Account Depletion and Coverage Risk

Why Account Depletion Matters

A universal life policy account is affected by several transactions over time: premiums may be allocated to the account, interest may be credited, and monthly charges may be deducted. When the account is large enough, these transactions produce an ordinary positive account value. When it is too small to meet a charge, the projection signals a funding problem and a potential threat to coverage.

The central calculation is a projection under clearly stated assumptions. The result depends on transaction timing: a premium credited before interest can earn interest during that month, while a charge deducted after interest does not. A projected shortfall identifies when the account cannot fund the scheduled deduction under the assumptions; it does not, by itself, specify the policy’s subsequent contract treatment.

Definition: Let $A_t$ be the account value immediately after the monthly charge at the end of month $t$, with $A_0$ the starting account value. Let $N_t$ be the net premium allocated to the account at the start of month $t$, $j_t$ the effective monthly credited rate, and $C_t$ the charge due at the end of that month. If the premium is credited first, interest is credited next, and the charge is deducted last, the algebraic projected balance is $$ A_t=(A_{t-1}+N_t)(1+j_t)-C_t. $$

This recurrence assumes the stated monthly rate and charge apply as specified, and that no other transaction occurs during the month. If the account cannot meet a charge, continuing the recurrence can produce a negative algebraic balance. That negative value is a measure of the unfunded amount in the projection, not necessarily an account value the contract permits. Contract provisions determine what happens next. In particular, account depletion is a warning about coverage risk, not a complete description of grace-period or termination rules.

Projecting the Account and Detecting a Shortfall

For each month, calculate the account available immediately before the charge. Under the timing above, this amount is $(A_{t-1}+N_t)(1+j_t)$. If it is at least $C_t$, the charge is fully funded and $A_t$ is nonnegative. If it is less than $C_t$, the difference is the shortfall:

\text{shortfall in month }t = C_t-(A_{t-1}+N_t)(1+j_t), \quad\text{when }C_t>(A_{t-1}+N_t)(1+j_t).

A projection should identify the first month in which the inequality holds. A later month’s algebraic balance can be useful for measuring the continuing funding gap, but it should not obscure when the first charge became unaffordable. If a contract limits the account to zero or changes how charges are handled after insufficiency, the algebraic projection is no longer a literal account-value projection beyond that point.

Worked Example: Finding the First Unfunded Charge

A hypothetical policy has a starting account value of $100, no premiums, a monthly credited rate of 1%, and charges of $35, $40, and $40 in months 1, 2, and 3. Interest is credited before each month’s charge. Find the first month with a shortfall and its amount. Method: project month by month and compare the amount available before each charge with that charge. The common mistake to avoid is identifying depletion from the starting balance without allowing for monthly interest and earlier deductions.

Month 1. The amount before the charge is $100(1.01)=101.00$. After the $35 charge, the account is $101.00-35.00=\$66.00$.

Month 2. The amount before the charge is $66.00(1.01)=66.66$. After the $40 charge, the account is $66.66-40.00=\$26.66$.

Month 3. The amount before the charge is $26.66(1.01)=26.9266$. The scheduled charge is $40, so the shortfall is $40-26.9266=\$13.0734$, or $13.07 to the nearest cent. Thus month 3 is the first month with an unfunded charge.

Check. The month 1 and month 2 amounts available exceed their charges, while the month 3 amount is less than its charge. The stated shortfall is the exact difference before rounding.

How Recurring Premiums Change Depletion

A regular premium can slow depletion or prevent it over a chosen projection period. The premium amount required depends on the target: maintaining a positive value, covering every scheduled charge over a stated period, or reaching a specified ending account value are different conditions. State the target and projection horizon before solving.

With a constant monthly credited rate $j$, a constant net premium $N$ paid at the start of each month, and constant end-of-month charge $C$, the recurrence is $A_t=(A_{t-1}+N)(1+j)-C$. Repeated substitution gives an ending value that is linear in $N$. The premium’s coefficient reflects the fact that each premium earns interest for the remaining part of the projection.

Derivation: Write $r=1+j$. Then $A_1=A_0r+Nr-C$, and $A_2=A_1r+Nr-C=A_0r^2+N(r^2+r)-C(r+1)$. Repeating the same substitution for $n$ months gives $$ A_n=A_0r^n+N\sum_{k=1}^{n}r^k-C\sum_{k=0}^{n-1}r^k. $$ The first sum contains each start-of-month premium grown through the crediting periods after it enters the account. The second contains each end-of-month charge grown through the periods after that deduction. This expression is valid for constant $j,N,C$ and the stated timing; if rates, premiums, or charges vary, use the month-by-month recurrence or its corresponding varying-parameter expansion.

Worked Example: Projecting a Fixed Net Premium

A hypothetical policy starts with $100. It receives a $20 net premium at the start of each month, earns 1% monthly interest, and has a $30 charge at each month-end. Find the account value after three months. Method: apply the recurrence sequentially. The common mistake to avoid is deducting the monthly charge before crediting interest when the problem specifies the reverse order.

Month 1. Premium added: $100+20=120$. After interest: $120(1.01)=121.20$. After the charge: $121.20-30= \$91.20$.

Month 2. Premium added: $91.20+20=111.20$. After interest: $111.20(1.01)=112.312$. After the charge: $112.312-30=\$82.312$.

Month 3. Premium added: $82.312+20=102.312$. After interest: $102.312(1.01)=103.33512$. After the charge: $103.33512-30=\$73.33512$, which is $73.34 to the nearest cent.

Check. Substitution into the three-month formula gives $100(1.01)^3+20[(1.01)+(1.01)^2+(1.01)^3]-30[1+(1.01)+(1.01)^2]$. The terms are $103.0301+61.20802-90.903=73.33512$, agreeing with the month-by-month result.

Calculating an Additional Premium for a Target

To calculate the premium needed to maintain an account value, first express the target as an inequality, such as $A_n\geq T$, where $T$ is the required value at the end of month $n$. Then solve for the net premium. If the premium is subject to a load, distinguish the gross premium paid from the net amount allocated to the account. With a proportional load $L$, the net premium is $N=(1-L)G$, where $G$ is gross premium.

Worked Example: Finding the Gross Premium for an Ending-Value Target

A hypothetical policy has $A_0=\$40$, a monthly credited rate of 1%, and a $20 charge at each month-end for three months. A level gross premium $G$ is paid at the start of each month. A 10% premium load is deducted, so $N=0.90G$ is allocated each month. Find the minimum gross premium, to the nearest cent, that produces an account value of at least $10 after month 3. Method: use the three-month recurrence formula, solve the target inequality for the net premium, and then account for the load. The common mistake to avoid is solving for the gross premium as though the entire payment enters the account.

Here $r=1.01$, $A_0=40$, and $C=20$. The sums are $r+r^2+r^3=1.01+1.0201+1.030301=3.060401$ and $1+r+r^2=1+1.01+1.0201=3.0301$. Therefore,

$$ A_3=40(1.030301)+N(3.060401)-20(3.0301) =41.21204+3.060401N-60.602 =-19.38996+3.060401N. $$

The target condition is $-19.38996+3.060401N\geq10$. Thus $3.060401N\geq29.38996$, so $N\geq29.38996/3.060401\approx9.603303$. Since $N=0.90G$, the unrounded gross threshold is $G\geq9.603303/0.90\approx\$10.670337$. A cent-denominated premium of $10.67 is not enough: it allocates $0.90(10.67)=\$9.603$, producing $A_3=-19.38996+3.060401(9.603)=\$9.999070803$, below the target. A gross premium of $10.68 allocates $0.90(10.68)=\$9.612$, producing $A_3=-19.38996+3.060401(9.612)=\$10.026614412$. The minimum cent-denominated gross premium is therefore $10.68 per month.

Check. The chosen premium produces more than $10, while the immediately lower cent amount produces less than $10. This verifies the rounding-up decision.

Coverage Risk and Common Traps

A scheduled charge that exceeds the amount available is a direct sign of account funding pressure. If the account is projected to approach zero, future charges may not be supportable without additional premiums or other contract provisions. Whether coverage continues, and on what terms, is determined by the policy; an account projection alone does not answer those contract questions.

Using the wrong transaction order. A premium, interest credit, and charge must be applied in the order specified. Changing the order changes the account value.
Confusing a shortfall with a negative account value. The shortfall is the charge minus the available amount. A negative algebraic balance is a projection device, not necessarily a value the contract records.
Ignoring the projection horizon. A premium sufficient to meet charges for one year may not achieve a target over a longer period, especially if charges or credited rates change.
Mixing gross and net premiums. Apply any load before using the premium in the account recurrence.
Treating account value as surrender value or death benefit. These are distinct policy quantities and may be affected by separate contract terms.

For an exam calculation, show the month-by-month values up to the first shortfall, state the available amount before the relevant charge, and compare it directly with the charge. For a premium requirement, state whether the answer is gross or net, identify the target date, and verify that the rounded premium actually meets the target.

Check Your Understanding

For each problem, use the transaction timing stated in the question. Round dollar answers to the nearest cent where appropriate.

A hypothetical account starts at $80, has no premiums, earns 0.5% monthly, and faces a $25 end-of-month charge for two months. What is the account value after month 2? (Answer: $30.68.)
A hypothetical account starts at $40. A net premium of $10 is paid at the start of each of two months, the monthly credited rate is 1%, and the end-of-month charge is $25. What is the account value after month 2? (Answer: $10.86.)
A hypothetical account starts at $60, earns no interest, receives no premiums, and has charges of $20, $30, and $35 at the ends of months 1, 2, and 3. In which month does the first shortfall occur, and how large is it? (Answer: Month 3; $25.00.)
A hypothetical account starts at $75, earns no interest, and has a $20 charge at each month-end for three months. Equal gross premiums are paid at the start of each month and are subject to an 8% load. What minimum gross premium, to the nearest cent, gives an ending account value of at least $30? (Answer: $5.44 per month.)
A hypothetical account starts at $100, receives no premiums, earns 2% monthly, and has charges of $30 and $40 at the ends of months 1 and 2. What is the account value after month 2? (Answer: $33.44.)

What You'll Learn