NPV Comparisons Across Contract Volumes

Why Contract Volume Matters

A product design may have a positive expected net present value (NPV) per policy but a small total NPV if only a few policies are sold. Conversely, a modest per-policy shortfall can become material across a large portfolio. The key is to distinguish the value for one policy from the value for a specified number of policies, and to check whether the costs and assumptions scale in the same way as the policy count.

This tutorial focuses on volume comparisons under a fixed set of profit assumptions. It uses a profit signature: the sequence of profits by duration. The signature may represent projected expected profits under a model; it is not a guarantee of the cash flows that will occur. NPV discounts those duration profits to a common valuation time.

Definition: Let $\pi_t$ be the profit attributed to duration $t$ per policy, with the profit occurring at time $t$. Given effective annual interest rate $i$, let $v=1/(1+i)$. For a contract with profits at times $t=0,1,\ldots,T$, the per-policy NPV is $$ \operatorname{NPV}_{1}=\sum_{t=0}^{T} v^t\pi_t. $$ For $N$ identical policies whose profits scale exactly in proportion to policy count, aggregate duration profit is $N\pi_t$, and aggregate NPV is $$ \operatorname{NPV}_{N}=\sum_{t=0}^{T}v^t(N\pi_t)=N\operatorname{NPV}_{1}. $$

The scaling result requires a common valuation basis and proportional profits: each policy has the same projected signature, and there are no additional costs or benefits that change with volume other than by multiplying the per-policy amounts. If a portfolio includes a fixed launch expense, a volume-dependent expense, or a different mix of policy types, its aggregate NPV must include those features explicitly. Simple multiplication may no longer describe the whole portfolio.

Why Proportional Scaling Works

The result follows from the linearity of the discounted sum. Under the identical-policy assumption, the profit at each duration is the per-policy profit multiplied by $N$. Since the discount factor for a given duration is the same for every policy, the factor $N$ can be taken outside the sum.

\operatorname{NPV}_{N} =\sum_{t=0}^{T}v^t(N\pi_t) =N\sum_{t=0}^{T}v^t\pi_t =N\operatorname{NPV}_{1}.

For two designs A and B sold in the same positive volume $N$, the difference between their aggregate NPVs is

\operatorname{NPV}_{N}^{A}-\operatorname{NPV}_{N}^{B} =N\left(\operatorname{NPV}_{1}^{A}-\operatorname{NPV}_{1}^{B}\right).

Because $N>0$, multiplication by $N$ preserves the sign of the difference. Therefore, if the per-policy NPVs are calculated on the same assumptions and valuation basis, their ranking is also the aggregate ranking when both designs are sold in the same volume and all relevant amounts scale proportionally. This does not say that the total NPV is unchanged: its magnitude changes with $N$. Nor does it guarantee that a ranking persists when the designs have different volumes or non-proportional costs.

Worked Example: Scaling One Profit Signature

A hypothetical contract has per-policy profits of $-\$100$ at issue, $\$110$ at the end of year 1, and $\$121$ at the end of year 2. The effective annual interest rate is 10%. Calculate the per-policy NPV and the aggregate NPV for 250 identical policies. Assume there are no fixed or volume-dependent costs beyond the stated profits.

Method. Discount each duration profit to time 0, then multiply each profit by the policy count to obtain the aggregate signature. The time-0 profit is not discounted.

Here $i=0.10$, so $v=1/1.10$. The per-policy present values are

\begin{aligned} t=0 &: \quad -100,\\ t=1 &: \quad 110v=\frac{110}{1.10}=100,\\ t=2 &: \quad 121v^2=\frac{121}{(1.10)^2}=\frac{121}{1.21}=100. \end{aligned}

Thus the per-policy NPV is

\operatorname{NPV}_{1}=-100+100+100=\$100.

The aggregate duration profits for 250 policies are $-\$25{,}000$, $\$27{,}500$, and $\$30{,}250$. Their present values are

-25{,}000,\qquad \frac{27{,}500}{1.10}=25{,}000,\qquad \frac{30{,}250}{1.21}=25{,}000.

Adding the discounted amounts gives an aggregate NPV of $-25{,}000+25{,}000+25{,}000=\$25{,}000$. This checks against direct scaling: $250(\$100)=\$25{,}000$. The common mistake this method avoids is scaling only the final answer while leaving an unscaled duration profit in the aggregate signature; each duration profit must be treated consistently.

Comparing Designs at the Same Volume

For two designs sold in a common volume, calculate each per-policy NPV using the same interest rate, time origin, and duration convention. Comparing those values establishes the ranking; multiplying both by the same $N$ gives the aggregate NPVs. The intermediate duration profits can still matter for understanding the source of the difference, but they do not change the proportional-scaling result.

Worked Example: Does the Ranking Change at 300 Policies?

Two hypothetical designs have the following per-policy profit signatures. Design A has profits of $-\$80$ at time 0, $\$55$ at the end of year 1, and $\$60$ at the end of year 2. Design B has profits of $-\$100$ at time 0, $\$70$ at the end of year 1, and $\$55$ at the end of year 2. The effective annual interest rate is 10%. Compare their per-policy NPVs and their aggregate NPVs if each design is issued to 300 policies. Assume no additional costs.

Method. First compare the discounted per-policy signatures; then apply the same volume to both designs.

For A, $v=1/1.10$, so

\begin{aligned} \operatorname{NPV}_{1}^{A} &=-80+\frac{55}{1.10}+\frac{60}{1.21}\\ &=-80+50+49.5867768595\ldots\\ &=\$19.5867768595\ldots. \end{aligned}

For B,

\begin{aligned} \operatorname{NPV}_{1}^{B} &=-100+\frac{70}{1.10}+\frac{55}{1.21}\\ &=-100+63.6363636364\ldots+45.4545454545\ldots\\ &=\$9.0909090909\ldots. \end{aligned}

A has the higher per-policy NPV by $19.5867768595\ldots-9.0909090909\ldots=\$10.4958677686\ldots$. At 300 policies, the aggregate NPVs are

\begin{aligned} \operatorname{NPV}_{300}^{A} &=300(19.5867768595\ldots) =\$5{,}876.03305785\ldots,\\ \operatorname{NPV}_{300}^{B} &=300(9.0909090909\ldots) =\$2{,}727.27272727\ldots. \end{aligned}

The aggregate difference is $\$3{,}148.76033058\ldots$, also equal to $300(\$10.4958677686\ldots)$. Rounded to the nearest cent, the aggregate NPVs are $\$5{,}876.03$ for A and $\$2{,}727.27$ for B. A remains the higher-NPV design because both per-policy values were multiplied by the same positive volume. The common mistake avoided here is treating a larger aggregate amount as evidence of a better design without first checking whether the volumes are equal.

When Volume Does Not Preserve the Ranking

Proportional scaling is not valid for costs that are incurred once for a whole product launch, rather than once per policy. A fixed cost is added to the portfolio NPV; it is not multiplied by the number of policies. Such costs can cause the preferred design to depend on volume. A separate issue arises when designs are compared at different volumes: their aggregate NPVs then combine per-policy value with the size of the business, so aggregate ranking need not match per-policy ranking.

A reliable comparison identifies which quantities are per-policy and which are portfolio-level. The following process makes that distinction explicit.

Calculate per-policy NPV.
Discount each duration profit using the stated rate and payment times. Use a consistent time origin for each design.

Identify the volume basis.
Check whether the designs are being compared at the same policy count or at different counts.

Separate scalable and fixed amounts.
Multiply per-policy amounts by the relevant count, but include each fixed portfolio cost only once.

State the comparison being made.
Report whether the conclusion concerns value per policy or total portfolio value; these answer different questions.

Worked Example: Fixed Costs and a Volume-Dependent Ranking

Two hypothetical product proposals have the following per-policy profit signatures at an effective annual interest rate of 10%. Proposal A has profits of $-\$100$ at time 0 and $\$143$ at the end of year 1. Proposal B has profits of $-\$50$ at time 0 and $\$77$ at the end of year 1. In addition, A requires a fixed launch expense of $\$10{,}000$ at time 0, and B requires a fixed launch expense of $\$1{,}000$ at time 0. Compare total NPV at 100 policies and at 1,000 policies. Assume each fixed expense is incurred once, regardless of volume.

Method. Calculate the per-policy NPVs, multiply those values by the policy count, and then subtract the one-time launch expense for each proposal.

Since $v=1/1.10$, the per-policy NPVs are

\begin{aligned} \operatorname{NPV}_{1}^{A} &=-100+\frac{143}{1.10} =-100+130 =\$30,\\ \operatorname{NPV}_{1}^{B} &=-50+\frac{77}{1.10} =-50+70 =\$20. \end{aligned}

For $N$ policies, the total NPVs, including the launch expenses, are therefore

\operatorname{NPV}^{A}_{\mathrm{total}}=30N-10{,}000, \qquad \operatorname{NPV}^{B}_{\mathrm{total}}=20N-1{,}000.

At $N=100$, A has total NPV $30(100)-10{,}000=-\$7{,}000$, while B has total NPV $20(100)-1{,}000=\$1{,}000$. B ranks higher by $\$8{,}000$. At $N=1{,}000$, A has total NPV $30(1{,}000)-10{,}000=\$20{,}000$, while B has total NPV $20(1{,}000)-1{,}000=\$19{,}000$. A ranks higher by $\$1{,}000$.

The ranking changes because the difference in total NPV is $10N-9{,}000$, not simply $N$ times the difference in per-policy NPVs. The designs tie when $10N-9{,}000=0$, or $N=900$; at that volume both total NPVs are $\$17{,}000$. The common mistake is to multiply the fixed launch expense by $N$, as though it were a per-policy cost, or to omit it entirely. Either error gives the wrong volume comparison.

Practical Checks and Common Traps

Volume comparisons are useful when assessing how a product design performs at a proposed scale, but the conclusion depends on what is held constant. If both designs are evaluated for the same count of identical policies and all profits are proportional, the NPV ranking is determined by per-policy NPV. If the counts differ, a higher aggregate NPV may simply reflect more policies. If portfolio-level expenses are present, the ranking can change with volume even when one design has the larger per-policy NPV.

In a model where each policy has an expected profit signature, the expected aggregate profit at each duration is $N$ times the per-policy expected profit when the policies share those assumptions. This expectation scaling does not require policy outcomes to be independent. It concerns expected values, not a claim that a realized portfolio will produce exactly $N$ times the result for a single policy. Dependence between outcomes may affect the variability of aggregate experience, but it does not alter the linear scaling of the expected value under the stated common assumptions.

Do not compare unlike measures. Label per-policy NPV and aggregate NPV separately, and state the policy count used.
Do not scale fixed amounts. A one-time project cost is included once, unless the scenario explicitly says it is incurred for every policy.
Do not assume a common ranking at unequal volumes. Different policy counts and non-proportional amounts can change aggregate rankings.
Do not discount time-0 amounts. A profit at time 0 has discount factor $v^0=1$.

Key takeaway: For identical policies with purely proportional profits, aggregate NPV equals policy count multiplied by per-policy NPV. A common positive count preserves the ranking of designs. Fixed costs, unequal volumes, or different policy mixes require a direct aggregate calculation.

Check Your Understanding

For each problem, distinguish the per-policy comparison from the aggregate comparison and account for the stated volume assumptions.

A hypothetical policy has profits of $-\$40$ at time 0 and $\$55$ at the end of year 1. The effective annual interest rate is 5%. Find its NPV and the aggregate NPV for 80 identical policies, with no additional costs. (Answer: Per-policy NPV $=\$12.380952\ldots$; aggregate NPV $=\$990.48$ to the nearest cent.)
At an effective annual interest rate of 10%, design X has a per-policy signature of $-\$60$ at time 0 and $\$88$ at year 1; design Y has $-\$80$ at time 0 and $\$99$ at year 1. If each is issued to 150 policies and there are no fixed costs, compare the aggregate NPVs. (Answer: X has per-policy NPV $\$20$ and aggregate NPV $\$3{,}000$; Y has per-policy NPV $\$10$ and aggregate NPV $\$1{,}500$, so X is higher by $\$1{,}500$.)
Two proposals have per-policy NPVs of $\$18$ and $\$15$, respectively. Their one-time fixed expenses are $\$2{,}400$ and $\$600$, respectively. At 100 policies for each proposal, find the total NPVs and the higher-ranked proposal. (Answer: The first proposal has total NPV $-\$600$; the second has total NPV $\$900$, so the second is higher by $\$1{,}500$.)
Design A has per-policy NPV $\$12$ and is sold to 500 policies. Design B has per-policy NPV $\$17$ and is sold to 300 policies. With no fixed costs, which design has the greater aggregate NPV, and which has the greater per-policy NPV? (Answer: A has the greater aggregate NPV, $\$6{,}000$ versus $\$5{,}100$; B has the greater per-policy NPV, $\$17$ versus $\$12$.)
Design A has per-policy NPV $\$26$ and a fixed expense of $\$7{,}200$; design B has per-policy NPV $\$18$ and a fixed expense of $\$800$. Both are evaluated at the same volume $N$. Find the break-even volume at which their total NPVs are equal, and state which design is higher above that volume. (Answer: They tie at $N=800$ policies, when both total NPVs are $\$13{,}600$; above 800 policies, A is higher.)

What You'll Learn