Comparing NPV Across Contract Terms

Why Contract Term Comparisons Require Alignment

A longer contract can produce profits in years when a shorter contract has already ended. It may also produce different profits during years common to both contracts: premiums, benefits, or expenses can differ because the term differs. A sound comparison therefore asks two separate questions: how do the profit entries differ while both contracts are in force, and what profit is added or removed outside that shared period?

This tutorial compares hypothetical contracts on a consistent per-policy basis. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The comparison uses the projected profit signatures as given; it does not assume that two otherwise similar designs necessarily have identical profits in their common years.

Definition: A contract term is the number of policy years for which the contract provides the specified coverage. Its profit signature is the sequence $(\pi_1,\pi_2,\ldots,\pi_n)$, where $\pi_t$ is the profit attributed to policy year $t$, under the stated projection basis. For this tutorial, each year’s profit is assumed to be received at the end of that year. With annual effective interest rate $i$, its net present value is $\operatorname{NPV}=\sum_{t=1}^{n}\pi_t v^t$, where $v=1/(1+i)$.

The end-of-year timing assumption matters. A profit in the first policy year is discounted one year, so it is multiplied by $v$, not by 1. Profits after a contract has ended are not included in that contract’s signature. If a particular profit test uses a different timing convention, the discount exponents must follow that convention instead.

Align the Signatures Before Comparing Them

Suppose contract A has term $n$ and contract B has the longer term $m$, with $m>n$. Write both signatures against the same duration scale. The first $n$ entries are the common-duration entries; the entries for B in years $n+1$ through $m$ are its added-duration profits. Subtracting the two NPVs gives:

\operatorname{NPV}_B-\operatorname{NPV}_A = \sum_{t=1}^{n}(\pi_{B,t}-\pi_{A,t})v^t + \sum_{t=n+1}^{m}\pi_{B,t}v^t.

The first sum is the common-duration contribution; the second is the extension contribution. The longer contract’s NPV advantage is their sum. If the common entries are identical, the first sum is zero and those entries cancel. If they differ, they must be compared and discounted; the longer contract’s extension alone does not give the full NPV difference.

Set the basis and timing.
Confirm that the signatures are measured on the same basis and that each profit is assigned to the same point in its policy year. State the annual effective interest rate used for discounting.

Align by duration.
Place first-year profit against first-year profit, second-year against second-year, and so on. Do not align entries by their position after removing years that occur only in one contract.

Separate the comparison.
Calculate discounted differences for common years. Then calculate discounted profits in the added years of the longer contract, or the removed years when comparing in the other direction.

Reconcile the result.
Add the two contributions and check that the result equals the longer contract’s NPV minus the shorter contract’s NPV.

Why the Decomposition Holds

The result follows directly from the definition of NPV and the fact that a finite sum can be split into two ranges. Write each contract’s NPV separately:

\operatorname{NPV}_A=\sum_{t=1}^{n}\pi_{A,t}v^t,\qquad \operatorname{NPV}_B=\sum_{t=1}^{n}\pi_{B,t}v^t+\sum_{t=n+1}^{m}\pi_{B,t}v^t.

Subtract the first expression from the second. In each common year $t$, the discounted difference is $\pi_{B,t}v^t-\pi_{A,t}v^t=(\pi_{B,t}-\pi_{A,t})v^t$. The terms that occur only in B remain in the second sum. Therefore the difference separates exactly into the common-duration and extension contributions. No additional assumption that shared profits are equal is needed; equality merely makes the common-duration sum zero.

Worked Comparisons

Worked Example: Identical Shared Profits and a Two-Year Extension

Two hypothetical contracts have annual profit signatures, in dollars, of $(100,80)$ for the two-year contract and $(100,80,12,12)$ for the four-year contract. Each profit is received at the end of its policy year. The annual effective interest rate is 10%. Find the four-year contract’s NPV advantage.

Method. Align the first two years. Their profits match, so their discounted differences are zero. Only the profits in years 3 and 4 contribute to the advantage.

The discount factors for years 3 and 4 are $v^3=1/(1.10)^3=1/1.331$ and $v^4=1/(1.10)^4=1/1.4641$. The extension contribution is:

\frac{12}{1.331}+\frac{12}{1.4641} =9.0157776+8.1961615 =17.2119391.

The common-duration contribution is $0$, and the four-year contract’s NPV advantage is $\$17.21$, rounded to the nearest cent. As a check, the two NPVs differ only by the present value of the year-3 and year-4 profits, which is the amount just calculated.

Common mistake avoided: Treating the year-3 profit as if it were received immediately would use no discounting. It is received three years from the valuation date, so its correct factor is $v^3$.

Worked Example: Different Profits in Shared Years

A hypothetical three-year contract has signature $(\$40,\$50,\$60)$. A four-year version has signature $(\$35,\$60,\$55,\$12)$. Profits are received at each policy year-end, and the annual effective interest rate is 10%. Calculate the four-year-minus-three-year NPV difference, identifying both contributions.

Method. Compare years 1 through 3 entry by entry, then add the year-4 profit of the longer contract. The differences in common years are $-\$5,\$10,-\$5$; the added year-4 profit is $\$12$.

The discounted common-duration contribution is:

\frac{-5}{1.10}+\frac{10}{(1.10)^2}+\frac{-5}{(1.10)^3} =-4.5454545+8.2644628-3.7565740 =-0.0375657.

The extension contribution, discounted four years, is:

\frac{12}{(1.10)^4}=\frac{12}{1.4641}=8.1961615.

Adding the contributions gives $-0.0375657+8.1961615=8.1585958$. Thus the four-year contract’s NPV exceeds the three-year contract’s NPV by $\$8.16$, rounded to the nearest cent. The common-duration contribution is slightly negative, so the total advantage is slightly less than the extension contribution.

Common mistake avoided: Counting only the year-4 profit would give about $\$8.20$, not the NPV difference. That calculation incorrectly assumes the first three years have equal profits; their discounted net effect is about $-\$0.04$.

Worked Example: A Small Net Difference from Offsetting Contributions

A hypothetical three-year contract has signature $(\$70,\$20,-\$10)$. A five-year version has signature $(\$65,\$30,-\$5,-\$20,\$10)$. Profits are received at policy year-end and the annual effective interest rate is 5%. Find the five-year-minus-three-year NPV difference and show its two components.

Method. The common-year differences are $-\$5,\$10,\$5$. The two added-year profits are $-\$20$ in year 4 and $\$10$ in year 5. Discount each amount by its own duration.

Using unrounded discount factors, the common-duration contribution is:

\frac{-5}{1.05}+\frac{10}{(1.05)^2}+\frac{5}{(1.05)^3} =-4.761904762+9.070294785+4.319187992 =8.627578015.

The extension contribution is:

\frac{-20}{(1.05)^4}+\frac{10}{(1.05)^5} =-16.454049496+7.835261665 =-8.618787831.

Therefore the total difference is $8.627578015-8.618787831=0.008790184$. Rounded only at the final step to the nearest cent, the five-year contract’s NPV advantage is $\$0.01$. The positive common-year contribution is almost exactly offset by the negative extension contribution.

Check. The answer is positive, so the five-year contract’s NPV is slightly higher. Rounding each contribution to cents gives $\$8.63-\$8.62=\$0.01$, consistent with the rounded total. Keeping full precision until the final step avoids a misleading conclusion when the contributions nearly cancel.

Common mistake avoided: A large positive common contribution does not by itself establish the longer contract’s advantage. The added years must also be included, even when their net contribution is negative.

Shortening a Term and Other Comparison Traps

If B is shorter than A, the same method applies with the comparison direction stated clearly. For example, when A has term $m$ and B has term $n<m$, the difference $\operatorname{NPV}_B-\operatorname{NPV}_A$ includes common-year differences through year $n$ and the negative of A’s discounted profits in years $n+1$ through $m$. A removed year is not silently discarded: its contribution matters because it was included in the longer contract’s NPV.

The most important traps are summarized below.

Trap	Incorrect approach	Correction
Assuming shared profits cancel	Use only the added years even though the signatures differ in common years.	Subtract the two entries in every common duration and discount each difference.
Miscounting the discount period	Discount year 3 by $v^2$, or treat it as an immediate payment.	For end-of-year profits, duration $t$ uses $v^t$.
Comparing unaligned entries	Pair entries based on their position after deleting years unique to one contract.	Pair by policy-year duration: year 1 with year 1, year 2 with year 2, and so forth.
Rounding too early	Round each discounted contribution aggressively before combining near-offsetting amounts.	Retain adequate precision through the calculation and round the final monetary answer.

An NPV comparison is meaningful only when the profit definitions, unit basis, valuation date, interest rate, and timing convention are comparable. If the signatures come from different assumptions or profit definitions, their difference may reflect those changes as well as the term. State what is held constant and interpret the result accordingly.

Key takeaway: For different contract terms, align profits by policy duration. The NPV difference is the discounted sum of changes in shared years plus the discounted profits in added years (or the negative of removed years). Shared entries cancel only when they are actually equal.

Check Your Understanding

For each problem, assume profits are received at the end of each policy year and use the stated annual effective interest rate. Round final monetary answers to the nearest cent.

A two-year contract has signature $(\$30,\$20)$, and a three-year version has signature $(\$30,\$20,\$8)$. At 5%, find the longer contract’s NPV advantage. (Answer: $\$6.91$)
A two-year contract has signature $(\$25,\$40)$, and a four-year contract has signature $(\$25,\$40,\$10,-\$5)$. At 10%, calculate the four-year-minus-two-year NPV difference, including both added years. (Answer: $\$4.10$)
A three-year contract has signature $(\$50,\$40,\$30)$, while a four-year version has $(\$45,\$44,\$25,\$12)$. At 5%, find the four-year-minus-three-year NPV difference. (Answer: $\$4.42$)
A three-year contract has signature $(\$60,\$30,\$20)$, while a five-year contract has $(\$55,\$35,\$18,\$10,-\$6)$. At 5%, calculate the common-duration contribution, the extension contribution, and the total NPV difference. (Answer: common-duration contribution $-\$1.95$; extension contribution $\$3.53$; total $\$1.57$)

What You'll Learn