Recovering Expense Assumptions from Gross Premiums

Why Recover Expense Assumptions?

A quoted gross premium reflects more than the expected present value of benefits. It may also provide for expenses paid at issue, expenses incurred when renewal premiums are collected, or expenses calculated as a percentage of premium. If the benefits, interest, mortality, and some expense assumptions are known, the quoted premium can be used to infer an unknown expense amount. The key is to match each expense to its correct timing and premium base.

This tutorial considers a level premium payable annually in advance while the policy is in force. The hypothetical contracts use a benefit paid at the end of the year of death. The methods also apply when the benefit present value is calculated using other standard life-contingency models, provided the timing and payment periods are stated consistently. This topic is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $P$ be a quoted level gross premium. Let $B$ be the expected present value of benefits and any known claim-related expenses. Let $\ddot{a}$ be the expected present value of the premium stream. Let $E_0$ be a fixed issue expense paid at time 0, $e$ a fixed expense incurred at each renewal premium, and $c$ the percentage of each gross premium consumed by a percentage expense. For an $n$-year contract with annual premiums in advance while alive, $e$ applies only to premiums after the first, so its present-value factor is $\ddot{a}-1$.

The Pricing Equation and Its Unknowns

For the timing just specified, the premium present value is $P\ddot{a}$. The present value of expenses is the issue expense $E_0$, plus renewal expenses $e(\ddot{a}-1)$, plus percentage expenses $cP\ddot{a}$. The equivalence principle equates the present value of premiums with the present value of benefits and expenses:

P\ddot{a}=B+E_0+e(\ddot{a}-1)+cP\ddot{a}.

A fixed expense is an amount in dollars, while a percentage expense varies with the premium. Therefore, the percentage expense is based on the present value of premiums, not on the benefit present value. Rearranging the equation also shows how the types of expenses affect the premium:

P=\frac{B+E_0+e(\ddot{a}-1)}{(1-c)\ddot{a}}.

This rearranged form applies when the listed expense bases are the only ones present and the same percentage $c$ applies to every premium, including the first. If a percentage expense applies only to renewal premiums, its factor must instead reflect only those premiums. Likewise, if the timing of a fixed expense differs from the timing assumed here, its present value must be adjusted.

Key takeaway: A single quoted premium gives one present-value equation. It can identify one unknown expense if the other assumptions are known, but it generally cannot identify two or more independent unknown expense assumptions on its own.

Why the Rearrangement Works

The equation follows by adding the expected present values of all cash flows on the insurer’s side of the equivalence principle. A premium is paid at time 0 if the policy is issued, and at each later premium date only if the insured is alive. Those payments have present-value factor $\ddot{a}$. The first premium is not a renewal premium, so renewal expenses have factor $\ddot{a}-1$, not $\ddot{a}$.

Write the premium present value.
The level premium $P$, payable on the stated premium dates while the policy is in force, has present value $P\ddot{a}$.

Add the present values of benefits and expenses.
These are $B$, $E_0$, $e(\ddot{a}-1)$, and $cP\ddot{a}$, under the stated timing and expense bases.

Equate the two sides and isolate $P$.
Subtract $cP\ddot{a}$ from both sides, giving $(1-c)P\ddot{a}=B+E_0+e(\ddot{a}-1)$. Dividing by $(1-c)\ddot{a}$ gives the premium expression above.

For expense recovery, the quoted premium $P$ and the other known quantities are inserted into the original equation. The remaining unknown can then be isolated. The following examples use a deliberately simple mortality model so the cash-flow factors can be calculated directly.

Recovering Fixed and Percentage Expenses

Worked Example: Inferring a Fixed Issue Expense

A hypothetical three-year term policy pays $10,000 at the end of the year of death. Annual premiums are payable in advance while the insured is alive, at times 0, 1, and 2. Assume an effective annual interest rate of 0%, and a conditional probability of death of 0.20 in each year. A quoted gross premium is $2,500 per year. A fixed renewal expense of $200 is incurred at each premium after the first, and there is no percentage expense. Find the fixed issue expense.

Method. Calculate the benefit and premium present values, then subtract the known renewal expenses from the total expense amount implied by the quote.

The probabilities of being alive at times 0, 1, and 2 are $1$, $0.80$, and $0.80^2=0.64$. With 0% interest, the premium present-value factor is:

\ddot{a}=1+0.80+0.64=2.44.

The probabilities of death in years 1, 2, and 3 are $0.20$, $0.80(0.20)=0.16$, and $0.64(0.20)=0.128$. Thus the benefit present value is:

B=10{,}000(0.20+0.16+0.128)=10{,}000(0.488)=\$4{,}880.

The present value of quoted premiums is $2{,}500(2.44)=\$6{,}100$. Therefore, total expenses have present value $6{,}100-4{,}880=\$1{,}220$. Renewal expenses apply only at times 1 and 2, with factor $0.80+0.64=2.44-1=1.44$. Their present value is $200(1.44)=\$288$. The inferred issue expense is:

E_0=1{,}220-288=\boxed{\$932.00}.

Check. Benefits plus expenses have present value $4{,}880+932+288=\$6{,}100$, equal to the quoted premium present value. The common mistake avoided here is charging a renewal expense against the first premium: doing so would incorrectly use $2.44$ instead of $1.44$ as its factor.

Worked Example: Inferring a Percentage Expense

Use the same hypothetical three-year term policy, 0% effective annual interest rate, 0.20 annual conditional death probability, and $10,000 end-of-year death benefit. The quoted gross premium remains $2,500 per year. Now assume a fixed issue expense of $322 and a fixed renewal expense of $200 at each premium after the first. The remaining expense is a percentage $c$ of every gross premium. Find $c$.

Method. Calculate the total expense present value implied by the premium quote. Subtract the known fixed expenses, then divide the remaining expense present value by the present value of gross premiums.

The prior cash-flow calculations give $\ddot{a}=2.44$ and $B=\$4{,}880$. The quoted premium present value is $2{,}500(2.44)=\$6{,}100$, so total expenses have present value $6{,}100-4{,}880=\$1{,}220$. The fixed expenses have present value:

322+200(2.44-1)=322+288=\$610.

The percentage expense therefore has present value $1{,}220-610=\$610$. Because it applies to every gross premium, its base is $2{,}500(2.44)=\$6{,}100$. Hence:

c=\frac{610}{6{,}100}=0.10=\boxed{10.0000\%}.

Check. Percentage expense plus fixed expenses is $610+610=\$1{,}220$, the total expense present value implied by the quote. The mistake avoided is dividing the percentage expense by the benefit present value or by a single annual premium instead of by the present value of all premiums on which the expense is charged.

Recovering a Renewal Expense and Assessing Identifiability

Worked Example: Inferring a Fixed Renewal Cost

A hypothetical two-year term policy pays $10,000 at the end of the year of death. The effective annual interest rate is 0%, and the conditional probability of death in each year is 0.20. Annual premiums of $2,500 are payable in advance while the insured is alive, at times 0 and 1. A fixed issue expense of $180 is known. There is no percentage expense. Find the fixed expense $e$ incurred at the renewal premium.

Method. Determine the excess of premium present value over benefit present value, subtract the known issue expense, and divide by the probability-weighted renewal-premium factor.

The probability of being alive at time 1 is $0.80$, so the premium factor is $\ddot{a}=1+0.80=1.80$. The death probabilities in years 1 and 2 are $0.20$ and $0.80(0.20)=0.16$. The benefit present value is $10{,}000(0.20+0.16)=\$3{,}600$. The premium present value is $2{,}500(1.80)=\$4{,}500$, so total expenses have present value $4{,}500-3{,}600=\$900$.

The issue expense accounts for $180 of that amount. The renewal expense has factor $\ddot{a}-1=0.80$, because the time-1 premium is paid only if the insured survives the first year. Thus:

e=\frac{900-180}{0.80}=\boxed{\$900.00}.

Check. The issue and renewal expense present values sum to $180+900(0.80)=\$900$. Adding the $3,600 benefit present value gives $4,500, the present value of quoted premiums. The common mistake avoided is treating the renewal expense as certain at time 1: its expected present value includes the 0.80 probability of survival to that premium date.

One quote does not generally determine several unknowns. For example, if both $E_0$ and $e$ are unknown, the equation $P\ddot{a}-B=E_0+e(\ddot{a}-1)$ supplies one relationship between them. Many pairs can satisfy that relationship. A second contract can provide another equation, but only if the new equation adds independent information. Merely repeating the same contract assumptions and premium factor will not separate the two unknowns.

When several quotes are available, write one present-value equation for each contract using that contract’s own benefit and premium factors. Solve the resulting system for the unknown expense parameters, then substitute the solutions into every equation. A negative inferred expense or a large unexplained difference may indicate inconsistent assumptions, a mismatched expense basis, or insufficient information; the quote alone does not establish which explanation is correct.

Common Mistakes

Using the wrong renewal factor. If the first premium is at time 0 and has no renewal expense, use $\ddot{a}-1$ for a per-renewal expense, not $\ddot{a}$.
Ignoring survival to later premium dates. The premium and renewal expense factors include the probability that the insured is alive when each payment is due.
Applying a percentage to the wrong base. If the percentage applies to every gross premium, its present value is $cP\ddot{a}$, not $cB$.
Confusing an annual expense with its present value. A fixed renewal expense $e$ is multiplied by the applicable expected present-value factor before it is compared with other present values.
Trying to recover multiple unknowns from one equation. One quote may constrain multiple expenses without identifying each one separately.

Key takeaway: Recover an expense by first calculating the benefit and premium present values under the stated assumptions. Then isolate the unknown using the correct expense timing and base, and verify the result by substituting it back into the equivalence equation.

Check Your Understanding

In each problem, use the stated hypothetical mortality and interest assumptions. Unless noted otherwise, benefits are paid at the end of the year of death and premiums are annual, in advance, while alive. Round dollar answers to the nearest cent and percentage answers to four decimal places.

A two-year policy pays $5,000 at the end of the year of death. The effective annual interest rate is 0%, and the conditional probability of death in each year is 0.10. The quoted gross premium is $650. A $100 fixed renewal expense applies at time 1 if the insured is alive, with no other expenses except a fixed issue expense. Find the issue expense. (Answer: $195.00)
A two-year policy pays $8,000 at the end of the year of death. The effective annual interest rate is 0%, and the annual conditional death probability is 0.25. The gross premium is $2,500. A fixed issue expense of $200 and a $100 renewal expense at time 1 are known. If a percentage expense applies to both premiums, what is its percentage of gross premium? (Answer: 13.7143%)
A three-year policy pays $12,000 at the end of the year of death. The effective annual interest rate is 0%, and the annual conditional death probability is 0.10. The gross premium is $1,600. A fixed issue expense is $300, and there is no percentage expense. Find the fixed expense at each renewal premium after the first. (Answer: $458.48)
A two-year policy pays $5,000 at the end of the year of death. The effective annual interest rate is 0%, and the annual conditional death probability is 0.20. The gross premium is $1,200. The issue expense is $90, and a percentage expense of 10% applies to both premiums. Find the fixed renewal expense at time 1 if alive. (Answer: $67.50)
Two hypothetical contracts each pay $10,000 at the end of the year of death, with a 0% effective annual interest rate and an annual conditional death probability of 0.20. The first contract lasts two years and has a gross premium of $2,500; the second lasts three years and has a gross premium of $2,450. Both have the same fixed issue expense $E_0$ and the same fixed renewal expense $e$ at each premium after the first, with no percentage expense. Find $E_0$ and $e$. (Answer: $E_0=\$652.50,\ e=\$309.38$)

What You'll Learn