Worked Term Insurance Profit Vector

Building a One-Year Term Insurance Profit Vector

A profit vector records profit under each possible outcome during a policy year. For a one-year term insurance contract, the central outcomes are death during the year and survival to the end of the year. The profit in each outcome depends not only on the death benefit, but also on when premiums and expenses are paid, how funds earn investment income, and whether a reserve must be held at year-end.

The calculations below use a per-policy basis: the policy is in force at the start of the year, and its premium is received at that time. A death benefit is paid if death occurs during the year. If the policy survives, a closing reserve may be required. All amounts are measured at the end of the year unless a different timing convention is stated.

Definition: A one-year term insurance profit vector is the collection of profits associated with the possible outcomes of a one-year term contract, calculated under stated cash-flow and reserve timing assumptions. In a two-outcome model, write the vector as $(\Pi_D,\Pi_S)$, where $\Pi_D$ is profit if death occurs during the year and $\Pi_S$ is profit if the policy survives to year-end.

Let $V_0$ be the opening reserve per policy, $P$ the premium received at the start of the year, $e_0$ a start-of-year expense, and $c$ the commission rate applied to the premium. Let $i$ be the effective annual investment return, $B$ the death benefit paid at year-end, and $V_1$ the closing reserve required for a surviving policy. Assume the policy is in force at time zero, with death probability $q$ and survival probability $p=1-q$. Assume there are no lapses, taxes, other benefits, or other cash flows during the year.

Under these assumptions, the cash available to invest immediately after the start-of-year cash flows is $V_0+P-e_0-cP$. It earns interest for the full year. The death claim is paid at year-end in the death outcome; the closing reserve is set aside at year-end only in the survival outcome.

$$ F_1=(V_0+P-e_0-cP)(1+i) $$

Here $F_1$ is the year-end fund before deducting the outcome-dependent claim or reserve. Profit is the amount left after those obligations are met. Therefore,

\Pi_D=F_1-B,\qquad \Pi_S=F_1-V_1

The expected profit per policy in force at the start of the year is the probability-weighted average of these conditional profits:

E[\Pi]=q\Pi_D+p\Pi_S =F_1-qB-pV_1

This formula assumes the same starting fund $F_1$ is available in either outcome and that the closing reserve $V_1$ is required only for survivors. If a death claim is paid before year-end, if an additional reserve is required on death, or if premiums are paid at a different time, the outcome profits must be changed to reflect those terms.

Why Expected Profit Has This Form

The expected-profit expression follows directly from the two mutually exclusive outcomes. With probability $q$, profit is $F_1-B$; with probability $p$, it is $F_1-V_1$. Taking the probability-weighted average gives

\begin{aligned} E[\Pi] &=q(F_1-B)+p(F_1-V_1)\\ &=(q+p)F_1-qB-pV_1\\ &=F_1-qB-pV_1, \end{aligned}

because $q+p=1$. Equivalently, the expected year-end obligations are the expected death claim $qB$ plus the expected closing reserve $pV_1$. The reserve is weighted by $p$, not by 1, because under the stated model it is held only when the policy survives.

Set the timing.
Identify when the premium, commission, expenses, claim, and closing reserve are paid or held. Do not calculate investment income until the cash-flow dates are clear.

Find the amount invested.
Add the opening reserve and premium, then subtract cash outflows paid at the start of the year.

Accumulate to the measurement date.
Apply the investment return for the period that each amount is actually invested.

Calculate each outcome profit.
Deduct the death claim in the death outcome and the closing reserve in the survival outcome.

Take the expectation if required.
Weight the conditional profits by their probabilities. Keep the conditional vector visible even when the final question asks only for expected profit.

Worked Profit Vectors

Worked Example: Calculating Death, Survival, and Expected Profit

A hypothetical one-year term policy has opening reserve $V_0=\$500$, start-of-year premium $P=\$1{,}200$, start-of-year expense $e_0=\$40$, and commission of 10% of premium. The annual effective investment return is $i=5\%$. The year-end death benefit is $B=\$100{,}000$, and the closing reserve for a survivor is $V_1=\$600$. The probability of death during the year is $q=0.01$. Calculate the profit vector and expected profit.

Method. First calculate the amount available for investment at the start of the year. The commission is $0.10(\$1{,}200)=\$120$, so the invested amount is

\$500+\$1{,}200-\$40-\$120=\$1{,}540.

Accumulating for one year gives the fund before the claim or closing reserve:

F_1=\$1{,}540(1.05)=\$1{,}617.

If death occurs, the year-end claim is deducted:

\Pi_D=\$1{,}617-\$100{,}000=-\$98{,}383.

If the policy survives, the closing reserve is set aside:

\Pi_S=\$1{,}617-\$600=\$1{,}017.

The survival probability is $p=1-0.01=0.99$. Thus the profit vector, in death-then-survival order, is $(-\$98{,}383,\ \$1{,}017)$, and expected profit is

\begin{aligned} E[\Pi] &=0.01(-\$98{,}383)+0.99(\$1{,}017)\\ &=-\$983.83+\$1{,}006.83\\ &=\$23.00. \end{aligned}

Check. The expected year-end obligations are $0.01(\$100{,}000)+0.99(\$600)=\$1{,}000+\$594=\$1{,}594$. The expected profit is therefore $\$1{,}617-\$1{,}594=\$23$, agreeing with the weighted calculation. The common mistake to avoid is deducting the full $\$600$ reserve in every outcome; it is required only for survivors.

Worked Example: Solving for a Break-Even Premium

A hypothetical one-year term policy has $V_0=\$100$, a start-of-year expense of $\$25$, commission equal to 20% of premium $P$, annual effective investment return $i=4\%$, death benefit $B=\$50{,}000$, and survivor closing reserve $V_1=\$750$. The death probability is $q=0.01$. The claim and reserve are both valued at year-end under the assumptions above. Find the premium that makes expected profit zero.

Method. The amount available for investment at the start of the year is

\$100+P-\$25-0.20P=\$75+0.80P.

At year-end, the fund is $1.04(\$75+0.80P)$. The expected death claim is $0.01(\$50{,}000)=\$500$. The expected closing reserve is $0.99(\$750)=\$742.50$. Thus expected obligations total

\$500+\$742.50=\$1{,}242.50.

Zero expected profit means expected year-end funds equal expected year-end obligations:

\begin{aligned} 1.04(\$75+0.80P)&=\$1{,}242.50,\\ \$75+0.80P&=\frac{\$1{,}242.50}{1.04}=\$1{,}194.711538\ldots,\\ 0.80P&=\$1{,}119.711538\ldots,\\ P&=\frac{\$1{,}119.711538\ldots}{0.80} =\$1{,}399.639423\ldots. \end{aligned}

Rounded to the nearest cent, the break-even premium is \$1,399.64. As a check, at this rounded premium the year-end fund is $1.04(\$75+0.80(\$1{,}399.64))=\$1{,}242.50048$. Subtracting expected obligations of $\$1{,}242.50$ leaves an expected surplus of $\$0.00048$, which rounds to $\$0.00$ to the nearest cent.

Common mistake. Do not forget to divide the premium contribution by $1-c=0.80$: each dollar of premium leaves only 80 cents available for investment after commission. Also, do not subtract $V_1$ as though every policy survives; its expected amount is $0.99V_1$.

Worked Example: Adjusting for a Midyear Death Claim

A hypothetical one-year term policy has $V_0=\$100$, premium $P=\$500$, start-of-year expense $e_0=\$20$, commission of 10% of premium, annual effective investment return $i=4\%$, death benefit $B=\$20{,}000$, survivor closing reserve $V_1=\$300$, and death probability $q=0.02$. The premium, expense, and commission are paid at the start of the year, but the death claim is paid exactly halfway through the year. Measure both outcome profits at year-end. Assume funds remaining after the midyear claim continue earning interest for the second half-year.

Method. The amount invested at the start is

\$100+\$500-\$20-0.10(\$500)=\$530.

The starting fund accumulates for the full year to

\$530(1.04)=\$551.20.

A claim paid halfway through the year is not a year-end cash flow. To compare it with year-end funds, accumulate it for the remaining half-year. Under an annual effective rate of 4%, the half-year accumulation factor is $(1.04)^{1/2}=1.0198039\ldots$. The claim’s year-end equivalent is therefore

\$20{,}000(1.04)^{1/2}=\$20{,}396.0781\ldots.

The death-outcome profit measured at year-end is

\Pi_D=\$551.20-\$20{,}396.0781\ldots=-\$19{,}844.8781\ldots.

The closing reserve is still deducted only in the survival outcome:

\Pi_S=\$551.20-\$300=\$251.20.

With $p=0.98$, expected profit is

\begin{aligned} E[\Pi] &=0.02(-\$19{,}844.8781\ldots)+0.98(\$251.20)\\ &=-\$396.8976\ldots+\$246.176\\ &=-\$150.7216\ldots. \end{aligned}

Expected profit is -\$150.72 to the nearest cent. This is a loss in expectation under the stated assumptions. The common timing error is to subtract the unaccumulated $\$20{,}000$ from year-end funds. That would treat the midyear payment as if it occurred at year-end and overstate year-end profit by $\$20{,}000((1.04)^{1/2}-1)$ in the death outcome.

Interpreting the Vector and Avoiding Common Traps

The two entries in the vector answer different questions from expected profit. $\Pi_D$ is profit conditional on death during the year; $\Pi_S$ is profit conditional on survival. Expected profit combines them using the probabilities appropriate to a policy in force at the start of the year. A large negative death-outcome profit can coexist with positive expected profit if death is sufficiently unlikely and survival profit is sufficiently large. Conversely, positive survival profit does not by itself establish that the contract has positive expected profit.

A useful check is to calculate expected profit in two ways whenever the cash flows permit it: first by weighting $\Pi_D$ and $\Pi_S$, and then by subtracting expected year-end obligations from the common year-end fund. The answers should agree. If they do not, check the survival weight on the closing reserve, the commission deduction, and the accumulation period for each cash flow.

Item	Under the stated base timing	Common error
Premium and commission	Deduct commission from premium cash available at the stated payment date	Invest the gross premium and also charge commission later without a timing adjustment
Investment income	Accumulate each amount for the period it is held	Apply a full year of interest to a midyear cash flow
Death claim	Deduct in the death outcome, at its stated payment time	Deduct it in both outcomes or ignore its payment timing
Closing reserve	Deduct for survivors when only survivors require the stated reserve	Deduct the full reserve unconditionally
Expected profit	Weight each conditional profit by its outcome probability	Report one conditional profit as though it were the expectation

These calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The key discipline is to state a timing convention before calculating, keep outcome-contingent amounts in their correct outcomes, and distinguish conditional profit from expected profit.

Key takeaway: Under start-of-year premium and expense timing, a year-end death claim, and a survivor-only closing reserve, accumulate the net opening funds for the year, subtract the death benefit in the death outcome, subtract the closing reserve in the survival outcome, and probability-weight those two profits only when expected profit is required.

Check Your Understanding

For each problem, use the cash-flow timing stated and report dollar answers to the nearest cent unless the answer is exact to whole dollars.

A one-year policy has $V_0=\$200$, $P=\$900$, a start-of-year expense of $\$30$, commission of 5% of premium, $i=3\%$, $B=\$40{,}000$, $V_1=\$400$, and $q=0.005$. The claim is paid at year-end and the reserve is held only on survival. Find the death profit, survival profit, and expected profit. (Answer: $-\$38{,}944.25$, $\$655.75$, and $\$457.75$, respectively.)
A hypothetical policy has $V_0=\$0$, a start-of-year expense of $\$50$, commission of 10% of premium $P$, $i=5\%$, $B=\$30{,}000$, $V_1=\$250$, and $q=0.02$. Both the death claim and survivor reserve are valued at year-end. Find the premium for zero expected profit. (Answer: $\$949.74$.)
A policy has $V_0=\$100$, $P=\$700$, a start-of-year expense of $\$20$, commission of 10% of premium, $i=4\%$, $B=\$10{,}000$, $V_1=\$150$, and $q=0.01$. The death claim is paid halfway through the year; measure death profit at year-end and let the remaining claim funds earn interest for the second half-year. Find the expected profit. (Answer: $\$487.92$.)
A one-year contract has $V_0=\$300$, $P=\$800$, start-of-year expense $\$25$, commission of 5% of premium, $i=6\%$, $q=0.01$, and survivor closing reserve $V_1=\$500$. The year-end death benefit is $B$. Find the death benefit that makes expected profit zero. (Answer: $\$60{,}210$.)

What You'll Learn