Expense Assumptions In Modified Reserves

Why Expense Assumptions Matter

A reserve calculation depends on more than premiums, benefits, interest, and mortality. Expenses also create cash outflows, and their timing matters: an acquisition expense may be incurred once at issue, a renewal expense may be incurred while a policy remains in force, and a claim expense may arise only when a claim occurs. Combining these items into one undifferentiated expense can put the right amount at the wrong time or attach it to the wrong event.

A modified reserve uses the premium and assumptions specified for its valuation basis. In particular, its expense assumptions need not equal the expenses expected under the contract’s operating assumptions. This tutorial is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Model and Expense Timing

Consider a discrete-time life insurance contract with annual durations $k=0,1,\ldots,n-1$. The reserve $V_k^m$ is measured immediately before the cash flows at the start of duration $k$. If the insured is alive and the policy is in force then, the valuation premium $P_k^m$ is received and the valuation renewal expense $e_k^r$ is paid. At the end of the year, death occurs with probability $q_k={}_1q_{x+k}$, and survival to the next duration occurs with probability $p_k={}_1p_{x+k}=1-q_k$. A death benefit $B_k$ and claim expense $c_k$ are paid at the end of the year if death occurs. Let $v=1/(1+i)$, where $i$ is the effective annual interest rate on the valuation basis.

An acquisition expense $e^a$, if present, is incurred once at issue, at the start of duration 0. The valuation premium $P_k^m$ is a receipt; expenses are outflows. For simplicity, the examples use level benefits and annual cash flows, but the amounts can vary by duration.

Definition: On this timing convention, the modified prospective reserve is the present value of future valuation-basis outgo less the present value of future valuation-basis premium income, conditional on the policy being in force at duration $k$. Acquisition, renewal, and claim expenses are separate outgo items: acquisition expense is paid at issue, renewal expense is paid at the start of a duration while the policy is in force, and claim expense is paid only if the claim event occurs.

Let $I_k^a$ equal 1 at issue and 0 at later durations. The reserve recursion is

V_k^m = I_k^a e^a+e_k^r-P_k^m + v\left[q_k(B_k+c_k)+p_kV_{k+1}^m\right], \qquad k=0,\ldots,n-1.

For a contract with no payments after the end of duration $n-1$, set $V_n^m=0$. If a maturity benefit is payable then, it must instead be included in the appropriate end-of-year cash flow. Any other payment timing requires a corresponding change to the recursion.

Why the Recursion Holds

The recursion follows by conditioning on what happens during one year. At the start of duration $k$, the acquisition expense, if applicable, and renewal expense are outflows, while the premium is an inflow. Their net amount is therefore $I_k^a e^a+e_k^r-P_k^m$. At year-end, death has probability $q_k$ and produces the payment $B_k+c_k$; survival has probability $p_k$ and leaves the insurer with the reserve $V_{k+1}^m$. The expected year-end amount is discounted one year. Adding the start-of-year net cash flow gives the stated recursion.

Place each expense in time.
Record acquisition expense at issue, renewal expense at the start of each applicable duration, and claim expense at the time of the claim.

Apply the relevant event condition.
Renewal expense is conditional on the policy being in force at the start of the year. Claim expense is conditional on a claim; in the life insurance model here, it is multiplied by the death probability.

Use the valuation basis consistently.
Use valuation premiums, interest, probabilities, benefits, and expenses in the reserve recursion. Do not substitute expected contract expenses unless the question specifies them as the valuation assumptions.

Work backward from the terminal condition.
Set the terminal reserve or maturity cash flow correctly, calculate the last duration first, and then continue to earlier durations.

Worked Reserve Calculations

Worked Example: Separating Issue, Renewal, and Claim Expenses

A hypothetical two-year term policy pays a death benefit of $5,000 at the end of the year of death. The annual valuation premium is $600, payable at the start of each year while the policy is in force. The valuation basis uses $i=10\%$, $q_0=0.10$, and $q_1=0.20$. The acquisition expense at issue is $300. There is no renewal expense in duration 0; the renewal expense at the start of duration 1 is $80. Claim expenses are $100 for death in duration 0 and $200 for death in duration 1. Find $V_0^m$.

Method. Use the prospective recursion, keeping expenses at their specified times. The common mistake to avoid is adding the claim expense to the year-end payment for every policy, rather than only for policies that result in a claim.

At the end of duration 1, the claim payment is $5,000 + $200 = $5,200. With $v=1/1.10$ and terminal reserve $V_2^m=0$,

V_1^m = 80-600+\frac{0.20(5{,}200)+0.80(0)}{1.10} = -520+\frac{1{,}040}{1.10} = 425.454545\ldots

At duration 0, the claim payment is $5,000 + $100 = $5,100. The acquisition expense is paid at issue, so

\begin{aligned} V_0^m &=300-600+\frac{0.10(5{,}100)+0.90V_1^m}{1.10}\\ &=-300+\frac{510+0.90(425.454545\ldots)}{1.10}\\ &=-300+\frac{892.909090\ldots}{1.10}\\ &=511.735537\ldots \end{aligned}

Therefore, $V_0^m=\$511.74$ to the nearest cent. As a check, the duration-1 reserve includes the renewal expense immediately, while the claim expense enters only through the death probability.

Worked Example: Valuation Expenses Differ From Expected Contract Expenses

A hypothetical policy has one year remaining. At the start of the year, it receives a $700 premium and incurs a renewal expense. At year-end, it pays a $5,000 death benefit if death occurs. The valuation basis has $i=10\%$, $q=0.20$, renewal expense $100, and claim expense $250. The expected contract expenses are instead $60 for renewal and $150 per claim. Calculate the reserve on the valuation expense basis, and compare it with the reserve that would result from using the expected contract expenses as the expense assumptions.

Method. Calculate each reserve with the same premium, benefit, interest, and mortality assumptions, changing only the expense assumptions. The common mistake to avoid is silently using the expected contract expenses in a reserve calculation that specifies a different valuation expense basis.

On the valuation expense basis, the claim payment conditional on death is $5,000 + $250 = $5,250. There is no reserve after the one-year contract ends, so

V_0^m = -700+100+\frac{0.20(5{,}250)}{1.10} = -600+954.545454\ldots = 354.545454\ldots

Using the expected contract expenses instead, the claim payment conditional on death is $5,000 + $150 = $5,150. The alternative calculation is

V_0^{\text{contract expenses}} = -700+60+\frac{0.20(5{,}150)}{1.10} = -640+936.363636\ldots = 296.363636\ldots

The difference is $354.545454\ldots - $296.363636\ldots = $58.181818\ldots, or $58.18 to the nearest cent. This also follows directly from the expense differences: the valuation renewal expense is $40 higher, and the valuation claim expense is $100 higher with probability 0.20, giving $40+0.20(100)/1.10=\$58.18$. The reserve uses the valuation assumptions specified for the calculation; the expected contract expenses are not interchangeable with them.

Worked Example: Duration-Dependent Claim Expenses

A hypothetical two-year term policy pays $8,000 at death. The annual valuation premium is $800, the effective annual interest rate is $i=5\%$, and the death probabilities are $q_0=0.10$ and $q_1=0.25$. The issue acquisition expense is $500. The renewal expense is $120 at the start of duration 1 and zero at issue. Claim expenses are $100 for a claim in duration 0 and $400 for a claim in duration 1. Calculate the issue reserve.

Method. Work backward, using the duration-specific claim expense with that duration’s death probability. The common mistake to avoid is applying the later claim expense to the earlier duration or charging it on survival.

At duration 1, the payment conditional on death is $8,000 + $400 = $8,400. With $V_2^m=0$ and $v=1/1.05$,

V_1^m = -800+120+\frac{0.25(8{,}400)}{1.05} = -680+2{,}000 = 1{,}320.

At duration 0, the payment conditional on death is $8,000 + $100 = $8,100. The survival probability is $p_0=1-0.10=0.90$. Include the acquisition expense at issue:

\begin{aligned} V_0^m &=500-800+\frac{0.10(8{,}100)+0.90(1{,}320)}{1.05}\\ &=-300+\frac{810+1{,}188}{1.05}\\ &=-300+1{,}902.857142\ldots\\ &=1{,}602.857142\ldots \end{aligned}

Thus the issue reserve is $1,602.86 to the nearest cent. The calculation checks out by assigning the $400 claim expense only to duration-1 deaths, and the $100 claim expense only to duration-0 deaths.

Common Expense-Reserve Traps

Expense calculations are particularly sensitive to signs and conditioning. In the preceding first example, the net start-of-duration-1 cash flow is renewal expense less premium: $80-600=-520$. The correct reserve is

V_1^m=-520+\frac{0.20(5{,}200)}{1.10}=425.454545\ldots

A sign error that treats the premium as an outflow gives $80+600=680$ instead of $-520$, and therefore gives the wrong result

680+\frac{0.20(5{,}200)}{1.10} = 1{,}625.454545\ldots

The error is $1,200, twice the $600 premium, because only the premium’s sign was reversed. It is not twice the present value of the premium: the premium is paid at the start of the year and is not discounted in this timing convention.

Also check whether an expense is conditional on being in force or on a claim. A renewal expense at the start of a year is not multiplied by the probability of surviving that year; the policy is in force when that start-of-year expense is paid. A claim expense at year-end is multiplied by the probability of the claim event. These distinctions are part of the model, not optional adjustments.

Key takeaway: Use a separate amount and timing for each expense type, and apply the valuation expense assumptions—not a different expected expense schedule—when calculating a reserve on the valuation basis. Check the premium sign, the event probability attached to each expense, and the terminal condition.

Check Your Understanding

For each problem, use the stated valuation expense assumptions and round monetary answers to the nearest cent.

A one-year policy has a $4,000 death benefit, a $500 start-of-year premium, renewal expense $50, and claim expense $200. The valuation basis has $i=10\%$ and $q=0.10$; there is no terminal reserve. Find the reserve at the start of the year. (Answer: $-\$68.18$)
A two-year term policy pays $6,000 at death. It has premium $700 at the start of each year, issue acquisition expense $200, and a $80 renewal expense at the start of duration 1. The valuation basis has $i=10\%$, $q_0=0.20$, and $q_1=0.10$; claim expenses are $100 in duration 0 and $200 in duration 1. Find the issue reserve. (Answer: $568.10)
A policy has one year remaining, with a $10,000 death benefit, premium $900, renewal expense $120, and claim expense $400. The valuation basis has $i=5\%$ and $q=0.25$. Find the reserve immediately before the start-of-year cash flows. (Answer: $1,696.19)
A one-year policy has a $5,000 death benefit, premium $600, $i=10\%$, and $q=0.20$. The valuation basis uses renewal expense $60 and claim expense $300; expected contract expenses are renewal $30 and claim expense $100. Calculate the reserve under each expense schedule and their difference. (Answer: valuation-expense reserve $423.64; contract-expense reserve $357.27; difference $66.36)
A two-year term policy pays $6,000 at death and has premium $750 at the start of each year. The issue acquisition expense is $250; the duration-1 renewal expense is $100. The valuation basis has $i=10\%$, $q_0=0.10$, and $q_1=0.20$; claim expenses are $0 in duration 0 and $200 in duration 1. Find the issue reserve. (Answer: $435.95)

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