Solve a Term-Insurance Premium-and-Reserve Chain

Why a Term-Insurance Calculation Is a Chain

A term-insurance premium and a reserve depend on the same contract cash flows, but they are valued at different points on the policy timeline. A reliable calculation starts by identifying who is alive at each premium date, when a death benefit is paid, and which future premiums remain at the reserve date. In particular, an annual death probability from a life table is conditional on survival to the start of that year; it is not automatically the probability of death in that year measured from issue.

The calculations below use a hypothetical policy with a level benefit payable at the end of the policy year of death and level premiums payable annually in advance while the insured is alive. This chain of methods is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Contract and timing assumptions: The term is $n$ years. The benefit $B$ is paid at the end of the year of death if death occurs within the term; there is no benefit at maturity. Premiums of amount $P$ are due at times $0,1,\ldots,n-1$, provided the insured is alive at the payment time. Interest is at a constant effective annual rate $i$, with $v=1/(1+i)$. Mortality follows the stated life table. Unless otherwise stated, reserves are measured immediately before the premium due at the specified duration.

From Annual Mortality to Contract-Year Probabilities

Write $q_{x+k}$ for the conditional probability that a life aged $x+k$, alive at that age, dies during the next year, and write $p_{x+k}=1-q_{x+k}$. With life-table numbers, $q_{x+k}=(\ell_{x+k}-\ell_{x+k+1})/\ell_{x+k}$. The probability, measured at issue, of surviving $k$ years and then dying in the following year is ${}_kp_xq_{x+k}$, where ${}_kp_x=\prod_{j=0}^{k-1}p_{x+j}$ and ${}_0p_x=1$.

Thus a death in policy year $k+1$ contributes a benefit present value of $Bv^{k+1}$ on that event. A premium due at time $k$ is paid only if the insured has survived $k$ years, and its expected present value is $Pv^k{}_kp_x$. These event probabilities and payment dates are the essential bookkeeping in the chain.

Net level premium by equivalence: For the contract described above, the expected present value of benefits equals the expected present value of premiums. Therefore,

B\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k} = P\sum_{k=0}^{n-1}v^k{}_kp_x, \qquad P= \frac{B\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k}} {\sum_{k=0}^{n-1}v^k{}_kp_x}.

The denominator is the expected present value of a unit premium stream due in advance. It includes a payment at issue, and it includes each later payment only if the insured is alive then. The numerator is the expected present value of the term death benefit. This equivalence principle gives a net premium: it includes the modeled benefit but not expenses or profit.

Prospective Reserves at a Contract Duration

At duration $k$, condition on the insured being alive at that time. Immediately before the premium due at time $k$, the remaining benefit can occur in policy years $k+1$ through $n$. The premium due at time $k$, as well as later premiums, is still a future cash flow at this instant. For $0\leq k<n$, the prospective reserve is therefore:

{}_kV_x = B\sum_{j=k}^{n-1}v^{j-k+1}{}_{j-k}p_{x+k}q_{x+j} - P\sum_{j=k}^{n-1}v^{j-k}{}_{j-k}p_{x+k}.

The first sum values remaining benefits from the reserve date; the second values premiums still due from that date, conditional on being alive at duration $k$. This convention makes the premium timing explicit. Immediately after the premium at time $k$ has been paid, the reserve is ${}_kV_x+P$, since that premium is no longer a future inflow to the insurer.

Read the contract dates.
Record the benefit payment date and every premium date, including whether the premium at the reserve duration is still due.

Build the survival path.
Use conditional annual $q$-values to calculate survival to each year and the probability of death in each policy year.

Calculate the issue premium.
Equate the expected present values of benefits and premiums, using the issue-age probabilities and the stated payment dates.

Revalue at each requested duration.
Condition on being alive then, value only the remaining contract cash flows, and include a premium due at that instant if the reserve is before premium.

Worked Calculations

Worked Example: Conditional Probabilities and a Two-Year Premium

A hypothetical two-year term policy has benefit $B=\$10{,}000$. The life table gives $\ell_x=2{,}500$, $\ell_{x+1}=2{,}450$, and $\ell_{x+2}=2{,}352$. The effective annual interest rate is $i=0\%$. Premiums are due at issue and at duration 1 if alive. Find the net level premium and the reserve immediately before the second premium.

Method. First calculate the conditional annual mortality rates from the table, then weight the second-year death probability by survival through year 1. The common error to avoid is treating the year-2 conditional death probability as an unconditional issue-age probability.

For year 1, $q_x=(2{,}500-2{,}450)/2{,}500=50/2{,}500=0.02$, so $p_x=0.98$. For year 2, $q_{x+1}=(2{,}450-2{,}352)/2{,}450=98/2{,}450=0.04$. The issue-age probability of death in year 2 is $0.98(0.04)=0.0392$, not $0.04$. With $i=0$, $v=1$.

The benefit expected present value is $\$10{,}000(0.02+0.0392)=\$10{,}000(0.0592)=\$592$. The expected number of premiums, in present-value terms, is $1+0.98=1.98$. Hence $P=\$592/1.98=\$298.989898\ldots$, or $\$298.99$ to the nearest cent.

At duration 1, conditional on survival to that time, the remaining benefit expected value is $\$10{,}000(0.04)=\$400$. The second premium of $\$298.989898\ldots$ is due at that instant, so the before-premium reserve is $\$400-\$298.989898\ldots=\$101.010101\ldots$, or $\$101.01$ to the nearest cent. After that premium is paid, the reserve is $\$400$, since no further premiums remain. The issue values balance, and the reserve calculation uses the conditional year-2 probability at the attained age.

Worked Example: Three-Year Reserves From a Life Table

A hypothetical three-year term policy pays $B=\$5{,}000$ at the end of the year of death. Its life table has $\ell_x=1{,}000$, $\ell_{x+1}=900$, $\ell_{x+2}=720$, and $\ell_{x+3}=540$. The effective annual interest rate is $i=0\%$. Annual premiums are due in advance while the insured is alive. Find $P$ and the reserves immediately before the premiums at durations 1 and 2.

Method. Convert each life-table decrement to a conditional annual death probability; then compute issue-age probabilities and value the cash flows. The trap is using the decrements as though each were a probability on the original 1,000 lives without accounting for prior survival.

The conditional death probabilities are $q_x=(1{,}000-900)/1{,}000=0.10$, $q_{x+1}=(900-720)/900=0.20$, and $q_{x+2}=(720-540)/720=0.25$. Survival to durations 1 and 2 from issue is $0.90$ and $0.90(0.80)=0.72$. Consequently, the issue-age probabilities of death in the three years are $0.10$, $0.90(0.20)=0.18$, and $0.72(0.25)=0.18$.

At zero interest, the benefit expected present value is $\$5{,}000(0.10+0.18+0.18)=\$5{,}000(0.46)=\$2{,}300$. The premium annuity factor is $1+0.90+0.72=2.62$. Thus $P=\$2{,}300/2.62=\$115{,}000/131=\$877.862595\ldots$, or $\$877.86$ to the nearest cent.

At duration 1 before the second premium, the remaining conditional probability of death in the next year is $0.20$, and the probability of surviving that year and dying in the final year is $0.80(0.25)=0.20$. The remaining benefit value is $\$5{,}000(0.20+0.20)=\$2{,}000$. The premium factor from duration 1 is $1+0.80=1.80$, so the reserve is $\$2{,}000-1.80(\$115{,}000/131)=\$55{,}000/131=\$419.847328\ldots$, or $\$419.85$.

At duration 2 before the third premium, only one year of coverage remains. The benefit value is $\$5{,}000(0.25)=\$1{,}250$, and the one due premium has value $P$. The reserve is $\$1{,}250-\$115{,}000/131=\$48{,}750/131=\$372.137404\ldots$, or $\$372.14$. As a check, at duration 1 the reserve plus the premium just received is $\$419.847328\ldots+\$877.862595\ldots=\$1{,}297.709924\ldots$. With zero interest, the expected amount needed at the end of that year is $0.20(\$5{,}000)+0.80(\$372.137405\ldots)=\$1{,}297.709924\ldots$, matching the funds available.

Worked Example: A Positive-Interest Reserve and Its Timing

A hypothetical two-year term policy has benefit $B=\$5{,}000$, conditional annual death probabilities $q_x=0.10$ and $q_{x+1}=0.30$, and effective annual interest $i=25\%$. Premiums are due at times 0 and 1 while alive. Find the net level premium and the duration-1 reserve immediately before the second premium.

Method. Discount each benefit and premium from its own payment date. The common timing error is to discount a year-2 benefit only one year from issue or to omit the duration-1 premium from the before-premium reserve.

Here $v=1/1.25=0.8$, and the probability of death in year 2 from issue is $0.90(0.30)=0.27$. The benefit expected present value is $\$5{,}000[0.10(0.8)+0.27(0.8^2)]=\$5{,}000(0.08+0.1728)=\$1{,}264$. The premium factor is $1+0.90(0.8)=1.72$. Therefore $P=\$1{,}264/1.72=\$31{,}600/43=\$734.883720\ldots$, or $\$734.88$ to the nearest cent.

At duration 1, conditional on being alive, the only remaining benefit has value $\$5{,}000(0.30)(0.8)=\$1{,}200$. The second premium is due now and is not discounted from the reserve date. Thus ${}_1V_x=\$1{,}200-\$31{,}600/43=\$20{,}000/43=\$465.116279\ldots$, or $\$465.12$. After that premium is paid, the reserve is $\$1{,}200$. A recursion check gives $({}_1V_x+P)(1.25)=\$1{,}200(1.25)=\$1{,}500$, equal to the expected year-end claim $0.30(\$5{,}000)=\$1{,}500$; there is no surviving contract reserve after the final year.

Why the Timeline and Probability Basis Matter

The premium formula and prospective reserve are two valuations of the same contract, conditional on different information and taken at different dates. At issue, the expected benefit value is balanced by the premium stream under the equivalence principle. At a later duration, the reserve conditions on survival to that duration and includes only cash flows still ahead. This is why a duration-$k$ reserve cannot generally be found by reusing the issue-age probabilities unchanged.

For example, if $q_{x+1}=0.04$, the conditional probability of death in the second year given survival to age $x+1$ is $0.04$. But the probability of both surviving the first year and dying in the second, measured at issue, is $p_xq_{x+1}$. Confusing those values changes the benefit expected present value and can also distort premium annuity factors. Always label whether a probability is conditional at the start of a policy year or unconditional from issue.

A second frequent error is to leave out the premium due at the reserve date. With a before-premium reserve, that payment is still a future inflow to the insurer and must be subtracted in the prospective valuation. With an after-premium reserve, it has already been paid and must not be counted again. State the reserve timing before writing the formula.

Key check: Verify that benefits are discounted from their payment dates, premiums are included only when due and payable, and survival weights match the date from which each probability is measured. At the end of the term, no term benefit or premium remains, so the terminal reserve is zero.

Check Your Understanding

For each hypothetical contract, identify the appropriate conditional survival weighting and use the stated before-premium reserve convention where a duration is requested.

A two-year policy has $q_x=0.10$, $q_{x+1}=0.25$, $B=\$1{,}000$, and $i=0\%$. Find the net level annual premium. (Answer: $\$171.05$)
A two-year policy has $q_x=0.05$, $q_{x+1}=0.10$, $B=\$20{,}000$, and $i=0\%$. Find the net level annual premium. (Answer: $\$1{,}487.18$)
For the contract in question 2, find the reserve immediately before the second premium, conditional on survival to duration 1. (Answer: $\$512.82$)
A three-year policy has conditional death probabilities $0.10,0.20,0.25$, benefit $\$5{,}000$, and $i=0\%$. Find the net level premium and the reserve immediately before the third premium. (Answer: $P=\$877.86$; ${}_2V_x=\$372.14$)
A two-year policy has conditional death probabilities $0.02$ and $0.04$, benefit $\$10{,}000$, and $i=0\%$. Find the issue-age probability of death in year 2, the net level premium, and the reserve immediately before the second premium. (Answer: $0.0392$; $P=\$298.99$; ${}_1V_x=\$101.01$)

What You'll Learn