Duration-Specific Reserves Under Changing Benefits

Why Reserves Must Be Calculated by Duration

A participating policy’s benefits need not be the same in every policy year. A death benefit may increase as bonuses vest, a maturity amount may include accumulated bonuses, and premiums or mortality probabilities may vary by duration. Consequently, a single level-benefit calculation can misstate the liability at one or more policy anniversaries.

A duration-specific reserve is the value, at a stated policy duration, of the policy’s remaining projected cash flows, conditional on the policy being in force at that duration. It is calculated from the cash flows and probabilities applying from that point forward—not by multiplying by the probability that the policy originally reached that duration.

Definition: $V_k$ is the prospective reserve at time $k$, immediately before the premium due at that time, conditional on the policy being in force. Premium $P_k$ is received at time $k$. During policy year $k+1$, death benefits are paid at time $k+1$; if the policy survives, any specified survival payment is also paid at that time, and the policy continues. The effective annual interest rate is $i$, and $v=1/(1+i)$.

Use $q_k$ for the probability of death during the year from time $k$ to $k+1$, conditional on being in force at time $k$, and $p_k=1-q_k$ for the probability of survival through that year. Let $B_{k+1}$ be the death benefit payable at the end of that year. Any specified payment made at the end of the year only if the policy survives is denoted $C_{k+1}$. A maturity benefit can instead be set as the boundary reserve at the maturity date. These quantities may all vary by duration.

The One-Year Reserve Recursion

Assume an annual model with deterministic effective annual interest $i$, the stated conditional death and survival probabilities, and the payment timing just defined. Premiums are received at the start of each policy year. There are no expenses, lapses, or other cash flows unless explicitly included. At maturity time $n$, if the maturity amount is $M_n$, set $V_n=M_n$, immediately before that amount is paid.

V_k=-P_k+v\left[q_kB_{k+1}+p_k\left(C_{k+1}+V_{k+1}\right)\right],\qquad k=0,\ldots,n-1.

At the final transition into maturity, take $C_n=0$ when $M_n$ is already represented by $V_n$. This convention prevents counting the maturity benefit twice. If a separate survival payment is made at an earlier anniversary, it belongs in $C_{k+1}$; it is not part of the death benefit.

Begin at the final duration.
Set the boundary reserve equal to the maturity amount due at that time, or to zero if no benefit is due.

List each year’s inputs.
For that duration, identify the premium, conditional death probability, death benefit, and any survival payment.

Work backward one year at a time.
Apply the recursion using that year’s inputs, then retain the resulting reserve as the continuation value for the preceding year.

Why the Recursion Holds

Immediately before the premium at time $k$, the insurer has reserve $V_k$. The premium $P_k$ is then received, so the amount available to accumulate for one year is $V_k+P_k$. At time $k+1$, death occurs with probability $q_k$, producing payment $B_{k+1}$. Survival occurs with probability $p_k$, producing the survival payment $C_{k+1}$ and leaving continuation reserve $V_{k+1}$.

The expected amount required at time $k+1$ is therefore $q_kB_{k+1}+p_k(C_{k+1}+V_{k+1})$. Equating this expected amount to the accumulated starting amount gives:

(V_k+P_k)(1+i)=q_kB_{k+1}+p_k(C_{k+1}+V_{k+1}).

Solving for $V_k$ gives the stated recursion. It is an equivalence calculation under the specified assumptions; it does not apply unchanged if cash-flow timing, interest accumulation, or the set of possible transitions differs.

Worked Calculations with Changing Benefits

Worked Example: Two Years with Different Death Benefits

A hypothetical two-year participating policy has an effective annual interest rate of 5%. Premiums of $40 and $30 are due at times 0 and 1, respectively. The conditional death probabilities are $q_0=0.10$ and $q_1=0.20$. Death benefits are $500 in the first year and $600 in the second year. The maturity benefit at time 2 is $700. There are no separate survival payments. Calculate $V_1$ and $V_0$.

Method. Start with the maturity boundary $V_2=700$, then apply the duration-specific recursion backward. The survival probabilities are $p_0=0.90$ and $p_1=0.80$, and $v=1/1.05$.

At time 1, use the second-year death benefit, probability, and premium:

V_1=-30+\frac{0.20(600)+0.80(700)}{1.05} =-30+\frac{120+560}{1.05} =-30+\frac{680}{1.05} =\frac{12970}{21} \approx 617.6190476.

At time 0, use the first-year death benefit and survival probability, with $V_1$ as the continuation reserve:

V_0=-40+\frac{0.10(500)+0.90(12970/21)}{1.05} =-40+\frac{50+11673/21}{1.05} =-40+\frac{4241/7}{1.05} =\frac{78940}{147} \approx 537.0068027.

Rounded only at the final step to the nearest cent, $V_1=\$617.62$ and $V_0=\$537.01$. The calculation checks the timing: the $30 premium is subtracted in $V_1$, while the time-2 maturity benefit enters through $V_2$. A common mistake is to use the second-year benefit or mortality probability in the first-year calculation merely because the calculation is being done backward.

Worked Example: A Survival Dividend and Increasing Bonus Entitlements

Consider a hypothetical two-year policy valued at an effective annual interest rate of 4%. Premiums of $50 at time 0 and $45 at time 1 are due. The conditional death probabilities are $q_0=0.05$ and $q_1=0.10$. The death benefit is $1,000 in year 1 and $1,200 in year 2, reflecting the specified bonus entitlement at each duration. A $20 cash dividend is paid at time 1 if the policy survives year 1. The maturity benefit at time 2 is $1,300, including the stated maturity entitlement. Find $V_1$ and $V_0$.

Method. Use $p_0=0.95$, $p_1=0.90$, and $v=1/1.04$. Set $V_2=1{,}300$. The time-1 dividend is a survival payment during the first year; it is not part of the year-1 death benefit.

For the second year, the death and survival outcomes have benefits of $1,200 and $1,300, respectively:

V_1=-45+\frac{0.10(1200)+0.90(1300)}{1.04} =-45+\frac{120+1170}{1.04} =-45+\frac{16125}{13} =\frac{15540}{13} \approx 1195.3846154.

For the first year, survival leads to both the $20 cash dividend and continuation reserve $V_1$:

V_0=-50+\frac{0.05(1000)+0.95(20+15540/13)}{1.04} =-50+\frac{50+15010/13}{1.04} =-50+\frac{195750}{169} =\frac{187300}{169} \approx 1108.2840237.

Rounded only at the final step to the nearest cent, $V_1=\$1{,}195.38$ and $V_0=\$1{,}108.28$. The check is that the dividend is weighted by the probability of survival, while the death benefit is weighted by the probability of death. Omitting the dividend understates the first-year survival cash flow; adding it to the death benefit incorrectly makes it payable on death.

Worked Example: Three Durations with Varying Mortality and Vested Bonuses

A hypothetical three-year policy has an effective annual interest rate of 10%. Premiums at times 0, 1, and 2 are $20, $25, and $30. The conditional death probabilities by year are $0.10$, $0.20$, and $0.25$. Death benefits are $100, $150, and $210 in years 1, 2, and 3; these amounts already include the specified vested bonuses at the relevant claim durations. The maturity benefit at time 3 is $240. There are no separate survival payments. Calculate reserves at durations 2, 1, and 0.

Method. The survival probabilities are $0.90$, $0.80$, and $0.75$, and $V_3=240$. Work backward using the inputs for each individual year.

V_2=-30+\frac{0.25(210)+0.75(240)}{1.10} =-30+\frac{52.5+180}{1.10} =-30+\frac{2325}{11} =\frac{1995}{11} \approx 181.3636364.

For duration 1, use the year-2 death benefit of $150 and continuation reserve $V_2$:

V_1=-25+\frac{0.20(150)+0.80(1995/11)}{1.10} =-25+\frac{30+1596/11}{1.10} =-25+\frac{19260}{121} =\frac{16235}{121} \approx 134.1735537.

For duration 0, use the year-1 death benefit of $100 and continuation reserve $V_1$:

V_0=-20+\frac{0.10(100)+0.90(16235/121)}{1.10} =-20+\frac{10+29223/242}{1.10} =-20+\frac{158215}{1331} =\frac{131595}{1331} \approx 98.8692705.

Rounded only at the final step to the nearest cent, the reserves are $V_2=\$181.36$, $V_1=\$134.17$, and $V_0=\$98.87$. As a check, each backward step uses a death benefit for the corresponding year and the survival-weighted reserve one duration later. A common mistake is to use the $100 first-year death benefit when calculating $V_1$; that ignores the increase in the specified entitlement to $150 in year 2.

Interpreting and Checking Duration-Specific Values

A reserve schedule is conditional on the policy being in force at each stated duration. Thus $V_1$ in a single-policy calculation is not multiplied by $p_0$: it is the reserve for a policy that has survived to time 1. If valuing a portfolio, the expected number of policies reaching a duration may be relevant to an aggregate liability, but that exposure adjustment is separate from the per-policy reserve.

For a reliable calculation, organize the inputs by policy year before applying the recursion. In particular, distinguish among:

Death benefits: use the amount payable on death in that specific year, including only bonus entitlements specified for that claim duration.
Survival payments: include payments made only on survival in the survival branch, separately from the continuation reserve.
Premiums: subtract the premium due at the valuation time under the convention used here.
Maturity benefits: use a consistent boundary condition and do not include the same maturity amount again as a separate survival payment.

The results are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The central exam discipline is to keep duration labels attached to every probability, premium, benefit, and bonus amount. A reserve is only as accurate as the schedule used to calculate it.

Check Your Understanding

For each problem, use the timing convention in this tutorial and work backward from the stated terminal reserve.

A one-year policy has $i=5\%$, $q_0=0.20$, a premium of $10 at time 0, a death benefit of $100, and a maturity benefit of $80. Find $V_0$. (Answer: $V_0=\$70.00$)
A two-year policy has $i=0\%$, $q_0=0.10$, $q_1=0.20$, premiums $P_0=\$20$ and $P_1=\$10$, death benefits of $100 and $150, and a maturity benefit of $200. Find $V_1$ and $V_0$. (Answer: $V_1=\$180.00,\;V_0=\$152.00$)
A one-year policy has $i=10\%$, $q_0=0.25$, a premium of $30, a death benefit of $200, a $12 survival dividend at time 1, and a maturity benefit of $300. Find $V_0$. (Answer: $V_0=\$228.18$)
A two-year policy has $i=5\%$, $q_0=0.10$, $q_1=0.20$, premiums $P_0=\$25$ and $P_1=\$20$, death benefits of $400 and $500, and a maturity benefit of $600. There are no separate survival payments. Find both reserves. (Answer: $V_1=\$532.38,\;V_0=\$469.42$)

What You'll Learn