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Exam ALTAM · Universal Life Profit and Reserves · Tutorial 635 of 999

Cost-of-Insurance Charges and Mortality Margins

Measure whether cost-of-insurance charges cover expected mortality costs, and calculate the resulting margin or shortfall by duration.

Advanced 10 min read

What You'll Learn

  • Calculate cost-of-insurance deductions from a charge rate and the net amount at risk.
  • Compare charges with expected death costs on a consistent valuation date.
  • Determine the break-even charge rate under an annual interest assumption.
  • Track how mortality margins change across policy durations.
  • Weight duration-specific margins for survival when calculating cohort results.
  • Avoid confusing a per-policy margin with an expected margin per original policy.

Why Compare Charges With Mortality Costs?

A universal life policy may deduct a cost-of-insurance (COI) charge from its account value while the policy is in force. That deduction is intended, in part, to fund the insurer’s mortality exposure. The amount exposed is the net amount at risk (NAR), not necessarily the policy’s full death benefit. The earlier material on net amount at risk explains how to identify that exposure for a specified death-benefit design.

A useful duration-by-duration comparison sets the expected mortality cost against the COI deduction on the same basis. The result is a mortality margin or shortfall under the assumptions used. It is not, by itself, a complete measure of policy profitability: expenses, lapses, other charges, and other cash flows are outside this comparison. This method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: For a policy in force at the start of a policy year, let \(M\) be the NAR during that year, \(q\) the conditional probability of death during the year, and \(c\) the COI charge in dollars per $1,000 of NAR. The charge is \(C=cM/1000\), and the expected death cost, when the death benefit attributable to the NAR is paid at year-end, is \(qM\). Both quantities are conditional on the policy being in force at the start of the year.

For the calculations below, assume annual policy-year steps, a charge deducted at the start of each year, and death claims attributable to the NAR paid at the end of the year. Assume the NAR remains fixed within that year. Let \(i\) be the effective annual rate at which the charge proceeds can accumulate. Ignore expenses, lapses, taxes, and other cash flows. These assumptions create a deliberately focused mortality comparison; if actual charges or benefits occur at different times, the timing must be adjusted.

Putting Charges and Expected Costs on the Same Date

The charge \(C\) is collected at the start of the year, whereas the expected death cost \(qM\) is measured at year-end. Accumulating the charge for one year gives \(C(1+i)\). Thus the end-of-year mortality margin, conditional on being in force at the start of the year, is

$$ \text{End-of-year margin}=C(1+i)-qM =M\left(\frac{c(1+i)}{1000}-q\right). $$

A positive value is a surplus on this simplified mortality comparison; a negative value is a shortfall. Equivalently, discount the expected death cost to the start of the year. The start-of-year margin is \(C-vqM\), where \(v=1/(1+i)\). The two measures agree in date value because

$$ C-vqM=v\bigl(C(1+i)-qM\bigr). $$

If comparing charge and mortality rates without interest adjustment, it is common to compare \(c/1000\) directly with \(q\). That comparison treats the charge and claim cost as being measured at the same time or ignores the timing value of money. Under the timing assumptions above, the end-of-year break-even charge rate satisfies

$$ \frac{c(1+i)}{1000}=q \quad\Longrightarrow\quad c=\frac{1000q}{1+i}. $$

This threshold applies only to the simplified one-year model. It does not imply that a charge at or above this rate makes the whole policy profitable.

Derivation: Conditional on being in force at the start of the year, death occurs during the year with probability \(q\). If the NAR is \(M\) and the claim is paid at year-end, the expected claim outgo at that date is \(qM+(1-q)0=qM\). The start-of-year charge \(C\), accumulated at rate \(i\), has year-end value \(C(1+i)\). Subtracting expected claim outgo from the accumulated charge gives \(C(1+i)-qM\). The calculation relies on the stated timing and fixed-NAR assumptions; a changing NAR or a claim paid at another time requires a corresponding cash-flow adjustment.

Worked Example: A Charge Above the Break-Even Rate

Worked Example: A Charge Above the Break-Even Rate

A hypothetical policy is in force at the start of a year with NAR \(M=\$80{,}000\). The annual death probability is \(q=0.004\), the COI charge is \(\$5.50\) per \(\$1{,}000\) of NAR, and the charge proceeds earn \(i=0.05\). Find the charge, expected death cost, and mortality margin at year-end. Use the annual timing assumptions stated above.

Method. Calculate the charge and expected claim cost conditional on being in force, then compare both at year-end.

The charge is

$$ C=5.50\left(\frac{80{,}000}{1{,}000}\right) =5.50(80)=\$440.00. $$

The expected death cost at year-end is

$$ qM=0.004(80{,}000)=\$320.00. $$

The accumulated charge is \(\$440.00(1.05)=\$462.00\). Therefore,

$$ \text{End-of-year margin}=462.00-320.00=\$142.00. $$

As a check, the start-of-year margin is \(\$440.00-0.004(80{,}000)/1.05=\$135.24\), rounded to cents. Accumulating \(\$135.24\) gives approximately \(\$142.00\), with the small rounding difference due to rounding the start-of-year figure. The unrounded start-of-year margin is \(440-304.7619\ldots\), which accumulates exactly to \(\$142.00\). The charge rate exceeds the timing-adjusted break-even rate \(1000(0.004)/1.05=\$3.81\) per \(\$1{,}000\). A common mistake is to subtract \(\$320\) directly from \(\$440\) and report \(\$120\) as the year-end margin; that subtracts amounts at different dates. The corrected year-end calculation accumulates the charge first.

Margins by Policy Duration

Both mortality and COI charge rates can vary by duration. For each duration, first use the rates and NAR applicable to a policy in force at that duration’s start. Then calculate its conditional margin. A positive margin in one year does not rule out shortfalls in later years. Nor does a per-policy margin tell how many policies are expected to generate it in a group.

Worked Example: Comparing Three Durations

A hypothetical block has a constant NAR of \(\$100{,}000\) per policy in force at the start of each year. The annual effective accumulation rate is \(i=0.04\). At durations 1, 2, and 3, respectively, the conditional death probabilities are \(0.003\), \(0.006\), and \(0.010\), while the COI charges are \(\$4\), \(\$7\), and \(\$11\) per \(\$1{,}000\) of NAR. Calculate the end-of-year margin per policy in force at each duration’s start.

Method. At each duration, compute \(C=c(100{,}000/1{,}000)=100c\), accumulate the charge by \(1.04\), and subtract \(q(100{,}000)\).

At duration 1, \(C=100(4)=\$400\), and the expected death cost is \(0.003(100{,}000)=\$300\). Thus the margin is \(\$400(1.04)-\$300=\$416-\$300=\$116.00\).

At duration 2, \(C=100(7)=\$700\), and the expected death cost is \(0.006(100{,}000)=\$600\). The margin is \(\$700(1.04)-\$600=\$728-\$600=\$128.00\).

At duration 3, \(C=100(11)=\$1{,}100\), and the expected death cost is \(0.010(100{,}000)=\$1{,}000\). The margin is \(\$1{,}100(1.04)-\$1{,}000=\$1{,}144-\$1{,}000=\$144.00\).

The results are \(\$116.00\), \(\$128.00\), and \(\$144.00\) per policy in force at the start of the respective duration. The calculation checks because the charge rates in all three years exceed their timing-adjusted break-even rates: approximately \(\$2.88\), \(\$5.77\), and \(\$9.62\) per \(\$1{,}000\). A common error is to use the duration-1 mortality rate for all three years; the conditional mortality assumption must be matched to the duration being valued.

From Per-Policy Margins to a Cohort Result

A duration-specific margin is conditional on the policy reaching that duration. To calculate expected margins per original policy, weight each duration’s margin by the probability that the policy is still in force at that duration’s start. With mortality as the only termination cause and conditional annual death probabilities \(q_1,q_2,\ldots\), the probability of reaching the start of duration \(k\) is the product of the preceding survival probabilities:

$$ {}_{{k-1}}p=\prod_{j=1}^{k-1}(1-q_j), \qquad \text{expected margin at duration }k ={}_{{k-1}}p\,\bigl(C_k(1+i)-q_kM_k\bigr). $$

If other termination causes affect whether a charge is collected, the persistence probability must include those causes as appropriate. The formula above does not model lapses; it illustrates mortality-only weighting. To add margins across durations, express them at a common valuation date. For example, an end-of-year margin at duration \(k\) is discounted by \(v^k\) to the original policy date when using a constant annual rate \(i\).

Worked Example: A Shortfall Emerging After a Positive First-Year Margin

A hypothetical two-year policy has \(i=0.05\). At duration 1, its NAR is \(\$120{,}000\), \(q_1=0.002\), and the COI charge is \(\$2\) per \(\$1{,}000\). At duration 2, its NAR is \(\$100{,}000\), \(q_2=0.005\), and the charge is \(\$4\) per \(\$1{,}000\). Assume mortality is the only termination cause. Find the conditional end-of-year margin in each duration, the expected undiscounted sum of end-of-year margins per original policy, and the present value at issue.

Method. Calculate each conditional margin first. Then weight the second-duration margin by the probability of surviving duration 1, and discount each expected year-end amount to issue.

In duration 1, the charge is \(2(120{,}000/1{,}000)=\$240\), and expected death cost is \(0.002(120{,}000)=\$240\). The end-of-year margin per policy in force is \(\$240(1.05)-\$240=\$12.00\).

In duration 2, the charge is \(4(100{,}000/1{,}000)=\$400\), and expected death cost is \(0.005(100{,}000)=\$500\). The conditional end-of-year margin is \(\$400(1.05)-\$500=-\$80.00\). The probability of reaching duration 2 is \(1-q_1=0.998\), so the expected duration-2 margin per original policy is \(0.998(-\$80)=-\$79.84\).

The expected undiscounted sum of the two year-end margins per original policy is \(\$12.00-\$79.84=-\$67.84\). To value at issue, discount duration 1’s margin by \(1.05\), and duration 2’s weighted margin by \(1.05^2=1.1025\):

$$ \frac{12}{1.05}+\frac{-79.84}{1.1025} =11.428571\ldots-72.417233\ldots =-\$60.99. $$

The result is a present-value shortfall of \(\$60.99\) per original policy, rounded to cents. The negative result is consistent with the later shortfall outweighing the positive first-year margin. A common mistake is to assign the full duration-2 shortfall to every original policy; only \(0.998\) of the original policies are expected to reach that duration under the stated assumptions.

Common Mistakes and Scope

  • Mixing valuation dates. A start-of-year deduction cannot be subtracted directly from a year-end expected claim without adjusting for interest.
  • Using face amount instead of NAR. The mortality cost in this comparison is based on the death benefit exposure represented by \(M\), not automatically the full face amount.
  • Mixing conditional and cohort quantities. A margin per policy in force at a duration’s start is not the same as the expected margin per original policy.
  • Treating the margin as total profit. This calculation isolates mortality charges and mortality costs. It omits expenses and other sources of income or outgo.
  • Assuming the charge is always collected. The model assumes the policy is in force at the charge date and does not allow for lapses or other terminations.

In practice, the calculation must match the contract and valuation basis: charge timing, claim timing, NAR definition, mortality basis, and applicable interest assumption all matter. If the NAR changes during a year, a single fixed-\(M\) calculation may not represent the actual exposure. If the charge is deducted at another time or claims are paid during the year, model those cash flows at their actual times rather than applying the annual formula unchanged.

Key takeaway: First calculate the COI deduction and expected mortality cost for the same duration and exposure. Then place them on a common date. For a cohort, weight each duration’s result by the probability of reaching it, and discount to a common valuation date before aggregating.

Check Your Understanding

Use the annual timing assumptions in the tutorial unless a question states otherwise. Amounts are per policy, and report dollar answers to the nearest cent.

  1. A policy has NAR \(\$50{,}000\), annual death probability \(0.005\), a charge of \(\$6\) per \(\$1{,}000\), and \(i=0.04\). Find the end-of-year mortality margin. (Answer: \(\$62.00\))
  2. A policy has NAR \(\$90{,}000\), annual death probability \(0.004\), a charge of \(\$3.60\) per \(\$1{,}000\), and \(i=0.05\). Find the end-of-year margin or shortfall. (Answer: \(-\$19.80\))
  3. For \(q=0.008\) and \(i=0.04\), find the break-even charge per \(\$1{,}000\) of NAR, rounded to the nearest cent. (Answer: \(\$7.69\) per \(\$1{,}000\))
  4. A two-year policy has constant NAR \(\$100{,}000\) and \(i=0\). In duration 1, \(q_1=0.01\) and the charge is \(\$11\) per \(\$1{,}000\). In duration 2, \(q_2=0.02\) and the charge is \(\$19\) per \(\$1{,}000\). Find the expected cumulative, undiscounted margin through duration 2 per original policy, assuming mortality is the only termination cause. (Answer: \(\$1.00\))
  5. A policy has NAR \(\$75{,}000\), \(q=0.006\), \(i=0.05\), and a charge of \(\$4\) per \(\$1{,}000\). Find the start-of-year mortality margin. (Answer: \(-\$128.57\))

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