Solving Gross Premiums with Multiple Expense Bases

Why Multiple Expense Bases Require Care

A gross premium must fund both the insured benefit and the expenses associated with issuing and administering the contract. When several expense types apply, the main challenge is not just adding them: each expense belongs on its own cash-flow base and at its own payment time. An expense paid once at issue is not weighted like one paid with every premium, and a claim expense is not weighted like either of those.

This tutorial develops a level-premium equation for a hypothetical $n$-year term insurance contract. Premiums are paid annually in advance while the insured is alive. The death benefit and any claim-related expenses are paid at the end of the year of death. The calculation uses the equivalence principle: the expected present value of premiums equals the expected present value of benefits and expenses. This approach is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: A gross premium is a premium whose expected present value is set to cover both the contract benefits and the expenses included in the pricing model. In this tutorial, the premium is level and annual. Expenses may include a fixed amount at issue, a fixed amount per premium payment, a percentage of each premium, and amounts incurred when a claim is paid.

Set Up the Cash-Flow Factors

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Write ${}_kp_x$ for the probability that a life aged $x$ survives $k$ years, and $q_{x+k}$ for the probability of dying during the following year given survival to age $x+k$. Thus, the probability of being alive at the $k$-th annual premium date is ${}_kp_x$, and the probability of dying in year $k+1$ is ${}_kp_xq_{x+k}$.

Define two factors for this contract. The premium factor $M$ is the expected present value of one dollar paid at each annual premium date while the insured is alive. The claim factor $D$ is the expected present value of one dollar paid at the end of the year of death during the term.

M=\sum_{k=0}^{n-1}v^k\,{}_kp_x, \qquad D=\sum_{k=0}^{n-1}v^{k+1}\,{}_kp_xq_{x+k}.

For example, the $k=0$ term in $M$ is 1 because the first premium is paid at issue. The first term in $D$ is $vq_x$ because a death in the first policy year produces a payment one year from issue. These factors reflect the stipulated timing; different premium or benefit timing requires different present-value factors.

Model assumptions: Premiums are paid at the start of each year while the insured is alive, for at most $n$ years. A death benefit $B$ is paid at the end of the year of death. A fixed issue expense $C_0$ is paid at issue. A fixed expense $e$ is incurred with each premium payment, and a fraction $\alpha$ of each gross premium is an expense. At claim, an expense $K+cB$ is paid, where $K$ is fixed and $c$ is a proportion of the benefit. There are no lapses or other cash flows in the model.

Deriving the Gross Premium Equation

Let $P$ be the annual gross premium. Its expected present value is $PM$. The fixed issue expense has present value $C_0$. Since both the fixed per-payment expense and the percentage-of-premium expense occur on premium dates, their expected present values are $eM$ and $\alpha PM$, respectively. The claim benefit and claim expenses have expected present value $(B+K+cB)D$. Equating the expected present values gives the gross premium equation.

PM=C_0+eM+\alpha PM+(B+K+cB)D.

The unknown premium appears on both sides because a percentage-of-premium expense depends on $P$. Collecting those terms gives the solution.

P(1-\alpha)M=C_0+eM+(B+K+cB)D, \qquad P=\frac{C_0+eM+(B+K+cB)D}{(1-\alpha)M}.

This rearrangement also explains the effect of a percentage loading: it reduces the part of each premium available to fund the other outgo. For a positive finite premium under this model, $M>0$ and $\alpha<1$ are required. If $\alpha\geq1$ and the other outgo is positive, the equation cannot produce a positive finite premium.

Calculate the premium factor.
Weight each possible premium by the probability the insured is alive on its due date and discount it to issue.

Calculate the claim factor.
Weight each possible claim by its death probability in that year and discount the year-end payment.

Place each expense on its correct base.
Use the premium factor for expenses incurred with premium payments, and the claim factor for expenses incurred when a claim is paid.

Solve and check.
Isolate $P$, then substitute it into the original expected-present-value equation to check the balance.

Worked Calculations

Worked Example: Fixed and Percentage Premium Expenses

A hypothetical two-year term policy has a death benefit of $10,000. The effective annual interest rate is 10%. The conditional death probabilities are $q_x=0.10$ in the first year and $q_{x+1}=0.20$ in the second year. A $100 issue expense is paid at issue, a $30 expense is incurred with each premium, a 5% expense is deducted from each gross premium, and a fixed $200 claim expense is paid with each death benefit. Find the annual gross premium. Use the method of forming separate premium and claim factors before solving for $P$.

Premium factor. The survival probability to the second premium date is $p_x=1-0.10=0.90$. With $v=1/1.10=10/11$,

M=1+v(0.90)=1+\frac{9}{11}=\frac{20}{11}.

Claim factor. The probability of death in the second year is $0.90(0.20)=0.18$. Therefore,

D=v(0.10)+v^2(0.18) =\frac{1}{11}+\frac{18}{121} =\frac{29}{121}.

The total payment at each death is $10,000+$200=$10,200. The expected present value of benefits and claim expenses is $10{,}200(29/121)=295{,}800/121$. The issue expense is $100, and the per-premium expense has expected present value $30(20/11)=600/11$. Thus, the numerator of the solved equation is

\frac{295{,}800}{121}+100+\frac{600}{11} =\frac{295{,}800+12{,}100+6{,}600}{121} =\frac{314{,}500}{121}.

The denominator is $(1-0.05)(20/11)=19/11$. Hence

P=\frac{314{,}500/121}{19/11} =\frac{314{,}500}{209} =1{,}504.7847\ldots

The annual gross premium is $1,504.78, rounded to the nearest cent. As a check, $P(1-0.05)M$ equals the numerator of the solved equation, so the expected present value available after the percentage expense covers the issue, per-payment, benefit, and claim expenses. The method avoids treating the percentage expense as a fixed dollar amount.

Worked Example: A Claim Expense Proportional to the Benefit

A hypothetical two-year term policy pays $8,000 at death. The effective annual interest rate is 5%; the first-year death probability is 0.05 and the second-year conditional death probability is 0.10. Expenses are $180 at issue, $25 with each premium, 8% of each gross premium, and 4% of the benefit at claim. Find the annual gross premium. The percentage claim expense is paid only when a death benefit is paid.

Premium factor. The survival probability to the second premium date is $0.95$, and $v=1/1.05=20/21$. Thus,

M=1+\frac{20}{21}(0.95) =1+\frac{19}{21} =\frac{40}{21}.

Claim factor. The probability of death in the second year is $0.95(0.10)=0.095$. Discounting the two year-end claim probabilities gives

D=\frac{20}{21}(0.05)+\left(\frac{20}{21}\right)^2(0.095) =\frac{1}{21}+\frac{76}{441} =\frac{97}{441}.

The claim expense is $0.04(8{,}000)=$320$, so the benefit plus claim expense per death is $8,320. Its expected present value is $8{,}320(97/441)=807{,}040/441$. The issue and per-premium expenses contribute $180+25(40/21)$. Combining these terms,

\frac{807{,}040}{441}+180+\frac{1{,}000}{21} =\frac{807{,}040+79{,}380+21{,}000}{441} =\frac{907{,}420}{441}.

After the 8% premium expense, the denominator is $(1-0.08)(40/21)=184/105$. Therefore,

P=\frac{907{,}420/441}{184/105} =\frac{4{,}537{,}100}{3{,}864} =1{,}174.1977\ldots

The annual gross premium is $1,174.20, rounded to the nearest cent. The claim-related percentage is applied to the benefit and weighted by $D$; it is not an additional percentage of the premium.

Worked Example: Combining All Four Expense Bases

A hypothetical three-year term policy has a $5,000 death benefit and an effective annual interest rate of 0%. The conditional probability of death is 0.10 in each year. The insurer incurs a $50 issue expense, a $20 expense with each premium, a 10% expense on each gross premium, and a claim expense of $100 plus 3% of the death benefit. Calculate the annual gross premium.

Premium factor. The probabilities of being alive at the three premium dates are $1$, $0.90$, and $0.90^2=0.81$. As $v=1$,

$$ M=1+0.90+0.81=2.71. $$

Claim factor. The probabilities of death in the three years are $0.10$, $0.90(0.10)=0.09$, and $0.81(0.10)=0.081$. With no discounting,

$$ D=0.10+0.09+0.081=0.271. $$

The claim expense is $100+0.03(5{,}000)=$250$, so the payment per death, including the benefit, is $5,250. Its expected present value is $5{,}250(0.271)=$1,422.75$. The other non-percentage expenses have present value $50+20(2.71)=50+54.20=$104.20$. The total numerator is therefore $1,526.95, and the denominator is $(1-0.10)(2.71)=2.439$.

P=\frac{1{,}526.95}{2.439} =626.0557\ldots

The annual gross premium is $626.06, rounded to the nearest cent. The check is $0.90(2.71)(626.0557\ldots)=$1,526.95$, matching the present value of all non-percentage outgo. The premium factor includes the issue-date premium as well as later premiums weighted by survival.

Common Errors and a Reliable Check

A frequent error is to calculate the premium without allowing for the percentage-of-premium expense. In the first worked example, omitting the 5% expense would give $P=(314{,}500/121)/(20/11)=1{,}429.55$ to the nearest cent. That amount does not fund the specified expense: the equation requires $P(1-0.05)M$, not $PM$, to cover the remaining outgo. Including the percentage expense gives the correct premium of $1,504.78.

Other errors often come from using the right amount with the wrong factor. Use $M$ for an expense incurred at each premium date and $D$ for an expense incurred at each death claim. A per-payment expense is weighted by survival to the premium date; a claim expense is weighted by the probability of death in the year and discounted to the end of that year. Also confirm whether a percentage applies to premiums or benefits: the two percentages multiply different cash flows and generally use different factors.

Key takeaway: Identify the amount, payment time, and cash-flow base for every expense before writing the equation. In this model, fixed issue expenses use no survival-weighted factor, per-premium expenses use $M$, and benefit and claim expenses use $D$. Isolate the premium percentage algebraically before calculating $P$.

Check Your Understanding

For each hypothetical contract, use the stated annual timing and solve for the annual gross premium. Round money to the nearest cent.

A one-year term policy has benefit $1,000, $i=0$, and death probability 0.10. Expenses are $20 at issue, $5 with the premium, 10% of premium, and $50 per claim. (Answer: $144.44)
A two-year term policy has benefit $4,000, $i=0$, and conditional death probabilities 0.20 and 0.10. Expenses are $60 at issue, $15 per premium, 5% of premium, and a fixed $100 per claim. (Answer: $722.22)
A two-year term policy has benefit $2,000, $i=0$, and conditional death probabilities 0.10 in each year. Expenses are $40 at issue, $10 per premium, 20% of premium, and 5% of benefit per claim. (Answer: $301.32)
A three-year term policy has benefit $1,000, $i=0$, and conditional death probabilities 0.20 in each year. Expenses are $30 at issue, $8 per premium, 10% of premium, and $25 per claim. (Answer: $250.33)
A two-year term policy has benefit $3,000, $i=0$, and conditional death probabilities 0.10 in each year. Expenses are $25 at issue, $12 per premium, 15% of premium, and a claim expense of $40 plus 2% of benefit. (Answer: $338.40)

What You'll Learn