Renewal Expenses in Level Gross Premiums

Why Renewal Expenses Need Survival Weighting

A policy may generate expenses not only when it is issued but also when it renews. For example, a hypothetical insurer might incur an administrative cost each year it processes an annual premium. Unlike an expense paid at issue, a renewal expense is incurred only if the policy is still in force at that renewal date. Its expected present value must therefore reflect both the chance of reaching the date and the time value of money.

This tutorial uses a discrete annual model. Premiums are payable at the beginning of each policy year while the insured is alive and the policy remains in force. A death benefit is paid at the end of the year of death. We assume no lapses, and we assume renewal expenses are paid at the same dates as renewal premiums. If a contract uses different dates or remains in force under other conditions, its cash flows and probabilities must be adjusted to match those terms.

Definition: A renewal expense is an expense incurred after issue when a policy remains in force to a later renewal date. If a level amount $c$ is incurred at each annual renewal, its expected present value is the sum of $c$ times the probability of being in force at each renewal date, discounted to issue. For an initial expense $E_0$ paid at issue, the expense amount is certain at time 0 and is not survival-weighted.

Let $G$ be the annual level gross premium, $E_0$ an initial expense, and $c$ the fixed expense at each renewal. The gross premium is determined by the equivalence principle: the expected present value of premiums equals the expected present value of benefits and expenses. Here, “gross” means that expenses are included. It does not, by itself, imply any additional profit or risk margin.

Present Value of Renewal Expenses

For a life aged $x$, let ${}_kp_x$ denote the probability of surviving $k$ years. With annual effective interest rate $i$, let $v=1/(1+i)$. A renewal at time $k$ occurs only if the policy is in force at that time, so an expense $c$ then has expected present value $c v^k {}_kp_x$.

For a whole-life policy with premiums due annually in advance while the insured is alive, the premium present-value factor is $\ddot{a}_x$. Its time-0 premium is due with certainty; renewal premiums are due at times $k=1,2,\ldots$ only if the insured has survived to those dates. The renewal expenses occur at precisely those later dates, so their present-value factor is the premium annuity factor with its time-0 payment removed.

\operatorname{EPV}(\text{renewal expenses}) =c\sum_{k=1}^{\infty}v^k\,{}_kp_x =c(\ddot{a}_x-1).

If the policy is an $n$-year term contract with premiums at times $0,1,\ldots,n-1$, there are only $n-1$ renewal dates. Write the premium factor explicitly as $\sum_{k=0}^{n-1}v^k{}_kp_x$. The renewal-expense factor is the same sum with its time-0 term removed. In particular, an expense is not charged at time $n$ under this payment pattern, because there is no premium due then.

\operatorname{EPV}(\text{renewal expenses}) =c\sum_{k=1}^{n-1}v^k\,{}_kp_x.

Deriving the Gross Premium Equation

The key step is to put every expected cash flow on the same basis: present values at issue. The premium at time $k$ is received only if the policy is in force at that date. Under the assumptions above, this is also the condition for incurring a renewal expense at time $k$.

Value premiums.
Multiply $G$ by the expected present-value factor for the premium dates. For whole-life annual premiums, this is $\ddot{a}_x$.

Value benefits.
Use the expected present value of the death benefit under the stated benefit and timing assumptions. For a unit whole-life benefit paid at the end of the year of death, this is $A_x$.

Add expenses.
Add $E_0$ for an expense paid at issue and $c(\ddot{a}_x-1)$ for fixed annual renewal expenses on whole-life coverage.

Solve for the gross premium.
Set expected premium present value equal to expected benefit and expense present value, then divide by the premium factor.

For a whole-life insurance benefit of amount $B$, the equivalence equation and solution are therefore:

G\ddot{a}_x =BA_x+E_0+c(\ddot{a}_x-1), \qquad G=\frac{BA_x+E_0+c(\ddot{a}_x-1)}{\ddot{a}_x}.

This is a direct consequence of the cash-flow timing. The expected present value of premiums is $G\sum_{k=0}^{\infty}v^k{}_kp_x=G\ddot{a}_x$. The initial expense is paid with certainty at issue, and the expected present value of later expenses is $c\sum_{k=1}^{\infty}v^k{}_kp_x$. Since the premium sum includes the time-0 term $1$, subtracting $1$ from $\ddot{a}_x$ leaves exactly the later payment dates. Equating inflows and outflows gives the displayed formula.

For an $n$-year term policy with annual premiums at times $0$ through $n-1$, a benefit $B$ paid at the end of the year of death, and renewal expense $c$, the equation is:

G\sum_{k=0}^{n-1}v^k{}_kp_x = B\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k} +E_0 +c\sum_{k=1}^{n-1}v^k{}_kp_x.

Here, $q_{x+k}$ is the probability of death within the next year for a life aged $x+k$. The benefit term accounts for death in each covered year, while the renewal term accounts only for reaching a renewal date. The same equivalence principle applies if the mortality probabilities vary by year; they should not be replaced by a constant mortality assumption unless the model specifies one.

Worked Example: Whole-Life Renewal Expenses

A hypothetical whole-life policy pays a death benefit of $10,000 at the end of the year of death. Premiums are annual in advance for life. Assume an annual effective interest rate of $5\%$, a constant annual death probability $q=0.10$, an initial expense of $200, and a renewal expense of $50 at each later annual premium date. Calculate the level gross premium. Give the final amount to the nearest cent.

Method. First calculate the whole-life premium and insurance factors under the specified constant mortality model. Then include the certain initial expense and the survival-weighted renewal expenses. This avoids counting renewal costs as if every policy necessarily reached every renewal date.

The annual survival probability is $p=1-q=0.90$, and $v=1/1.05=20/21$. The common ratio in the whole-life annuity sum is $vp=(20/21)(0.90)=6/7$. Thus:

\ddot{a}_x=\sum_{k=0}^{\infty}(vp)^k =\frac{1}{1-6/7}=7.

The unit death benefit is paid one year after each year’s survival-and-death event. Its present-value factor is:

A_x=\sum_{k=0}^{\infty}v^{k+1}p^kq =\frac{vq}{1-vp} =\frac{(20/21)(0.10)}{1/7} =\frac{2}{3}.

The benefit present value is $10{,}000(2/3)=\$6{,}666.6667$. The renewal-expense factor is $7-1=6$, so renewal expenses have present value $50(6)=\$300$. Adding the initial expense gives total expected outgo of $\$6{,}666.6667+\$300+\$200=\$7{,}166.6667$. Therefore:

G=\frac{7{,}166.6667}{7} =\$1{,}023.8095\ldots \approx \boxed{\$1{,}023.81}.

Check. At the unrounded premium, expected premium present value is $7G=\$7{,}166.6667$, equal to the benefit and expense present value. The renewal expense is less than $50$ times an infinite certain payment stream because survival and discounting reduce its expected present value.

Worked Example: Three-Year Term with Constant Mortality

A hypothetical three-year term policy pays $8,000 at the end of the year of death. Annual premiums are due at times 0, 1, and 2, while the insured is alive. Assume an annual effective interest rate of $5\%$, constant annual death probability $q=0.20$, an initial expense of $100, and a $40 renewal expense at times 1 and 2 if the policy remains in force. Find the level gross premium and round only the final result to cents.

Method. Use the finite premium-annuity sum for the three premium dates. The renewal expense sum has only two terms because there is no renewal at issue and no premium date at time 3.

Here $p=0.80$, $v=20/21$, and $vp=16/21$. The premium factor is:

\sum_{k=0}^{2}(vp)^k =1+\frac{16}{21}+\frac{256}{441} =\frac{1033}{441} =2.3424036.

The unit death-benefit present value is the annual death probability times $v$, multiplied by the premium-factor sum:

\sum_{k=0}^{2}v^{k+1}p^kq =vq\sum_{k=0}^{2}(vp)^k =\frac{4}{21}\frac{1033}{441} =\frac{4132}{9261} =0.4461719.

Thus the benefit present value is $8{,}000(0.4461719\ldots)=\$3{,}569.377\ldots$. The renewal-expense factor is $1033/441-1=592/441=1.3424036$, so the renewal expense present value is $40(592/441)=\$53.6961$. Total expected outgo is:

\$3{,}569.377\ldots+\$100+\$53.6961\ldots =\$3{,}723.073\ldots

Divide by the premium factor:

G=\frac{3{,}723.073\ldots}{2.3424036\ldots} =\$1{,}589.424\ldots \approx \boxed{\$1{,}589.42}.

Check. Multiplying the unrounded premium by $2.3424036\ldots$ returns the total expected outgo of $\$3{,}723.073\ldots$. The expense at time 2 is weighted by survival through two years, not by the probability of surviving only one year.

Worked Example: Changing Mortality and the Renewal-Date Condition

A hypothetical three-year term policy pays $5,000 at the end of the year of death. Premiums are due at times 0, 1, and 2. Assume zero interest, annual death probabilities of $0.10$, $0.20$, and $0.25$ in years 1, 2, and 3, respectively, an initial expense of $80, and a $30 renewal expense at times 1 and 2 if the policy is still in force. Calculate the level gross premium to the nearest cent.

Method. Build the survival probabilities from the year-specific mortality rates. Then value the death benefit, premium stream, and renewal expenses on the same time-0 basis. The zero interest assumption means $v=1$, but it does not remove the need to weight payments by survival.

Survival to time 1 is $0.90$. Survival to time 2 is $0.90(1-0.20)=0.90(0.80)=0.72$. The premium factor is consequently $1+0.90+0.72=2.62$. The probabilities of death in the three years are $0.10$, $0.90(0.20)=0.18$, and $0.72(0.25)=0.18$. Their sum is $0.46$, so the expected benefit present value is $5{,}000(0.46)=\$2{,}300$.

Renewal expenses are paid at times 1 and 2 only if the policy has survived to those dates. Their expected present value is $30(0.90+0.72)=\$48.60$. Including the initial expense, total expected outgo is $\$2{,}300+\$48.60+\$80=\$2{,}428.60$. Therefore:

G=\frac{2{,}428.60}{2.62} =\$926.9465\ldots \approx \boxed{\$926.95}.

Check. The unrounded premium times $2.62$ is $\$2{,}428.60$. If the two renewal expenses were incorrectly treated as certain, their value would be $60 rather than $48.60, overstating the premium by $(60-48.60)/2.62=\$4.35$ to the nearest cent.

Common Mistake: Counting Every Renewal as Certain

A recurring expense is not automatically a certain expense stream. Under the stated model, a policy that ends through death cannot generate later renewal costs. The correct term at renewal date $k$ is $c v^k{}_kp_x$, not $c v^k$. Omitting ${}_kp_x$ values the expense as though the policy were guaranteed to remain in force.

A second timing error is to include an expense at a date when the contract has no renewal. For example, a term policy with premiums at times $0$ through $n-1$ has renewal expenses only at times $1$ through $n-1$, if expenses coincide with renewal premiums. Check the payment dates before writing the sum. An expense that is payable at a different event, such as on death, needs a different probability and timing.

The central practical test is to list each cash flow with its date and condition. An initial expense is certain at issue; a renewal expense requires survival and continuation to its date; a premium is received only when due under the contract. Keeping these conditions visible makes the gross premium equation easier to construct and audit. This treatment of recurring expenses is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Key takeaway: For level gross premiums, include each fixed renewal expense at its actual date and multiply by the probability the policy remains in force then. With annual whole-life premiums and matching renewal expenses, the renewal-expense factor is $\ddot{a}_x-1$, not $\ddot{a}_x$.

Check Your Understanding

For each problem, use the stated premium and expense timing. Round money to the nearest cent where needed.

A hypothetical whole-life policy has a $1,000 end-of-year-of-death benefit, annual premiums in advance, zero interest, and constant annual death probability $q=0.20$. There is no initial expense and the renewal expense is $10 at each later premium date. Find the level gross premium. (Answer: $208.00)
A two-year term policy has a $2,000 end-of-year-of-death benefit, zero interest, death probabilities $0.10$ in year 1 and $0.20$ in year 2, a $50 initial expense, and a $15 renewal expense at time 1 if the policy is in force. Find the level gross premium. (Answer: $328.16)
A three-year term policy has a $4,000 end-of-year-of-death benefit, zero interest, constant annual death probability $q=0.25$, a $100 initial expense, and a $20 expense at each later annual premium date. Find the level gross premium. (Answer: $1,054.59)
A hypothetical whole-life policy has a $5,000 end-of-year-of-death benefit, annual premiums in advance, an annual effective interest rate of $5\%$, and constant annual death probability $q=0.10$. The initial expense is $120 and the renewal expense is $25 at each later premium date. Find the level gross premium. (Answer: $514.76)
A three-year term policy has a $1,000 end-of-year-of-death benefit, zero interest, annual death probabilities $0.10$, $0.20$, and $0.25$, a $40 initial expense, and a $12 renewal expense at times 1 and 2 if the policy remains in force. Find the level gross premium. (Answer: $198.26)

What You'll Learn