Why Expense Classification Matters
A gross premium must fund more than the insured benefit. It may also need to cover costs incurred when a policy is issued, costs that recur while it remains in force, costs tied to collecting premiums, and costs incurred when a claim is paid. To include each cost correctly, identify both when it occurs and what determines its amount.
These are separate questions. “Acquisition” and “renewal” describe expense timing or purpose, while “per policy” and “per premium” describe the basis on which an expense amount is set. A single expense can therefore be described on both dimensions: for example, a fixed per-policy renewal expense paid at an anniversary.
The labels do not determine the payment date by themselves. An acquisition cost might be paid at issue or later, and a renewal cost might be paid at an anniversary or when a renewal premium is collected. The contract assumptions must specify the actual timing. Likewise, do not infer that every policy expense is incurred only when a premium is paid: a contract can incur an administration cost at a policy anniversary even if no premium is due then.
From an Expense to Its Expected Present Value
Use the same interest and contingency assumptions for expenses as for benefits and premiums. Let \(i\) be the effective annual interest rate and \(v=1/(1+i)\). If a fixed expense \(C\) is paid at time \(t\) only when event \(E\) occurs, its expected present value is \(Cv^tP(E)\). If the amount is random, replace \(C\) by its conditional expected amount given that the expense-triggering event occurs.
Here \(I_E\) equals 1 when \(E\) occurs and 0 otherwise. The equality follows because \(C\), \(v^t\), and the payment time are fixed, so \(E[Cv^t I_E]=Cv^tE[I_E]=Cv^tP(E)\). For several expenses, add their expected present values. This uses linearity of expectation; the expense events do not need to be independent.
For a life policy, let \({}_kp_x\) denote the probability that a life aged \(x\) survives \(k\) years, and \({}_kq_x\) the probability of death within \(k\) years. For a discrete model with death benefits paid at year-end, the probability of death in year \(k\) is \({}_{k-1}p_x q_{x+k-1}\). That is the probability to use for an expense incurred at that death, not simply the conditional death probability \(q_{x+k-1}\).
Classifying Common Expense Bases
| Expense description | Amount basis | Typical timing and contingency |
|---|---|---|
| Policy setup or underwriting cost | Fixed amount per new policy | At issue, whether or not a claim later occurs |
| Policy continuation or administration cost | Fixed amount per policy or policy-year | At specified anniversaries while the policy is in force |
| Premium collection cost | Fixed amount or percentage per premium | When each premium is collected |
| Claim handling cost | Fixed amount, percentage of benefit, or claim-dependent amount | When the claim occurs or is settled |
The table gives common patterns, not universal rules. An amount described as “per policy-year” might be incurred at the start or end of the year; that distinction changes its present value and survival contingency. Similarly, a fixed expense “per premium” is different from a percentage-of-premium expense: the former has a fixed dollar amount per collection, while the latter changes with the premium amount.
Worked Example: Separating Five Expense Components
A hypothetical two-year term policy has annual premiums in advance at times 0 and 1, provided the insured is alive. The conditional death probabilities are \(q_0=0.20\) in year 1 and \(q_1=0.25\) in year 2. The effective annual interest rate is \(i=10\%\). Expenses are: $30 at issue; a $9 renewal cost at time 1 if alive; $4 of policy administration at each premium date while in force; $1 per premium collected; and $15 of claim handling at the end of the year of death. Find the expected present value of all expenses.
Method. Classify each cost, identify the event that triggers payment, and discount each expected payment. The method avoids treating conditional mortality rates as unconditional death probabilities.
The discount factor is \(v=1/1.10=0.9090909\), and \(v^2=0.8264463\). The probability of surviving year 1 is \(p_0=1-0.20=0.80\). The probability of death in year 2, measured from issue, is \(0.80(0.25)=0.20\).
Acquisition expense: $30 is paid at time 0 regardless of survival, so its present value is \(30\).
Renewal expense: $9 is paid at time 1 only if alive, so its expected present value is \(9(0.80)v=9(0.80)(0.9090909)=\$6.5455\).
Per-policy administration: $4 is paid at times 0 and 1 if the policy is in force at those dates. Its expected present value is \(4[1+(0.80)v]=4[1+0.7272727]=\$6.9091\).
Per-premium expense: $1 is incurred at each premium collection, which occurs at the same dates and on the same survival conditions. Its expected present value is \(1[1+(0.80)v]=\$1.7273\).
Claim expense: The unconditional probabilities of death in years 1 and 2 are \(0.20\) and \(0.80(0.25)=0.20\), respectively. The expected present value is \(15[0.20v+0.20v^2]=15[0.1818182+0.1652893]=\$5.2066\).
Adding the five components gives \(30+6.5455+6.9091+1.7273+5.2066=\$50.3885\), or $50.39 to the nearest cent. As a check, the acquisition cost is the only amount paid with certainty at time 0; every later expense is weighted by the probability that its triggering event occurs.
Putting Expense Terms into a Gross Premium Equation
A gross premium equation balances the expected present value of premiums with the expected present value of benefits and expenses, on the stated pricing assumptions. The expense classification tells you which cash flows belong on the expense side and how to weight them. It does not, by itself, determine the premium-payment schedule; that schedule must also be specified.
For example, if a level gross premium \(G\) is paid at times 0 and 1 only while the insured is alive, its expected present value is \(G[1+{}_1p_xv]\). If an expense equals a proportion \(c\) of every collected premium, its expected present value is \(cG[1+{}_1p_xv]\), because it has the same payment dates and survival conditions as those premiums. A fixed per-premium expense \(e\), by contrast, contributes \(e[1+{}_1p_xv]\), not a percentage of \(G\).
An expense that is paid regardless of whether the policy is renewed is not automatically multiplied by a survival probability. Its actual triggering condition controls. For instance, an issue cost payable at time 0 has present value equal to its full amount, while a fixed anniversary expense payable only to a surviving policy has present value equal to its amount times the probability of survival to that anniversary and the appropriate discount factor.
Worked Example: Keeping a Premium-Percentage Expense Separate
A hypothetical two-year term policy has a death benefit of $1,000 paid at the end of the year of death. Premiums are level and payable in advance at times 0 and 1 while the insured is alive. The conditional death probabilities are 0.10 in year 1 and 0.20 in year 2, and \(i=5\%\). Expenses are $50 at issue, $8 at time 1 if alive, $20 plus 1% of the death benefit for each claim, and 3% of each premium collected. Write the gross premium equation and calculate the present-value coefficients and expense amounts. Do not solve for the premium.
Method. First convert conditional mortality to unconditional death probabilities; then put each expense on the basis that determines its amount. This avoids treating a percentage-of-premium cost as a fixed dollar amount.
Here \(v=1/1.05=0.9523810\), \(v^2=0.9070295\), and the probability of surviving year 1 is \(0.90\). The unconditional probabilities of death in years 1 and 2 are \(0.10\) and \(0.90(0.20)=0.18\).
The present-value factor for premiums payable at times 0 and 1 is \(1+0.90v=1+0.8571429=1.8571429\). Thus, for level gross premium \(G\), premium present value is \(1.8571429G\). The 3% premium expense has present value \(0.03(1.8571429G)=0.0557143G\).
The claim expense per death is \(20+0.01(1{,}000)=\$30\). Its expected present value is \(30[0.10v+0.18v^2]=30[0.0952381+0.1632653]=\$7.7551\). The $8 renewal expense has expected present value \(8(0.90)v=\$6.8571\), and the issue expense has present value $50.
The death-benefit present value is \(1{,}000[0.10v+0.18v^2]=\$258.5034\). Therefore the gross premium equation is
The fixed expense present values sum to \(50+6.8571+7.7551=\$64.6122\). The premium-percentage expense remains a coefficient of \(G\), rather than being added as though it were a known fixed dollar amount. The equation is balanced by placing benefits and all expense present values on the right and premium present value on the left.
Common Exam Traps
- Using a conditional death probability as if it were unconditional. In the first example, year-2 death probability from issue is \(0.80(0.25)=0.20\), not \(0.25\).
- Ignoring when a payment is made. A $6 cost at the beginning of a policy year is not discounted like a $6 cost at the end of that year.
- Confusing a per-policy amount with a per-premium amount. A fixed policy expense can be incurred even when no premium is collected; a per-premium expense is triggered by collection.
- Turning a percentage loading into a fixed dollar cost. If an expense is \(cG\), its value depends on the unknown gross premium \(G\) and belongs as a term in the equation.
- Counting costs on policies that are no longer in force. Apply survival probabilities only where the stated condition requires the policy to be in force at the expense date.
For any new expense item, write down its amount basis, payment date, and triggering event before calculating. This simple three-part record helps ensure that an expense is neither omitted nor counted under the wrong probability or premium basis. The expense classification and present-value setup are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, identify the relevant expense basis and use the stated timing and assumptions.
- A hypothetical two-year contract has \(i=0\), conditional death probabilities 0.20 in year 1 and 0.25 in year 2, an $18 issue cost, a $7 expense at time 1 if alive, and a $4 expense at each year-end death claim. Find the expected present value of these expenses. (Answer: \(18+7(0.80)+4[0.20+0.80(0.25)]=\$25.20\).)
- A policy has premiums \(G\) at time 0 and at time 1 if the insured survives year 1, with \({}_1p_x=0.75\) and \(i=0\). A 5% expense is incurred on every premium collected. Express its expected present value as a multiple of \(G\). (Answer: \(0.05G[1+0.75]=0.0875G\).)
- A three-year policy has a $9 per-policy administration expense at the beginning of each policy year while in force. The probabilities of being in force at times 0, 1, and 2 are 1, 0.90, and 0.72. If \(i=5\%\), calculate the expense present value to the nearest cent. (Answer: \(9[1+0.90/1.05+0.72/(1.05)^2]=\$22.59\).)
- A hypothetical policy pays a claim expense consisting of $12 plus 2% of a $400 benefit, whenever death occurs. The unconditional probabilities of death in years 1 and 2 are 0.10 and 0.15, and \(i=0\). Find the expected present value of the claim expenses. (Answer: \([12+0.02(400)](0.10+0.15)=\$5.00\).)
- A policy has \(i=0\), with probabilities of being in force at times 0, 1, and 2 equal to 1, 0.90, and 0.72. Compare a $6 expense paid at times 1 and 2 only if in force with a $6 per-policy expense paid at times 0, 1, and 2 if in force. Find both expected present values and their difference. (Answer: anniversary expense \(=6(0.90+0.72)=\$9.72\); beginning-of-year expense \(=6(1+0.90+0.72)=\$15.72\); difference \(=\$6.00\).)