Prospective Reserves for Account-Based Policies

Why Prospective Reserves Need an Account Projection

A universal life policy can have future premiums, deductions from its account, expenses, and death benefits that depend on the account value. A reserve calculation must put these items on one consistent timeline. It must also distinguish the insurer’s future cash flows from the account-value projection used to determine benefits and charges.

The account value is not automatically an extra payment by the insurer at the valuation date. Instead, it affects future cash flows when a benefit or charge is defined in terms of that value. For example, under a hypothetical account-plus-amount-at-risk death benefit, the account value helps determine the claim. Adding the account value again as a separate claim would count it twice.

Definition: A prospective reserve at a valuation date is the expected present value of future insurer outgo, conditional on the policy being in force at that date, less the expected present value of future insurer income. Outgo can include benefits and expenses; income can include gross premiums and charges deducted from the account.

The examples use annual periods and a specified annual cash-flow convention. Premiums and account deductions occur at the start of a year; the account then earns the credited rate for that year; death claims occur at year-end. Expenses are paid at the start of the year. The credited rate governs account accumulation, while the valuation interest rate discounts insurer cash flows. They need not be equal.

Set Up the Account and the Insurer Cash Flows

Let $A_k$ be the account value at the start of year $k$, conditional on the policy surviving to that time. Let $P_k$ be the gross premium paid then, $L_k$ the premium load deducted from that premium, and $C_k$ other deductions from the account, such as cost-of-insurance and expense charges. Under the stated timing convention, a simple account projection is

A_{k+1}=(A_k+P_k-L_k-C_k)(1+j_k),

where $j_k$ is the credited rate for the year. This is an account projection, not a reserve recursion. It specifies an account value along the survival path; the prospective reserve is then calculated from the insurer’s future cash flows.

A crucial accounting distinction is that the gross premium is income to the insurer. The premium load is part of that premium, not a second receipt. By contrast, a charge deducted from the account is a separate transfer to the insurer, so it is included as income if the model treats it as such. The load still matters because it reduces the amount added to the account.

For an annual mortality model, let $q_k$ be the probability of death during year $k$, conditional on being alive and in force at the start of that year, and let $p_k=1-q_k$. With no lapses in the model, the probability of being in force at the start of year $k$, given in force at time 0, is the product of the earlier $p$ values.

Annual prospective-value formula: Suppose premiums, account charges, and expenses are paid at the start of each year, and death benefits are paid at the end of the year of death. If $E_k$ is the expense at the start of year $k$, $B_{k+1}$ is the projected death benefit at the end of that year, and $v=1/(1+i)$ uses the valuation interest rate $i$, then $$ V_0=\sum_{k=0}^{n-1}v^k\left(\prod_{r=0}^{k-1}p_r\right)(E_k-P_k-C_k) +\sum_{k=0}^{n-1}v^{k+1}\left(\prod_{r=0}^{k-1}p_r\right)q_kB_{k+1}. $$ An empty product, as in the $k=0$ terms, is 1. Include any maturity or surrender payments only if the model specifies them. This formula assumes no lapses and the stated annual timing; a model with different payment timing or termination events needs corresponding terms.

The premium load $L_k$ appears in the account projection because it reduces the premium credited to the account. It does not appear separately in the formula as income: the full gross premium $P_k$ is already counted. This distinction prevents the common error of counting a portion of premium twice.

Why the Prospective Formula Works

Each future cash flow occurs only if the policy is still in force when it is due. A start-of-year payment in year $k$ therefore has probability equal to the chance of surviving the first $k$ years. A death claim in year $k$ has probability equal to that survival probability multiplied by the conditional probability $q_k$ of dying during that year.

Identify each cash flow and its date.
Record expenses and income at the start of the relevant year, and death claims at the end of the year of death.

Weight by the probability of payment.
A start-of-year item is weighted by survival to that year. A death claim is weighted by survival to the year and death during it.

Discount and take the signed sum.
Discount each expected amount to the valuation date. Add outgo and subtract income.

More formally, represent a cash flow by its amount multiplied by an indicator that it occurs. Its expected present value is the discounted amount multiplied by the probability that the indicator is 1. Adding these expected present values is valid by linearity of expectation, whether or not the cash flows are independent. Subtracting the expected value of income from that of outgo gives the prospective reserve.

Worked Valuations

Worked Example: Account-Plus-Amount-at-Risk Benefit

A hypothetical policy is valued at time 0 with account value $A_0=\$10{,}000$. There is one year remaining. At time 0, the gross premium is $\$100$, the premium load is $\$5$, the cost-of-insurance charge is $\$40$, the expense charge is $\$30$, and the insurer’s expense is $\$30$. The account earns 4% during the year. The death benefit at year-end is the account value plus a fixed $\$50{,}000$ amount at risk. The annual probability of death is $q_0=0.02$, and the valuation interest rate is $i=4\%$. There is no payment if the policy survives the year.

Method. Project the account on the survival path, use it in the death benefit, and then calculate the expected present value of the insurer’s outgo less income.

The account at the end of the year is

A_1=(10{,}000+100-5-40-30)(1.04) =10{,}025(1.04)=10{,}426.

Thus the year-end death benefit is $\$10{,}426+\$50{,}000=\$60{,}426$. Its expected present value is

\frac{0.02(60{,}426)}{1.04} =\frac{1{,}208.52}{1.04} =\$1{,}162.03846\ldots.

At time 0, outgo less income is the expense less the gross premium and the two account charges:

30-100-40-30=-\$140.

Therefore,

V_0=1{,}162.03846\ldots-140 =\$1{,}022.03846\ldots \approx \boxed{\$1{,}022.04}.

The calculation is consistent with the cash-flow timeline: the account deductions reduce the projected account, and those same charges are income to the insurer. The $\$5$ load reduces the account but is not added again to income because it is already part of the $\$100$ gross premium. The reserve is rounded to the nearest cent.

Worked Example: Two-Year Reserve With Survival-Contingent Cash Flows

A hypothetical policy has account value $\$5{,}000$ at time 0 and two years remaining. At the start of each year in which the policy is in force, the gross premium is $\$500$, the premium load is $\$50$, and the expense is $\$25$. The cost-of-insurance charge is $\$20$ in year 0 and $\$25$ in year 1; the expense charge is $\$15$ each year. The account earns 5% annually. The death benefit at each year-end is the account value at that year-end plus a fixed $\$20{,}000$ amount at risk. Mortality probabilities are $q_0=0.02$ and $q_1=0.03$. The valuation interest rate is $i=4\%$. There are no lapses or maturity payments.

Method. Project account values along the survival path, then weight each year’s expenses, income, and claim by the probability the policy is in force when the cash flow occurs.

The year-1 account value is

A_1=(5{,}000+500-50-20-15)(1.05) =5{,}415(1.05)=5{,}685.75.

The year-2 account value, conditional on surviving year 0, is

A_2=(5{,}685.75+500-50-25-15)(1.05) =6{,}095.75(1.05)=6{,}400.5375.

The death benefits are $B_1=\$25{,}685.75$ and $B_2=\$26{,}400.5375$. The probabilities of death in years 0 and 1 are, respectively, $0.02$ and $0.98(0.03)=0.0294$. Their expected present values are

\frac{0.02(25{,}685.75)}{1.04} =\frac{513.715}{1.04} =\$493.95673\ldots,

\frac{0.0294(26{,}400.5375)}{1.04^2} =\frac{776.1758025}{1.0816} =\$717.61816\ldots.

At time 0, expense less income is $25-500-20-15=-\$510$. At time 1, the corresponding amount is $25-500-25-15=-\$515$, paid only if the policy survives year 0. Its expected present value is $0.98(-515)/1.04=-\$485.28846\ldots$. Combining these amounts gives

V_0=-510-485.28846\ldots+493.95673\ldots+717.61816\ldots =\$216.28643\ldots \approx \boxed{\$216.29}.

The year-1 premium and charges are not unconditional: they are included only for a policy that survives to time 1. The year-2 account value is used only in the year-2 death benefit, which is also weighted by survival to that year. The reserve is rounded to the nearest cent.

Worked Example: Level Face Death Benefit and Account-Based Charges

A hypothetical policy is valued with one year remaining and account value $\$18{,}000$. It has a level face death benefit of $\$30{,}000$, so the amount at risk at the start of the year is $\$12{,}000$. The gross premium is $\$250$, the premium load is 5% of the premium, the cost-of-insurance rate is $0.002$ per dollar of amount at risk, and the expense charge is $\$12$. The insurer’s expense is $\$20$. The account earns 3%; the death probability is $0.015$; and the valuation interest rate is 5%. There is no survival payment.

Method. For this level-face design, use the fixed face amount as the death benefit. Use the account value to determine the amount at risk and its related charge, not to add an extra amount to the claim.

The cost-of-insurance charge is $0.002(12{,}000)=\$24$, and the premium load is $0.05(250)=\$12.50$. The projected account at year-end, conditional on survival, is

A_1=(18{,}000+250-12.50-24-12)(1.03) =18{,}201.50(1.03)=18{,}747.545.

The death claim is $\$30{,}000$, not $\$30{,}000+A_1$. Its expected present value is $0.015(30{,}000)/1.05=\$428.57143\ldots$. At time 0, expense less income is

20-250-24-12=-\$266.

The prospective reserve is therefore

V_0=428.57143\ldots-266 =\$162.57143\ldots \approx \boxed{\$162.57}.

This example separates two roles of account value: it affects the amount at risk and hence the charge, while the specified level-face benefit remains fixed under the stated design. The premium load is reflected in the account projection but is not counted as income in addition to the gross premium. The reserve is rounded to the nearest cent.

Common Errors and Why the Distinctions Matter

Counting the account twice. If a benefit is already defined as account value plus an amount at risk, do not add the account value again as a separate claim. For a level-face benefit, do not add the account value to the face amount unless the contract’s specified benefit formula calls for it.
Counting a premium load as extra income. The gross premium is the insurer receipt. A load deducted from it changes the account projection; it is not a second receipt on top of the gross premium.
Forgetting to project charges through the account. If a charge is deducted from the account, subtract it when projecting account values. If the model treats it as income to the insurer, include it once in the reserve cash flows as well.
Using the credited rate to discount claims automatically. The credited rate projects the account. The valuation interest rate discounts the insurer’s expected cash flows; use the same rate only when the assumptions specify that they are equal.
Ignoring survival to a future payment date. A premium or expense due at the start of year $k$ is payable only if the policy is in force then. A later death claim requires both survival to the year and death during it.

These distinctions matter because an account-based product can produce a plausible account projection while still yielding an incorrect reserve if cash flows are misclassified, counted twice, or placed at the wrong time. A prospective valuation makes the assumptions visible: the benefit design, charge basis, payment timing, mortality, credited rate, and discount rate all contribute separately.

Key takeaway: Project the account to determine account-dependent benefits and charges, but calculate the reserve from insurer cash flows. Count gross premiums once, include account deductions consistently, weight payments by the chance the policy is in force, and discount at the valuation rate.

Check Your Understanding

For each problem, use the stated annual timing convention: premiums and charges are paid at the start of the year, the account earns the credited rate, and death claims are paid at year-end. Assume no lapses or survival payments unless stated.

A one-year policy has $A_0=\$8{,}000$, gross premium $\$200$, premium load $\$10$, cost-of-insurance charge $\$18$, expense charge $\$7$, and expense $\$25$. Its account earns 4%. The death benefit is the year-end account value plus $\$12{,}000$; $q_0=0.01$, and $i=5\%$. Find the prospective reserve. (Answer: $-\$4.84$)
A one-year policy has $A_0=\$9{,}000$, gross premium $\$300$, premium load $\$15$, cost-of-insurance charge $\$11$, expense charge $\$10$, and expense $\$20$. The account earns 3%. The level-face death benefit is $\$20{,}000$; $q_0=0.02$, and $i=4\%$. Find the prospective reserve. (Answer: $\$83.62$)
A two-year policy has account value $\$1{,}000$, earns 0%, and has no premiums, charges, or expenses. Its death benefit is the account value plus a fixed $\$5{,}000$ amount at risk. The conditional annual death probability is 0.10 in each year, the valuation interest rate is 0%, and there is no survival payment. Find the reserve at time 0. (Answer: $\$1{,}140.00$)
A one-year policy has $A_0=\$12{,}000$, gross premium $\$400$, premium load $\$20$, cost-of-insurance charge $\$30$, expense charge $\$10$, and expense $\$25$. Its account earns 4%. The death benefit is the year-end account value plus a fixed $\$8{,}000$ amount at risk; $q_0=0.03$, and $i=5\%$. Find the prospective reserve. (Answer: $\$180.25$)

What You'll Learn