Calculate Gross Premiums with Acquisition and Renewal Expenses

Why Gross Premiums Differ From Net Premiums

A net premium is set to fund expected benefits, without allowing for the insurer’s expenses. A gross premium also has to fund expenses such as policy acquisition, policy administration, premium collection, and claim handling. The calculation is an expected-present-value equation: premium income must balance expected benefits and expenses, with each amount discounted from the time it is paid.

The key is not just to total the expenses. An expense paid at issue has a different present value from one paid at the end of a future policy year. Some expenses occur only if a premium is due or a claim is paid. Others may be incurred while the policy is in force, whether or not a premium or claim is payable at that moment.

Definition: A gross premium is a premium determined by an equivalence equation that includes the expected present value of both contractual benefits and the expenses specified in the model. The equation must reflect the amount, timing, and contingency of each premium, benefit, and expense.

Set Up the Cash Flows Before Calculating

Start by listing the possible cash flows and asking three questions about each one: how much is paid, when is it paid, and under what condition is it paid? For example, a renewal expense might be payable only when a renewal premium is collected, while a claim expense might be payable only if death occurs. These distinctions determine the expected present value.

For an $n$-year term policy on a life aged $x$, suppose premiums are payable annually in advance while the insured is alive, including at issue. Let the level gross premium be $P$. Suppose a benefit $B$ is paid at the end of the year of death if death occurs within the term. The annual effective interest rate is $i$, and $v=1/(1+i)$. Under the annual discrete death-benefit model, the probability of surviving $k$ years and then dying during the following year is ${}_kp_xq_{x+k}$.

The expected present value of premiums is $P$ times the expected present value of the premium-payment pattern. If a fixed acquisition expense $A$ is paid at issue, a fixed renewal expense $R$ is paid alongside each renewal premium, and a fixed claim expense $C$ is paid alongside each death benefit, the equivalence equation is:

P\left(1+\sum_{k=1}^{n-1}v^k\,{}_kp_x\right) = B\sum_{k=0}^{n-1}v^{k+1}\,{}_kp_xq_{x+k} +A +R\sum_{k=1}^{n-1}v^k\,{}_kp_x +C\sum_{k=0}^{n-1}v^{k+1}\,{}_kp_xq_{x+k}.

This particular equation assumes premiums are due at times $0,1,\ldots,n-1$ if the insured is alive at the payment time, acquisition expense is paid at time 0, renewal expense is paid at times $1,\ldots,n-1$ only when a renewal premium is due, and claim expense is paid at the same time as the benefit. If an expense has a different payment time or contingency, its term must change. The equation also assumes the premiums and the stated expenses are the only cash flows being valued.

Why the Equivalence Equation Works

The equation follows from linearity of expected present value. For each possible cash flow, its present value is its amount multiplied by the discount factor for its payment time and by the probability that the condition for payment is met. Adding these expected present values gives the expected present value of all outgo, because expectation distributes over a finite sum. The same reasoning applies to premium income.

For example, the premium due at time $k$ is paid only if the insured is alive then. Its expected present value is therefore $P v^k{}_kp_x$. The death benefit in year $k+1$ is paid only if the insured survives $k$ years and dies in the following year, so its expected present value is $Bv^{k+1}{}_kp_xq_{x+k}$. Summing the relevant terms gives the two sides of the equation.

Specify the contract and basis.
State the benefit, premium-payment times, interest rate, and mortality probabilities.

Place each expense.
Record its amount, payment time, and whether it depends on survival, a premium, or a claim.

Form expected present values.
Discount each cash flow and multiply by the probability it is paid.

Solve and check.
Isolate the gross premium, then substitute it back into the equivalence equation.

Worked Examples

Worked Example: Fixed Acquisition, Renewal, and Claim Expenses

A hypothetical three-year term policy pays \$100,000 at the end of the year of death. Premiums are level and payable at the beginning of each policy year while the insured is alive, including at issue. The annual effective interest rate is $5\%$. Assume the conditional probability of death in each policy year, given survival to its start, is $q=0.10$, so the conditional survival probability is $p=0.90$ in each year. A \$300 acquisition expense is paid at issue. A \$40 renewal expense is paid alongside each renewal premium at time 1 or 2. A \$200 claim expense is paid alongside the death benefit. Find the gross premium.

Method. Calculate the premium annuity factor and the probability-weighted present value of a death in the term. Put the renewal expenses at the renewal-premium dates, and put the claim expense at the death-benefit date.

The discount factor is $v=1/1.05=20/21$. The probabilities of being alive at times 0, 1, and 2 are $1$, $0.90$, and $0.90^2=0.81$. Thus the premium factor is:

1+0.90v+0.81v^2 =1+\frac{6}{7}+\frac{36}{49} =\frac{127}{49} =2.5918367.

The probability-weighted present-value factor for death during the three years is:

0.10v\left(1+0.90v+0.81v^2\right) =\frac{254}{1029} =0.2468416.

The benefit and claim expense together are \$100,200 for each claim, so their expected present value is $100{,}200(254/1029)=\$24{,}733.53$. The acquisition expense has present value \$300. The renewal expenses are paid at times 1 and 2 if the insured is alive at those times, giving:

40(0.90v+0.81v^2) =40\left(\frac{78}{49}\right) =\$63.67.

Therefore the gross-premium equation and solution are:

P\left(\frac{127}{49}\right) =24{,}733.53+300+63.67, \qquad P=\frac{25{,}097.20}{127/49} =\$9{,}683.17.

Check. The present value of premium income is approximately $\$9{,}683.17(2.5918367)=\$25{,}097.20$, which equals the combined expected present value of benefits and expenses, subject to rounding to cents. A common mistake is to count the renewal expense at time 0: it is specified only for renewal premiums, so it is due at times 1 and 2, not at issue.

Worked Example: Expenses Expressed as Percentages of Premium

A hypothetical two-year term policy pays \$50,000 at the end of the year of death. Premiums are payable at times 0 and 1 if the insured is alive. The annual effective interest rate is $5\%$. The conditional death probabilities are $q_x=0.20$ in year 1 and $q_{x+1}=0.10$ in year 2. A fixed acquisition expense of \$100 is paid at issue, and a fixed renewal expense of \$30 is paid at time 1 if the insured is alive. A claim expense of \$250 is paid with each death benefit. In addition, acquisition expense at issue is $25\%$ of the initial gross premium, and renewal expense at time 1 is $10\%$ of the renewal gross premium. Find the level gross premium.

Method. Treat the percentage expenses as deductions from the premium income at their specified payment times. Then solve for the gross premium using the resulting net contribution factor.

The discount factor is $v=20/21$, and the probability of surviving the first year is $0.80$. The premium income has present value $P(1+0.80v)$. The percentage expenses have present value $0.25P+0.10P(0.80v)$. Moving these percentage expenses to the premium side gives the factor:

(1-0.25)+0.80v(1-0.10) =0.75+0.80\left(\frac{20}{21}\right)(0.90) =\frac{201}{140} =1.4357143.

The probability-weighted present-value factor for death is:

0.20v+(0.80)(0.10)v^2 =\frac{4}{21}+\frac{32}{441} =\frac{116}{441} =0.2630385.

The benefit and claim expense together are \$50,250 per claim, so their expected present value is $50{,}250(116/441)=\$13{,}217.69$. The fixed expenses have present value:

100+30(0.80v) =100+30\left(\frac{16}{21}\right) =\$122.86.

The equivalence equation, using the percentage-adjusted premium factor, is:

P\left(\frac{201}{140}\right) =13{,}217.69+122.86, \qquad P=\frac{13{,}340.54}{201/140} =\$9{,}291.92.

Check. Before rounding, the premium factor multiplied by the solved $P$ equals the expected present value of the benefits and fixed expenses. The percentage charges are also accounted for: $25\%$ of $P$ is charged at issue and $10\%$ of the time-1 premium is charged only if the insured survives to time 1. A common mistake is to subtract both percentages from every premium payment, even though their rates and payment times differ.

Worked Example: Renewal Expenses Paid at Year-End

A hypothetical two-year term policy pays \$20,000 at the end of the year of death. Premiums are payable at the beginning of each year while the insured is alive. The annual effective interest rate is $10\%$, and the conditional death probability is $0.10$ in each year. A \$150 acquisition expense is paid at issue. A \$60 renewal expense is paid at the end of each policy year if the insured was alive at the start of that year. A \$100 claim expense is paid at the end of the year of death. Find the level gross premium.

Method. Use an annuity-due factor for premiums, but value renewal expenses at year-end. For the second year’s renewal expense, payment requires survival to the start of year 2.

Here $v=1/1.10=10/11$, and the survival probability to time 1 is $0.90$. The premium factor is:

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

The present-value factor for death in the two years is:

0.10v(1+0.90v) =0.10\left(\frac{10}{11}\right)\left(\frac{20}{11}\right) =\frac{20}{121}.

The benefit and claim expense together are \$20,100 per claim, so their expected present value is $20{,}100(20/121)=\$3{,}322.31$. The renewal expenses are paid at the ends of years 1 and 2, not at the premium dates. Their expected present value is:

60v(1+0.90v) =60\left(\frac{10}{11}\right)\left(\frac{20}{11}\right) =\frac{7{,}200}{121} =\$59.50.

Adding the issue expense and solving gives:

P\left(\frac{20}{11}\right) =3{,}322.31+59.50+150, \qquad P=\$1{,}942.23.

Check. Using unrounded values, the outgo is $402{,}000/121+7{,}200/121+150=427{,}350/121$, and multiplying this by $11/20$ gives $427{,}350/220=\$1{,}942.50$. Thus the rounded premium must be \$1,942.50.

Timing and Interpretation

The third example highlights why every term should be checked. The death-benefit and claim-expense factor is $20/121$, while the year-end renewal-expense factor is $120/121$: the expense is paid for each year that begins with the policy in force, whether or not death occurs during that year. Consequently, the outgo is:

\frac{402{,}000}{121}+\frac{7{,}200}{121}+150 =\frac{427{,}350}{121}.

Dividing by the premium factor $20/11$ gives $427{,}350/220=\$1{,}942.50$. The corrected check also illustrates a useful discipline: substitute exact or unrounded intermediate values before rounding the final premium. In the earlier display, the renewal-expense multiplication must use $60(10/11)(20/11)=12{,}000/121$, not $7{,}200/121$. The corrected renewal value is $\$99.17$, and the outgo is $402{,}000/121+12{,}000/121+150$; this again gives $P=\$1{,}960.50$. The correct final premium is therefore \$1,960.50.

A reliable gross-premium calculation uses the cash-flow descriptions rather than a memorized expense formula. In particular, do not assume a renewal expense has the same timing as a premium simply because both relate to policy administration. The model must say when the expense is paid.

Common Mistakes to Avoid

Putting every expense at issue. An expense paid at a later date must be discounted from that date and weighted by the probability it is incurred.
Using the wrong survival probability. A renewal payment at time $k$ generally requires survival to time $k$; a claim payment in year $k+1$ requires survival $k$ years and death in the next year.
Applying a percentage expense to the wrong premium. A charge on the initial premium and a charge on renewal premiums have different expected present values.
Confusing payment time with incurral time. If an expense is paid at year-end, discount it to year-end even if it relates to an earlier administrative event.
Rounding too early. Keep full precision in the present-value factors and round the premium only at the end.

Key takeaway: A gross premium is found by equating expected present values, not by adding a guessed expense loading to a net premium. For every expense, specify its amount, payment date, and payment condition; then include its corresponding expected present value in the equation.

Check Your Understanding

For each problem, write the expected-present-value equation before solving. Unless stated otherwise, premiums are payable at the beginning of each year while the insured is alive. Round final premiums to the nearest cent.

A two-year term policy pays \$10,000 at the end of the year of death. The annual effective interest rate is $0\%$, and the conditional death probability is $0.10$ each year. A \$100 acquisition expense is paid at issue and a \$20 renewal expense is paid at time 1 if the insured survives. Find the gross premium. (Answer: \$1,062.11)
A two-year term policy pays \$20,000 at the end of the year of death. The annual effective interest rate is $5\%$, with conditional death probabilities $0.20$ in year 1 and $0.10$ in year 2. A \$50 acquisition expense is paid at issue, a \$40 renewal expense is paid at time 1 if the insured survives, and a \$100 claim expense is paid with the benefit. Find the level gross premium. (Answer: \$3,046.45)
A two-year term policy pays \$10,000 at the end of the year of death. The interest rate is $0\%$, and the conditional death probability is $0.10$ each year. There is a \$50 fixed issue expense and a \$100 claim expense per death. In addition, the issue expense is $20\%$ of the initial gross premium, and renewal expense is $10\%$ of the time-1 gross premium if the insured survives. Find the level gross premium. (Answer: \$1,241.61)
A two-year term policy pays \$15,000 at the end of the year of death. The annual effective interest rate is $10\%$, with conditional death probabilities $0.10$ in year 1 and $0.20$ in year 2. Premiums are due at times 0 and 1 while alive. A \$120 acquisition expense is paid at issue, a \$50 expense is paid at the end of year 1 if the policy was in force at the start of that year, and a \$100 claim expense is paid with the benefit. Find the level gross premium. (Answer: \$1,529.95)
A three-year term policy pays \$30,000 at the end of the year of death. The annual effective interest rate is $5\%$, and the conditional death probability is $0.10$ in each year. Premiums are payable annually in advance while alive. A \$200 acquisition expense is paid at issue; a \$25 renewal expense is paid alongside each renewal premium; and a \$150 claim expense is paid with each death benefit. Find the level gross premium. (Answer: \$3,116.78)

What You'll Learn