Why a Projection Table Needs a Transaction Order
A universal life policy account projection follows the account from one duration to the next. For each period, the table records premiums and deductions, credits interest according to the stated timing, and reflects any withdrawal. The closing account value for one period becomes the opening account value for the next.
The order matters. A withdrawal made at the end of a year does not reduce the balance that earns interest during that year. A charge deducted before interest does reduce that balance. The policy contract determines the timing; a projection must follow the stated provisions rather than assume one universal order.
For the examples below, assume premiums are received at the beginning of each policy year. The premium load and other stated charges are deducted before interest is credited. The credited rate applies to the remaining balance for the full year. Any stated withdrawal occurs at year-end, after interest. Assume the policy remains in force throughout each projected period. Charges and amounts are shown in dollars; final account values are rounded to the nearest cent.
Under this timing convention, let \(V_{t-1}\) be the opening account value for year \(t\), \(P_t\) the gross premium, \(\lambda_t\) the premium-load rate, \(E_t\) the expense charge, \(C_t\) the insurance and other charges deducted before interest, \(j_t\) the effective credited rate, and \(W_t\) the year-end withdrawal. The load is a proportion of the gross premium, so the net premium added to the account is \(P_t(1-\lambda_t)\).
This recurrence is a compact way to audit the table, not a substitute for checking each transaction. If the contract instead deducts a charge after interest or processes a withdrawal midyear, the formula must be changed to reflect that timing.
Building the Table One Period at a Time
A useful table layout makes the order visible. Include a column for each material transaction rather than combining all deductions into one unexplained total. A typical row contains:
- duration and opening account value;
- gross premium, load, and net premium;
- expense and insurance charges;
- balance before interest, interest credited, and balance after interest;
- withdrawal and closing account value; and
- the stated contractual benefit in force, where relevant.
A benefit column is not another account transaction. For example, a level death benefit is the stated amount payable under the specified contract provisions if a covered death occurs. The projected account value is the account balance while the policy remains in force. They should not be substituted for one another in the table.
Worked Example: One Year With a Premium Load
A hypothetical policy begins the year with an account value of $2,000 and a level death benefit of $40,000. At the beginning of the year, the owner pays a $500 premium. The premium load is 4% of gross premium, the expense charge is $10, and the insurance charge is $30. The account earns a 3% effective annual credited rate on the balance after these deductions. There is no withdrawal. Find the closing account value and prepare the period’s transaction row.
Method. Apply the load to the gross premium, deduct all stated charges before interest, then credit interest. The level death benefit is recorded separately and does not enter this account-value calculation.
The premium load is \(500(0.04)=\$20.00\). Therefore, the net premium credited to the account is \(500-20=\$480.00\). The balance before interest is:
Interest credited is \(2{,}440(0.03)=\$73.20\). With no withdrawal, the closing account value is \(2{,}440+73.20=\$2{,}513.20\).
| Duration | Opening value | Gross premium | Load | Expense charge | Insurance charge | Interest | Withdrawal | Closing value | Level benefit |
|---|---|---|---|---|---|---|---|---|---|
| 1 | $2,000.00 | $500.00 | $20.00 | $10.00 | $30.00 | $73.20 | $0.00 | $2,513.20 | $40,000.00 |
Check. The account increases by the net premium of $480.00, then decreases by $40.00 in charges, and increases by $73.20 of interest: \(2{,}000+480-40+73.20=2{,}513.20\). A common mistake is to subtract the load twice, once from the premium and again as if it were a separate fixed charge. It is already reflected in the $480 net premium.
Withdrawals and Multiple Durations
A withdrawal’s position in the transaction sequence determines whether it earns interest during the period. In the convention used here, the withdrawal is taken after the interest credit. For a multi-year projection, use the prior year’s closing account value as the next year’s opening value. Do not restart from the original value or apply interest to the same opening amount each year.
Worked Example: A Year-End Withdrawal
A hypothetical policy opens the year with $8,000. The owner pays a $1,000 beginning-of-year premium subject to a 5% load. The expense charge is $25, the insurance charge is $75, and the credited rate is 4%. The owner makes a $400 withdrawal at year-end. The contractual level death benefit in force is $50,000. Calculate the closing account value and show the transaction row.
Method. Find the net premium, deduct the charges, credit interest to that balance, and only then subtract the year-end withdrawal.
The premium load is \(1{,}000(0.05)=\$50.00\), leaving a net premium of $950.00. Before interest, the account is \(8{,}000+950-25-75=\$8{,}850.00\). The interest credit is \(8{,}850(0.04)=\$354.00\), so the balance after interest is $9,204.00. Subtract the $400.00 withdrawal:
| Duration | Opening value | Gross premium | Load | Expense charge | Insurance charge | Interest | Withdrawal | Closing value | Level benefit |
|---|---|---|---|---|---|---|---|---|---|
| 1 | $8,000.00 | $1,000.00 | $50.00 | $25.00 | $75.00 | $354.00 | $400.00 | $8,804.00 | $50,000.00 |
Check. The full calculation is \(8{,}000+950-25-75+354-400=8{,}804\). The common timing error is to subtract the withdrawal before calculating interest. That wrong ordering gives \((8{,}850-400)(1.04)=\$8{,}788.00\), which is $16.00 too low because it omits the $400 withdrawal’s $16.00 interest effect. The stated year-end timing requires the correct value of $8,804.00.
Worked Example: Projecting an Insurance Charge From Opening Net Amount at Risk
A hypothetical policy has an opening account value of $3,000 and a level death benefit of $40,000. The insurance charge is $2 per $1,000 of opening net amount at risk, defined for this example as the death benefit less the opening account value. The owner pays a $600 beginning-of-year premium with a 5% load. The expense charge is $20, the credited rate is 5%, and a $100 withdrawal occurs at year-end. Find the insurance charge and closing account value.
Method. Calculate the charge from the specified opening net amount at risk, then follow the same transaction order as in the preceding examples. This example explicitly defines the charge basis; a contract using another basis would require that basis instead.
Opening net amount at risk is \(40{,}000-3{,}000=\$37{,}000\), or 37 units of $1,000. The insurance charge is \(37(2)=\$74.00\). The premium load is \(600(0.05)=\$30.00\), so the net premium is $570.00. The balance before interest is:
Interest is \(3{,}476(0.05)=\$173.80\). The balance after interest is $3,649.80. After the year-end withdrawal, the closing account value is \(3{,}649.80-100=\$3{,}549.80\).
| Duration | Opening value | Gross premium | Load | Expense charge | Insurance charge | Interest | Withdrawal | Closing value | Level benefit |
|---|---|---|---|---|---|---|---|---|---|
| 1 | $3,000.00 | $600.00 | $30.00 | $20.00 | $74.00 | $173.80 | $100.00 | $3,549.80 | $40,000.00 |
Check. The account reconciliation is \(3{,}000+570-20-74+173.80-100=3{,}549.80\). The $74.00 charge uses the opening account value, as specified. Using the closing value or the balance after premium to calculate the charge would change the stated charge basis and would not answer this problem.
A Two-Year Projection and the Link Between Rows
The opening-value column is the key continuity check in a duration-by-duration table. Each row should satisfy the transaction equation, and the opening value in a row after the first should equal the previous row’s closing value. In the following hypothetical projection, the insurance charge is stipulated as a fixed $30 each year; it is not recalculated from net amount at risk.
Worked Example: Carrying the Account Forward for Two Years
A policy opens at $2,000. In each of two years, a $500 premium is paid at the beginning of the year, the premium load is 4%, the expense charge is $10, and the insurance charge is $30. The credited rate is 3% each year, and a $100 withdrawal occurs at the end of each year. The level death benefit remains $40,000. Project both closing account values.
Year 1. The load is \(500(0.04)=\$20.00\), so the net premium is $480.00. The pre-interest balance is \(2{,}000+480-10-30=\$2{,}440.00\). Interest is \(2{,}440(0.03)=\$73.20\). After interest and the withdrawal, the closing account value is \(2{,}440+73.20-100=\$2{,}413.20\).
Year 2. The opening value is the year 1 closing value, $2,413.20. The net premium is again $480.00. The pre-interest balance is \(2{,}413.20+480-10-30=\$2{,}853.20\). Interest is \(2{,}853.20(0.03)=\$85.596\). Before rounding, the year-end account value is \(2{,}853.20+85.596-100=\$2{,}838.796\), which rounds to $2,838.80.
| Duration | Opening value | Gross premium | Load | Expense charge | Insurance charge | Pre-interest balance | Interest | Withdrawal | Closing value | Level benefit |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | $2,000.00 | $500.00 | $20.00 | $10.00 | $30.00 | $2,440.00 | $73.20 | $100.00 | $2,413.20 | $40,000.00 |
| 2 | $2,413.20 | $500.00 | $20.00 | $10.00 | $30.00 | $2,853.20 | $85.596 | $100.00 | $2,838.80 | $40,000.00 |
Check. The second row opens at the first row’s closing value, and each row includes the $100 withdrawal after interest. The unrounded year 2 calculation is retained until the final cent rounding. A common error is to calculate year 2 interest on the original $2,000 rather than on the year 2 pre-interest balance of $2,853.20.
Why the Recurrence Holds—and What It Does Not Say
The account recurrence follows directly from the transaction sequence. Start with the opening value \(V_{t-1}\). Add the portion of the premium remaining after the load, \(P_t(1-\lambda_t)\), and subtract each charge deducted before interest. The resulting balance is the amount earning the credited rate. A balance \(B\) earning effective rate \(j_t\) becomes \(B+j_tB=B(1+j_t)\). Finally, subtract a withdrawal made at year-end. Thus:
This is an account accumulation calculation under the stated timing assumptions. It does not by itself calculate a probability-weighted value of claims, an insurer reserve, or the cash amount payable on a particular death or surrender. Likewise, a projected benefit column reports the benefit specified by the hypothetical contract; it is not automatically added to or subtracted from the account balance.
Common Projection Errors
- Applying the load to the wrong amount. A 4% load on a $500 gross premium is $20, leaving $480 for the account. Do not apply 4% to the account value.
- Using the wrong transaction order. Interest is earned on the post-charge balance in the examples here, while year-end withdrawals occur after interest. Follow the problem’s stated timing if it differs.
- Failing to carry forward the account. Each period’s opening value after the first must equal the prior period’s closing value.
- Confusing benefit with account value. Show the contractual benefit separately; do not treat it as a premium, deduction, or account balance unless a stated transaction requires that treatment.
- Using an unstated charge basis. If an insurance charge is defined using opening net amount at risk, calculate it from that specified basis. Do not silently substitute a different account value or timing basis.
- Rounding too early. Retain intermediate precision where practical and round the reported money to cents at the final step, unless the contract explicitly specifies transaction-by-transaction rounding.
Check Your Understanding
For each problem, use the transaction timing stated in the question and round the final account value to cents.
- A policy opens at $1,200. A beginning-of-year premium of $300 has a 3% load. Expense and insurance charges are $12 and $18, respectively; the credited rate is 4%, and there is no withdrawal. Find the closing account value. (Answer: $1,519.44)
- A policy opens at $5,000 and receives a beginning-of-year premium of $800 with a 5% load. The expense charge is $20, the insurance charge is $40, and the credited rate is 2%. A $250 withdrawal occurs at year-end. Find the closing account value. (Answer: $5,564.00)
- A policy opens at $4,200 with a $30,000 level death benefit. Its insurance charge is $1.50 per $1,000 of opening net amount at risk, defined as benefit less opening account value. A $500 premium has a 4% load, the expense charge is $15, the credited rate is 3%, and there is no withdrawal. Find the insurance charge and closing account value. (Answer: Insurance charge $38.70; closing account value $4,765.09)
- A policy opens at $2,000. In each of two years it receives a $500 beginning-of-year premium with a 4% load, incurs a $10 expense charge and a $30 insurance charge before interest, earns 3%, and has a $100 year-end withdrawal. Find the closing account value at the end of each year. (Answer: Year 1 $2,413.20; year 2 $2,838.80)
- A policy opens at $10,000 with a $50,000 level death benefit. It receives no premium. The expense charge is $50, and the insurance charge is $2.50 per $1,000 of opening net amount at risk, defined as benefit less opening account value. The credited rate is 4%, and a $300 withdrawal is made at year-end. Find the insurance charge and closing account value. (Answer: Insurance charge $100.00; closing account value $9,944.00)