Endowment Insurance Retrospective Reserves

Why Endowment Reserves Need Two Benefit Types

An endowment insurance policy combines two possible benefit payments: a death benefit if the insured dies during the contract term, and a maturity benefit if the insured survives to the end. A retrospective reserve must account for death claims already paid before the valuation date. At maturity, it must also reflect the fact that surviving policyholders are about to receive the maturity benefit.

The key is to specify the valuation timing. Throughout this tutorial, assume a hypothetical policy issued at age $x$ with term $n$, effective annual interest rate $i$, and level net annual premium $P$. Premiums are payable at times $0,1,\ldots,n-1$, provided the insured is alive at the payment time. A death benefit $B$ is paid at the end of the policy year of death. A maturity benefit $M$ is paid at time $n$ if the insured survives to time $n$. The reserve at duration $k$ is measured immediately before any premium due at time $k$, and, when $k=n$, immediately before the maturity benefit is paid.

Definition: For this endowment contract, the retrospective reserve at duration $k$ is the accumulated value at time $k$ of past premiums less past death claims, divided by the probability of survival to time $k$. It is the accumulated net amount attributable, on average, to a policyholder who has survived to the valuation date. The maturity benefit at time $n$ is not a past payment when the reserve is measured immediately before it is paid.

Write $v=1/(1+i)$. For each policy year $j$, the probability of death during that year is ${}_{j-1}p_x q_{x+j-1}$, and the probability of being alive at the premium time $j$ is ${}_jp_x$. These probabilities distinguish premiums and claims that apply to the entire original group from the reserve for the subgroup alive at duration $k$.

Net Premium for an Endowment Contract

The net premium is determined by the equivalence principle: at issue, the expected present value of premiums equals the expected present value of benefits. The following expression writes those values directly in terms of survival and death probabilities, without requiring a separate endowment notation.

P\sum_{j=0}^{n-1} v^j\,{}_jp_x = B\sum_{j=1}^{n} v^j\,{}_{j-1}p_x q_{x+j-1} + M v^n\,{}_np_x.

The premium at time $j$ is paid only if the insured is alive then, which gives the premium term ${}_jp_x$. A death claim in year $j$ is paid at the end of that year, so it is discounted for $j$ years. The maturity benefit is paid at time $n$ only for those surviving to that time.

Net premium formula: If premiums are level and payable annually in advance while the insured is alive, divide the expected present value of all benefits by the expected present value of the premium annuity: $$ P= \frac{ B\displaystyle\sum_{j=1}^{n} v^j\,{}_{j-1}p_x q_{x+j-1} +M v^n\,{}_np_x }{ \displaystyle\sum_{j=0}^{n-1} v^j\,{}_jp_x }. $$ This formula assumes the stated annual timing, no expenses, and benefits $B$ and $M$ payable at the ends of the relevant periods. Different payment timing requires different discount factors.

Retrospective Reserve Formula

At time $k$, each premium paid at time $j<k$ has accumulated for $k-j$ years. Its expected contribution per policy issued is weighted by the probability ${}_jp_x$ that it was paid. A death claim paid at time $j\leq k$ has accumulated for $k-j$ years and is weighted by the probability of death during year $j$. Subtracting the expected accumulated claims from the expected accumulated premiums gives the accumulated amount per policy issued. Dividing by ${}_kp_x$ converts it to an amount per policyholder who has survived to time $k$.

Theorem: Under the annual timing and benefit assumptions above, for $0\leq k\leq n$, the retrospective reserve immediately before any premium due at time $k$ is $$ {}_kV_x= \frac{ P\displaystyle\sum_{j=0}^{k-1}{}_jp_x(1+i)^{k-j} - B\displaystyle\sum_{j=1}^{k}{}_{j-1}p_xq_{x+j-1}(1+i)^{k-j} }{ {}_kp_x }. $$ An empty sum is zero, so ${}_0V_x=0$. The expression assumes ${}_kp_x>0$. It includes death claims paid at time $k$, but does not include a maturity benefit paid at time $n$, because the reserve at $n$ is measured immediately before that payment.

A short derivation makes the conditioning explicit. Let the first sum in the numerator be the expected accumulated premiums per policy issued, and the second be the expected accumulated death claims per policy issued. The expected accumulated net amount for the original group is therefore their difference. Only the fraction ${}_kp_x$ of that group survives to time $k$. Dividing the group amount by this fraction gives the average accumulated amount for a survivor, which is the retrospective reserve.

Accumulate each premium.
A premium paid at time $j$ grows by $(1+i)^{k-j}$ to time $k$, and is paid with probability ${}_jp_x$.

Accumulate each death claim.
A claim paid at time $j$ grows by $(1+i)^{k-j}$ to time $k$, and occurs with probability ${}_{j-1}p_xq_{x+j-1}$.

Condition on survival.
Subtract accumulated claims from accumulated premiums, then divide by ${}_kp_x$.

The formula gives ${}_0V_x=0$, because no premiums or claims have accumulated at issue. At $k=n$, it gives the reserve just before the maturity payment. For a net premium calculated by the equivalence principle, that terminal reserve is $M$. Immediately after the maturity benefit is paid, the contract has ended and the reserve is zero. These are different valuation instants, not conflicting results.

Worked Applications

Worked Example: Equal Death and Maturity Benefits

A hypothetical two-year endowment pays $B=M=\$1{,}640$. Assume a constant annual mortality probability $q_{x+j}=0.20$, effective annual interest $i=0.25$, and annual premiums payable in advance while alive. Calculate the net premium and the reserve at durations 1 and 2.

Method. First calculate the survival and death probabilities. Here $p=1-q=0.80$, so survival to years 1 and 2 is $0.80$ and $0.64$. Death in years 1 and 2 has probabilities $0.20$ and $0.80(0.20)=0.16$. The discount factor is $v=1/1.25=0.80$.

The expected present value of benefits is $1{,}640(0.20)(0.80)+1{,}640(0.16)(0.80^2)+1{,}640(0.64)(0.80^2)$. The three terms are $\$262.40$, $\$167.936$, and $\$671.744$, giving $\$1{,}102.08$. The expected present value of premiums per dollar of premium is $1+0.80(0.80)=1.64$. Thus $P=1{,}102.08/1.64=\$672.00$.

At duration 1, the accumulated premium is $\$672$, and expected claims paid by that date are $0.20(\$1{,}640)=\$328$. Divide the difference by the $0.80$ who survive: ${}_1V_x=(672-328)/0.80=\$430.00$.

At duration 2, accumulated premiums per policy issued are $\$672(1.25^2)+0.80(\$672)(1.25)=\$1{,}050+\$672=\$1{,}722$. Accumulated claims are $0.20(\$1{,}640)(1.25)+0.16(\$1{,}640)=\$410+\$262.40=\$672.40$. Therefore, ${}_2V_x=(1{,}722-672.40)/0.64=\$1{,}640.00$.

Check. The terminal reserve immediately before the maturity benefit equals $M=\$1{,}640$. The common mistake avoided here is omitting the year-2 death claims from the history at duration 2.

Worked Example: Different Death and Maturity Benefits

A hypothetical two-year contract pays $\$1{,}000$ on death during the term and $\$1{,}500$ on survival to maturity. Assume $q_{x+j}=0.20$, $i=0.25$, and annual premiums in advance while alive. Find the net premium and the duration-1 reserve.

Method. The survival probabilities are ${}_1p_x=0.80$ and ${}_2p_x=0.64$. Death probabilities in years 1 and 2 are $0.20$ and $0.16$, and $v=0.80$. The expected present value of the death benefits is $\$1{,}000(0.20)(0.80)+\$1{,}000(0.16)(0.80^2)=\$160+\$102.40=\$262.40$. The maturity benefit has expected present value $\$1{,}500(0.64)(0.80^2)=\$614.40$.

The total expected present value of benefits is $\$262.40+\$614.40=\$876.80$. The premium annuity factor is $1+0.80(0.80)=1.64$, so $P=\$876.80/1.64=\$534.634146\ldots$, or $\$534.63$ to the nearest cent.

At duration 1, the accumulated premium is $P$, and expected death claims paid by time 1 are $0.20(\$1{,}000)=\$200$. Conditioning on the $0.80$ who survive gives ${}_1V_x=(534.634146\ldots-200)/0.80=\$418.292683\ldots$, or $\$418.29$ to the nearest cent.

Check. The maturity benefit affects the premium even though no maturity benefit has yet been paid at duration 1. The reserve calculation subtracts only past death claims; it does not subtract the future maturity payment as though it were already paid.

Worked Example: Varying Mortality by Policy Year

A hypothetical three-year endowment pays $\$1{,}000$ on death and $\$2{,}000$ at maturity. The annual death probabilities are $0.10$, $0.10$, and $0.10$, conditional on being alive at the start of each year. Assume $i=0.10$ and annual premiums in advance. Calculate the net premium and the reserve at duration 2.

Method. The survival probabilities at times 0, 1, 2, and 3 are $1$, $0.90$, $0.81$, and $0.729$. The probabilities of death in years 1, 2, and 3 are $0.10$, $0.09$, and $0.081$. Also $v=1/1.10$.

The premium annuity factor is $1+0.90v+0.81v^2=1+9/11+81/121=301/121=2.487603\ldots$. The expected present value of death benefits is $\$1{,}000(0.10v+0.09v^2+0.081v^3)=\$301{,}000/1{,}331=\$226.1450\ldots$. The expected present value of the maturity benefit is $\$2{,}000(0.729)v^3=\$1{,}458{,}000/1{,}331=\$1{,}095.4170\ldots$.

Thus the total benefit present value is $\$1{,}759{,}000/1{,}331$, and $P=(1{,}759{,}000/1{,}331)/(301/121)=\$1{,}759{,}000/3{,}311=\$531.2594\ldots$.

At duration 2, accumulated premiums per policy issued are $P(1.10^2)+0.90P(1.10)=2.20P=\$1{,}168.7708\ldots$. Accumulated death claims are $0.10(\$1{,}000)(1.10)+0.09(\$1{,}000)=\$110+\$90=\$200$. Therefore, ${}_2V_x=(1{,}168.7708\ldots-200)/0.81=\$1{,}196.0133\ldots$, or $\$1{,}196.01$ to the nearest cent.

Check. The denominator is $0.81$, the probability of surviving to the valuation date, not the probability of surviving through the full three-year term. The death probabilities used for the claims are unconditional probabilities from issue.

Timing Traps and Interpretation

The retrospective method is especially sensitive to the instant at which the reserve is measured. At duration $k<n$, the formula above is immediately before the premium due at $k$, so the premium at time $k$ is not among the past premiums. If a problem instead asks for the reserve immediately after that premium, the premium has just been received and the value changes accordingly. State the timing before selecting the terms in the accumulated cash-flow sums.

Do not count future maturity benefits as past cash flows. The maturity payment influences the net premium from issue, but it is not subtracted in the retrospective numerator before payment.
Include a death claim paid at the valuation time. At duration $k$, the claim for death during year $k$ is paid at time $k$, so it belongs in the accumulated-claims sum.
Do not divide by the wrong survival probability. The denominator is ${}_kp_x$, survival to the valuation date. It is not survival to maturity unless $k=n$.
Separate the two maturity instants. Immediately before the maturity benefit, a net endowment reserve is $M$; immediately after payment, the contract has ended and its reserve is zero.

Key takeaway: For an endowment retrospective reserve, accumulate premiums received and death claims paid through the valuation date, then condition on survival to that date. At maturity, the reserve just before the payment equals the maturity benefit under net-premium equivalence.

Check Your Understanding

Use the stated annual timing: level net premiums are payable in advance while alive, death benefits are paid at year-end, and the reserve is measured before any premium due at the valuation time.

A two-year endowment has $B=M=\$800$, annual death probability $0.25$, and $i=0$. Find the net premium and the duration-1 reserve. (Answer: $P=\$457.14$; ${}_1V_x=\$342.86$)
A two-year endowment pays $\$1{,}000$ on death and $\$1{,}500$ at maturity. Let the annual death probability be $0.10$ and $i=0.10$. Find the duration-1 reserve. (Answer: ${}_1V_x=\$603.54$)
A three-year endowment pays $\$1{,}000$ on death and $\$2{,}000$ at maturity. The annual death probability is $0.10$ in each year and $i=0.10$. Find the net premium and duration-2 reserve. (Answer: $P=\$531.26$; ${}_2V_x=\$1{,}196.01$)
For a net-premium endowment with maturity benefit $M=\$2{,}400$, state the reserve immediately before the maturity payment and immediately after it. Explain why the two values differ. (Answer: Before payment, ${}_nV_x=\$2{,}400$; after payment, the reserve is $\$0$, because the contract has ended.)

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