Sign Convention Errors in NPV Calculations

Why Sign Conventions Matter

A profit signature is useful only if each entry uses the same cash-flow perspective and sign convention. In this tutorial, the perspective is the insurer’s: premiums received are positive cash flows, while claims and expenses paid are negative cash flows. The net profit at each time is the algebraic sum of the cash flows at that time.

A sign error can persist through an otherwise correct calculation. For example, if an expense is first recorded as a negative cash flow and then subtracted again when forming profit, the expense incorrectly increases profit. A reliable check is to reconstruct the net signature from its underlying premiums, claims, and expenses, then compare its discounted value with the separate present values of those gross items.

Definition: Under the insurer cash-flow convention, premiums received have a positive sign, and claims and expenses paid have negative signs. The profit signature is the sequence of net cash flows $P_0,P_1,\ldots,P_n$ at times $0,1,\ldots,n$. At effective annual interest rate $i$, with $v=1/(1+i)$, its net present value is $\operatorname{NPV}=\sum_{t=0}^{n}P_t v^t$. A cash flow at time 0 is not discounted.

The entries $P_t$ can be actual amounts for a deterministic contract or expected amounts under a model. If a signature contains expected cash flows, its assumptions about mortality, persistency, and payment timing must be consistent with the quantities used to form it. The sign convention does not replace those assumptions; it determines how the resulting inflows and outflows are combined.

Constructing and Checking a Signature

For each time $t$, let $G_t$ denote premiums received, $C_t$ claims paid, and $E_t$ expenses paid, with $G_t,C_t,E_t$ stated as nonnegative amounts. Then the signed net profit is $P_t=G_t-C_t-E_t$. Alternatively, if the cash flows are already signed, add them directly. Do not subtract a negative outgo a second time.

Item from insurer perspective	Signed amount	Contribution to net profit
Premium received	Positive	Increases profit
Claim paid	Negative	Reduces profit
Expense paid	Negative	Reduces profit

The timing convention is equally important. If a premium is received at the end of year 1, it is discounted by $v$; if it is received immediately at time 0, it is not discounted. Assigning the right sign to a cash flow does not correct a mistake in its timing.

List gross cash flows by time.
Record premiums received, claims paid, and expenses paid at the time each is assumed to occur. Keep amounts nonnegative in this list if you plan to apply the formula $G_t-C_t-E_t$.

Form each net profit.
Calculate $P_t=G_t-C_t-E_t$, or add cash flows that already carry their signs. Check that an outgo reduces the net amount.

Discount by time.
Multiply $P_t$ by $v^t$, using the stated interest rate and the correct time index.

Reconcile independently.
Calculate the present value of premiums less the present values of claims and expenses. It must equal the NPV from the net signature.

Why Gross-Flow Reconciliation Works

The reconciliation is a direct consequence of the distributive property. At each time, net profit is premiums less claims and expenses. Multiplying by the same discount factor and adding over all times preserves that relationship.

\begin{aligned} \operatorname{NPV} &=\sum_{t=0}^{n}(G_t-C_t-E_t)v^t\\ &=\sum_{t=0}^{n}G_t v^t -\sum_{t=0}^{n}C_t v^t -\sum_{t=0}^{n}E_t v^t. \end{aligned}

Thus the NPV of the net profit signature must agree with the present value of premiums received minus the present value of claims and expenses paid, provided the same cash flows, timings, and interest basis are used in both calculations. This check detects inconsistent signs and can also reveal an omitted or mistimed cash flow. It does not, by itself, establish that the underlying cash-flow assumptions are appropriate.

Worked Example: Correcting and Reconciling a Two-Year Signature

A hypothetical contract has a deterministic cash-flow schedule. At time 0, the insurer pays an expense of $100. At times 1 and 2, it receives a premium of $300 and pays total claims and expenses of $130 at each time. Use an effective annual interest rate of $i=10\%$. Assume no mortality or other contingencies: the stated cash flows occur with certainty, so the distribution is degenerate at those amounts. Find the profit signature and NPV, then reconcile using gross cash flows.

Method and signs. Use the insurer perspective. The time-0 expense is $-100$. At each of times 1 and 2, profit is premium less outgo, $300-130=170$. Therefore the signature is $(-100,170,170)$, not $(-100,300,-130)$ unless the latter is understood as separate signed cash-flow components to be added.

The discount factor is $v=1/1.10=10/11$, and $v^2=100/121$. The time-0 amount is undiscounted. Thus $\operatorname{NPV}=-100+170(10/11)+170(100/121)$. Putting terms over 121 gives $(-12100+18700+17000)/121=23600/121$, or $195.04 to the nearest cent.

Gross-flow check. The present value of premiums is $300(10/11)+300(100/121)=(33000+30000)/121=63000/121$, or $520.66. The present value of outgoes is $100+130(10/11)+130(100/121)=(12100+14300+13000)/121=39400/121$, or $325.62. Their difference is $(63000-39400)/121=23600/121=$195.04$, agreeing with the signature NPV.

Check. Both methods give $23600/121$; the cent amounts also reconcile, since $520.66 minus $325.62 is $195.04. The common mistake avoided is using an incorrect common denominator or discounting the time-0 expense.

Worked Example: Finding an Expense Sign Error

A hypothetical deterministic policy has premiums of $250 at times 0, 1, and 2. Claims are $80 at time 1 and $120 at time 2. Expenses are $100 at time 0, $30 at time 1, and $20 at time 2. Use $i=5\%$. Assume no mortality contingency and that all amounts listed occur with certainty (a degenerate cash-flow distribution). A draft calculation records the signature as $(150,200,110)$. Diagnose it, calculate the corrected NPV, and reconcile it.

Method: rebuild each year from gross items. At time 0, profit is $250-100=150$. At time 1, profit is $250-80-30=140$, not 200. At time 2, profit is $250-120-20=110$. The corrected signature is $(150,140,110)$. The draft added the time-1 premium and then effectively failed to reduce it by both outgoes.

Here $v=1/1.05=20/21$, and $v^2=400/441$. The corrected signature gives $\operatorname{NPV}=150+140(20/21)+110(400/441)$. Writing each term over 441 gives $(66150+58800+44000)/441=168950/441=$383.11$, rounded to the nearest cent.

Gross-flow check. Premiums have present value $250+250(20/21)+250(400/441)=(110250+105000+100000)/441=315250/441$. Claims have present value $80(20/21)+120(400/441)=(33600+48000)/441=81600/441$. Expenses have present value $100+30(20/21)+20(400/441)=(44100+12600+8000)/441=64700/441$. Subtracting outgoes gives $(315250-81600-64700)/441=168950/441=$383.11$, as required.

Check. The reconciled NPV agrees exactly before rounding. The mistake avoided is assuming the draft signature is trustworthy without rebuilding its entries from the underlying cash flows.

Worked Example: Checking a Signature With Negative Profits

A hypothetical deterministic contract has an expense of $50 at time 0, a premium of $120 and a claim of $20 at time 1, and a claim of $15 plus an expense of $5 at time 2. Use $i=10\%$. Assume there is no mortality contingency and each stated cash flow is certain. Calculate the signature NPV and verify it by gross cash flows.

Method: net the cash flows at each time. At time 0, profit is $-50$. At time 1, it is $120-20=100$. At time 2, there is no premium, so profit is $-15-5=-20$. The signature is $(-50,100,-20)$. In particular, the final entry is negative because both cash flows are outgoes.

With $v=10/11$ and $v^2=100/121$, $\operatorname{NPV}=-50+100(10/11)-20(100/121)$. The numerator over 121 is $-6050+11000-2000=2950$. The NPV is $2950/121=$24.38$, rounded to the nearest cent.

Gross-flow check. The premium present value is $120(10/11)=13200/121$. Claims have present value $20(10/11)+15(100/121)=(2200+1500)/121=3700/121$. Expenses have present value $50+5(100/121)=(6050+500)/121=6550/121$. Therefore the reconciled NPV is $(13200-3700-6550)/121=2950/121=$24.38$.

Check. The time-2 discount factor is $100/121$, and both calculations give the same positive NPV. The common mistake avoided is treating a negative signature entry as positive merely because the overall NPV is positive.

Common Sign and Timing Traps

A useful diagnosis separates two questions: what is the cash flow’s sign from the chosen perspective? and when does it occur? A correct answer to only one question is not enough. For example, a claim paid at the end of year 2 is negative and receives the factor $v^2$; it is not positive simply because it is being subtracted in a written expression.

Subtracting a signed outgo twice. If the claim is recorded as $-C_t$, add it to the other signed flows. Do not compute premium minus $(-C_t)$, which would increase profit.
Changing convention mid-calculation. A policyholder-perspective signature reverses every insurer-perspective sign. Reversing only claims or only profits is inconsistent.
Discounting time 0. The factor at time 0 is $v^0=1$, not $v$.
Confusing a positive NPV with positive profits in every year. A positive NPV can include negative entries in the signature, as the third example demonstrates.

If all cash flows are consistently reversed, the NPV also reverses sign. Its magnitude is unchanged. This can be a useful check when switching perspectives, but it does not excuse mixing perspectives within one calculation.

Key takeaway: First establish one perspective, then assign signs to every gross cash flow and combine amounts at their actual times. Calculate NPV from the net signature and verify it by discounting premiums, claims, and expenses separately. The two results must agree before rounding.

Check Your Understanding

For each problem, use the insurer perspective and show enough intermediate work to check the signs and discount factors. Unless stated otherwise, the cash flows are deterministic, with no mortality contingency.

A signature at times 0, 1, and 2 is $(40,-70,50)$. At $i=10\%$, calculate its NPV to the nearest cent. (Answer: $2140/121=\$17.69$.)
At time 0 an insurer pays $25 of expenses. At time 1 it receives a $180 premium, pays a $60 claim, and pays $20 of expenses. At time 2 it pays a $75 claim and $5 of expenses, with no premium. At $i=5\%$, find the signature and NPV to the nearest cent. (Answer: Signature $(-25,100,-80)$; NPV $=-1025/441=-\$2.32$.)
A contract has a time-0 expense of $30; at time 1 it receives a $100 premium, pays a $25 claim, and incurs $15 of expenses; at time 2 it receives an $80 premium, pays a $40 claim, and incurs $10 of expenses. At $i=10\%$, calculate the corrected signature and NPV. (Answer: Signature $(-30,60,30)$; NPV $=5970/121=\$49.34$.)
A signature is $(-120,80,90,-30)$ at times 0 through 3. At $i=5\%$, calculate the NPV to the nearest cent. (Answer: $110280/9261=\$11.91$.)
A draft lists a time-0 expense of $40, a time-1 premium of $90, a time-1 claim of $30, and a time-2 claim of $70 plus expense of $5. There are no other flows and $i=0\%$. State the signed signature and NPV. (Answer: Signature $(-40,60,-75)$; NPV $=-\$55.00$.)

What You'll Learn