Net Present Value as the Sum of a Profit Signature

From a Profit Signature to Net Present Value

A profit signature records how expected profit emerges over a policy’s durations. Net present value (NPV) reduces that sequence to one amount at the issue date by discounting each duration’s profit and adding the results. This makes the timing of each entry essential: an amount arising later contributes less to NPV when the effective annual interest rate is positive.

The signature entries used here are unconditional: each is the expected profit at a duration per policy originally issued, allowing for the chance that the policy is no longer in force. A profit stated only for policies in force at the start of a duration is conditional and must be weighted by the probability of being in force before it is used as a signature entry.

Definition: For a policy with annual profits emerging at the end of each duration, let $\phi_t$ be the unconditional expected profit at the end of duration $t$, per policy issued. The sequence $(\phi_1,\phi_2,\ldots)$ is the profit signature. At effective annual interest rate $i$, with $v=1/(1+i)$, the NPV through duration $n$ is

\operatorname{NPV}_0=\sum_{t=1}^{n}v^t\phi_t.

For a whole-life contract, the sum extends over all durations and is used when the discounted series converges.

The subscript 0 indicates valuation at issue. The exponent $t$ follows from the end-of-duration timing convention: profit at the end of duration $t$ is $t$ years from issue. If a problem specifies a different emergence date, use that date in the discount factor rather than automatically using $v^t$.

Why Discounted Signature Entries Are Added

Each signature entry is a separate expected profit amount attached to a particular time. Under the end-of-duration convention, an amount $\phi_t$ at time $t$ has issue-date value $v^t\phi_t$. The NPV is the sum of these duration-specific present values. This is a direct application of discounting and addition; it does not require the signature to be level or all its entries to be positive.

Establish the units and timing.
Check whether profits are dollars or thousands of dollars, and identify the duration-end date represented by each entry.

Obtain unconditional entries.
If profits are conditional on being in force at the start of a duration, multiply by the probability of being in force at that time.

Discount each entry separately.
For an end-of-duration-$t$ entry, multiply by $v^t=(1+i)^{-t}$.

Sum and interpret.
Add the discounted amounts, then report the NPV in the units stated by the problem.

For a simple survival model, let ${}_{t-1}p_x$ be the probability that a life aged $x$ survives to the start of duration $t$. If the conditional expected profit for that duration is $\pi_t$ per policy in force at its start, and there are no other ways for a policy to leave the in-force group, then

\phi_t={}_{t-1}p_x\,\pi_t, \qquad \operatorname{NPV}_0=\sum_{t=1}^{n}v^t\,{}_{t-1}p_x\,\pi_t.

This conversion assumes the conditional profit is defined for the policy in force at the start of the duration and includes the relevant profit contingencies during that duration. If the supplied profits are already unconditional per policy issued, applying an in-force probability again would understate the NPV.

Worked Finite-Term Examples

Worked Example: Discounting a Given Three-Year Signature

A hypothetical three-year contract has the following unconditional profit signature, per policy issued, in thousands of dollars: $(-4,6,2)$. Each entry emerges at the end of its stated duration. At an effective annual interest rate of $4\%$, calculate the issue-date NPV in dollars. State the method and identify a likely error.

Method. Since the signature is already unconditional, discount each entry directly by $v^t=(1.04)^{-t}$. The amounts are in thousands, so convert the final result to dollars.

The three discounted entries, in thousands, are

\begin{aligned} \text{Duration 1: }&\frac{-4}{1.04}=-3.8461538,\\ \text{Duration 2: }&\frac{6}{(1.04)^2}=\frac{6}{1.0816}=5.5473373,\\ \text{Duration 3: }&\frac{2}{(1.04)^3}=\frac{2}{1.124864}=1.7779927. \end{aligned}

Therefore,

\operatorname{NPV}_0=-3.8461538+5.5473373+1.7779927 =3.4791762\text{ thousand dollars}.

Converting to dollars gives $3.4791762\times 1{,}000=\$3{,}479.1762$, or \$3,479.18 to the nearest cent. The check is that the first loss is discounted one year, the second profit two years, and the third profit three years. A common error is to discount all three entries by the same factor, or to shift each exponent by one. Neither matches the stated end-of-duration dates.

Worked Example: Converting Conditional Profits Before Discounting

A hypothetical three-year survival product has annual mortality probability $q=0.10$, independent from year to year, with no lapses. At the start of a year, the conditional expected profit is $-\$105$ in year 1, $\$100$ in year 2, and $\$100$ in year 3, per policy in force. These profits include the contingencies occurring during each year. Profits emerge at year-end, and the effective annual interest rate is $5\%$. Find the NPV per policy issued.

Method. First weight each conditional profit by the probability the policy is in force at the beginning of that duration. Then discount the resulting unconditional signature entries.

The in-force probabilities at the three duration starts are $1$, $0.90$, and $0.90^2=0.81$. Thus the signature per policy issued is

\phi_1=-105,\qquad \phi_2=0.90(100)=90,\qquad \phi_3=0.81(100)=81.

Discounting duration by duration at $i=0.05$ gives

\begin{aligned} \operatorname{NPV}_0 &=\frac{-105}{1.05}+\frac{90}{(1.05)^2}+\frac{81}{(1.05)^3}\\ &=-100+81.6326531+69.9708455\\ &=51.6034986. \end{aligned}

The NPV is \$51.60 per policy issued, rounded to the nearest cent. The first term is exactly $-\$100$, since $105/1.05=100$; the later amounts are positive and sum to more than that loss.

Common mistake avoided. Discounting the conditional profits without weighting them would give

\frac{-105}{1.05}+\frac{100}{(1.05)^2}+\frac{100}{(1.05)^3} =-100+90.7029478+86.3837594 =77.0867072.

That calculation treats every issued policy as if it were still in force at the start of years 2 and 3. The correct values for those durations are $90$ and $81$, not $100$ and $100$, so the correct NPV is $\$51.60$.

Whole-Life Signatures

A whole-life signature can have infinitely many entries. Its NPV is then an infinite discounted sum, provided the sum converges. A common tractable case has a constant conditional expected profit for each policy in force, together with a constant annual probability of remaining in force. The resulting discounted terms form a geometric series.

Worked Example: Whole-Life NPV with Constant Annual Mortality

A hypothetical whole-life policy has no lapses and annual death probability $q=0.10$. Its expected year-1 profit per policy issued is $-\$100$. From year 2 onward, the conditional expected profit is $\$30$ per policy in force at the start of the year, including that year’s mortality contingencies. Profits emerge at the end of each year. The effective annual interest rate is $5\%$. Find the NPV per policy issued.

Method. Keep the first-year signature entry separate. For each later duration, multiply the conditional profit by the probability of surviving to its start, then discount. Sum the resulting geometric series.

The first-year entry is $\phi_1=-100$. For duration $t\geq 2$, the probability of being in force at the start is $0.9^{t-1}$, so

\phi_t=30(0.9)^{t-1}.

The first two parts of the NPV are

\frac{-100}{1.05}=-95.2380952, \qquad \frac{\phi_2}{(1.05)^2} =\frac{30(0.9)}{(1.05)^2} =\frac{27}{1.1025} =24.4897959.

For subsequent durations, the ratio of one discounted term to the previous discounted term is $0.9/1.05=6/7$, which is less than 1. Thus the later contributions sum to

24.4897959\left(1+\frac67+\left(\frac67\right)^2+\cdots\right) =\frac{24.4897959}{1-6/7} =171.4285714.

Adding the year-1 contribution gives

\operatorname{NPV}_0=-95.2380952+171.4285714 =76.1904762.

The NPV is \$76.19 per policy issued, rounded to the nearest cent. The geometric sum applies only to durations 2 onward; including the first-year loss in that series would use the wrong pattern of profits and survival weights.

Checks and Common Exam Traps

A reliable NPV calculation keeps three questions separate: what amount is the signature entry, when does it emerge, and what units are being reported? In particular, distinguish a profit per policy issued from a profit conditional on being in force. The first is already an unconditional entry; the second generally needs an in-force probability.

Applying survival weights twice. If the problem gives unconditional signature entries per policy issued, discount those entries directly. Do not multiply by survival probabilities again.
Omitting survival weights. If a profit is conditional on being in force at a duration’s start, convert it to an unconditional entry before discounting.
Using the wrong exponent. Under end-of-duration timing, duration $t$ is discounted by $v^t$, not $v^{t-1}$.
Mixing units. If entries are stated in thousands of dollars, the sum is also in thousands until it is converted to dollars.
Summing a whole-life series without checking convergence. For a geometric series, the absolute value of the common ratio must be less than 1.

Key takeaway: NPV is the sum of the discounted unconditional profit signature. Establish the timing and units, convert conditional profits to unconditional entries when necessary, discount duration $t$ by $v^t$ under year-end emergence, and then add.

Check Your Understanding

For each question, use the stated timing convention and report the final NPV in the requested units.

A hypothetical two-year product has unconditional signature entries of $\$12{,}000$, $-\$5{,}000$, and $\$4{,}000$ at the ends of years 1, 2, and 3, respectively. At $i=6\%$, find the NPV to the nearest cent. (Answer: \$10,229.25)
A contract has two years of possible profit. Conditional expected profit is $\$40$ per policy in force at each year’s start. Annual survival probability is $0.80$, with no lapses, and $i=4\%$. Profits emerge at year-end. Find NPV per policy issued to the nearest cent. (Answer: \$68.05)
A three-year policy has annual death probability $0.20$, no lapses, and conditional profits of $-\$80$, $\$50$, and $\$70$ per policy in force at each year’s start. Profits emerge at year-end and $i=10\%$. Find NPV per policy issued to the nearest cent. (Answer: $-\$6.01$)
A hypothetical whole-life contract has no lapses, annual survival probability $0.95$, and conditional expected profit of $\$18$ per policy in force at the start of each year, including that year’s contingencies. Profits emerge at year-end and $i=7\%$. Find NPV per policy issued. (Answer: \$150.00)
A two-entry unconditional signature is $(-\$20{,}000,\$15{,}000)$, with entries emerging at the ends of years 1 and 2. At $i=5\%$, find NPV to the nearest cent. (Answer: $-\$5,442.18$)

What You'll Learn