Mortality Charge Rates

Why the Rate Basis Matters

A universal life policy may assess a mortality charge monthly even when its rate schedule is stated annually. To calculate the charge correctly, first identify what the quoted annual rate means. A dollar charge rate per $1,000 of exposure per year is not the same thing as an annual probability of death. The two have different units and generally require different conversions.

The rate also depends on the contract details used by the schedule. A table may vary by attained age, contract duration, or both. Using the right conversion with the wrong age or duration still produces the wrong charge.

Definition: An annual dollar mortality charge rate is a stated amount of charge per unit of exposure for one year, such as dollars per $1,000 of exposure per year. A periodic dollar mortality charge rate is the corresponding charge per unit of exposure for a shorter period, such as one month. An annual mortality probability, such as $q_x$, is instead the probability that a life aged $x$ dies within one year, subject to the mortality model and age convention.

For the examples below, the exposure amount is given directly. It is used only to scale the stated charge rate; no death-benefit formula or account-value transaction is being developed. Unless an example says otherwise, charge rates are assumed to apply uniformly within the year and to be assessed at the stated periodic frequency.

Converting a Dollar Charge Rate to a Periodic Rate

Suppose an annual dollar charge rate is $c$ dollars per $1,000 per year, and the contract assesses charges $m$ times per year. If the contract specifies a pro-rata allocation of the annual rate, the periodic rate is $c/m$ dollars per $1,000 per period. If the exposure is $E$ dollars, the number of $1,000 units is $E/1{,}000$, so the charge for one period is:

\text{periodic charge} = \frac{c}{m}\left(\frac{E}{1{,}000}\right).

For monthly charges, $m=12$. This division is appropriate for a dollar charge rate that is defined on a per-year basis and prorated evenly across the year. It is not automatically the correct conversion for every quantity called an “annual mortality rate.” A policy specification may instead quote a monthly rate directly, prescribe another conversion, or provide a rate schedule already expressed for the assessment period.

Rate-basis check: Before converting, record both the meaning and the units of the annual figure. For example, “$2.40 per $1,000 per year” is a dollar charge rate; “0.012 probability of death in one year” is a probability. Dividing either number by 12 without checking its definition can be incorrect.

Why the Pro-Rata Conversion Works

The pro-rata rule follows from the stated uniform-accrual assumption. If the annual dollar charge rate is $c$, an equal allocation across $m$ periods assigns $c/m$ of that annual charge rate to each period. The total allocated rate over the year is therefore:

\underbrace{\frac{c}{m}+\frac{c}{m}+\cdots+\frac{c}{m}}_{m\text{ periods}} = m\left(\frac{c}{m}\right) = c.

Multiplying each periodic rate by the same exposure preserves the same result for charges: the sum of the $m$ periodic charges equals the annual charge $c(E/1{,}000)$. This justification relies on an even allocation and a constant exposure for the periods being compared. If the exposure changes, the periodic charges must be calculated using the applicable exposure for each period; their sum need not equal a single annual charge calculated from one fixed exposure.

Worked Example: Monthly Rate from an Annual Dollar Charge Rate

A hypothetical contract states an annual mortality charge rate of $2.40 per $1,000 of exposure per year. It assesses the charge monthly. For this calculation, the exposure is $80,000 throughout the month. Find the monthly rate and charge.

Assumptions. The stated rate is a dollar charge rate, allocated evenly across 12 monthly periods. The exposure is fixed at $80,000. No interest rate or discounting is used. The rate schedule is deterministic; no probability distribution is required for this charge calculation.

Method. Convert the annual dollar rate to a monthly dollar rate, then multiply by the number of $1,000 exposure units.

There are $80{,}000/1{,}000=80$ exposure units. The monthly rate per unit is:

\frac{\$2.40}{12}=\$0.20\text{ per } \$1{,}000\text{ per month}.

Thus the monthly charge is:

\$0.20(80)=\$16.00.

Check. The annual charge at the same exposure is $\$2.40(80)=\$192.00$. Twelve monthly charges of $\$16.00$ total $\$192.00$, as required. The common mistake of multiplying the annual rate by 80 without first converting to a monthly rate would produce $192.00 for one month—an annual amount, not a monthly charge.

Choosing the Applicable Age and Duration

A mortality charge schedule may be indexed by age and contract duration. Let $x$ denote the age specified by the contract’s age convention for the relevant period, and let $t$ denote the contract duration specified by the schedule. A rate written $c_{x,t}$ is then selected using both indices before any period conversion is made.

The applicable age is not automatically the issue age. It may be an attained age at the start of a contract year, or an age determined under a stated age-nearest-birthday or age-last-birthday convention. Similarly, duration may count completed years, policy years, or another contract-defined measure. The contract or problem statement controls. When the schedule depends on duration, finding the right age alone is not enough.

Locate the assessment period.
Determine which monthly or other periodic charge is being calculated and when it is assessed.

Identify the schedule indices.
Apply the stated age convention and determine the contract duration for that period.

Read the rate and its units.
Confirm whether the schedule gives an annual dollar charge rate, a periodic dollar rate, or a mortality probability.

Convert only if needed.
For an annual dollar rate prorated across $m$ periods, divide by $m$; do not convert a rate that is already periodic.

Apply the stated exposure.
Multiply the periodic dollar rate by the number of exposure units, using the exposure specified for that period.

Worked Example: Looking Up Age and Duration Before Converting

A hypothetical monthly charge schedule gives annual dollar rates per $1,000. In policy year 4, the applicable age is 50 and the rate for age 50, duration 4 is $1.80 per $1,000 per year. The exposure for the month is $125,000. Find the monthly charge.

Assumptions. The contract specifies age 50 and duration 4 for this assessment, and the schedule’s annual dollar rate is prorated evenly over 12 monthly periods. Exposure is $125,000 for the month. No interest rate or discounting is used; the schedule is deterministic, so no probability distribution is needed.

Method. Select the rate using both specified indices, convert that annual dollar rate to a monthly rate, and apply it to the exposure.

The schedule lookup is the age-50, duration-4 entry: $1.80 per $1,000 per year. The monthly rate is:

\frac{\$1.80}{12}=\$0.15\text{ per } \$1{,}000\text{ per month}.

The exposure contains $125{,}000/1{,}000=125$ units. Therefore:

\$0.15(125)=\$18.75.

Check. At the same exposure, the annual charge is $\$1.80(125)=\$225.00$, and 12 monthly charges of $\$18.75$ total $225.00. Using the issue age instead of the specified age, or selecting a different duration row, would select a different table entry even if the arithmetic afterward were flawless.

When the Annual Figure Is a Probability

An annual mortality probability cannot generally be converted to a monthly probability by simply dividing by 12. If $q$ is the probability of death during a year and the contract or model assumes a constant force of mortality within that year, the conditional probability of death in each month, given survival to the start of that month, is:

q_{\text{month}}=1-(1-q)^{1/12}.

The reason is that survival through all 12 months must equal annual survival, $1-q$. If the conditional monthly survival probability is $1-q_{\text{month}}$, the probability of surviving 12 such months is $(1-q_{\text{month}})^{12}$. Equating this to $1-q$ and solving gives the formula. This conversion assumes a constant force, or equivalently equal conditional monthly survival probabilities. If the problem specifies a different within-year mortality pattern, use that pattern instead.

A probability conversion and a dollar charge conversion answer different questions. The pro-rata dollar-rate rule divides a stated annual charge rate by 12. The probability rule converts annual survival to an equivalent monthly conditional probability. Which applies depends on the definition of the schedule, not on a preference for one calculation.

Worked Example: Converting an Annual Mortality Probability

A hypothetical group has an annual mortality probability of $q=0.12$ for the applicable age and duration. Assume a constant force of mortality within the year. There are initially 1,000 independent lives, each with a fixed death benefit of $10,000. Find the equivalent conditional monthly death probability and the expected total death benefits over the year.

Assumptions. Annual death probability is 0.12; conditional monthly death probabilities are equal, so the monthly death-time model is geometric with parameter $q_{\text{month}}$. Lives are independent. The initial cohort is 1,000, with benefit $10,000 per death. No interest rate or discounting is used.

Method. Convert the annual probability using the survival-equivalence formula, then use the 12-month survival probability to calculate expected deaths. Keep the unrounded monthly probability in the calculation.

Annual survival probability is $1-0.12=0.88$. Therefore:

q_{\text{month}} =1-0.88^{1/12} \approx 0.0105962.

To check the conversion, monthly survival raised to the 12th power is:

(1-q_{\text{month}})^{12}=0.88, \qquad 1-(1-q_{\text{month}})^{12}=0.12.

Thus expected deaths during the year are $1{,}000(0.12)=120$. Equivalently, the expected deaths in month $k$ are $1{,}000(1-q_{\text{month}})^{k-1}q_{\text{month}}$; summing over all 12 months gives $1{,}000[1-(1-q_{\text{month}})^{12}]=120$. Expected total death benefits are:

120(\$10{,}000)=\$1{,}200{,}000.

Check. The annual probability applied directly to the initial cohort also gives 120 expected deaths and $1,200,000 in expected benefits. Dividing 0.12 by 12 would give 0.0100, which would not reproduce annual survival: $1-(1-0.0100)^{12}\approx0.1136$, not 0.12. The final monthly probability, rounded to four decimals, is 0.0106.

Common Exam Traps

Confusing units. A dollar charge rate per $1,000 per year and an annual death probability are not interchangeable.
Dividing every annual figure by 12. This is appropriate for a pro-rated annual dollar charge rate, but not for an annual probability when an equivalent monthly probability is required.
Using issue age by default. Use the age defined for the charge period under the stated contract convention.
Ignoring duration. If the rate is indexed by both age and duration, read the row and column that match both.
Rounding too early. Retain the unrounded monthly probability through the annual survival check and round only the requested final result.

Key takeaway: Select the contractually applicable age and duration, identify whether the annual schedule entry is a dollar rate or a probability, and then apply the matching conversion. The rate’s units determine the calculation.

Check Your Understanding

For each problem, identify the rate basis before calculating.

A hypothetical policy has an annual dollar mortality charge rate of $3.60 per $1,000 per year. Charges are monthly, and the exposure is $42,000. Find the monthly charge under even pro-rata allocation. (Answer: $12.60)
For a hypothetical policy’s third policy year, the contract specifies age 64 and duration 3. The corresponding annual dollar rate is $4.80 per $1,000 per year. Monthly exposure is $75,000. Find the monthly charge. (Answer: $30.00)
An annual mortality probability is 0.06, with a constant force of mortality within the year. Find the equivalent monthly conditional death probability, rounded to four decimals. (Answer: 0.0051)
A quarterly contract charge uses an annual dollar rate of $1.92 per $1,000 per year and exposure of $300,000. Under even pro-rata allocation, find one quarterly charge. (Answer: $144.00)
An annual mortality probability is 0.08. Under a constant force of mortality within the year, give the monthly conditional probability formula and its value rounded to four decimals; verify that 12 months reproduce the annual probability. (Answer: $1-0.92^{1/12}\approx0.0069$; $1-(1-0.0069244)^{12}=0.0800$, using the unrounded monthly value)

What You'll Learn