula uses annual effective yield and discounting by \((1+y)^{-t}\).
- Check the cash-flow model. The formula assumes fixed cash flows. If the cash flows depend on yield, differentiating only the discount factors is not sufficient.
- Do not confuse price with convexity. Convexity \(P''/P\) is normalized by price; a change in price does not by itself establish the direction of a change in convexity.
The central comparison is therefore not simply “which security pays later?” It is: what are the discounted cash-flow weights at the specified yield, and how are those weights distributed over payment times? That question connects a security’s payment structure directly to its convexity.
Check Your Understanding
Assume fixed positive cash flows and annual effective compounding in each question. Convexity means \(P''(y)/P(y)\).
- A security makes a single $800 payment at year 4. The annual effective yield is 5%. Find its convexity. (Answer: \(20/1.1025=18.1406\), to four decimal places.)
- At a 10% annual effective yield, a security has discounted cash-flow weights 0.4 at year 1 and 0.6 at year 3. Find its convexity. (Answer: \([0.4(2)+0.6(12)]/1.21=6.6116\), to four decimal places.)
- At a 10% annual effective yield, Security A has all its discounted weight at year 2. Security B has weights 0.5 at year 1 and 0.5 at year 3. Both have price $1,000. Find each convexity and their difference. (Answer: A = 4.9587, B = 5.7851, and B exceeds A by 0.8264, to four decimal places.)
- At a 10% annual effective yield, a schedule has weights 0.2, 0.3, and 0.5 at years 1, 2, and 3. A second schedule is a one-year shift of every payment, so its weights are unchanged. Find both convexities. (Answer: original = \(8.2/1.21=6.7769\); shifted = \(14.8/1.21=12.2314\), to four decimal places.)