Why Projection Checks Matter
A universal life account projection can look reasonable while still being wrong. A premium may be credited at the wrong time, a charge may be subtracted twice, or interest may be applied to a balance that includes amounts not yet eligible to earn interest. The resulting account value can be materially different even when every individual input appears plausible.
The reliable approach is to write down the policy’s transaction order, label each balance by when it applies, and then reconcile the account from one valuation point to the next. The calculations below use hypothetical policies and explicit timing assumptions. Actual contract provisions can specify a different order, so a projection must follow the provisions being modeled.
The stated formula is not a universal contract rule. For example, a policy could credit interest before a charge, apply different timing to a premium received during the year, or specify a different charge base. The formula applies only when the timing assumptions above match the projection being checked. The examples state their assumptions directly.
A Consistent Audit Sequence
Before calculating, identify the opening account, each cash flow, each charge, the rate and period for which interest is credited, and the definition of any benefit being tested. A useful audit keeps the pre-interest balance separate from the credited interest and the closing account.
State whether premiums and charges occur at the beginning, during, or end of the period, and how long each eligible balance earns interest.
Add premiums credited to the account and subtract loads and charges. Check that each item appears once and has the correct sign.
For each charge, identify the specified base and confirm that it is measured at the correct time. Do not substitute a convenient balance.
Credit interest to the balance and over the period specified by the timeline, not automatically to every dollar shown in the transaction list.
Check the closing account against the pre-interest balance plus interest, and calculate benefits separately using their stated definitions.
Worked Account Reconciliation
The recurrence follows from a simple balance argument. Under the stated annual convention, the account immediately before interest is the opening account plus the premium, less the three deductions. If that balance is \(B_t\), then interest for the year is \(i_t B_t\). Therefore the closing value is \(B_t+i_tB_t=B_t(1+i_t)\). This derivation also identifies the eligible interest base: it is the balance remaining after the specified pre-interest transactions.
Worked Example: Finding the Pre-Interest Balance
A hypothetical policy begins the year with an account value of $1,500. It receives a $500 gross premium at the beginning of the year. The premium load is 6% of gross premium; an expense charge is $20; and an insurance charge is $80. All charges are deducted before the account earns a 2% annual effective credited rate. Assume no other transactions, and that the account remains positive. Find the year-end account value. Also calculate the result if a colleague credits interest on the opening value plus gross premium and then deducts all charges.
Method. Calculate the load from gross premium, reconcile the balance before interest, and then apply the annual accumulation factor. This avoids treating gross premium as if the load had been deposited in the account.
The load is \(0.06(500)=\$30\). The total deductions are \(\$30+\$20+\$80=\$130\). Thus the balance eligible to earn interest is
Interest credited is \(0.02(1{,}870)=\$37.40\). The year-end account value is therefore \(1{,}870+37.40=\$1{,}907.40\), or equivalently \(1{,}870(1.02)=\$1{,}907.40\).
Under the specified erroneous convention, the colleague first accumulates the opening account plus the gross premium: \((1{,}500+500)(1.02)=\$2{,}040\). The colleague then deducts the same $130 of charges, giving \(2{,}040-130=\$1{,}910.00\). This is $2.60 higher than the correct result because the $130 of charges did not earn interest under the stated timing convention. The check is that the correct pre-interest balance is $1,870 and its credited interest is $37.40; together they give $1,907.40.
Charge Bases and Benefit Definitions
A correct sign is not enough: a charge must also be calculated from the right base. For instance, a hypothetical contract might specify an insurance charge as a rate per $1,000 of net amount at risk, with net amount at risk defined for that charge as face amount less account value at the beginning of the year. That charge is not calculated using the account after the premium, and it is not calculated using the death benefit unless the contract defines the base that way. Always state the measurement date and formula before computing.
Likewise, account value and death benefit are distinct quantities. A policy’s stated benefit formula determines the claim; the account value does not automatically equal the death benefit. In the examples below, benefits are calculated only where a definition is explicitly supplied.
Worked Example: Checking the Insurance-Charge Base
A hypothetical policy has a $100,000 face amount and a beginning-of-year account value of $18,000. For this example, the insurance charge is based on beginning-of-year net amount at risk, defined as face amount less that opening account value. The charge rate is $2.40 per $1,000 of net amount at risk. A $4,000 gross premium is paid at the beginning of the year; the premium load is 5% of gross premium; the expense charge is $120; and the account earns 4% annual effective interest after deductions. Find the year-end account value. Assume no other charges or transactions and no lapse.
Method. Calculate the insurance charge from its stated beginning-of-year base, then reconcile all pre-interest items. This avoids using the post-premium account value to calculate a charge that is explicitly based on the opening value.
Beginning-of-year net amount at risk is \(100{,}000-18{,}000=\$82{,}000\), or 82 units of $1,000. The insurance charge is \(82(2.40)=\$196.80\). The premium load is \(0.05(4{,}000)=\$200\).
The pre-interest balance is
Interest is \(0.04(21{,}483.20)=\$859.328\). The unrounded year-end account value is \(21{,}483.20+859.328=\$22{,}342.528\). Rounded to the nearest cent, it is $22,342.53. As a check, \(21{,}483.20(1.04)=22{,}342.528\), consistent with the balance-plus-interest calculation.
Another frequent error is to use a death benefit formula without first identifying the benefit option. For example, if a hypothetical contract explicitly defines the death benefit as the greater of a $100,000 specified amount and account value at the claim time, and account value then is $104,000, the claim under that definition is \(\max(100{,}000,104{,}000)=\$104{,}000\). A different contract definition could produce a different claim. Do not infer a benefit formula from the account projection alone.
Interest Timing and Midyear Transactions
An annual accumulation factor applies to an amount only for the period in which that amount is eligible to earn interest. If the account is credited in subperiods, project each subperiod in order. A premium received midway through the year should not receive a full year of interest unless the contract’s crediting rule explicitly provides that treatment.
Worked Example: A Midyear Premium and Two Crediting Factors
A hypothetical account begins the year at $12,000. A $150 charge is deducted at the start of the year. The account then earns a factor of 1.02 for the first half-year. At midyear, a $1,000 premium is credited with no load. The resulting account earns a factor of 1.03 for the second half-year. There are no other transactions. Find the year-end account value and compare it with the result from incorrectly giving the midyear premium both subperiod factors.
Method. Project chronologically, applying each stated factor only to the balance present during its subperiod. This avoids giving the midyear premium interest for the first half-year.
After the beginning charge, the balance is \(12{,}000-150=\$11{,}850\). After the first-half factor, it is \(11{,}850(1.02)=\$12{,}087\). Add the midyear premium to obtain \(12{,}087+1{,}000=\$13{,}087\). The second-half credit produces
The correct year-end account value is $13,479.61. If the premium is incorrectly included before the first factor, the result is \((11{,}850+1{,}000)(1.02)(1.03)=12{,}850(1.0506)=\$13{,}500.21\). The erroneous result is $20.60 higher. The difference is the extra first-half growth on the $1,000 premium: \(1{,}000(1.02)(1.03)-1{,}000(1.03)=\$20.60\). This confirms that the error is a timing error, not a rounding difference.
Common Projection Errors
A projection check should focus on the model’s assumptions as well as its arithmetic. The same list of transactions can lead to different valid results under different contract timing rules; a calculation is erroneous when it does not follow the rule being modeled.
| Potential error | Why it fails | Correction |
|---|---|---|
| Adding a premium load to the account | The load is deducted from gross premium before the net amount enters the account. | Calculate the load from its defined base and subtract it once. |
| Applying interest to gross premium | The portion removed as a load may not be part of the interest-bearing account. | Use the balance eligible to earn interest under the stated timing. |
| Subtracting charges after crediting interest | This gives pre-interest charges an extra period of growth. | Follow the specified transaction order; under the annual convention here, deduct them before interest. |
| Using the wrong charge base | The charge may depend on a specified opening account, benefit amount, or other contractual measure. | Write the base and measurement time before calculating the charge. |
| Giving a midperiod premium too much interest | The premium was not present for the earlier subperiod. | Split the projection at the transaction date. |
| Calling account value the death benefit | The claim depends on the policy’s benefit definition, not the account projection alone. | Apply the stated benefit formula at the claim time. |
A useful independent check is to recalculate the closing account in two equivalent ways when the timing is uniform: (1) multiply the pre-interest balance by \(1+i\), and (2) add the pre-interest balance to \(i\) times that balance. If the results differ, inspect the arithmetic. For a projection with midyear transactions, perform this check separately within each subperiod rather than forcing a single annual factor onto all cash flows.
These checks are useful because errors in a policy account projection can affect later decisions about funding, coverage, or a claim calculation. They do not replace contract terms: verify the timing, charge definitions, crediting basis, and benefit formula before accepting a numerical result. This material is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each hypothetical projection, use the stated timing and charge definitions. Round money to the nearest cent where necessary.
- A policy opens the year with $8,000. A $1,200 gross premium is paid at the beginning of the year, with a 4% load. Expense and insurance charges of $30 and $72 are deducted before 3% annual effective interest. Find the year-end account value. (Answer: $9,321.50)
- A hypothetical contract defines the insurance-charge base as beginning-of-year face amount less beginning-of-year account value. The face amount is $150,000, the opening account is $24,000, and the rate is $1.80 per $1,000 of base. What is the insurance charge? (Answer: $226.80)
- An account has $5,000 at the start of the year. A $100 charge is deducted immediately. The first-half crediting factor is 1.01, a $800 premium is credited at midyear without a load, and the second-half factor is 1.02. Find the year-end account value. (Answer: $5,863.98)
- A policy’s explicitly stated death benefit is a level $75,000. At the claim time, account value is $71,500. Under this definition, what is the death benefit? (Answer: $75,000)
- A policy opens with an account value of $30,000 and receives a $5,000 gross premium at the beginning of the year. Its premium load is 5% of gross premium, its expense charge is $100, and its insurance charge is $1.50 per $1,000 of beginning-of-year net amount at risk, defined as a $100,000 face amount less the opening account. All deductions occur before 2.5% annual effective interest. Find the closing account value. (Answer: $35,408.63)