Introduction
A universal life projection is a sequence of account transactions, not a single annual interest calculation. The order of premiums, loads, charges, withdrawals, and interest matters: changing that order can change both the projected account value and whether coverage continues. A complete projection also distinguishes the policy account from the death benefit and follows the contract’s lapse provisions when the account cannot meet a charge.
This workshop uses hypothetical contract terms to combine those mechanics. The timing conventions are deliberately stated in each problem; actual contracts may use different conventions. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
A Monthly Projection Framework
Let \(B_{m-1}\) be the account value immediately before the transactions at the start of month \(m\). Let \(P_m\) be the gross premium received then, \(\lambda\) the premium-load proportion, \(C_m\) the monthly charge, and \(j\) the effective monthly credited rate. In the basic convention used below, the insurer deducts the load, deducts the monthly charge, and then credits interest. A withdrawal \(W_m\), when specified, is taken at month-end after interest.
This recurrence applies only when the balance is sufficient to meet the charge. If it is not, do not blindly continue the formula into a negative account value. The contract’s insufficient-value and grace-period provisions determine what happens next.
For a simplified projection in which the same net premium is added at the start of the first month, the same charge \(c\) is deducted at the start of each of \(n\) months, and there are no withdrawals, set \(q=1+j\) and \(Q=q^n\). Iterating the monthly recurrence gives a useful check:
Here \(P^{\mathrm{net}}\) is the net premium added at the start of month 1. The formula is not a substitute for tracking transaction timing: it assumes the premium is posted once, each charge is taken before that month’s interest, and no policy event interrupts the projection.
Worked Example: Flexible Premiums, Monthly Charges, and a Withdrawal
Worked Example: Flexible Premiums, Monthly Charges, and a Withdrawal
A hypothetical policy starts with an account value of $1,000. Each policy year has a premium at the start of month 1, a 5% load, and a monthly credited rate of 0.5%. Charges are deducted at the start of each month before interest. The monthly charge is $20 in year 1, $30 in year 2, and $40 in year 3. Gross premiums are $1,200 in year 1, $600 in year 2, and zero in year 3. At the end of year 2, the owner withdraws $300. Project the account at each year-end, assuming no charge is insufficient to be deducted.
Method. Use the monthly recurrence, or its geometric-series check for each year. The monthly accumulation factor over 12 months is \(Q=(1.005)^{12}=1.0616778\), and the factor multiplying a fixed monthly charge is \(H=1.005(Q-1)/0.005=12.3972402\). These factors retain the convention that charges are taken before monthly interest.
Year 1. The net premium is \(1,200(1-0.05)=\$1,140\). The account immediately after the premium is \(1,000+1,140=\$2,140\). Therefore, \(B_{12}=2,140Q-20H=\$2,024.05\), to the nearest cent.
Year 2. The net premium is \(600(1-0.05)=\$570\), so the amount before the year’s monthly charges and credits is \(2,024.0457+570=\$2,594.0457\). The account before the year-end withdrawal is \(2,594.0457Q-30H=\$2,382.12\), to the nearest cent. After the $300 withdrawal, the year-end account is \(\$2,082.12\).
Year 3. There is no premium, so the account after the year’s charges and credits is \(2,082.1236Q-40H=\$1,714.65\), to the nearest cent.
Check. All three projected year-end balances are positive, consistent with the stated assumption that each monthly charge can be deducted. The withdrawal reduces the year-2 account after that year’s interest has been credited; it is not a premium load or a monthly charge.
Common mistake avoided. Applying the 5% load to the account value as well as to the premium is incorrect. The load applies to the gross premium: in year 1, it is \(0.05(1,200)=\$60\), leaving $1,140 to be credited.
Benefit Options and the Amount at Risk
The account and death benefit are related by the benefit option, but they are not interchangeable. For the following simplified hypothetical definitions, let \(S\) be the specified amount and \(B\) the account value at the time of death. Option A pays the greater of \(S\) and \(B\); Option B pays \(S+B\). The contract’s cost-of-insurance charge is based on the net amount at risk (NAR), which is the death benefit less the account value, but not below zero. These definitions are modeling assumptions for the examples, not universal contract terms.
Under these definitions, the Option B NAR is \(S\), because subtracting the account value from \(S+B\) leaves \(S\). Under Option A, a larger account reduces the NAR while the account remains below the specified amount. An actual contract may also include benefit corridors, other charge rules, or different definitions; use the stated contract terms rather than assuming these simplified formulas.
Worked Example: Comparing Charges and Death Benefits
A hypothetical policy has a $50,000 specified amount and an account value of $4,000 immediately before a monthly charge. The monthly credited rate is 0.4%. The monthly charge is a $6 administrative charge plus a cost-of-insurance charge equal to 0.02% of the month’s NAR. Compare Option A and Option B, using the definitions above. Assume death occurs at month-end, after the charge and interest credit.
Option A charge. The NAR is \(50,000-4,000=\$46,000\). The cost-of-insurance charge is \(0.0002(46,000)=\$9.20\), so the total monthly charge is \(6+9.20=\$15.20\). The account after charge and interest is \((4,000-15.20)(1.004)=\$4,000.7392\), or \(\$4,000.74\).
Option B charge. The NAR is \(\$50,000\). The cost-of-insurance charge is \(0.0002(50,000)=\$10.00\), so the total charge is \(\$16.00\). The account after charge and interest is \((4,000-16)(1.004)=\$3,999.936\), or \(\$3,999.94\).
Death benefits at month-end. Under Option A, the account is below $50,000, so the benefit is \(\max(50,000,4,000.7392)=\$50,000.00\). Under Option B, the benefit is \(50,000+3,999.936=\$53,999.94\). The different NARs also produce different charges.
Check and common mistake avoided. The calculations use the beginning-of-month account value to determine that month’s NAR and charge, and the month-end account value for the death claim. Crediting interest before deducting the charge would give a different result; it would not follow the stated transaction order.
Withdrawals, Insufficient Value, and Lapse
A premium holiday means no premium is paid for the period; it does not mean monthly charges stop. A withdrawal similarly reduces the account without automatically changing the specified amount or the charge basis. The contract may impose other consequences, but those must be supplied by its terms. For projection purposes, record a withdrawal at the exact time stated and then test the remaining account against the next charge.
If the account cannot meet a monthly charge, distinguish insufficient value from lapse. A contract may provide a grace period during which coverage continues if the required amount is paid by a deadline. The account should not be carried as a negative value unless the specified contract mechanics explicitly require such a representation. The grace-period length, required payment, and lapse date are determined by the contract assumptions.
Worked Example: A Premium Holiday, Withdrawal, and Grace Period
A hypothetical policy begins with a $250 account value. A $100 gross premium is paid at the start of month 1, subject to a 5% load. The monthly charge is $25 and the monthly credited rate is 0.5%. No further premium is paid. At the start of month 6, the owner withdraws $200 before that month’s charge. If the account cannot meet a charge, it is set to zero and a two-month grace period begins; coverage continues during grace, but the policy lapses at the end of the second grace month if no payment cures the deficiency. No payment is made during grace. Find the account through month 6 and determine the lapse timing.
Months 1 to 5. The net premium in month 1 is \(100(1-0.05)=\$95\). The month-1 account is \((250+95-25)(1.005)=\$321.60\). Subsequent months have no premium: month 2 is \((321.60-25)(1.005)=\$298.083\); month 3 is \((298.083-25)(1.005)=\$274.448415\); month 4 is \((274.448415-25)(1.005)=\$250.695657\); and month 5 is \((250.695657-25)(1.005)=\$226.824135\).
Month 6. After the $200 withdrawal, the account available for the charge is \(226.824135-200=\$26.824135\). It can meet the $25 charge. The remaining \(\$1.824135\) earns 0.5% interest, giving an end-of-month account of \(1.824135(1.005)=\$1.833256\).
Months 7 and 8. At the month-7 charge date, \(\$1.833256\) is less than the \(\$25\) charge, so the account is insufficient and the two-month grace period begins. The account is not recorded as \(-\$23.166744\); it is set to zero under the stated assumption. With no payment in month 8, the deficiency remains uncured, and coverage ends at the end of month 8. The policy does not lapse at the first insufficient-charge date because the hypothetical contract provides a two-month grace period.
Check and common mistake avoided. The month-1 load is $5, leaving $95 of the premium for the account. Incorrectly crediting the full $100 would produce \((250+100-25)(1.005)=\$326.63\), rather than the correct $321.60. The $5.03 difference then carries into later months.
A Reliable Projection Check
Before accepting a multi-year result, reconcile each year’s opening and closing values and verify that every transaction is in the right place. A useful sequence is:
Calculate the credited premium as \(P(1-\lambda)\); do not load the existing account.
Use the account value, benefit option, NAR definition, and charge rate at the time stipulated by the contract.
Use the stated monthly rate and timing. Convert an annual rate to a monthly rate only when the contract’s convention requires it.
Apply withdrawals at their stated time, then check whether the account meets the next charge. Follow the stated grace-period rule before declaring lapse.
A projection that ends with a positive account is not automatically proof that coverage remained in force: every monthly charge must also have been met, or any shortfall must have been handled under the stated grace provisions. Conversely, a premium holiday alone does not establish lapse. The result depends on the evolving account and the contractual lapse conditions.
Check Your Understanding
For each hypothetical problem, use the transaction timing and contract assumptions stated in the question.
- An account is $1,200 immediately before a month’s transactions. A $500 premium is received at the start of the month with a 4% load. A $18 charge is deducted, followed by a 0.5% monthly credit. Find the ending account. (Answer: $1,670.31)
- A policy has specified amount $40,000 and account value $9,000. Under the simplified Option B definition in this tutorial, find the death benefit and NAR. (Answer: death benefit $49,000; NAR $40,000)
- An account of $2,000 receives a $1,000 gross premium at the start of month 1 with a 6% load. The monthly charge is $12, deducted before interest, for 12 months; the monthly credited rate is 0.5%. There are no withdrawals. Find the account after 12 months, to the nearest cent. (Answer: $2,972.57)
- An account of $30 faces a $35 monthly charge. The monthly credited rate is 0.4%, but the charge is deducted before interest. Under a hypothetical contract that sets an insufficient account to zero and grants a grace period, does the account earn interest on a negative balance, and has coverage necessarily lapsed at that charge date? (Answer: No negative balance earns interest; no, the stated grace period means lapse is not necessarily immediate.)
- A policy has a $50,000 specified amount and a $48,000 account under simplified Option A. Find the NAR and death benefit. If the account then increases to $52,000, find the revised NAR and death benefit under the same definition. (Answer: initially NAR $2,000 and death benefit $50,000; afterward NAR $0 and death benefit $52,000)