Deriving the Reserve Recursion for Fund-Linked Benefits

Why a Reserve Recursion Is Useful

A prospective reserve is the value, at a specified date, of future benefit outgo less future premium income, assessed on a stated valuation basis. For a fund-linked policy, future benefits depend on the projected policy fund. A period-by-period recursion connects that prospective value to the benefit payments and the reserve one period later.

The timing convention matters. A premium credited before investment growth produces a different fund from one credited afterward; a charge deducted before growth has a different effect from one deducted at the end of the period. The derivation below fixes those timings first, then applies the same convention throughout. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $V_t$ be the prospective reserve at time $t$, immediately before the premium due at $t$ is credited to the policy fund and before the charge due at $t$ is deducted. In this tutorial, $P_t$ is the premium credited to the fund, $C_t$ is a charge deducted from the fund, and the reserve is future benefit value less future credited premiums. Charges affect the projected fund and therefore benefits; they are not also counted as separate income in this reserve definition.

State the Timing and Valuation Basis

Consider one-year periods. At time $t$, the policy is in force, the credited premium is $P_t$, and the charge is $C_t$. Both are applied at the start of the period. The remaining fund earns the period’s investment factor $G_t$, and any death benefit is paid at the end of the period. A policy that survives continues to time $t+1$; if that is the maturity date, it instead receives its maturity benefit.

F_{t+1}=G_t(F_t+P_t-C_t)

Here $F_t$ is the fund immediately before the time-$t$ premium and charge. The factor $G_t$ may be fixed or may vary by valuation scenario. Assume a one-year effective valuation interest rate $i$, with $v=1/(1+i)$. The investment factor $G_t$ need not equal $1+i$: the valuation discount rate and the policy fund’s investment experience have distinct roles.

Let $q_t$ be the probability of death during the period and $p_t=1-q_t$ the probability of survival, on the valuation mortality basis. Benefits and reserves may depend on the investment scenario, so expectations must be taken over the relevant joint assumptions. For now, assume there are no expenses, lapses, or other payments; if they are part of the contract valuation, they must be included as appropriately timed cash flows.

Recursion: If death in the period produces benefit $B^D_{t+1}$, while survival produces a continuation reserve $V_{t+1}$, then

V_t=-P_t+v\,\mathbb{E}_t\!\left[ \mathbf{1}_{D}B^D_{t+1} +\mathbf{1}_{S}V_{t+1} \right].

The indicators $\mathbf{1}_{D}$ and $\mathbf{1}_{S}$ identify death and survival during the period. If survival leads directly to maturity, replace $V_{t+1}$ in the survival term with the maturity benefit $B^M_{t+1}$. The charge $C_t$ enters through the fund projection and thus through the benefits and continuation reserve that depend on that fund.

Why the Recursion Holds

The prospective value can be separated according to what happens during the first year. The premium due immediately at time $t$ reduces the net prospective value by $P_t$. Every benefit or continuation value after that point is one year in the future, so its value at $t$ is its expected amount multiplied by $v$. Death and survival are mutually exclusive and exhaustive outcomes. Their discounted values therefore add to give the recursion.

\begin{aligned} V_t &=-P_t+ \mathbb{E}_t\!\left[ v\left(\mathbf{1}_{D}B^D_{t+1} +\mathbf{1}_{S}V_{t+1}\right) \right] \\ (1+i)(V_t+P_t) &=\mathbb{E}_t\!\left[ \mathbf{1}_{D}B^D_{t+1} +\mathbf{1}_{S}V_{t+1} \right]. \end{aligned}

This is a one-period prospective valuation identity, not an assumption that the policy fund itself earns $i$. Fund-linked benefits are evaluated using the fund projection and the specified valuation assumptions; the resulting future amounts are discounted at the valuation rate. The expectation must preserve any dependence between investment outcomes, mortality, benefits, and continuation reserves.

Set the reserve date.
Record whether $V_t$ is before or after the premium and charge. Here it is before both.

Project the fund.
Apply the credited premium and charge in the stated order, then apply the investment factor.

List end-of-period outcomes.
Calculate the death benefit and, where relevant, the survival continuation reserve or maturity benefit.

Take the expected value and discount.
Use the specified joint probabilities for investment and decrement outcomes, then subtract the current credited premium.

Worked Applications

Worked Example: One-Year Benefit with Two Investment Outcomes

A hypothetical policy has a current fund of $200. At time 0, a $40 premium is credited and a $10 charge is deducted; the resulting fund earns a one-year factor $G$. The valuation basis has $i=5\%$, and the probability of death during the year is $q_0=0.10$, independent of investment performance. The death benefit is 1.2 times the year-end fund; a survivor receives a maturity benefit equal to that fund. The investment factor is 0.90 with probability 0.25 and 1.10 with probability 0.75. Find $V_0$.

Method. Project the fund in each investment outcome, average the contingent benefit using the mortality probabilities, discount one year, and subtract the premium due at time 0.

The fund exposed to investment performance is $200+40-10=230$. The expected investment factor is

\mathbb{E}[G]=0.25(0.90)+0.75(1.10)=0.225+0.825=1.05.

Because death and survival are independent of investment, the expected benefit multiplier on the year-end fund is

$$ 0.10(1.2)+0.90(1.0)=0.12+0.90=1.02. $$

Thus the expected year-end benefit is $230(1.05)(1.02)=246.33$. The prospective reserve is

V_0=-40+\frac{246.33}{1.05} =-40+234.60 =\$194.60.

Check. The expected benefit discounted one year is $234.60, and subtracting the $40 premium gives the stated reserve. The charge has reduced the projected fund; it has not also been subtracted as a separate income item. This avoids counting the same charge twice under the stated reserve definition.

Worked Example: Recursing Through Two Policy Years

A hypothetical policy has an initial fund of $100. The effective valuation interest rate is $i=5\%$. The investment factor is a deterministic 1.10 each year, and the probability of death in each year is 0.10. At time 0, the credited premium is $50 and the charge is $5. At time 1, the credited premium is $20 and the charge is $5. Death benefits equal the fund at the end of the year of death; a survivor at the end of year 2 receives the fund as maturity benefit. Find $V_0$.

Method. Project the fund to maturity, value the final-year benefits to obtain $V_1$, then use the year-0 recursion. Death benefits in each year use the fund at that year’s end.

After the time-0 premium and charge, the fund is $100+50-5=145$. After one year of investment growth,

$$ F_1=145(1.10)=159.50. $$

At time 1, after the $20 premium and $5 charge, the fund is $159.50+20-5=174.50$. The year-2 fund is therefore

$$ F_2=174.50(1.10)=191.95. $$

At time 1, both death in year 2 and survival to maturity produce a $191.95 benefit. Since those outcomes exhaust the possibilities, their expected benefit is $191.95$. The reserve immediately before the time-1 premium is

V_1=-20+\frac{191.95}{1.05} =-20+182.8095238 =162.8095238.

At time 0, death in year 1 produces a $159.50 benefit with probability 0.10; survival leads to $V_1$ with probability 0.90. Thus

\begin{aligned} V_0 &=-50+\frac{0.10(159.50)+0.90(162.8095238)}{1.05}\\ &=-50+\frac{15.95+146.5285714}{1.05}\\ &=-50+154.7414966 =\$104.74. \end{aligned}

The reserve is rounded to cents only at the final step. Check. The final-year calculation uses the time-1 premium and charge to determine the maturity fund, while the time-0 calculation uses the time-0 fund to determine the first-year death benefit. This avoids applying a later premium to an earlier death claim.

Worked Example: Joint Investment and Mortality Scenarios

A hypothetical one-year policy has an initial fund of $300. A $30 premium is credited and a $6 charge is deducted at the start of the year. On the valuation basis, the low-return scenario has probability 0.40, investment factor 0.80, and death probability 0.30. The high-return scenario has probability 0.60, investment factor 1.20, and death probability 0.10. These are specified as joint valuation scenarios. Death pays 1.3 times the year-end fund; survival to maturity pays 0.9 times the year-end fund. The valuation rate is $i=4\%$. Find $V_0$.

Method. Calculate the expected benefit separately in each joint scenario, weight by the scenario probability, discount, and subtract the current premium. Do not average mortality and investment assumptions separately when they are dependent.

The fund before investment is $300+30-6=324$. In the low-return scenario, the year-end fund is $324(0.80)=259.20$. The expected benefit conditional on that scenario is

259.20\left[0.30(1.3)+0.70(0.9)\right] =259.20(0.39+0.63) =264.384.

Weighted by the scenario probability, this contributes $0.40(264.384)=105.7536$. In the high-return scenario, the year-end fund is $324(1.20)=388.80$, and the conditional expected benefit is

388.80\left[0.10(1.3)+0.90(0.9)\right] =388.80(0.13+0.81) =365.472.

This contributes $0.60(365.472)=219.2832$. The unconditional expected benefit is $105.7536+219.2832=325.0368$, so

V_0=-30+\frac{325.0368}{1.04} =-30+312.5353846 =\$282.54.

Check. The two weighted scenario contributions sum to the expected benefit before discounting. Averaging return and mortality separately would ignore their stated dependence and give a different result.

Common Errors and Interpretation

The recursion is useful because it replaces a long future projection with a one-period calculation that can be repeated. It also makes the cash-flow timing visible. For fund-linked products, the reserve is sensitive both to contractual outcomes and to how premiums and charges alter the fund from which those outcomes are determined.

Averaging dependent assumptions separately. In the third example, using the average investment factor $0.40(0.80)+0.60(1.20)=1.04$ and average death probability $0.40(0.30)+0.60(0.10)=0.18$ would produce a benefit estimate of $324(1.04)[0.18(1.3)+0.82(0.9)]=327.52512$. That is not the correct $325.0368 expected benefit because the joint scenarios associate different mortality probabilities with different returns.
Applying a charge at the wrong time. Under the convention used here, the charge is deducted before investment growth. If a charge is instead specified at the end of the year, the fund projection must reflect that different timing; it cannot be silently moved across the investment factor.
Mixing reserve dates. The formula $V_t=-P_t+v\,\mathbb{E}_t[\cdots]$ uses a reserve before the time-$t$ premium. A reserve defined after that premium would require a different current-period expression.
Discounting at the fund return by default. The investment factor projects the policy fund. The valuation interest rate discounts the expected future cash flows. They are equal only if the stated basis makes them equal.

Key takeaway: Fix the reserve date and cash-flow timing first. Project the fund with the specified premium, charge, and investment factor; value death, survival, and maturity outcomes on the stated joint basis; then discount and subtract the premium included in the reserve definition.

Check Your Understanding

For each problem, use the stated timing and reserve convention: the reserve is before the current premium, the premium is credited at the start of the period, charges are deducted at the start, and benefits are paid at year-end.

A one-year policy has fund $80, premium $12, charge $2, deterministic investment factor 1.10, death probability 0.05, death benefit 1.25 times fund, and maturity benefit 0.95 times fund. At $i=5\%$, calculate $V_0$. (Answer: $78.99)
A two-year policy has initial fund $60, deterministic investment factor 1.10, annual death probability 0.10, and $i=10\%$. At time 0, premium and charge are $10 and $2; at time 1, they are $8 and $3. Death benefits equal the year-end fund, and survival to year 2 gives the maturity fund. Calculate $V_0$. (Answer: $55.55)
A one-year policy has fund $100, premium $20, charge $5, and $i=5\%$. In a low scenario of probability 0.50, $G=0.90$ and death probability is 0.20; in a high scenario of probability 0.50, $G=1.10$ and death probability is 0.10. Death pays 1.2 times fund and maturity pays 0.8 times fund. Calculate $V_0$. (Answer: $73.97)
A one-year policy has fund $100, premium $20, deterministic investment factor 1.08, and $i=4\%$. The charge of $4 is deducted either before or after investment growth; death and maturity both pay the year-end fund. Calculate the reserve under each charge timing. (Answer: Before growth, $100.46; after growth, $100.77)
A one-year policy has fund $40, premium $10, charge $2, and investment factor 1.05. Death probability is 0.20; death pays 1.5 times fund and maturity pays 0.9 times fund. At $i=5\%$, calculate the reserve. (Answer: $38.96)

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