Expense Timing in Gross Reserve Recursions

Why Expense Timing Matters

A gross reserve reflects not only future benefits and premiums but also the insurer’s future expenses. The timing of an expense matters because a dollar paid now does not have the same present value as a dollar paid at the end of a year. It also matters whether an expense is due for every policy exposed during the year or only for policies that survive to the next premium date.

This tutorial develops an annual gross reserve recursion that keeps those distinctions explicit. The model is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. All products and numerical values below are hypothetical.

Definition: A gross prospective reserve is the expected present value of future benefits and expenses, less the expected present value of future gross premiums, conditional on the policy being in force at the valuation time. In this tutorial, $V_k$ denotes that reserve immediately before the premium, if any, due at time $k$.

Set the Timing Convention Before Writing the Recursion

Consider a policy during the year from time $k$ to time $k+1$. At time $k$, if the insured is alive, a premium $P_k$ is received and a premium-date expense $e_k$ is paid. An acquisition expense $A$, if any, is paid at issue, time $0$. During the year, death occurs with probability $q_k$, and survival to time $k+1$ has probability $p_k=1-q_k$. Assume an effective annual interest rate $i$, with $v=1/(1+i)$.

For the main recursion, assume a death benefit $B_{k+1}$ is paid at time $k+1$ if death occurs during the year. A year-end expense $c_k$ is also paid at time $k+1$ for every policy exposed during that year, whether death or survival occurs. If the insured survives, the policy continues with reserve $V_{k+1}$, measured just before the next premium. This is a discrete annual timing model: it does not distinguish the exact date of death within the year.

The expense categories have different roles. An acquisition expense is a time-0 cash flow, so it appears only in the initial reserve equation. A premium-date expense is paid when a premium is due, so it appears alongside that premium. A year-end expense is discounted from time $k+1$. The symbols $e_k$ and $c_k$ can vary by policy year.

Recursion: Under the stated assumptions, for $k=0,\ldots,n-1$, $$ V_k = \mathbf{1}_{\{k=0\}}A + e_k - P_k + v\left[q_k(B_{k+1}+c_k)+p_k(c_k+V_{k+1})\right]. $$ Equivalently, $$ V_k = \mathbf{1}_{\{k=0\}}A + e_k - P_k + v\left[q_kB_{k+1}+c_k+p_kV_{k+1}\right]. $$ Here $\mathbf{1}_{\{k=0\}}$ equals 1 at issue and 0 otherwise. $V_n$ is the reserve at the end of the final coverage period, after any final benefit and expense cash flows specified by the model.

Why the Recursion Holds

The reserve at time $k$ is a present value at time $k$. At that instant, the expense due is added as an outgo and the premium received is subtracted as an inflow. The issue expense is included only when $k=0$. Future cash flows are valued by conditioning on death or survival during the year.

If death occurs, the end-of-year outgo is $B_{k+1}+c_k$. If survival occurs, the end-of-year expense is $c_k$, and the continuing liability is $V_{k+1}$. Both branches are discounted by $v$. Thus the expected end-of-year value is

v\left[q_k(B_{k+1}+c_k)+p_k(c_k+V_{k+1})\right].

Because $q_k+p_k=1$, the year-end expense contributes $q_kc_k+p_kc_k=c_k$ inside the expectation. This gives the equivalent compact form. The premium-date expense $e_k$ is not discounted in this equation because it is paid at the valuation time; nor is the premium $P_k$.

The assumption that $c_k$ is paid whether death or survival occurs is essential to the compact formula. If instead the expense is paid only when the insured survives, its contribution inside the discounted expectation is $p_kc_k$, not $c_k$. If it is incurred only on death, its contribution is $q_kc_k$. Always model the expense’s eligibility condition as well as its payment date.

Worked Applications

Worked Example: Comparing Issue and Year-End Expenses

A hypothetical one-year term policy has death benefit $B_1=\$500$, annual death probability $q_0=0.20$, effective annual interest rate $i=0.25$, and premium $P_0=\$50$. The terminal reserve is $V_1=0$. Compare the reserve at issue when a $\$40$ expense is paid at issue with the reserve when a $\$40$ expense is paid at year-end for every policy exposed during the year. Also find the reserve with no expense.

Method and assumptions. Here $p_0=1-0.20=0.80$, and $v=1/(1+0.25)=0.80$. There is no premium-date expense, so $e_0=0$.

No expense. The present value of the expected death benefit is $v q_0 B_1=0.80(0.20)(500)=\$80$. The reserve is $V_0=0-50+80=\$30$.

Expense paid at issue. The expense is not discounted because it is paid at time 0: $V_0=40-50+0.80(0.20)(500)=40-50+80=\$70$.

Expense paid at year-end. The year-end expense is included with the end-of-year cash flows: $V_0=0-50+0.80[0.20(500)+40]=-50+0.80(140)=-50+112=\$62$.

The year-end expense raises the reserve by $\$32$ relative to the no-expense case, its present value $0.80(40)$. The issue expense raises it by the full $\$40$. The calculation avoids treating a year-end outgo as though it were paid immediately.

Worked Example: Two-Year Reserves with Renewal and Year-End Expenses

A hypothetical two-year term policy pays $\$400$ at the end of the year of death. The annual death probabilities are $q_0=0.25$ and $q_1=0.50$, and the effective annual interest rate is $i=0.25$. The premium is $\$100$ at the start of each year while the policy is in force. There is a $\$20$ acquisition expense at issue, a $\$10$ expense with the premium at time 1, and year-end expenses of $\$15$ in the first year and $\$5$ in the second year. Year-end expenses are paid whether death or survival occurs. Let $V_2=0$. Find $V_1$ and $V_0$.

Method and assumptions. The discount factor is $v=1/1.25=0.80$. First calculate the reserve at time 1, then use it in the time-0 recursion. The acquisition expense is not included again at time 1.

At time 1, $p_1=1-0.50=0.50$. The death branch has end-of-year outgo $\$400+\$5=\$405$; the survival branch has the $\$5$ expense and terminal reserve zero. Therefore,

\begin{aligned} V_1 &=10-100+0.80\left[0.50(400+5)+0.50(5+0)\right]\\ &=-90+0.80(202.50+2.50)\\ &=-90+164\\ &=\$74. \end{aligned}

At time 0, $p_0=1-0.25=0.75$. The expected end-of-year value before discounting is $0.25(400+15)+0.75(15+74)=103.75+66.75=\$170.50$. Thus,

\begin{aligned} V_0 &=20-100+0.80(170.50)\\ &=-80+136.40\\ &=\$56.40. \end{aligned}

As a check, the compact expression inside the time-0 brackets is $q_0B_1+c_0+p_0V_1=0.25(400)+15+0.75(74)=\$170.50$, matching the branch calculation. The common mistake avoided here is carrying the issue expense into $V_1$: it was paid at time 0, so it affects $V_0$, not the future cash flows valued at time 1.

Worked Example: Solving for a Premium with Expenses at Different Times

A hypothetical one-year policy has death probability $q_0=0.10$, death benefit $\$300$, and effective annual interest rate $i=0.05$. It has an acquisition expense of $\$12$ at issue, a premium-date expense of $\$3$ at issue, and a year-end expense of $\$5$ payable whether death or survival occurs. Find the gross premium $P_0$ that gives $V_0=0$, with $V_1=0$.

Method and assumptions. The survival probability is $p_0=0.90$, and $v=1/1.05=20/21$. Use the issue and premium-date expenses at time 0, but discount the year-end benefit and expense.

\begin{aligned} 0=V_0 &=12+3-P_0+\frac{20}{21}\left[0.10(300)+5+0.90(0)\right]\\ &=15-P_0+\frac{20}{21}(35)\\ &=15-P_0+\frac{100}{3}. \end{aligned}

Solving gives $P_0=15+100/3=145/3=\$48.333\ldots$. To the nearest cent, the gross premium is $\$48.33$. Substitution checks the result: $15-145/3+100/3=0$. The calculation keeps the two time-0 expenses undiscounted and discounts the year-end outgo.

How Timing Changes Reserves

For a fixed set of benefits, premiums, and mortality assumptions, expense timing changes the present value of the outgo. In the recursion, an issue expense increases $V_0$ dollar-for-dollar and has no direct effect on later prospective reserves. A premium-date expense increases the reserve at that date dollar-for-dollar. Its effect on an earlier reserve is discounted and multiplied by the probability that the policy remains in force until that date.

A year-end expense is discounted from the end of the year. Under the main model, it applies to every policy exposed, so it contributes $vc_k$ to $V_k$. For example, its effect on the time-0 reserve from an expense in year 1 is $vc_0$. A later renewal expense may be due at the same calendar time as an expense described as “year-end.” If the cash flow, eligibility condition, and payment time are all identical, the recursion treats them identically; the label alone does not create a different value.

A useful comparison is:

Expense type	When it is paid	Contribution at time $k$
Acquisition expense $A$	At issue, time 0	Included only when $k=0$
Premium-date expense $e_k$	At time $k$, with the premium	$e_k$, not discounted
Year-end expense $c_k$	At time $k+1$, under the stated exposure assumption	$vc_k$

If expenses are payable only for policies that survive, the year-end cost is not $vc_k$ under this annual model; it is $vp_kc_k$. That change can matter materially when mortality is substantial. The model must match the contractual or actuarial assumption before the recursion is applied.

Common Timing Trap

In the first worked example, the no-expense reserve is $\$30$. A tempting but incorrect calculation for the $\$40$ year-end expense is to add the full $\$40$ immediately, obtaining $30+40=\$70$. That treats the expense as an issue-date payment. The correct increase is its present value, $v(40)=0.80(40)=\$32$, so the reserve is $30+32=\$62$.

Another error is to multiply a year-end expense by the death probability merely because a death benefit appears in the same recursion. In the stated model, the expense is paid whether death or survival occurs, so its probability-weighted amount is $q_kc_k+p_kc_k=c_k$. Use $q_kc_k$ only if the expense is payable only on death, and $p_kc_k$ only if it is payable only on survival.

Key takeaway: Fix the reserve’s valuation time, identify who incurs each expense, and place each cash flow at its actual payment time. Then condition on death and survival and discount only cash flows occurring after the valuation time.

Check Your Understanding

For each problem, use the reserve convention and timing assumptions stated in the question. Amounts are hypothetical.

A one-year policy has $q_0=0.10$, benefit $\$600$, $i=0.10$, premium $\$40$, issue expense $\$8$, and year-end expense $\$11$ payable on death or survival. Given $V_1=0$, find $V_0$. (Answer: $\$32.55$)
A one-year policy has $q_0=0.20$, benefit $\$300$, $i=0.25$, premium $\$70$, issue expense $\$5$, premium-date expense $e_0=\$4$, and year-end expense $\$10$ payable on either outcome. Given $V_1=0$, find $V_0$. (Answer: $-\$5.00$)
A two-year policy has $q_0=0.10$, $q_1=0.20$, benefit $\$500$ in either year of death, $i=0.10$, and premiums of $\$100$ at times 0 and 1 if in force. The issue expense is $\$10$, the time-1 premium-date expense is $\$6$, and year-end expenses are $\$5$ and $\$8$, each payable on death or survival in its respective year. Given $V_2=0$, calculate $V_1$ and $V_0$. (Answer: $V_1=\$4.18$, $V_0=-\$36.58$)
A two-year policy has $q_0=0.25$, $q_1=0.20$, death benefit $\$400$ in either year, $i=0.25$, and an unknown level premium $P$ at times 0 and 1 if in force. The issue expense is $\$16$, the time-1 premium-date expense is $\$4$, and year-end expenses are $\$6$ and $\$10$, payable on death or survival. Given $V_2=0$, find $P$ such that $V_0=0$. (Answer: $\$91.50$)

Expense type	When it is paid	Contribution at time \(k\)
Acquisition expense \(A\)	At issue, time 0	Included only when \(k=0\)
Premium-date expense \(e_k\)	At time \(k\), with the premium	\(e_k\), not discounted
Year-end expense \(c_k\)	At time \(k+1\), under the stated exposure assumption	\(vc_k\)

What You'll Learn