Workshop: Constructing a Signature from a Profit Vector

From Annual Profits to a Profit Signature

A profit vector gives the profit at each policy duration, but those profits do not necessarily apply to every policy originally issued. Some policies may lapse or otherwise leave the group before later durations. A profit signature accounts for this persistency by expressing each duration’s expected profit per policy originally issued. Discounting those expected duration profits then gives the net present value at issue.

This distinction matters when a problem supplies profits per policy in force at the start of each year. The year-one profit applies to the whole issued group, while later profits apply only to policies that persist. The signature combines these two ingredients—per-policy profit and probability in force—before discounting.

Definition: Let $\pi_t$ be the profit in policy year $t$ per policy in force at the start of that year. Let $c_t$ be the probability that a policy issued at time 0 is in force at the start of year $t$, with $c_1=1$. The duration-$t$ profit signature component is $\phi_t=c_t\pi_t$. If each year’s profit is paid at the end of that year and the effective annual discount rate is $i$, the NPV per policy issued is $\sum_{t=1}^{n}v^t\phi_t$, where $v=1/(1+i)$.

The signature is therefore not necessarily the same as the supplied profit vector. If the vector already gives expected profits per policy originally issued, it should not be multiplied by persistency again. Always establish the meaning of the input before applying a weighting.

Constructing the Persistency Weights

If $r_t$ is the conditional probability of persisting from the start of year $t$ to the start of year $t+1$, then the probability of being in force at the start of year $t$ is the product of the earlier annual persistency probabilities. In particular, $c_1=1$ and $c_t=\prod_{j=1}^{t-1}r_j$ for $t\geq2$. The rate for year $t$ does not reduce the number in force at the start of year $t$; it affects persistency into year $t+1$.

c_1=1,\qquad c_t=\prod_{j=1}^{t-1}r_j\quad (t\geq2),\qquad \phi_t=c_t\pi_t,\qquad \operatorname{NPV}_0=\sum_{t=1}^{n}v^t\phi_t.

The formula assumes that $\pi_t$ is the applicable profit for a policy in force at the start of year $t$, and that the profit experience represented by $\pi_t$ does not vary with the policy’s prior persistency history. If profits are already weighted by survival, or if different persisting groups have different conditional profits, the simple product $\phi_t=c_t\pi_t$ must be adjusted to match the problem’s definitions.

The timing assumption also matters. Here, duration-$t$ profit is received at the end of year $t$, so it is discounted by $v^t$. If a problem explicitly places a profit at another time, use that stated timing rather than automatically using $v^t$.

Worked Construction and NPV Calculations

Worked Example: A Three-Year Profit Vector

A hypothetical contract has year-end profits per policy in force at the start of each year of $-\$100$, $\$150$, and $\$200$. The conditional probabilities of persisting from year 1 to year 2 and from year 2 to year 3 are 0.80 and 0.75, respectively. Profits occur at each year-end, and the effective annual interest rate is 10%. Construct the profit signature and calculate NPV per policy issued.

Method. First calculate the probability in force at the start of each year. Then multiply each conditional per-policy profit by that probability. Finally, discount the resulting signature at the year-end timing.

At the start of year 1, every issued policy is in force, so $c_1=1$. At the start of year 2, $c_2=0.80$. At the start of year 3, both persistency conditions must hold, so $c_3=0.80(0.75)=0.60$.

The signature components are $\phi_1=1(-\$100)=-\$100$, $\phi_2=0.80(\$150)=\$120$, and $\phi_3=0.60(\$200)=\$120$. Thus the profit signature is $(-\$100,\ \$120,\ \$120)$.

With $v=1/1.10$, the NPV is calculated from the unrounded present values:

\operatorname{NPV}_0 =\frac{-100}{1.10}+\frac{120}{1.10^2}+\frac{120}{1.10^3} =-90.9090909+99.1735537+90.1577761 =98.4222389.

The NPV is $\$98.42$ per policy issued, rounded to the nearest cent. The check is that the year-two and year-three signature components are each smaller than their corresponding conditional profits because only a fraction of the original policies are in force by those durations. The common mistake avoided here is using 0.75 as the year-three in-force probability rather than multiplying the two successive persistency probabilities.

Worked Example: Using Annual Persistency Rates Across Four Durations

A hypothetical four-year product has profits per policy in force at the start of each year of $\$100$, $-\$50$, $\$80$, and $\$40$. The conditional annual persistency probabilities are 0.90, 0.80, and 0.75, in that order. Each profit is paid at year-end, and the effective annual interest rate is 10%. Find the signature and NPV per policy issued.

Method. Convert the conditional rates to cumulative probabilities in force at the beginning of each duration. Apply those probabilities to the profit vector, then discount the signature.

The in-force probabilities are $c_1=1$, $c_2=0.90$, $c_3=0.90(0.80)=0.72$, and $c_4=0.90(0.80)(0.75)=0.54$. The corresponding signature components are:

\phi_1=1(100)=100,\quad \phi_2=0.90(-50)=-45,\quad \phi_3=0.72(80)=57.6,\quad \phi_4=0.54(40)=21.6.

Thus the signature is $(\$100,\ -\$45,\ \$57.60,\ \$21.60)$. The present values, computed without rounding the components before summing, are:

\begin{aligned} \operatorname{NPV}_0 &=\frac{100}{1.10}+\frac{-45}{1.10^2} +\frac{57.6}{1.10^3}+\frac{21.6}{1.10^4}\\ &=90.9090909-37.1900826+43.2746056+14.7530906\\ &=111.7467045. \end{aligned}

The NPV is $\$111.75$ per policy issued, rounded to the nearest cent. The negative year-two component remains negative after persistency weighting; persistency changes its magnitude, not its sign. As a check, the four discounted contributions sum to a positive NPV even though the year-two contribution is a loss.

Worked Example: Reconciling a Signature with an Issued Block

A hypothetical insurer issues 1,000 identical policies. The profit vector per policy in force at the start of each year is $-\$250$, $\$110$, and $\$160$. Conditional annual persistency is 0.90 in each of the first two years. Profits are received at year-end, and the effective annual interest rate is 5%. Calculate the signature per policy issued and the NPV of the block.

Method. Weight the per-policy profits by the start-of-year in-force probabilities, discount the signature to issue, and multiply the resulting per-policy NPV by the number of policies issued.

The in-force probabilities are $c_1=1$, $c_2=0.90$, and $c_3=0.90^2=0.81$. Therefore the signature is $(-\$250,\ \$99,\ \$129.60)$, since $0.90(\$110)=\$99$ and $0.81(\$160)=\$129.60$.

Using $v=1/1.05$, the unrounded NPV per policy issued is:

\begin{aligned} \operatorname{NPV}_0 &=\frac{-250}{1.05}+\frac{99}{1.05^2}+\frac{129.6}{1.05^3}\\ &=-238.0952381+89.7959184+111.9533520\\ &=-36.3459677. \end{aligned}

The block NPV is $1{,}000(-36.3459677)=-36{,}345.9677$, or $-\$36{,}345.97$ to the nearest cent. Multiplying by 1,000 at the end is equivalent to multiplying every signature component by 1,000 before discounting, by linearity. The negative result is consistent with the size of the initial year-one loss relative to the later, persistency-weighted profits.

Why Persistency Weighting Works

The signature formula follows from the expected contribution of a duration. Let $I_t$ equal 1 if a policy issued at time 0 is in force at the start of year $t$, and 0 otherwise. If the year-$t$ profit is $\pi_t$ whenever the policy is in force, its contribution is $I_t\pi_t$. Since $E[I_t]=P(I_t=1)=c_t$, the expected contribution per issued policy is $E[I_t\pi_t]=c_t\pi_t$. This is exactly the signature component $\phi_t$.

Discounting and summing those expected contributions gives the expected present value of the policy’s profit. The sum is valid whether or not persistency events across years are independent: the relevant probability $c_t$ is the probability of being in force at the start of year $t$, obtained from the specified conditional rates or supplied directly.

Interpret the profit vector.
Confirm whether each entry is conditional on being in force or is already expressed per policy originally issued.

Find the in-force probabilities.
Use 1 for the first year; multiply successive conditional persistency probabilities for later start-of-year probabilities.

Construct the signature.
For each duration, multiply the conditional profit by the probability in force at the relevant time.

Discount and reconcile.
Use the stated profit timing, sum unrounded present values, and scale by the policy count only if a block NPV is required.

Common Errors and Checks

A frequent error is treating annual conditional persistency rates as if they were cumulative probabilities in force. In the four-year example, incorrectly using 0.90, 0.80, and 0.75 directly for years two through four would produce the incorrect signature $(100,-45,64,30)$. Its NPV at 10% would be approximately $\$122.29$, rather than the correct $\$111.75$. The year-three probability should include persistence through both earlier transitions, and the year-four probability should include all three.

A second error is discounting the year-one profit by $v^0=1$. Under the end-of-year convention used here, year-one profit is received one year after issue and is discounted by $v^1$. Discount by $v^0$ only if the problem places that profit at time 0. A third error is rounding each present value before adding them. Keep full precision through the sum and round the final requested amount only.

Input or timing statement	Correct treatment
Conditional persistency rates by year	Multiply successive rates to get the probability in force at each later duration.
Cumulative probability in force supplied directly	Use that probability as the duration weight; do not multiply it again.
Profit per policy in force at start of year $t$	Weight by the probability in force at the start of year $t$.
Profit received at end of year $t$	Discount by $v^t$, where $v=1/(1+i)$.

Key takeaway: A profit signature converts conditional annual profits into expected profits per policy issued. Build the start-of-duration persistency probabilities carefully, apply each weight once, and discount according to the stated cash-flow timing.

Check Your Understanding

For each problem, construct the signature before calculating NPV. Unless stated otherwise, profits are per policy in force at the start of the year and are paid at year-end.

A two-year contract has profits of $\$50$ and $\$80$ per policy in force. Persistency from year 1 to year 2 is 0.80, and the effective annual interest rate is 0%. Find the signature and NPV per policy issued. (Answer: signature $(\$50,\ \$64)$; NPV $\$114.00$.)
A three-year contract has profits of $\$100$, $\$40$, and $-\$20$. Conditional persistency into years 2 and 3 is 0.90 and 0.80, respectively. At an effective annual interest rate of 10%, find the signature and NPV per policy issued. (Answer: signature $(\$100,\ \$36,\ -\$14.40)$; NPV $\$109.84$.)
A three-year contract has profits of $\$120$, $\$50$, and $\$30$. The probabilities in force at the starts of years 1, 2, and 3 are supplied directly as 1.00, 0.96, and 0.88. At an effective annual interest rate of 5%, find the signature and NPV per policy issued. (Answer: signature $(\$120,\ \$48,\ \$26.40)$; NPV $\$180.63$.)
A block of 250 policies has per-policy profits of $-\$40$, $\$70$, and $\$90$. Conditional persistency into years 2 and 3 is 0.80 and 0.75. At an effective annual interest rate of 5%, find the signature per policy issued and the block NPV. (Answer: signature $(-\$40,\ \$56,\ \$54)$; block NPV $\$14{,}836.41$.)

What You'll Learn