Modified Reserves For Term And Endowment Benefits

Why the Benefit Type Matters

A modified reserve is calculated from the cash flows specified by a valuation basis, which need not be identical to the contract’s original cash flows. A benefit adjustment may change the death benefit, the maturity benefit, or both; a premium adjustment changes the future premium income included in the reserve. To compare two contracts or valuation bases meaningfully, keep the interest and mortality assumptions consistent and identify exactly which cash flows have changed.

The distinction between term assurance and endowment assurance is especially important at the end of the coverage period. Term assurance pays a death benefit if death occurs during the term and has no maturity payment. Endowment assurance pays a death benefit on death during the term and a maturity benefit if the insured survives to the end. Even when their death benefits and modified premiums are identical, their reserves can differ substantially because only the endowment has a survival payment.

Definition: A modified prospective reserve is the present value, at the valuation date, of future benefits specified by the modified valuation basis, less the present value of future premiums specified by that basis. The reserve is conditional on the insured being alive and the policy being in force at the valuation date. In this tutorial, reserves at times $k=0,\ldots,n-1$ are measured immediately after any premium then due; the terminal reserve is measured immediately before any maturity settlement.

Use an annual model with effective annual interest rate $i$, $v=1/(1+i)$, and annual death probabilities $q_{x+k}$, with ${}_{1}p_{x+k}=1-q_{x+k}$. Benefits are paid at the end of the year of death. Premiums are paid at the start of a year while the insured is alive, at times $0,\ldots,n-1$. There are no expenses or other decrements in the examples. The modified death benefit during policy year $k+1$ is $\widetilde B_k$, the modified premium due at time $k$ is $\widetilde P_k$, and the modified maturity benefit is $\widetilde M$. Tildes indicate valuation-basis amounts; they do not imply that the policyholder’s contractual amounts have changed.

Prospective Formulas for Term and Endowment Assurance

For a reserve at time $k$, count each future death benefit only if the insured survives to the start of its year and then dies during that year. Count a future premium only if the insured survives to its payment date. Since the reserve is measured just after any premium at time $k$, premiums at later times are future premium income and are subtracted from the benefit value.

Result: For an $n$-year modified term assurance, the terminal reserve is ${}_nV^{T}=0$, and for $k<n$, $$ {}_kV^{T} = \sum_{j=1}^{n-k} v^j\,{}_{j-1}p_{x+k}\,q_{x+k+j-1}\,\widetilde B_{k+j-1} - \sum_{j=1}^{n-k-1} v^j\,{}_jp_{x+k}\,\widetilde P_{k+j}. $$ For an $n$-year modified endowment assurance with maturity benefit $\widetilde M$, the terminal reserve is ${}_nV^{E}=\widetilde M$, and for $k<n$, $$ {}_kV^{E} = {}_kV^{T} + v^{n-k}\,{}_{n-k}p_{x+k}\,\widetilde M, $$ where ${}_kV^{T}$ uses the same modified death benefits and premiums. The maturity term is the present value of the modified maturity benefit, payable only if the insured survives to time $n$.

The premium sum ends at $n-k-1$, not $n-k$: with coverage ending at time $n$, the specified premiums are due at times $0$ through $n-1$. At time $k$, the remaining premium payment dates are $k+1,\ldots,n-1$. A different premium schedule can be handled by summing only over the dates on which premiums are actually due.

Why the Reserve Recursion Holds

The prospective formulas can be checked one policy year at a time. Let ${}_kV$ be the reserve just after the premium at time $k$, and let ${}_{k+1}V$ be the reserve just after the next premium if the insured survives. Immediately before that next premium, the reserve needed for the remaining liability is ${}_{k+1}V-\widetilde P_{k+1}$: the premium is an inflow that reduces the net liability. During the year, death leads to payment of $\widetilde B_k$, while survival leads to that remaining net liability. Discounting the expected end-of-year amount gives the recursion.

{}_kV = v\left(q_{x+k}\widetilde B_k + {}_{1}p_{x+k}\bigl({}_{k+1}V-\widetilde P_{k+1}\bigr)\right), \qquad k<n-1.

At the last year, there is no premium at time $n$. Thus the terminal condition determines the last step: ${}_{n-1}V^{T}=v q_{x+n-1}\widetilde B_{n-1}$ for term assurance, whereas ${}_{n-1}V^{E}=v(q_{x+n-1}\widetilde B_{n-1}+{}_{1}p_{x+n-1}\widetilde M)$ for endowment assurance. This is exactly where the maturity benefit enters. The recursion is not a formula for a continuous-time model or for benefits paid at a different time within the year; the payment timing and premium timing must match the model.

Specify the valuation cash flows.
Write down modified death benefits, any maturity benefit, and every premium payment date and amount.

Set the terminal reserve.
Use zero for term assurance and the maturity benefit for endowment assurance immediately before settlement.

Work backward.
Apply the annual recursion, subtracting the premium due at the next time on the survival branch.

Compare on a like-for-like basis.
When death benefits, premiums, interest, and mortality are the same, the endowment reserve exceeds the term reserve by the discounted survival-contingent maturity benefit.

Worked Comparisons

Worked Example: A Modified Two-Year Term Assurance

A hypothetical two-year term assurance is valued at an effective annual interest rate of $10\%$. At the start of each covered year, the conditional probability of death during that year is $0.10$; hence the survival probability is $0.90$. The modified death benefit is $\$120$ at the end of either year, and a modified premium of $\$30$ is due at times 0 and 1 if the insured is alive. Calculate the reserves immediately after the premiums at times 1 and 0. Assume there is no reserve floor.

Method. Set the term terminal reserve to zero, then work backward using the recursion. Here $v=1/1.10=0.9090909$. At time 1, no future premium remains, and the only possible payment is the year-2 death benefit:

{}_1V^{T} = v(0.10\times120+0.90\times0) = \frac{12}{1.10} = 10.9090909.

At time 0, the year-1 death benefit is paid with probability $0.10$. On survival to time 1, the $\$30$ premium is received, so the net liability after that premium is ${}_1V^{T}-30$:

{}_0V^{T} = \frac{0.10(120)+0.90(10.9090909-30)}{1.10} = \frac{12-17.1818182}{1.10} = -4.7107438.

Rounded to cents, ${}_1V^{T}=\$10.91$ and ${}_0V^{T}=-\$4.71$. The negative issue-time reserve is consistent with the specified modified premium income exceeding the prospective value of benefits at that time; a floor should not be imposed unless the valuation rules require one. As a check, the future benefit value at time 0 is $12/1.10+0.09(120)/1.10^2=\$19.8347$, and the expected present value of the future premium is $0.90(30)/1.10=\$24.5455$. Their difference is $-\$4.7108$, agreeing up to displayed rounding.

Common mistake avoided: adding the future premium to the reserve recursion. It is premium income, so its value is subtracted on the survival branch.

Worked Example: Add a Maturity Benefit on the Same Basis

Use the same hypothetical mortality, interest rate, two-year term, modified death benefit of $\$120$, and premiums of $\$30$ at times 0 and 1. Now suppose the contract is an endowment assurance with a modified maturity benefit of $\$150$ payable at time 2 if the insured survives. Calculate the reserves at times 1 and 0 and compare them with the term reserves.

Method. The terminal reserve is $\$150$, rather than zero. At time 1, the death benefit is paid with probability $0.10$, and the maturity benefit is payable on survival with probability $0.90$:

{}_1V^{E} = \frac{0.10(120)+0.90(150)}{1.10} = \frac{147}{1.10} = 133.6363636.

At time 0, subtract the premium due at time 1 on the survival branch:

{}_0V^{E} = \frac{0.10(120)+0.90(133.6363636-30)}{1.10} = \frac{105.2727272}{1.10} = 95.7024793.

Rounded to cents, the endowment reserves are ${}_1V^{E}=\$133.64$ and ${}_0V^{E}=\$95.70$. The difference from the term reserve at time 1 is $133.6363636-10.9090909=\$122.7273$. This is the time-1 value of the maturity benefit, $0.90(150)/1.10$. At time 0, the difference is $95.7024793-(-4.7107438)=\$100.4132$, equal to $0.81(150)/1.10^2$, the present value of the maturity payment conditional on survival for both years. These checks isolate the maturity benefit’s contribution without changing any other basis item.

Common mistake avoided: treating the maturity benefit as payable with certainty. It is payable only if the insured survives the full term, so its probability is $0.81$ at time 0, not 1.

Worked Example: Modified Benefits and an Equivalence Premium

A hypothetical two-year endowment assurance has annual conditional death probability $0.10$, annual survival probability $0.90$, and effective annual interest rate $10\%$. Its modified death benefit is $\$80$ in either year, and its modified maturity benefit is $\$120$. Find the level annual modified premium due at times 0 and 1 that makes the expected present value of premiums equal to the expected present value of benefits. Then calculate the reserve immediately after the time-0 premium and check it by recursion.

Method. The probability of death in year 1 is $0.10$, death in year 2 is $0.90(0.10)=0.09$, and survival to maturity is $0.90^2=0.81$. With $v=10/11$, the benefit present value at issue is:

\begin{aligned} \operatorname{EPV}(\text{benefits}) &=80(0.10)v+80(0.09)v^2+120(0.81)v^2\\ &=\frac{1600}{121}+\frac{9720}{121} =\frac{11320}{121} =\$93.5537. \end{aligned}

The expected present value of a level premium $P$, due at times 0 and 1 while alive, is $P(1+0.90v)=P(20/11)$. Therefore:

P=\frac{11320/121}{20/11} =\frac{11320}{220} =\$51.4545.

The premium is $\$51.45$ to the nearest cent. At time 1, after the second premium has been paid, the remaining one-year endowment reserve is:

{}_1V^E = \frac{0.10(80)+0.90(120)}{1.10} = \frac{116}{1.10} = 105.4545455.

At time 0, the time-1 premium is subtracted on survival:

{}_0V^E = \frac{0.10(80)+0.90(105.4545455-51.4545455)}{1.10} = \frac{56.60}{1.10} = \$51.4545.

The result checks: at issue after the first premium, the reserve equals the level premium in this equivalence-premium setup. The computation uses the modified death and maturity benefits throughout; substituting contractual benefits that differ from these valuation amounts would calculate a different reserve.

Common mistake avoided: valuing the premium stream as two certain payments. The time-1 premium is payable only if the insured survives, so its present value is $0.90vP$, not $vP$.

Common Traps and Practical Checks

The comparison is useful because it separates benefit-type effects from other modifications. If term and endowment assurance use the same death benefits, premiums, interest, and mortality assumptions, their reserves differ only by the present value of the endowment’s survival-contingent maturity benefit. If premiums or death benefits also differ, first calculate each reserve with its own complete modified cash-flow stream; do not attribute the whole difference to the maturity payment.

Omitting the maturity payment. For endowment assurance, the terminal reserve is the maturity amount immediately before settlement, not zero.
Using the wrong survival probability. A maturity benefit valued $n-k$ years before the end is weighted by ${}_{n-k}p_{x+k}$.
Adding rather than subtracting future premiums. The prospective reserve is benefits less premiums. On the survival branch of the recursion, the premium reduces the net liability.
Including a premium after coverage ends. Under the schedule used here, there is no premium at time $n$. Include a payment at that date only if the stated contract or valuation basis explicitly requires it.
Mixing bases. A reserve calculated with modified benefits must not combine them with unmodified premiums unless that mixed basis is explicitly intended.

These distinctions are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. For any calculation, state the reserve timing, benefit timing, premium dates, and terminal condition before applying a formula. Those details determine whether the calculation is a term reserve, an endowment reserve, or a different valuation altogether.

Check Your Understanding

Use annual benefits paid at year-end and reserves immediately after any premium due at that time. Unless a question says otherwise, assume no expenses or reserve floor.

A one-year term assurance has $i=5\%$, death probability $0.20$, death benefit $\$100$, and no future premium after the valuation date. Find its reserve at time 0. (Answer: $\$19.05$)
A two-year term assurance has $i=10\%$, conditional death probability $0.10$ each year, death benefit $\$100$, and a $\$20$ premium due at time 1 if alive. Calculate ${}_1V^T$ and ${}_0V^T$. (Answer: ${}_1V^T=\$9.09;\ {}_0V^T=\$0.17$)
A two-year endowment has $i=10\%$, conditional death probability $0.10$ each year, death benefit $\$100$, maturity benefit $\$100$, and a $\$25$ premium due at time 1 if alive. Calculate ${}_1V^E$ and ${}_0V^E$. (Answer: ${}_1V^E=\$90.91;\ {}_0V^E=\$63.02$)
For a one-year contract, $i=5\%$ and the death probability is $0.25$. The modified death benefit is $\$80$, and the modified maturity benefit is $\$120$. With no future premium, calculate the term reserve and the endowment reserve at time 0, and give their difference. (Answer: term $\$19.05$; endowment $\$104.76$; difference $\$85.71$)
A two-year term assurance has $i=5\%$, conditional death probability $0.20$ each year, death benefit $\$100$, and level premiums due at times 0 and 1 while alive. Find the level equivalence premium. (Answer: $\$19.05$)

What You'll Learn