The Policy Account Recurrence

Why a Policy Account Needs a Recurrence

A universal life account changes as premiums are credited, charges are deducted, and interest is credited. A year-by-year recurrence turns those movements into a calculation that can be repeated and audited. The essential discipline is to identify when each cash flow occurs: a charge deducted before interest reduces the amount that earns interest, while a charge deducted after interest does not.

This tutorial uses an annual account model. It does not determine the amounts of premiums, charges, or credited rates; those are inputs. The recurrence tracks how the specified inputs change the account value under a clearly stated timing convention.

Definition: Let $V_t$ be the policy account value immediately before the transactions at the start of policy year $t+1$. Let $N_t$ be the amount credited to the account at that start of year, $C_t$ the charge deducted at that same time, and $i_t$ the effective annual rate credited to the balance over the year. If the policy remains in force and the account is not exhausted, then the account value at the start of the next year is $$ V_{t+1}=(V_t+N_t-C_t)(1+i_t). $$

Here, $N_t$ means the premium amount that actually enters the account, not necessarily the gross amount paid by the policyholder. Charges are shown separately because this recurrence assumes they are deducted from the account. All amounts should use the same currency and time period.

The credited rate $i_t$ is an effective rate for the full year in this annual model. It may vary by policy year. The recurrence assumes the start-of-year transactions occur before interest is credited, and that there are no other account transactions during the year. If a policy has an intervening withdrawal, a midyear charge, or a different crediting schedule, this annual recurrence must be adjusted to reflect that timing.

Deriving the Beginning-of-Year Recurrence

The recurrence follows directly from the account balance in two stages. First, apply the transactions at the start of the year: add the account premium and subtract the charge. Second, apply the annual interest factor to the resulting balance. The amount earning interest is therefore $V_t+N_t-C_t$, not $V_t+N_t$ and not $V_t+N_t-C_t$ after interest has already been credited.

Start with the opening account value.
Immediately before the year’s transactions, the account contains $V_t$.

Apply the start-of-year cash flows.
Credit $N_t$ and deduct $C_t$, leaving $V_t+N_t-C_t$.

Credit interest for the year.
Multiplication by $1+i_t$ gives the next opening value, $V_{t+1}$.

The same logic gives an accumulated expression over several years. For $n$ complete policy years, each start-of-year net amount $N_k-C_k$ earns interest during year $k+1$ and every subsequent year through year $n$. Thus

V_n = V_0\prod_{r=0}^{n-1}(1+i_r) + \sum_{k=0}^{n-1}(N_k-C_k)\prod_{r=k}^{n-1}(1+i_r).

This expression can be justified by repeated substitution. For one year it is exactly $V_1=(V_0+N_0-C_0)(1+i_0)$. If the expression holds through year $n$, the next recurrence multiplies all accumulated amounts by $1+i_n$ and adds the new net amount $(N_n-C_n)(1+i_n)$. That produces the same product pattern with one more year, so the expression holds for each positive integer $n$. It is often easier and less error-prone to calculate with the year-by-year recurrence than to expand the products directly.

Worked Applications

Worked Example: Projecting an Account Over Three Years

A hypothetical policy has an opening account value of $1,000. At the beginning of year 1, $600 is credited to the account and a $100 charge is deducted. At the beginnings of years 2 and 3, the account receives $500 and $0, respectively; the charges are $120 and $137. The annual credited rates are 4%, 5%, and 3%, respectively. Calculate the account value at the beginning of year 4.

Method. Apply the stated beginning-of-year recurrence separately in each year. The net amount earning interest is the opening value plus the account premium less the charge.

Year 1: The balance before interest is $V_0+N_0-C_0=1{,}000+600-100=1{,}500$. Interest is $1{,}500(0.04)=60$, so $V_1=1{,}500+60=\$1{,}560$.

Year 2: The balance before interest is $1{,}560+500-120=1{,}940$. Interest is $1{,}940(0.05)=97$, so $V_2=1{,}940+97=\$2{,}037$.

Year 3: The balance before interest is $2{,}037+0-137=1{,}900$. Interest is $1{,}900(0.03)=57$, so $V_3=1{,}900+57=\$1{,}957$.

The account value at the beginning of year 4 is $1,957.00. The arithmetic check is that each year’s ending value equals its pre-interest balance multiplied by that year’s interest factor. A common error is to apply the 3% rate to the year-2 opening value before deducting the year-3 charge; that would give interest on money that was no longer in the account during year 3.

Worked Example: Comparing Charge Timing

A hypothetical policy starts a year with an account value of $2,500. A $400 account premium is credited at the beginning of the year, the annual credited rate is 8%, and a $100 charge is specified. Calculate the year-end account value if the charge is deducted at the beginning of the year, then calculate it if the same charge is instead deducted at the end of the year.

Method. Keep the cash flows and rate fixed, but use a recurrence that matches each stated transaction time.

Charge at the beginning of the year: The amount earning interest is $2{,}500+400-100=2{,}800$. The interest credited is $2{,}800(0.08)=224$, giving $V_1=2{,}800+224=\$3{,}024$.

Charge at the end of the year: The account earns interest on $2{,}500+400=2{,}900$. The balance after interest is $2{,}900(1.08)=3{,}132$. Deducting the charge at year-end gives $V_1=3{,}132-100=\$3{,}032$.

The end-of-year deduction produces an account value $8 higher. This is exactly the interest on $100 for one year: $100(0.08)=8$. The common mistake is to use the beginning-of-year formula for a charge that is actually deducted after interest, thereby failing to credit the account with the interest earned on that $100. In general, with a year-end charge $D_{t+1}$, the appropriate expression for this timing is $V_{t+1}=(V_t+N_t)(1+i_t)-D_{t+1}$.

Worked Example: Inferring a Charge and Checking a Later Value

A hypothetical policy begins year 1 with $3,000. At the beginning of that year, $500 is credited and $200 is deducted as a charge. The credited rate is 4%, and the recorded value at the beginning of year 2 is $3,432. In year 2, no premium is credited, the annual credited rate is 5%, and the recorded value at the beginning of year 3 is $3,507. Assuming charges are deducted at the beginning of each year, find the year-2 charge and verify both recorded values.

Method. First verify the known year-1 transactions. Then reverse the year-2 interest factor to find the balance that must have earned interest; compare that balance with the year-2 opening value to infer the charge.

Year 1 check: The amount before interest is $3{,}000+500-200=3{,}300$. After 4% interest it is $3{,}300(1.04)=\$3{,}432$, matching the recorded year-2 opening value.

Year 2 inference: The balance before interest must be $3{,}507/1.05=\$3{,}340$. The year begins with $3,432 and no premium, so the charge must satisfy $3{,}432-C_1=3{,}340$. Therefore $C_1=\$92$.

Verification: Applying that charge gives $(3{,}432-92)(1.05)=3{,}340(1.05)=\$3{,}507$, exactly the recorded value. The inferred charge is $92.00. A common mistake is to subtract the charge from the recorded year-end value and call the result the pre-interest balance; the charge is at the beginning of the year, so it must be inferred before applying the interest factor.

Timing, Limits, and Checks

A recurrence is only as reliable as its timing convention. The beginning-of-year model used here assumes that the premium allocation and charge are both applied before interest for that year. If the problem specifies a different time for either transaction, write the sequence of events first and then construct the equation. For example, a charge at year-end is subtracted after the account has earned interest, as in the second worked example.

The recurrence also describes an account that remains able to meet its specified deductions. If $V_t+N_t-C_t$ is negative, the formula would produce a negative balance, which is not automatically a valid policy account value. The actual contract may specify that coverage terminates, that a payment is required, or that a different rule applies. Without such a rule, do not extend the recurrence through account exhaustion as though a negative account were available.

Check the opening-value definition. $V_t$ is measured before the transactions assigned to that year.
Check the amount earning interest. Under the convention here, it is $V_t+N_t-C_t$.
Check the rate period. An effective annual rate applies once to the full annual balance in this model.
Check units and signs. Premiums entering the account increase it; charges deducted from it decrease it.
Check the endpoint. The result $V_{t+1}$ is the value at the beginning of the following policy year, before that year’s transactions.

These checks matter because a small timing error can persist through every later projection year. For example, a charge incorrectly treated as year-end rather than beginning-of-year changes the next opening value; that changed value then affects the base on which the next year’s interest is calculated.

Key takeaway: For a beginning-of-year account premium and charge followed by annual interest, use $V_{t+1}=(V_t+N_t-C_t)(1+i_t)$. State what the account value represents, place each transaction on the timeline, and apply interest only to the balance that is actually present during the interest period.

Check Your Understanding

Use the stated transaction timing in each problem. Amounts are in dollars, and each rate is an effective annual credited rate.

A policy opens a year with $800. At the beginning of the year, $300 is credited and a $50 charge is deducted; the rate is 3%. Find the next opening account value. (Answer: $1,081.50)
A policy opens a year with $2,400. There is no account premium, a $120 charge is deducted at the beginning of the year, and the rate is 4%. Find the next opening value. (Answer: $2,371.20)
A policy opens with $1,000, receives a $250 account premium at the beginning of the year, earns 6%, and has an $80 charge deducted at year-end. Find the year-end value. (Answer: $1,245.00)
A policy opens with $1,500, receives $400, and has a $100 beginning-of-year charge. Its next opening value is $1,872. Find the annual credited rate. (Answer: 4.00%)
A policy opens with $1,000. In year 1 it receives $200, has a $50 beginning-of-year charge, and earns 5%. In year 2 it receives $100, has a $25 beginning-of-year charge, and earns 4%. Find the opening value at the beginning of year 3. (Answer: $1,333.80)

What You'll Learn