Level Gross Premiums with Initial Expenses

Why an Initial Expense Changes the Premium

A level gross premium must fund the policy benefits and the expenses included in its pricing basis. An expense paid at issue is different from a future expense: it is incurred immediately, so its present value is the full amount at time 0. The premium income used to fund it, however, may arrive over several years and only while the insured is alive.

This tutorial isolates a one-time issue expense. Unless stated otherwise, examples have no other expenses. Premiums are annual and payable in advance while the insured is alive; a premium at time 0 is therefore certain. The premium is determined by the equivalence principle: the expected present value of premiums equals the expected present value of benefits and expenses. This method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: An initial expense is an expense incurred at policy issue, at time 0. If its amount is $e_0$, its present value is $e_0$. A level gross premium, denoted here by $G$, is a constant premium amount at each scheduled payment date. The premium is gross because the equivalence equation includes expenses as well as benefits.

The Gross Premium Equation

Let an insured aged $x$ have a policy with annual premiums payable at times $0,1,\ldots,n-1$, provided the insured is alive at each payment date. Let $B_k$ be the benefit payable at the end of policy year $k$ if death occurs in that year. Use an effective annual interest rate $i$, with $v=1/(1+i)$. The probability of surviving $k$ years is ${}_kp_x$, and the probability of death during year $k+1$, conditional on survival to its start, is $q_{x+k}$.

The expected present value of the premiums is the level amount $G$ multiplied by the expected present value of the premium payment pattern. The expected present value of death benefits is found by multiplying each possible payment by its probability and discount factor. With only an initial expense $e_0$, the equivalence equation is:

G\sum_{k=0}^{n-1}v^k\,{}_kp_x = \sum_{k=0}^{n-1}B_{k+1}v^{k+1}\,{}_kp_x q_{x+k} +e_0.

The $k=0$ premium term is $G$, since ${}_0p_x=1$. A benefit paid at the end of a year is discounted for that full year. If there is also a maturity benefit, its expected present value must be added to the benefit side of the equation. If the premium payment period ends before the benefit coverage ends, only scheduled premium dates belong in the premium sum.

Write the premium timing.
List each premium date and the probability that the insured is alive to pay on that date.

Find benefit present values.
For each possible benefit, multiply its amount by the probability it is paid and discount to time 0.

Add the issue expense.
Add the full issue expense to the benefit present value because the expense is paid at time 0.

Divide by the premium factor.
Set $G$ times the expected present value of one unit of premium equal to total expected present value of outgo, then solve for $G$.

Why the Premium Factor Uses Survival Probabilities

A premium due at time $k$ is paid only if the insured has survived to that date. Its expected present value is therefore $Gv^k{}_kp_x$. Summing those terms gives the premium present value. This explains why the premium factor is not simply an annuity-certain factor: future payments are conditional on survival.

\begin{aligned} \text{EPV of premiums} &=G+Gv\,{}_1p_x+Gv^2\,{}_2p_x+\cdots+Gv^{n-1}\,{}_{n-1}p_x\\ &=G\sum_{k=0}^{n-1}v^k\,{}_kp_x. \end{aligned}

An issue expense does not receive a survival probability or a discount factor in this setup. It is paid at issue with certainty. Thus, adding it to the numerator of the premium equation is appropriate, while multiplying it by a future survival probability is not.

Worked Example: Three-Year Term Insurance with an Issue Expense

A hypothetical three-year term policy pays $10,000 at the end of the year of death. Annual premiums are due at times 0, 1, and 2 while the insured is alive. The effective annual interest rate is $i=25\%$, and the probability of death in each year, conditional on survival to that year, is $q=0.20$. An issue expense of $320 is paid at time 0. Find the level gross premium.

Method. Use the equivalence equation, keeping the premium payment factor separate from the expected present value of benefits. Here $v=1/1.25=0.8$ and $p=1-q=0.8$.

The probability of death in year 1 is $0.20$. The probability of death in year 2 is $0.80(0.20)=0.16$, and the probability of death in year 3 is $0.80^2(0.20)=0.128$. Discounting each benefit to time 0 gives:

\begin{aligned} \text{EPV of benefits} &=10{,}000\left[0.20(0.8)+0.80(0.20)(0.8)^2 +0.80^2(0.20)(0.8)^3\right]\\ &=10{,}000[0.160000+0.102400+0.065536]\\ &=10{,}000(0.327936)=\$3{,}279.36. \end{aligned}

The premiums are due at time 0 with certainty, at time 1 with probability $0.80$, and at time 2 with probability $0.80^2=0.64$. Thus, the premium factor is:

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

The total expected present value of outgo is $3{,}279.36+320=\$3{,}599.36$. Therefore:

G=\frac{3{,}599.36}{2.0496} =\$1{,}756.128\ldots \approx \boxed{\$1{,}756.13}.

The check is $1{,}756.128\ldots(2.0496)=\$3{,}599.36$, matching the benefit and issue-expense present value. The result is rounded to the nearest cent. The common mistake this method avoids is treating all three premiums as certain: the premiums at times 1 and 2 are payable only if the insured survives to those dates.

Endowment Benefits and a Limited Premium Period

The same principle applies when the policy can pay a benefit on survival. A maturity benefit is multiplied by the probability of surviving to maturity and discounted from the maturity date. It is not counted as a death benefit. Similarly, when premiums are payable for fewer years than the coverage lasts, the premium factor ends at the last scheduled premium date, even though benefits may remain payable later.

Worked Example: Two-Year Endowment with an Issue Expense

A hypothetical two-year endowment pays $9,000 at the end of the year of death, or $9,000 at the end of year 2 if the insured survives. Premiums of level amount $G$ are payable at times 0 and 1 while the insured is alive. Assume $i=25\%$, $q=0.10$ in each year, and an issue expense of $450 at time 0. Find $G$.

Method. Add the present values of the two possible death benefits and the maturity benefit, then divide total outgo by the two-payment premium factor. Here $v=0.8$ and $p=0.9$.

The year 1 death probability is $0.10$, and the year 2 death probability is $0.90(0.10)=0.09$. The probability of surviving to maturity is $0.90^2=0.81$. The benefit present value is:

\begin{aligned} \text{EPV of benefits} &=9{,}000\left[0.10(0.8)+0.90(0.10)(0.8)^2 +0.81(0.8)^2\right]\\ &=9{,}000[0.0800+0.0576+0.5184]\\ &=9{,}000(0.6560)=\$5{,}904.00. \end{aligned}

The premium at time 0 is certain. The premium at time 1 is paid with probability $0.90$, so the premium factor is $1+0.90(0.8)=1.72$. Total outgo has expected present value $5{,}904+450=\$6{,}354$. Hence:

G=\frac{6{,}354}{1.72} =\$3{,}694.186\ldots \approx \boxed{\$3{,}694.19}.

Checking, $3{,}694.186\ldots(1.72)=\$6{,}354$. The result is rounded to the nearest cent. A frequent error is to omit the maturity payment or to discount it only one year; it is payable at the end of year 2 and must be weighted by $0.81v^2$.

Worked Example: Four-Year Coverage with Only Two Premiums

A hypothetical four-year term policy pays $30,000 at the end of the year of death. Only two annual premiums are payable: at issue and at the end of year 1 if the insured is then alive. The conditional probability of death is $0.10$ in each year, and the effective annual interest rate is $i=5\%$. An issue expense of $700 is paid at issue. Find the level premium.

Method. Calculate the four-year benefit present value, but include only the two scheduled premium dates in the denominator. Let $v=1/1.05=20/21$, $p=0.9$, and $pv=(0.9)(20/21)=6/7$.

For death in year $k+1$, the probability is $0.1(0.9)^k$ and the discount factor is $v^{k+1}$. The four discounted death-benefit terms form a geometric sequence with first term $0.1v=2/21$ and ratio $pv=6/7$:

\begin{aligned} \text{EPV of benefits} &=30{,}000\left[\frac{2}{21} \left(1+\frac67+\left(\frac67\right)^2+\left(\frac67\right)^3\right)\right]\\ &=30{,}000\left[\frac{2}{21} \left(1+\frac67+\frac{36}{49}+\frac{216}{343}\right)\right]\\ &=30{,}000\left(\frac{2}{21}\cdot\frac{1105}{343}\right)\\ &=30{,}000\left(\frac{2210}{7203}\right) =\$9{,}204.498\ldots. \end{aligned}

The premium factor is $1$ for the issue-date premium plus $pv=6/7$ for the year-1 premium, provided the insured survives that year. Thus, it is $1+6/7=13/7$. The total outgo present value is $9{,}204.498\ldots+700=\$9{,}904.498\ldots$, so:

G=\frac{9{,}904.498\ldots}{13/7} =\$5{,}333.191\ldots \approx \boxed{\$5{,}333.19}.

As a check, multiplying the unrounded premium by $13/7$ gives the total outgo present value. The answer is rounded to the nearest cent. The common mistake avoided here is using four premium terms simply because coverage lasts four years; the contract schedules only two premiums.

Common Errors and a Reliable Check

An initial expense is easy to place in the equation, but errors in timing can materially change the result. In particular, confusing a certain issue-date expense with a survival-contingent future cash flow, or confusing premium dates with benefit dates, changes the equation rather than just its arithmetic.

Discounting the issue expense. An expense paid at time 0 has present value $e_0$, not $ve_0$.
Applying survival to the issue expense. The expense is incurred at issue, so it is not multiplied by a probability of surviving a year.
Counting premiums after death. A premium due at time $k$ is included only with probability ${}_kp_x$ when the contract requires the insured to be alive on that date.
Using the coverage term as the premium term. These periods need not match. Follow the stated premium schedule.
Omitting a survival benefit. For an endowment, include the maturity benefit with its survival probability and discount factor.

A useful final check is to multiply the solved premium by its premium factor. That product must equal the expected present value of all benefits and expenses included in the pricing basis. Also check that every payment has the correct date, discount factor, and probability of payment.

Key takeaway: With only a one-time expense $e_0$ at issue, the level gross premium is the expected present value of benefits plus $e_0$, divided by the expected present value of one unit of scheduled premiums. The numerator describes outgo; the denominator reflects premium timing and survival.

Check Your Understanding

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

A hypothetical one-year term policy pays $10,000 at the end of the year if death occurs. The annual effective interest rate is $25\%$, the death probability is $0.10$, one premium is due at issue, and the issue expense is $200. Find the level gross premium. (Answer: $1,000.00)
A two-year term policy pays $4,000 at the end of the year of death. Annual premiums are due at times 0 and 1 while alive. Assume $i=0$, a conditional death probability of $0.25$ in each year, and an issue expense of $100. Find the level gross premium. (Answer: $1,057.14)
A hypothetical two-year endowment pays $6,000 at death during the term or at the end of year 2 on survival. Premiums are due at times 0 and 1 while alive. Assume $i=25\%$, a conditional death probability of $0.10$ in each year, and a $300 issue expense. Find the level gross premium. (Answer: $2,462.79)
A whole life policy pays $5,000 at the end of the year of death. Annual premiums are payable in advance for life. Assume $i=25\%$, a conditional death probability of $0.20$ every year, and an issue expense of $250. Find the level gross premium. (Answer: $2,090.00)
A three-year endowment pays $10,000 at death during the term or at the end of year 3 on survival. Premiums are payable at times 0, 1, and 2 while alive. Assume $i=25\%$, conditional death probabilities $0.20$, $0.10$, and $0.25$ in years 1, 2, and 3, respectively, and a $500 issue expense. Find the level gross premium. (Answer: $2,998.10)

What You'll Learn