Solve an Endowment Chain from Life Table to Gross Premium

Introduction

An endowment policy pays a benefit if the insured dies during a specified term and also pays a benefit if the insured survives to the end of that term. Calculating its premium requires keeping both outcomes in view: death in each policy year and survival through the full term. A complete calculation chain also has to connect mortality-table data to the benefit value, account for expenses, and value future cash flows at the reserve date.

The method here uses an annual discrete model: premiums are payable at the beginning of each policy year while the insured is alive; death benefits are paid at the end of the year of death; and the survival benefit is paid at the end of the term. These timing assumptions are part of the model, not details that can be changed without changing the calculations. This chain is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: An $n$-year endowment insurance with benefit amount $B$ pays $B$ at the end of the year of death if death occurs within the term, or pays $B$ at time $n$ if the insured survives the term. With level annual premiums payable in advance while alive, premiums are due at times $0,1,\ldots,n-1$, provided the insured is alive at the corresponding time.

From a Life Table to the Endowment Value

Suppose the life table gives $\ell_x$, the number of lives at exact age $x$. The one-year death probability at age $x$ and the one-year survival probability are $q_x=(\ell_x-\ell_{x+1})/\ell_x$ and $p_x=\ell_{x+1}/\ell_x=1-q_x$. In later policy years, use the corresponding age-specific values. In particular, ${}_kp_x=\ell_{x+k}/\ell_x$ is the probability of surviving $k$ years from age $x$, and the probability of dying during policy year $k+1$ is ${}_kp_xq_{x+k}$.

Let the effective annual interest rate be $i$, with $v=1/(1+i)$. Under the stated end-of-year benefit timing, the present value per dollar of benefit is the sum of the discounted death-benefit probabilities and the discounted probability of survival to maturity:

\sum_{k=0}^{n-1} v^{k+1}{}_kp_xq_{x+k}+v^n{}_np_x.

The first term for each policy year discounts the payment one year from the start of that year. The last term is separate: the maturity benefit is payable only on survival to time $n$. Since death during the term and survival to maturity are mutually exclusive and cover all outcomes, their probabilities sum to one.

Worked Example: From Life-Table Counts to a Net Premium

A hypothetical three-year endowment issued at age 40 pays $10,000 at the end of the year of death during the term, or $10,000 at time 3 if the insured survives. The life table has $\ell_{40}=1{,}000$, $\ell_{41}=800$, $\ell_{42}=640$, and $\ell_{43}=512$. Assume an effective annual interest rate of $25\%$, so $v=1/1.25=0.8$. Annual premiums are level, payable at times 0, 1, and 2 while alive. Find the benefit present value and the level net premium.

Method and mortality probabilities. First calculate the conditional death probabilities from each successive life-table row:

q_{40}=\frac{1{,}000-800}{1{,}000}=0.2,\qquad q_{41}=\frac{800-640}{800}=0.2,\qquad q_{42}=\frac{640-512}{640}=0.2.

Thus the probabilities of dying in years 1, 2, and 3 are $0.2$, $0.8(0.2)=0.16$, and $0.8^2(0.2)=0.128$, respectively. The probability of surviving three years is $0.8^3=0.512$. These four outcome probabilities add to $0.2+0.16+0.128+0.512=1$, a useful check.

Benefit present value. Discount each death benefit from its payment date and the maturity benefit from time 3:

\begin{aligned} \operatorname{EPV}(\text{benefits}) &=10{,}000\left[ 0.2(0.8)+0.16(0.8^2)+0.128(0.8^3)+0.512(0.8^3) \right]\\ &=10{,}000(0.16+0.1024+0.065536+0.262144)\\ &=10{,}000(0.59008)=\$5{,}900.80. \end{aligned}

Premium present value and net premium. The time-0 premium is certain, the time-1 premium is payable with probability $0.8$, and the time-2 premium with probability $0.64$. Their expected present-value factor is

$$ 1+0.8(0.8)+0.64(0.8^2)=1+0.64+0.4096=2.0496. $$

By the equivalence principle for a net premium, the expected present value of premiums equals that of benefits. Therefore $P^{\mathrm{net}}(2.0496)=5{,}900.80$, giving $P^{\mathrm{net}}=\$2{,}879.00$ to the nearest cent. The unrounded quotient is approximately $2,879.0008. This is a checkable result: multiplying the rounded premium by 2.0496 gives approximately $5,900.80.

The calculation assumes the life-table probabilities apply to the insured, interest is fixed at the stated rate, and all payments follow the stated annual timing. It is not the appropriate formula for benefits paid immediately on death or premiums payable continuously.

Why the Endowment Formula Works

The benefit formula follows by partitioning possible outcomes. A death in policy year $k+1$ has probability ${}_kp_xq_{x+k}$, and its benefit is paid $k+1$ years after issue. Its expected present value is therefore $Bv^{k+1}{}_kp_xq_{x+k}$. Survival to time $n$ has probability ${}_np_x$, and its benefit is discounted by $v^n$. Adding the mutually exclusive outcome values gives the formula.

For premiums payable in advance, the time-$k$ premium is paid only if the insured is alive then. Its expected present value is $P v^k{}_kp_x$. Summing over $k=0,\ldots,n-1$, then equating premium and benefit present values, gives the level net premium:

P^{\mathrm{net}} = \frac{ B\left(\displaystyle\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k} +v^n{}_np_x\right)} {\displaystyle\sum_{k=0}^{n-1}v^k{}_kp_x}.

This is a net premium: it covers the modeled benefits but not expenses. The denominator changes if premiums are payable at the end of each year rather than in advance, or if premiums stop before the benefit term ends. The payment pattern and survival conditions must match the contract being valued.

Adding Expenses to Find the Gross Premium

A gross premium includes expenses as well as benefits. An acquisition expense paid at issue contributes its full amount at time 0. A renewal expense paid at a later premium date contributes only when the policy is in force at that date, and must be discounted from that date. Equating the expected present value of gross premiums with benefits plus expenses gives the gross premium under the equivalence principle.

Worked Example: Including Acquisition and Renewal Expenses

Use the same hypothetical three-year contract: benefit $10,000, $\ell_{40}=1{,}000$, $\ell_{41}=800$, $\ell_{42}=640$, $\ell_{43}=512$, and effective annual interest $25\%$, so $v=0.8$. Premiums are payable at times 0, 1, and 2 while alive. Suppose there is a $300 acquisition expense at time 0 and a $100 renewal expense at each of times 1 and 2 if the insured is alive. Find the level gross premium.

Method. The benefit present value, already computable directly from the stated mortality and interest assumptions, is $5,900.80. The expected present value of the renewal expenses is $100(0.8)(0.8)+100(0.64)(0.8^2)=100(0.64+0.4096)=\$104.96$. Adding the certain acquisition expense gives total expense present value $300+104.96=\$404.96$.

The gross premium present-value factor is still $2.0496$, because the premium dates and survival conditions have not changed. If $P^{\mathrm{gross}}$ is the level gross premium, equivalence gives

P^{\mathrm{gross}}(2.0496)=5{,}900.80+404.96=6{,}305.76, \qquad P^{\mathrm{gross}}=\frac{6{,}305.76}{2.0496} =\$3{,}076.58.

The calculation is checked by multiplying the rounded result by 2.0496, which gives approximately $6,305.76. The method avoids a common expense error: the time-1 and time-2 renewal expenses are conditional on survival and discounted, whereas the issue expense is certain at time 0.

Prospective Gross Reserve and Cash-Flow Timing

A prospective reserve at duration $k$, conditional on the insured being alive then, is the expected present value at time $k$ of future benefits and future expenses minus the expected present value of future premiums. Only cash flows after the reserve valuation point belong in this calculation. Whether a cash flow at the valuation date is included depends on whether the reserve is measured immediately before or immediately after that payment.

Worked Example: A Reserve Immediately After a Premium Date

For the three-year contract, suppose the gross premium is $3,076.58 before rounding, specifically $6{,}305.76/2.0496$. The $100 renewal expense and premium due at time 1 have both just been paid, and the insured is alive. The $10,000 endowment benefit is payable on death during the three-year term or survival to time 3. The mortality probabilities for each remaining year are $q=0.2$, $p=0.8$, and the effective annual interest rate is $25\%$. Find the prospective gross reserve at time 1 immediately after the time-1 premium and expense have been paid.

Future benefits. Conditional on being alive at time 1, death in the next year has probability $0.2$ and is paid at time 2. Death in the second remaining year has probability $0.8(0.2)=0.16$ and is paid at time 3. Survival to time 3 has probability $0.8^2=0.64$ and also leads to payment at time 3. From time 1, the present value of the benefits is

10{,}000\left[0.2(0.8)+0.16(0.8^2)+0.64(0.8^2)\right] =10{,}000(0.16+0.1024+0.4096)=\$6{,}720.

Future premiums and expenses. The time-1 premium and expense are already paid, so neither is a future cash flow. There is one remaining premium and renewal expense, each payable at time 2 if the insured survives from time 1 to time 2. Each has expected present-value factor $0.8(0.8)=0.64$ at time 1. The reserve is therefore

\begin{aligned} {}_1V_{40}^{\mathrm{gross}} &=6{,}720-(3{,}076.5808)(0.64)+(100)(0.64)\\ &=6{,}720-1{,}969.011712+64\\ &=\$4{,}814.99. \end{aligned}

The answer is rounded to the nearest cent. As a timing check, the time-2 premium and expense remain future at the time-1 valuation date; they are discounted one year and weighted by survival. A wrong calculation would subtract the full $3,076.5808 premium and add the full $100 expense at time 1, as if the time-2 cash flows were due immediately. That gives $6{,}720-3{,}076.5808+100=\$3{,}743.42$, which is inconsistent with the stated payment dates and reserve point.

A Reliable Calculation Sequence

For an integrated endowment problem, use the following sequence. It makes the dependencies explicit and helps prevent mortality, interest, expense, and timing assumptions from being mixed together.

State the contract timeline.
Mark the issue date, premium dates, death-benefit dates, maturity date, expense dates, and reserve date. Identify whether the reserve is immediately before or after any payment at that date.

Convert life-table counts to probabilities.
Calculate each relevant $q_x$, $p_x$, and multi-year survival probability using the life-table ages for that policy year. Check that death during the term plus survival to maturity has total probability one.

Value benefits and premiums.
Discount each benefit from its payment date and each premium from its due date, applying the appropriate survival probability to conditional cash flows.

Add expenses, then value the reserve.
Include each expense at its actual date and condition. At the reserve date, include only future benefits, expenses, and premiums, valued from that date conditional on survival to it.

Common Traps

Using the issue-age death probability for every year. If mortality changes by age, use $q_{x+k}$ for policy year $k+1$, not $q_x$ throughout.
Omitting the maturity benefit. An endowment pays on survival to the end of the term as well as on death during the term.
Discounting a benefit from the wrong date. Under the annual model here, a death in policy year $k+1$ is paid at time $k+1$, so its discount factor is $v^{k+1}$.
Using the wrong premium factor. Premiums payable in advance use survival to the premium date; they are not valued as though payable at each year-end.
Mixing reserve timing with payment timing. A premium due at time 2 is future at a time-1 reserve date. Do not describe it as paid at time 1 or subtract it as an immediate cash flow.

Key takeaway: An endowment chain is a sequence of consistent valuations: life-table counts determine outcome probabilities; the payment dates determine discount factors; expenses change the gross premium; and the reserve includes only cash flows that remain future at its stated valuation time.

Check Your Understanding

Use annual discrete timing as stated in each question. Round dollar answers to the nearest cent.

A two-year endowment pays $5,000 on death during the term or survival to maturity. The life table has $\ell_{50}=500$, $\ell_{51}=400$, and $\ell_{52}=300$. Assume $i=25\%$. Find the expected present value of benefits. (Answer: $3,360.00)
For the same two-year contract and assumptions as Question 1, premiums are level and payable at times 0 and 1 while alive. Find the level net premium. (Answer: $2,048.78)
A hypothetical two-year endowment pays $5,000 on death in the term or survival to maturity. Assume $q=0.2$ in each year, $i=25\%$, premiums payable at times 0 and 1 while alive, a $80 acquisition expense at time 0, and a $20 renewal expense at time 1 if alive. Find the gross premium. (Answer: $2,105.37)
For the contract in Question 3, find the prospective gross reserve at time 1 immediately before the second premium and renewal expense are paid, conditional on survival to time 1. (Answer: $1,914.63)

What You'll Learn