Premium Patterns and Profit Emergence

Why Premium Timing Changes Profit Emergence

Two hypothetical contracts can have the same term and benefit but different premium patterns. One may charge the same amount every year; another may charge less initially and more later. Comparing only the total scheduled premiums can hide important differences: a later premium may not be collected if the policy has already terminated, and a later profit is discounted for longer. The resulting duration profit signature shows when expected profit emerges, while the net present value (NPV) summarizes its discounted value.

This tutorial uses a deliberately specified annual model so the effect of premium timing is visible. The examples are hypothetical and use discrete annual mortality; they are not descriptions of actual insurance products. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A premium pattern is the sequence of premium amounts scheduled at successive durations. A duration profit signature is the sequence of expected profits by duration, stated on a consistent basis. Here, $\pi_t$ denotes expected profit at the end of year $t$ per policy originally issued. The NPV at issue is $\sum_{t=1}^{n}v^t\pi_t$, where $v=1/(1+i)$ and $i$ is the effective annual discount rate.

Annual Model and Signature Formula

Assume a hypothetical $n$-year term contract. A policy is in force at the start of each year unless death occurred in an earlier year. Conditional on being in force at a year’s start, the probability of death during that year is $q$, and the probability of surviving to its end is $1-q$. Deaths are paid at the end of the year. There are no lapses. A premium $P_t$ is collected at the end of year $t$ only if the policy survives to that date. An expense $e$ is incurred at the end of the year for every policy in force at its start, whether it survives or dies. A death benefit $B$ is paid at year-end if death occurs during that year. There is no maturity payment.

Write $p_{t-1}=(1-q)^{t-1}$ for the probability of being in force at the start of year $t$. Conditional on being in force at that start, the expected profit during year $t$ is $(1-q)P_t-e-qB$: the premium is received only on survival, the expense is always incurred, and the death benefit is paid with probability $q$. Multiplying by the probability of reaching the start of the year gives the signature per issued policy.

\pi_t=p_{t-1}\big[(1-q)P_t-e-qB\big],\qquad t=1,\ldots,n.

Because profit is measured at year-end, its discount factor is $v^t$, not $v^{t-1}$. Thus the NPV is

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

These formulas depend on the stated timing. If premiums were payable at the beginning of the year, or if expenses or benefits were paid at different times, the cashflows would need to be placed at those times before discounting. Likewise, if premiums were payable on death as well as survival, the premium term in the expected profit would change.

How the Premium Pattern Enters NPV

The part of $\pi_t$ that depends on the premium is $p_{t-1}(1-q)P_t$. Since $p_{t-1}(1-q)=(1-q)^t$, the premium’s contribution to NPV is $P_t(1-q)^tv^t$. This gives a direct way to see the effect of moving premium amounts across durations.

\operatorname{NPV} =\sum_{t=1}^{n}P_t(1-q)^tv^t -\sum_{t=1}^{n}p_{t-1}(e+qB)v^t.

The second sum is unchanged when only the premium pattern changes. The first sum weights each scheduled premium by both survival to the premium date and discounting to time zero. Therefore equal total scheduled premiums do not generally imply equal NPV. They imply equal NPV only when their weighted premium sums are equal, assuming the other cashflows and assumptions are unchanged.

This also explains why equal NPV does not mean equal profit emergence. Two patterns can have the same weighted premium sum while producing different profits in individual years. The signature retains this duration-by-duration information; NPV compresses it into one discounted total.

Worked Example: A Level-Premium Signature

A hypothetical three-year term contract charges a level premium of $10 at the end of each year if the policyholder survives to that date. The death benefit is $20, payable at year-end on death, and the end-of-year expense is $1 for every policy in force at the start of the year. Assume a constant annual death probability $q=0.10$, no lapses, and an effective annual interest rate $i=0$. Find the duration signature and NPV per issued policy.

Method. First find the probability of reaching each year’s start. Then calculate conditional expected profit for one policy in force at that start and multiply by that probability. With $i=0$, $v=1$, so the NPV is the sum of signature entries.

The probabilities of being in force at the starts of years 1, 2, and 3 are $1$, $0.90$, and $0.90^2=0.81$. In any year, the conditional expected profit is

$$ (1-0.10)(10)-1-(0.10)(20)=9-1-2=6. $$

Thus the unconditional signature entries are

\pi_1=1(6)=6,\qquad \pi_2=0.90(6)=5.40,\qquad \pi_3=0.81(6)=4.86.

The NPV is $6+5.40+4.86=\$16.26$. As a check, the expected premiums are $0.90(10)+0.81(10)+0.729(10)=\$24.39$. Expected expenses plus death benefits are $(1+0.10(20))(1+0.90+0.81)=3(2.71)=\$8.13$. Their difference is $24.39-8.13=\$16.26$, agreeing with the signature sum.

Common mistake avoided: multiplying the year-2 conditional profit by 1 rather than $0.90$ would report profit per policy in force at the start of year 2, not per policy originally issued.

Worked Example: Same Scheduled Total, Different Emergence

Keep the term, mortality, benefit, expense, and interest assumptions of the previous example, but use premiums of $6, $10, and $14 at the ends of years 1, 2, and 3, respectively, if the policy survives. The scheduled premiums still sum to $30. Calculate the signature and NPV, and compare them with the level-premium contract.

Method. Apply the conditional expected profit formula separately at each duration; the premium amount varies by year. The probability of reaching each year’s start remains $1$, $0.90$, and $0.81$, and the expected expense plus death benefit per policy in force at a year’s start remains $1+0.10(20)=3$.

At duration 1, conditional expected profit is $0.90(6)-3=2.40$, so $\pi_1=1(2.40)=2.40$. At duration 2, it is $0.90(10)-3=6$, so $\pi_2=0.90(6)=5.40$. At duration 3, it is $0.90(14)-3=9.60$, so $\pi_3=0.81(9.60)=7.776$. Therefore,

\operatorname{NPV}=2.40+5.40+7.776=\$15.576.

The non-level pattern produces less profit early and more in the final year than the level pattern: its signature is $(2.40,5.40,7.776)$, compared with $(6,5.40,4.86)$. Its NPV is also lower by $16.26-15.576=\$0.684$, despite the same scheduled total of $30.

Check this difference by valuing expected premiums. For the non-level pattern they are $0.90(6)+0.81(10)+0.729(14)=5.40+8.10+10.206=\$23.706$. The expected outgo remains $8.13, so the NPV is $23.706-8.13=\$15.576$. Later scheduled premiums have lower survival weights, which accounts for the difference from the level pattern.

Common mistake avoided: comparing only the scheduled totals ignores that the year-3 premium is received only from policies surviving three years.

Payment Timing Can Shift Profit Without Changing NPV

The next example isolates a different point. A premium pattern can move profit from one duration to another yet preserve NPV if the weighted premium contributions are equal. This is a statement about value on the specified basis; it does not imply that the patterns have the same cashflow risk or the same year-by-year profitability.

Worked Example: Equal NPV with a Later-Weighted Pattern

Consider a hypothetical three-year contract with premiums due at year-end on survival, death benefit $B=\$10$, expense $e=\$1$ each year, and constant annual death probability $q=0.20$. The effective annual interest rate is $i=0.25$, so $v=0.80$. Compare a level premium pattern $(10,10,10)$ with $(5,17.8125,10)$. Find both signatures and NPVs.

Method. Calculate the signature under each pattern, then discount each year-end profit by $v^t$. Here the probabilities of reaching the starts of the three years are $1$, $0.80$, and $0.64$, and the expected outgo per policy in force at a year’s start is $e+qB=1+2=3$.

For the level pattern, conditional expected profit in each year is $0.80(10)-3=5$. The signature is therefore $(5,0.80(5),0.64(5))=(5,4,3.2)$. Its discounted profits are $5(0.80)=4$, $4(0.80^2)=2.56$, and $3.2(0.80^3)=1.6384$. Hence its NPV is $4+2.56+1.6384=\$8.1984$.

For the second pattern, year 1 conditional profit is $0.80(5)-3=1$, giving $\pi_1=1$. In year 2, it is $0.80(17.8125)-3=14.25-3=11.25$, giving $\pi_2=0.80(11.25)=9$. Year 3 has the same premium as the level pattern, so $\pi_3=0.64(0.80(10)-3)=0.64(5)=3.2$. Its signature is $(1,9,3.2)$, and its discounted profits are $1(0.80)=0.80$, $9(0.64)=5.76$, and $3.2(0.512)=1.6384$. Its NPV is also $0.80+5.76+1.6384=\$8.1984$.

The signatures differ substantially: the second pattern has less year-1 profit and more year-2 profit. To check why the NPVs agree, the premium weights are $(0.80)^2=0.64$, $(0.80)^4=0.4096$, and $(0.80)^6=0.262144$. The level pattern’s weighted premium total is $10(0.64+0.4096+0.262144)=\$13.11744$. The other pattern gives $5(0.64)+17.8125(0.4096)+10(0.262144)=3.2+7.296+2.62144=\$13.11744$. The discounted outgo is identical, so the NPVs must agree.

Common mistake avoided: concluding that equal NPV means equal duration profits. NPV is a discounted total; the signature shows where those profits emerge.

Reading and Comparing Premium Signatures

A useful comparison separates three ideas that are easy to conflate:

Scheduled premium total: the sum of the amounts listed in the contract, without adjusting for survival or timing.
Expected premium value: each scheduled amount weighted by the probability it is payable and discounted to the valuation date.
Profit emergence: the signature entry at each duration, which also reflects expenses and benefits occurring at that duration.

In the annual model here, the expected discounted premium at duration $t$ is $P_t(1-q)^tv^t$. If an amount $x$ is removed from the premium due at duration $s$ and added to the premium due at a later duration $t$, the change in NPV is

x\big[(1-q)^tv^t-(1-q)^sv^s\big].

When $q>0$ or $i>0$ and $t>s$, the later weight is smaller, so this shift reduces NPV. The signature also shows the direct duration effect: reducing the earlier premium lowers profit at duration $s$, while increasing the later premium raises profit at duration $t$, subject to the survival weight at that later date.

This comparison matters in profit testing because a single NPV can conceal an early strain or a concentration of profit late in the contract. Reading the signature alongside NPV gives both the timing profile and the discounted total. The formula should not be applied mechanically if the premium due date, benefit timing, or expense basis differs from the assumptions stated above; those cashflows must first be placed in the correct durations and states.

Key takeaway: Compare premium patterns using the duration signature and the discounted NPV, not the scheduled premium total alone. Weight each premium for the chance it is actually payable and for its payment time. Equal NPV can coexist with quite different profit emergence.

Check Your Understanding

For each problem, use the stated end-of-year timing and report the signature first, followed by NPV.

A two-year hypothetical contract has $q=0.10$, $B=\$12$, $e=\$1$, $i=0$, and premiums of $8 and $12 payable only on survival to each year-end. Find the signature and NPV. (Answer: signature $(\$5.00,\ \$7.74)$; NPV $=\$12.74$.)
A three-year hypothetical contract has $q=0.10$, $B=\$20$, $e=\$1$, $i=0$, and premiums $(8,10,12)$. Find its signature and NPV. (Answer: signature $(\$4.20,\ \$5.40,\ \$6.318)$; NPV $=\$15.918$.)
A three-year hypothetical contract has $q=0.10$, $B=\$15$, $e=\$2$, $i=0$, and level premiums of $10 payable on survival at each year-end. Find its signature and NPV. (Answer: signature $(\$5.50,\ \$4.95,\ \$4.455)$; NPV $=\$14.905$.)
For a three-year contract with $q=0.20$, $B=\$10$, $e=\$1$, and $i=0.25$, premiums are payable only on survival at year-end. The level pattern is $(10,10,10)$. If the alternative pattern is $(5,P_2,10)$, what $P_2$ gives the same NPV, and what is the alternative signature? (Answer: $P_2=\$17.8125$; signature $(\$1.00,\ \$9.00,\ \$3.20)$; both NPVs are $\$8.1984$.)
Under the assumptions in question 2, a candidate calculates year-3 expected premium as $12(0.90^2)$. Explain the error and give the correct year-3 expected premium. (Answer: the premium is payable only if the policy survives through year 3, so use $12(0.90^3)=\$8.748$, not $12(0.90^2)=\$9.72$.)

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