Fixed Per-Premium Expense Loadings

Why a Fixed Expense Changes the Premium

A level premium pays for expected benefits and expenses. When an insurer incurs a fixed dollar expense each time a premium is collected, the expense is tied to the number and timing of premium payments. The expected present value of those expenses therefore depends on the same survival-contingent payment pattern as the premiums.

This tutorial considers hypothetical contracts priced by the equivalence principle: the expected present value of gross premiums equals the expected present value of benefits plus expenses. The focus is a fixed expense of $e$ dollars charged at every premium payment, with the expense paid at the same time as that premium. Interest and mortality assumptions are stated for each calculation.

Definition: Let $G$ be a level gross premium and $e$ the fixed expense charged at each premium payment. Let $N$ be the random present value of one dollar paid at every time a premium is due. Define $\Pi=E[N]$, the expected present value factor for the premium stream. If $B$ is the expected present value of benefits, then the equivalence principle gives $G\Pi=B+e\Pi$, and hence $G=B/\Pi+e$, provided the same payments trigger both premium and expense.

Building the Premium Present Value Factor

Suppose premiums are annual and payable in advance while the insured is alive. A premium is due at time 0, and later premiums are due at integer times $k$ if the insured is alive then. With effective annual interest rate $i$, let $v=1/(1+i)$. The random present value of one dollar of premiums is the sum of $v^k$ over the premium dates that actually occur. Its expected value is

\Pi=\sum_{k}v^k P(\text{a premium is due at time }k).

For example, if premiums are due at times 0 and 1 only while the insured is alive, then the first premium is certain and the second is due only if the insured survives the first year. Thus $\Pi=1+v\,{}_1p_x$. This is an expected present value factor, not the present value on every individual policy path.

The fixed expense has the same payment pattern under the assumptions here. On any particular path, the expense present value is $e$ times the realized present value of one dollar of premiums. Taking expectations therefore gives an expense present value of $e\Pi$; the premium present value is $G\Pi$. This proportionality after taking expectations is the basis of the gross premium equation.

Why the Gross Premium Equation Holds

The equivalence principle equates expected present values. It does not require the premiums and expenses on each possible path to equal the expected present values calculated across all paths. Write $N$ for the realized discounted count of premium payments. Then the realized premium present value is $GN$, and the realized fixed-expense present value is $eN$. By linearity of expectation,

E[GN]=G E[N]=G\Pi,\qquad E[eN]=e E[N]=e\Pi.

If the expected present value of benefits is $B$, the equivalence equation is $G\Pi=B+e\Pi$. When $\Pi>0$, solving for $G$ gives the following result.

Theorem: If a fixed expense $e$ is incurred at every premium payment, and expenses occur at exactly the same times as premiums, then the level gross premium is $$ G=\frac{B}{\Pi}+e. $$ Here $B/\Pi$ is the level net premium for the same benefit and premium-payment pattern. The result does not apply unchanged if the expense is incurred at different times, applies to only some premium payments, or has an amount other than $e$ at each payment; those situations require separate expense present values.

The equation has a useful interpretation: each premium must cover the net premium amount needed for benefits, plus the fixed expense incurred when that premium is collected. This additive result depends on the expense being a fixed amount per payment. It is not a general formula for every type of expense.

Worked Example: Two-Year Term Insurance

A hypothetical two-year term policy pays $1,000 at the end of the year of death. Annual premiums are payable in advance at times 0 and 1 while the insured is alive. The effective annual interest rate is 25%, so $v=1/1.25=0.8$. The conditional probabilities of death in years 1 and 2 are $q_1=0.10$ and $q_2=0.20$. A fixed expense of $12 is incurred at each premium payment. Find the gross premium.

Method and assumptions. Use the equivalence principle. The insured is alive at time 0; survival through year 1 has probability $p_1=1-q_1=0.90$. The second-year death probability from issue is $p_1q_2=0.90(0.20)=0.18$.

Expected present value of benefits. A first-year death benefit is paid at time 1 with probability 0.10. A second-year death benefit is paid at time 2 with probability 0.18:

$$ B=1000(0.10)(0.8)+1000(0.18)(0.8^2) =80+115.20 =195.20. $$

Premium factor. The premium at time 0 is certain. The premium at time 1 is payable only if the insured survives year 1:

\Pi=1+(0.90)(0.8)=1+0.72=1.72.

Gross premium. The present value of expenses is $12(1.72)=20.64$. Therefore,

G=\frac{195.20+20.64}{1.72} =\frac{215.84}{1.72} =125.488372\ldots

Rounding the premium to the nearest cent gives $G=\$125.49$. The check is $1.72(125.488372\ldots)=215.84=195.20+20.64$, before rounding. The common mistake avoided here is treating the premium factor as if it were the same realized payment count on every policy path.

Applying the Equation to Different Benefits

The benefit design changes $B$, while the premium schedule determines $\Pi$. The expense adjustment remains $e$ when every scheduled premium triggers exactly one expense of amount $e$. The next example illustrates that the same premium-factor method applies when a benefit is payable at maturity as well as on death.

Worked Example: Two-Year Endowment Insurance

A hypothetical two-year endowment policy pays $1,000 at the end of year 1 if the insured dies in that year. If the insured survives year 1, it pays $1,000 at the end of year 2, whether the insured dies during year 2 or survives to maturity. Annual premiums are payable in advance while alive, at times 0 and 1. The effective annual interest rate is 25% ($v=0.8$); the conditional death probabilities are $q_1=0.10$ and $q_2=0.20$. Each premium incurs a fixed $15 expense. Find the gross premium.

Method and assumptions. Separate the first-year death benefit from the payment at the end of year 2. The year-2 payment is certain conditional on surviving year 1; it does not depend on whether death or survival occurs during year 2.

Expected present value of benefits. The year-1 benefit has probability 0.10. The year-2 payment has probability 0.90, since it is paid whenever the insured survives the first year:

$$ B=1000(0.10)(0.8)+1000(0.90)(0.8^2) =80+576 =656. $$

Premium factor and expense present value. The time-0 premium is certain, and the time-1 premium is due with probability 0.90. Thus $\Pi=1+0.90(0.8)=1.72$, and the expense present value is $15(1.72)=25.80$.

Gross premium.

G=\frac{656+25.80}{1.72} =\frac{681.80}{1.72} =396.395348\ldots

Rounded to the nearest cent, the gross premium is $396.40. As a check, $396.395348\ldots(1.72)=681.80$, which equals the benefit present value of $656 plus the expense present value of $25.80. Counting only death during year 2 as a reason for the year-2 payment would incorrectly omit the endowment payment to survivors.

Worked Example: Three-Year Term Insurance

A hypothetical three-year term policy pays $2,000 at the end of the year of death. Annual premiums are payable in advance at times 0, 1, and 2 while the insured is alive. The effective annual interest rate is 25% ($v=0.8$); the conditional death probability is 0.20 in each of the three years. A fixed expense of $20 is incurred at every premium payment. Find and check the gross premium.

Method and assumptions. Survival through one and two years has probabilities $0.80$ and $0.80^2=0.64$. The probabilities of death in years 1, 2, and 3 are $0.20$, $0.80(0.20)=0.16$, and $0.64(0.20)=0.128$.

Expected present value of benefits.

\begin{aligned} B&=2000\big[(0.20)(0.8)+(0.16)(0.8^2)+(0.128)(0.8^3)\big]\\ &=2000(0.160000+0.102400+0.065536)\\ &=2000(0.327936)=655.872. \end{aligned}

Premium factor. Premiums are due at time 0 with certainty, at time 1 if alive then, and at time 2 if alive then:

\Pi=1+(0.80)(0.8)+(0.64)(0.8^2) =1+0.64+0.4096 =2.0496.

Gross premium and check. The expense present value is $20(2.0496)=40.992$. Since $655.872/2.0496=320$,

G=\frac{655.872+40.992}{2.0496} =\frac{696.864}{2.0496} =340.

The gross premium is $340.00. Its expected present value is $340(2.0496)=696.864$, exactly the sum of the benefit present value, $655.872, and expense present value, $40.992.

Common Mistake: Treating an Expectation as a Path Value

In the two-year term example, $\Pi=1.72$ is an expected discounted payment count across possible outcomes. On a path where the insured dies in year 1, only the time-0 premium is paid, so the realized premium present value is $G$, not $1.72G$. On a path where the insured survives year 1, the realized premium present value is $G(1+0.8)=1.8G$. The corresponding expense present values on those paths are $12$ and $12(1.8)=21.60$.

The incorrect calculation is to claim that the first-year-death path has premium present value $1.72G$, or expense present value $12(1.72)=20.64$. Those are expected present values over all paths, not the values on that one path. The correction is to calculate path values using the payments that actually occur, then take expectations. Here, $0.10(1)+0.90(1.8)=1.72$, so the expected premium factor is 1.72; the expected expense present value is consequently $12(1.72)=20.64$.

A second useful check is dimensional: $B/\Pi$ and $e$ are both dollars per premium, so they can be added to form $G$. The expense present value $e\Pi$, by contrast, is a present value and must be added to $B$ before dividing by $\Pi$.

Key takeaway: First identify the premium dates and the probability each payment is due. Use them to calculate $\Pi$, calculate the benefit present value $B$, and then apply $G=(B+e\Pi)/\Pi=B/\Pi+e$. Interpret $G\Pi$ and $e\Pi$ as expected present values, not as values realized on every policy path.

Check Your Understanding

For each hypothetical contract, use the equivalence principle and round the gross premium to the nearest cent when necessary.

A two-year term policy pays $2,000 at the end of the year of death. Annual premiums are in advance at times 0 and 1 while alive. The effective annual interest rate is 25%; conditional death probabilities are 0.10 in each year. A fixed $10 expense is charged at each premium. Find the gross premium. (Answer: $170.00)
A three-year term policy pays $4,000 at the end of the year of death. Annual premiums are in advance while alive. The effective annual interest rate is 0%; the conditional death probability is 0.25 in each year. A fixed expense of $25 is charged at each premium. Find the gross premium. (Answer: $1,025.00)
A two-year endowment pays $3,000 at the end of year 1 if death occurs then; if the insured survives year 1, it pays $3,000 at the end of year 2 whether the insured dies during year 2 or survives. Annual premiums are in advance while alive. The effective annual interest rate is 25%; conditional death probabilities are 0.20 in each year. A fixed expense of $18 is charged at each premium. Find the gross premium. (Answer: $1,247.27)
A two-year endowment pays $1,500 at the end of year 1 if death occurs in year 1, and otherwise pays $1,500 at the end of year 2. Annual premiums are in advance while alive. The effective annual interest rate is 25%; conditional death probabilities are 0.10 in year 1 and 0.20 in year 2. A fixed expense of $10 is charged at each premium. Find the gross premium. (Answer: $582.09)
A three-year term policy pays $5,000 at the end of the year of death. Annual premiums are in advance while alive. The effective annual interest rate is 25%; conditional death probabilities in years 1, 2, and 3 are 0.10, 0.20, and 0.25. A fixed expense of $30 is charged at each premium. Find the gross premium. (Answer: $688.84)

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