Deriving Expense-Adjusted Premium Equations

Why Expense Timing Belongs in the Equation

A gross premium must fund more than the insured benefit. It must also fund the insurer’s expenses. The key is to value each expected payment at the time it occurs: an expense at issue has a different present value from an expense due only if the policyholder survives to a later premium date. Expenses proportional to the premium create an additional issue: their amount is not known until the premium is solved.

The equivalence principle sets the expected present value of premiums equal to the expected present value of benefits and expenses. It is a pricing equation, not a claim that every policy will produce exactly zero profit. The calculation depends on the specified interest, mortality, benefit, premium, and expense assumptions.

Definition: Let $P$ be a level gross premium, and let $B$ be the expected present value of benefits. Let $D$ be the expected present value of one unit of each scheduled premium, $C$ the expected present value of expenses that do not depend on $P$, and $L$ the expected present value of proportional expenses per unit of $P$. The gross premium equivalence equation is $$ P D = B + C + P L. $$ Here $C$ and $L$ must reflect the timing and conditions under which the expenses are incurred.

Build the Present Values from the Cash-Flow Schedule

Consider a policy issued at age $x$, with premiums due at times $k=0,1,\ldots,n-1$, provided the insured is then alive. Premiums are paid at the beginning of each policy year. Under effective annual interest rate $i$, write $v=1/(1+i)$. The expected present value of one unit of premium at each scheduled date is

D=\sum_{k=0}^{n-1} v^k\,{}_kp_x.

The $k=0$ term is 1 because the policy is in force at issue. Each later term includes both discounting and the probability that the insured is alive for that premium. If a premium is not payable at a particular date, that date must not be included.

For example, suppose the benefit is $b$, payable at the end of the year of death during an $n$-year term. Its expected present value is

B=b\sum_{k=0}^{n-1}v^{k+1}\,{}_kp_x\,q_{x+k}.

The factor ${}_kp_xq_{x+k}$ is the probability of surviving $k$ years and then dying in the following year. The additional factor $v^{k+1}$ discounts the end-of-year benefit. Other benefit patterns require their own payment times and probabilities; do not use this term-insurance expression automatically for a survival benefit or a different payment schedule.

A fixed expense has a specified dollar amount that does not change with $P$. A proportional expense equals a stated fraction of the premium, such as $0.08P$. If it is charged on the same dates as the premiums, its present value is tied to $D$. If its timing or condition differs, calculate its present value separately.

Deriving the Expense-Adjusted Premium Formula

The equivalence equation follows by adding the expected present values of the insurer’s outflows and equating them to the expected present value of premium income. If each premium is $P$, its expected present value is $PD$. Benefits contribute $B$, fixed expenses contribute $C$, and proportional expenses contribute $PL$. Thus $PD=B+C+PL$. Collecting the terms that contain $P$ gives

P(D-L)=B+C, \qquad P=\frac{B+C}{D-L},

provided $D-L\ne 0$. A finite positive premium requires $B+C>0$ and $D-L>0$. If all proportional expenses are a common fraction $\alpha$ of each premium, and are charged at the premium dates, then $L=\alpha D$, so

P=\frac{B+C}{(1-\alpha)D}, \qquad 0\leq \alpha<1.

This form shows why it is generally incorrect to calculate a premium without the proportional expense and then simply add $\alpha$ times that result. The expense is a fraction of the gross premium, including the amount needed to fund the expense itself.

List the dated cash flows.
Record when each benefit, premium, and expense is due, and whether survival or another condition is required.

Find each expected present value.
Use the stated interest and mortality assumptions, applying the appropriate probability to each contingent payment.

Separate premium-independent and proportional expenses.
Put the former into $C$; put the coefficient of $P$ into $L$.

Solve and check.
Calculate $P=(B+C)/(D-L)$, verify the denominator is positive for a positive premium, and substitute $P$ back into the original equation.

Worked Applications

Worked Example: Initial and Per-Premium Fixed Expenses

A hypothetical three-year term policy pays $10,000 at the end of the year of death. Premiums are level and due at the start of each year while the insured is alive, for at most three years. Assume $i=5\%$, and assume the conditional probability of surviving each year is $0.90$, so ${}_kp_x=0.90^k$ and $q_{x+k}=0.10$ for each covered year. There is a $200 expense at issue and a fixed $30 expense at every premium date, including issue. Find the gross premium.

Method. Calculate the death-benefit present value and the premium annuity factor, then include each fixed expense at its actual premium date. The $30 expense is not multiplied by $P$.

The premium factor is

D=1+\frac{0.90}{1.05}+\frac{0.90^2}{1.05^2} =1+\frac{6}{7}+\frac{36}{49} =\frac{127}{49} \approx 2.5918.

The benefit is payable at the end of the year of death. Its expected present value is

\begin{aligned} B &=10{,}000\left( \frac{0.10}{1.05} +\frac{0.90(0.10)}{1.05^2} +\frac{0.90^2(0.10)}{1.05^3}\right)\\ &=10{,}000\left(\frac{2}{21}+\frac{4}{49}+\frac{24}{343}\right) =10{,}000\left(\frac{254}{1029}\right) \approx \$2{,}468.42. \end{aligned}

The issue expense has present value $200. The fixed $30 expense occurs whenever a premium is due, so its present value is $30D$. Equating premium income to benefits and expenses gives

PD=B+200+30D, \qquad P=\frac{B+200}{D}+30 =\frac{2{,}468.4169+200}{127/49}+30 \approx \$1{,}059.55.

Check. At the unrounded premium, $PD$ equals $B+200+30D$; the premium covers the benefit, acquisition expense, and three expected fixed premium-date expenses. Rounded to the nearest cent, the gross premium is $1,059.55 per scheduled premium. A common mistake is to put the $200 issue expense into the per-premium expense factor; it occurs once at issue, not at every premium date.

Worked Example: A Percentage of Each Premium

A hypothetical two-year term policy pays $10,000 at the end of the year of death. Premiums are due at times 0 and 1 while the insured is alive. Assume $i=10\%$, annual survival probability $0.80$, and hence annual death probability $0.20$. The insurer incurs a $100 expense at issue and an expense equal to 10% of each premium at that premium’s due date. Find the gross premium.

Method. Compute $B$ and $D$, then include the proportional expense as $0.10PD$. The percentage expense is charged at exactly the same dates and under the same survival conditions as the premiums.

The premium factor is

D=1+\frac{0.80}{1.10} =1+\frac{8}{11} =\frac{19}{11} \approx 1.7273.

The benefit present value is

\begin{aligned} B &=10{,}000\left( \frac{0.20}{1.10} +\frac{0.80(0.20)}{1.10^2}\right)\\ &=10{,}000\left(\frac{2}{11}+\frac{16}{121}\right) =10{,}000\left(\frac{38}{121}\right) \approx \$3{,}140.50. \end{aligned}

The equivalence equation and its solution are

\begin{aligned} PD&=B+100+0.10PD,\\ P&=\frac{B+100}{0.90D} =\frac{3{,}140.4959+100}{0.90(19/11)} \approx \$2{,}084.53. \end{aligned}

Check. Using the unrounded premium, $0.90PD=B+100$, so the part of premium present value left after the 10% expense funds the benefit and issue expense. Rounded to the nearest cent, the gross premium is $2,084.53. A common mistake is to use $B/D+100$; that charges the fixed expense as though it were a premium and omits the percentage expense.

Worked Example: Renewal Expense at a Later Date

A hypothetical two-year pure endowment pays $1,000 at time 2 only if the insured survives two years. Premiums are due at times 0 and 1 while the insured is alive. Assume $i=0\%$ and independent annual survival probabilities of $0.50$. There is a $60 expense at issue and a fixed $20 renewal expense at time 1 if the insured is alive then. In addition, 5% of each premium is spent at that premium’s due date. Find the gross premium.

Method. Value the renewal expense using its own timing and survival condition. Here it occurs at time 1 only if the insured survives the first year; it is not an expense at issue.

The expected present value of premiums per dollar is

$$ D=1+(0.50)=1.50. $$

The pure endowment benefit is payable only on survival to time 2, so

B=1{,}000(0.50)^2=\$250.

With zero interest, the issue expense has present value $60, and the renewal expense has present value $20(0.50)=\$10$. Thus the premium-independent expenses have present value $C=60+10=\$70$. The proportional expenses are charged with the premiums, so their present value is $0.05PD$. Therefore,

\begin{aligned} PD&=250+70+0.05PD,\\ P&=\frac{320}{0.95(1.50)} =\frac{320}{1.425} \approx \$224.56. \end{aligned}

Check. The unrounded premium satisfies $0.95PD=320$, the present value of benefits and fixed expenses. Rounded to the nearest cent, the gross premium is $224.56. Treating the renewal expense as payable at issue would incorrectly assign it a present value of $20 instead of $20(0.50)=\$10$.

Timing Rules and Exam Traps

A proportional expense belongs in $L$ only after its amount and payment condition have been identified. If a fraction $\alpha_k$ of each premium is charged at time $k$, then

L=\sum_{k=0}^{n-1}\alpha_k v^k\,{}_kp_x.

When the percentage is constant, $\alpha_k=\alpha$, this reduces to $L=\alpha D$. If the percentage is instead charged at a different time, or only when a different event occurs, its present value need not be $\alpha PD$. Calculate the expected present value of that expense from its own cash-flow schedule.

Likewise, a fixed expense due at a future date may depend on the policy remaining in force. A renewal expense due at time $k$ only if the insured is alive then has expected present value equal to its amount multiplied by $v^k{}_kp_x$. An expense due regardless of survival has a different probability factor. Identify the condition in the problem rather than assuming every expense follows the premium pattern.

A useful final check is to substitute the solved premium into the original, unsimplified equation. Confirm that premium present value equals benefit present value plus both categories of expense. Also check that expenses have not been counted twice—for example, as both a fixed amount in $C$ and a loading in $L$.

Key takeaway: Derive the gross premium from dated expected cash flows. Fixed expenses enter as their own present value; premium-proportional expenses enter as a coefficient of $P$. The solution $P=(B+C)/(D-L)$ is valid only when the quantities use the correct timing and conditions and the denominator supports a positive premium.

Check Your Understanding

For each problem, set up the cash flows before solving. Round monetary answers to the nearest cent.

A hypothetical two-year term policy pays $5,000 at the end of the year of death. Premiums are due at times 0 and 1 while the insured is alive. Assume $i=5\%$, annual survival probability $0.80$, and annual death probability $0.20$. There is a $40 issue expense and a $15 fixed expense at each premium date. Find the gross premium. (Answer: $990.08)
A hypothetical two-year term policy pays $1,000 at the end of the year of death. Assume $i=0\%$, annual survival probability $0.80$, and two premiums due at times 0 and 1 while alive. The issue expense is $60, and 10% of each premium is spent at its due date. Find the gross premium. (Answer: $259.26)
A hypothetical three-year term policy pays $5,000 at the end of the year of death. Assume $i=5\%$, annual survival probability $0.90$, and three premiums due at times 0, 1, and 2 while alive. There is a $25 issue expense. The expense proportions at those three premium dates are 5%, 8%, and 8%, respectively. Find the gross premium. (Answer: $521.52)
A hypothetical two-year pure endowment pays $1,000 at time 2 if the insured survives two years. Premiums are due at times 0 and 1 while alive. Assume $i=5\%$ and annual survival probability $0.90$. There is a $50 issue expense, a fixed $30 expense at time 1 if the insured is alive, and a 10% expense on each premium at its due date. Find the gross premium. (Answer: $484.86)

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