Why Expense Charge Structure Matters
A universal life policy can deduct expenses in different ways. A charge may be a fixed dollar amount each period, a percentage of a stated account-value base, or an amount that changes with policy duration. These structures are not interchangeable: even if two schedules have the same total charges over several years, they can produce different account values because charges are deducted at different times or calculated on different bases.
For a projection, the key is to identify the contract’s stated charge, its frequency, and the value to which it applies. Then apply the specified charge in the stated sequence. The examples below assume that premiums have already been allocated to the account, the expense charge is deducted next, and interest is credited afterward. This is an explicit assumption for these calculations, not a universal ordering rule.
Let \(A_{t-1}\) be the account value at the start of period \(t\), \(C_t\) the premium allocated to the account for that period, and \(E_t\) the expense charge deducted in the period. If the charge is deducted after the allocation and interest is credited afterward, then
where \(i_t\) is the effective interest rate for that period. If the projection uses the same effective annual rate \(i\) every year, set \(i_t=i\). The account-value recurrence changes if the contract specifies a different transaction sequence or crediting period; a projection must follow those terms rather than assume this particular sequence.
Three Expense Charge Structures
A fixed charge is the most direct to project: subtract the stated dollar amount for the period. For an account-value-based charge, first identify its exact base. For example, a rate might apply to the account value after a premium allocation but before the charge. It should not be applied automatically to the account value at the start of the year or to the value after the charge.
A duration-varying schedule can be written as a sequence of charges \(E_1,E_2,\ldots\), or as a formula depending on duration. “Duration” here means the period of the policy specified in the charge schedule. A step-up at renewal, a declining charge, or a scheduled increase after an initial period can all produce duration-varying charges. Do not infer a pattern between stated durations unless the problem gives a rule for doing so.
| Structure | How to determine the charge | Projection point to check |
|---|---|---|
| Fixed per period | Use the stated dollar amount for each period | Confirm the period and whether the amount changes by duration |
| Account-value-based | Multiply the stated rate by the specified base | Use the precise pre-charge or other stated account value |
| Duration-varying | Use the amount or rate assigned to the current duration | Match each charge to the correct policy period |
Worked Example: Projecting Three Charge Structures
A hypothetical policy begins with an account value of $1,000. A net premium of $400 is allocated at the start of each of the next three years. The effective annual interest rate is 4%. Compare three expense-charge schedules: (A) a fixed $30 each year; (B) 2% of the account value immediately after premium allocation and before the charge; and (C) duration-specific fixed charges of $25, $35, and $45 in years 1, 2, and 3. In all cases, deduct the charge after premium allocation and credit interest afterward.
Method. For each year, calculate the pre-charge base \(A_{t-1}+400\), determine the charge under the applicable schedule, subtract it, and multiply the remainder by 1.04. Keep full precision during the calculations and round account values to cents only at the end.
Schedule A: fixed $30 per year. Year 1 starts with a pre-charge base of \(1{,}000+400=1{,}400\). After the $30 charge, the account is $1,370, so
Year 2 has a pre-charge base of \(1{,}424.80+400=1{,}824.80\). Thus
Year 3 has a pre-charge base of \(1{,}866.592+400=2{,}266.592\). Therefore
The year-3 account value is $2,326.06.
Schedule B: 2% of the pre-charge base. Year 1’s base is $1,400, so the charge is \(0.02(1{,}400)=28\). The account becomes \(1{,}372(1.04)=1{,}426.88\). In year 2, the base is \(1{,}426.88+400=1{,}826.88\), and the charge is \(0.02(1{,}826.88)=36.5376\). Thus
In year 3, the base is \(1{,}861.956096+400=2{,}261.956096\). The charge is \(0.02(2{,}261.956096)=45.23912192\), giving
The year-3 account value is $2,305.39.
Schedule C: duration-varying charges. In year 1 the charge is $25, giving \((1{,}400-25)(1.04)=1{,}430\). In year 2 the base is \(1{,}430+400=1{,}830\); after the $35 charge, the value is \(1{,}795(1.04)=1{,}866.80\). In year 3 the base is \(1{,}866.80+400=2{,}266.80\); after the $45 charge,
The year-3 account value is $2,310.67. The results differ because the amount and calculation basis of the charge differ. A common mistake is to apply the 2% in schedule B to the opening account value; the stated base is instead the value after premium allocation and before the charge.
Why Earlier Charges Have a Larger Projection Effect
A charge deducted earlier reduces the account available to earn interest in subsequent periods. This can be made precise when the allocated premiums and interest rate are fixed, the same sequence is used each period, and all charges are known independently of the account value. Starting from
substitution gives \(A_1=A_0(1+i)+(C_1-E_1)(1+i)\). Substituting the recurrence for \(A_2\) then gives
Continuing this substitution through \(n\) periods yields
The formula follows because a net amount added or deducted in period \(t\) remains in the account for \(n-t\) further periods of interest crediting. Thus a fixed-dollar expense charge \(E_t\) reduces the terminal account by \(E_t(1+i)^{n-t}\). The result applies directly to charges known in advance, such as a fixed schedule. For a charge defined as a percentage of the account, the charge depends on the projected account value; calculate it period by period using the recurrence rather than treating it as a predetermined dollar sequence.
Worked Example: Comparing a Level Charge With a Duration Schedule
A hypothetical policy has an opening account value of $800 and receives a net allocated premium of $300 at the start of each of three years. Interest is credited at an effective annual rate of 5%, after the expense charge. Compare a level $30 annual charge with duration-specific charges of $20, $20, and $50.
Method. Project each schedule using the same account recurrence. This isolates the effect of the timing of the charges. Both schedules total $90 over the three years.
For the level schedule, year 1 gives \((800+300-30)(1.05)=1{,}070(1.05)=1{,}123.50\). Year 2’s pre-charge base is \(1{,}123.50+300=1{,}423.50\), so
Year 3’s base is \(1{,}463.175+300=1{,}763.175\), giving
For the duration schedule, year 1 gives \((800+300-20)(1.05)=1{,}080(1.05)=1{,}134\). In year 2 the base is \(1{,}134+300=1{,}434\), so \((1{,}434-20)(1.05)=1{,}414(1.05)=1{,}484.70\). In year 3 the base is \(1{,}484.70+300=1{,}784.70\), giving
The duration schedule’s year-3 value is $1,821.44, compared with $1,819.83 for the level schedule. The difference before rounding is \(1{,}821.435-1{,}819.83375=1.60125\), or $1.60 to the nearest cent. The schedules have the same total charge, but the duration schedule deducts less in the first two years and more in the third. The earlier reductions under the level schedule would otherwise have had more time to earn interest. The correction is that the later, larger charge in the duration schedule does not necessarily outweigh the accumulated effect of earlier charges; project both schedules to compare them.
Charge Bases, Periods, and Common Traps
An account-value-based charge requires extra care because the charge amount changes as the account changes. A statement such as “2% of account value” is incomplete unless the applicable account value and the period are clear. In an exam problem, use the base stated in the question and do not replace it with a more familiar base. If a charge is monthly but premiums and interest are given annually, put the cash flows and rates on consistent time intervals before projecting. For example, an effective annual rate \(i\) corresponds to an effective monthly rate \((1+i)^{1/12}-1\), not generally \(i/12\).
The most common errors are applying the correct rate to the wrong value, assigning a duration charge to the wrong policy year, and rounding intermediate account values. A percentage charge based on the pre-charge account should be calculated before subtracting the charge; it is not a percentage of the post-charge value. Keep unrounded values through each recurrence and round only the requested result.
Worked Example: Applying a Percentage Charge to Its Stated Base
A hypothetical policy has an opening account value of $600 and receives a $250 net premium allocation at the start of the year. An expense charge equal to 3% of the post-allocation, pre-charge account value is deducted. The effective annual interest rate is 4%, credited after the charge. Find the charge and the end-of-year account value.
Method. First calculate the stated charge base: \(600+250=850\). The charge is \(0.03(850)=25.50\). The remaining account before interest is \(850-25.50=824.50\). Therefore
The charge is $25.50 and the end-of-year account value is $857.48. The check is that interest is applied to the post-charge account, not to $850. A common wrong calculation is \(824.50(1.04)=857.46\); direct multiplication gives \(824.50+0.04(824.50)=824.50+32.98=857.48\).
Expense-charge structures matter because they influence projected account values, and an error in the charge base or schedule carries forward into every later period. In longer projections, maintain a row for each period containing the opening account, premium allocation, charge base, expense charge, post-charge balance, interest credit, and closing account. This makes it easier to identify a duration mismatch or a charge applied at the wrong stage.
Check Your Understanding
For each problem, follow the stated charge base and transaction sequence. Round account values to cents.
- A hypothetical policy has an opening account value of $900, receives a $200 net allocation, and incurs a fixed $15 charge. Interest at an effective annual rate of 3% is credited after the charge. Find the end-of-year account value. (Answer: $1,117.55)
- A policy has a $1,200 post-allocation, pre-charge account value. Its expense charge is 2.5% of that base, and the remaining account earns 4% for the year. Find the charge and ending value. (Answer: Charge $30.00; ending value $1,216.80)
- A policy starts with $500 and receives a $100 allocation at the start of each year. Charges are $12 in year 1 and $24 in year 2. The effective annual interest rate is 5%, credited after each charge. Find the year-2 account value. (Answer: $728.07)
- A policy starts with $1,000 and receives $100 at the start of each of two years. Compare a fixed $20 charge each year with a charge equal to 2% of the post-allocation, pre-charge account each year. Interest is 5% annually after charges. Which schedule gives the larger year-2 account, and by how much? (Answer: Fixed-charge account $1,274.70; percentage-charge account $1,267.63; fixed charge gives $7.07 more.)
- A policy starts with $700 and receives $150 at the start of each of two years. Compare fixed charges of $12 per year with duration-specific charges of $5 in year 1 and $19 in year 2. Interest is 4% annually after charges. Find both year-2 account values and their difference. (Answer: Fixed-charge account $1,049.90; duration-schedule account $1,050.19; duration schedule is higher by $0.29.)