Reconciling Annual Profit By Cash Flow And Reserves

Why Reconcile an Annual Profit Entry?

An annual profit entry combines several items that can be easy to misalign: premiums and expenses, investment income, benefits, and the movement in reserves. A result can look plausible while still using the wrong reserve sign, earning interest on a cash flow at the wrong time, or charging a closing reserve to policies that have already left the portfolio.

The reconciliation gives each item a clear place in the annual account. It also provides an independent check: the profit calculated from the listed entries must agree with the change from opening funds to ending assets and liabilities. This reconciliation is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: An annual profit entry for a specified period is the amount remaining after accounting for the opening reserve, cash receipts, investment income, cash payments, and closing reserve on a consistent basis. A reserve release is a positive profit item; a reserve increase is a negative profit item.

The Reconciliation Under a Clear Timing Convention

Begin with a simple annual model. At time 0, each policy is in force with opening reserve $V_0$. A premium $P$ is received and an expense $E$ is paid at time 0. These amounts, together with the opening reserve, are invested for the full year at effective annual interest rate $i$. At time 1, claims and surrender benefits are paid, and reserves are held for policies still in force. The convention must be adapted if cash flows occur at different times.

On a per-policy basis, the funds available at the start of the year are $V_0+P-E$. The investment income earned on these funds is $i(V_0+P-E)$. If $C$ denotes total benefits paid per starting policy and $V_1^{*}$ denotes the closing reserve per starting policy, including zero reserve for policies that have exited, the reconciliation is:

\text{Profit} =V_0+P-E+i(V_0+P-E)-C-V_1^{*}.

Equivalently, the investment income is included as a separate profit item and the reserve movement is written as opening reserve minus closing reserve:

\text{Profit} =P-E+\text{investment income}-C+(V_0-V_1^{*}).

These are two forms of the same account, not two separate sources of profit. The first displays the opening reserve as part of the starting funds; the second groups it with the reserve movement. In either form, an increase in the reserve reduces profit, while a decrease releases funds and increases profit.

Why the Formula Holds

The equation follows from a funds balance. Opening assets are represented by the opening reserve. Add cash received, subtract cash paid, and add investment income to obtain the funds at year end before benefit payments and the closing reserve are accounted for. What remains after those obligations is the profit for the year.

Establish starting funds.
At time 0, opening funds plus the premium less the expense equal $V_0+P-E$.

Accrue investment income.
Because these funds are invested for a full year at effective annual rate $i$, year-end funds before payments are $(V_0+P-E)(1+i)$.

Settle benefits and retain closing reserves.
Subtract benefits $C$ paid during the year and the reserve $V_1^{*}$ required at the end of the year.

Identify the residual.
The residual is $(V_0+P-E)(1+i)-C-V_1^{*}$, which expands to the reconciliation formula above.

This derivation depends on the stated timing convention. If a premium is received at year end, it is not invested for the full year; if an expense is paid at year end, it does not reduce the investment base for the full year. The formula remains a cash-flow balance, but investment income must reflect the actual timing assumption.

Worked Examples

Worked Example: Expected Profit for a One-Year Cohort

A hypothetical insurer has 100 identical policies in force at the start of a year. Per policy, the opening reserve is $900, a premium of $300 is received at the start of the year, and an expense of $40 is paid at the start. The effective annual interest rate is 5%. Each policy has a 0.02 probability of death during the year and otherwise survives. A death benefit of $1,200 is paid at year end; a surviving policy has a closing reserve of $950. Find expected profit per starting policy and for the cohort.

Method. Use the time-0 cash-flow convention and calculate expected benefits and closing reserves per policy that was in force at the start.

The funds invested for the full year per policy are:

$$ 900+300-40=1{,}160. $$

Investment income per policy is $0.05(1{,}160)=\$58$. Expected death benefits per starting policy are $0.02(1{,}200)=\$24$. The probability of survival is $1-0.02=0.98$, so the expected closing reserve per starting policy is $0.98(950)=\$931$.

Therefore, expected annual profit per starting policy is:

900+300-40+58-24-931=\$263.

For 100 policies, expected cohort profit is $100(263)=\$26{,}300$. To check, year-end funds before benefits and reserves are $100(1{,}160)(1.05)=\$121{,}800$. Expected benefits are $100(24)=\$2{,}400$, and expected closing reserves are $100(931)=\$93{,}100$. The residual is $121{,}800-2{,}400-93{,}100=\$26{,}300$, as required.

Common mistake avoided: charging a $950 closing reserve for every starting policy would ignore the fact that a policy that dies does not remain in force for the closing reserve. The reserve exposure is the surviving population.

Worked Example: Reconciling a Realized Cohort Account

A hypothetical cohort has 50 policies at the start of the year. Each has an opening reserve of $600. A premium of $500 is received and an expense of $30 is paid per policy at the start of the year. The effective annual interest rate is 4%. At year end, 2 policies have died, each generating a $1,000 death benefit; 3 other policies have lapsed, each generating a $550 surrender payment; 45 policies remain in force with a closing reserve of $650 each. Calculate total cohort profit.

Method. Aggregate the opening reserves and time-0 cash flows, calculate interest on the full-year invested amount, then subtract actual payments and reserves for policies still in force.

Opening reserves total $50(600)=\$30{,}000$. Premiums total $50(500)=\$25{,}000$, and expenses total $50(30)=\$1{,}500$. The invested amount is therefore:

30{,}000+25{,}000-1{,}500=\$53{,}500.

Investment income is $0.04(53{,}500)=\$2{,}140$. Death benefits total $2(1{,}000)=\$2{,}000$, and surrender payments total $3(550)=\$1{,}650$. The closing reserve is $45(650)=\$29{,}250$. Thus:

30{,}000+25{,}000-1{,}500+2{,}140-2{,}000-1{,}650-29{,}250 =\$22{,}740.

The independent check is to accumulate the invested amount: $53{,}500(1.04)=\$55{,}640$. Subtract actual benefits and surrender payments, $2{,}000+1{,}650=\$3{,}650$, and the closing reserve, $\$29{,}250$. The residual is $55{,}640-3{,}650-29{,}250=\$22{,}740$, matching the profit entry.

Common mistake avoided: the closing reserve is based on 45 continuing policies, not all 50 original policies. The death benefits and surrender payments are recorded for the policies that exited; those policies are not also assigned a closing reserve unless the stated contract or valuation basis specifically requires one.

Worked Example: Finding an Exposure-Basis Error

A hypothetical one-year policy has opening reserve $400. A $200 premium is received and a $20 expense is paid at the start of the year. The effective annual interest rate is 5%. Death occurs with probability 0.10 and produces a $1,000 benefit at year end. If the policy survives, its closing reserve is $450. A completed entry reports profit of $59 per starting policy. Check the entry.

Method. Recalculate the expected account per starting policy, weighting the closing reserve by the probability that the policy remains in force.

The full-year invested amount is $400+200-20=\$580$, giving investment income of $0.05(580)=\$29$. The expected death benefit is $0.10(1{,}000)=\$100$. The survival probability is $0.90$, so the expected closing reserve per starting policy is $0.90(450)=\$405$. The correct profit is:

400+200-20+29-100-405=\$104.

The reported $59 can be reproduced by subtracting the full $450 reserve from the expected account: $400+200-20+29-100-450=\$59$. That calculation is wrong because it applies a survivor reserve to every starting policy, including the 10% expected to die. The error is $450-405=\$45$, and the corrected profit is $59+45=\$104$.

Check. Year-end funds before benefits and reserves are $580(1.05)=\$609$. Expected benefit plus expected closing reserve is $100+405=\$505$. The remaining amount is $609-505=\$104$.

Timing, Signs, and Exposure Basis

A reconciliation is reliable only when each item uses the same timing and exposure basis. For a cohort, let $N$ be the number of policies at the start. If $N_j$ policies finish the year in state $j$, each with closing reserve $V_{1,j}$, then the total closing reserve is:

\sum_j N_j V_{1,j}.

This amount is not generally $N$ times the reserve for a continuing policy. Deaths, lapses, surrenders, and other exits may generate payments and remove policies from the closing-reserve exposure. Use the benefits actually associated with exits and the reserves associated with the states remaining at the valuation time.

The sign convention also deserves an explicit check. If the reserve rises from $V_0$ to $V_1^{*}$, then $V_0-V_1^{*}$ is negative and reduces profit. If the reserve falls, the difference is positive and releases funds. Writing “reserve movement” without defining its direction is a common source of sign errors.

Investment income must be based on invested funds and the time they are invested. Under the example convention, $i(V_0+P-E)$ is appropriate because the opening reserve, premium, and expense adjustment are all present at time 0. It would not be appropriate to apply a full year of interest to a premium received at year end. Nor should interest already included in an accumulated fund balance be added again as a separate amount.

A Practical Audit of a Completed Entry

Use the following sequence before accepting a profit result:

State the unit. Confirm whether each figure is per policy at the start, per policy in a later state, or for the whole cohort.
Set the timeline. Identify when premiums, expenses, benefits, and reserve valuations occur.
Check the interest base. Include only funds invested over the relevant period and apply the stated interest assumption.
Check benefit and reserve exposures. Count payments for the policies that trigger them and closing reserves only for policies represented in the closing valuation.
Recompute the balance. Add opening reserve, receipts, and investment income; subtract payments and closing reserves. Compare the result with the reported profit.

If the two calculations disagree, isolate the difference by item rather than adjusting the final profit. In particular, verify whether a reserve is being used as a cash payment or as a closing liability; these are different entries. This discipline makes the annual account auditable even when its components come from different assumptions or calculations.

Key takeaway: Annual profit reconciles when opening funds and correctly timed receipts and investment income, less payments and closing liabilities, leave exactly the reported residual. The reserve sign, cash-flow timing, and policy exposure must all be consistent.

Check Your Understanding

For each problem, use the stated timing convention and reconcile the account before checking the answer.

A policy has opening reserve $300, a $100 premium and $10 expense paid at the start of the year, and an effective annual interest rate of 4%. Death probability is 0.05, the death benefit is $800, and the reserve for a survivor is $320. Find expected profit per starting policy. (Answer: $61.60)
A cohort of 20 policies has opening reserves of $400 each. Each pays a $150 premium and incurs a $10 expense at the start of the year. The effective annual interest rate is 5%. At year end, one policy pays a $900 death benefit, two policies each receive a $430 surrender payment, and 17 survivors each have a $420 reserve. Find total cohort profit. (Answer: $2,440)
A policy has opening reserve $1,000. An expense of $30 is paid at the start of the year, and a premium of $200 is received at the end. The effective annual interest rate is 6%, there are no benefits, and the closing reserve is $1,050. Find the annual profit. (Answer: $178.20)
A policy has opening reserve $650, a $250 premium and $25 expense at the start of the year, and an effective annual interest rate of 4%. Death probability is 0.08, the death benefit is $1,500, and a survivor has closing reserve $700. Find expected profit per starting policy. (Answer: $146.00)
A completed entry uses investment income of $20 for a policy with opening reserve $500, a $100 premium, and a $20 start-of-year expense. The effective annual interest rate is 4%. Death probability is 0.02, the death benefit is $1,000, and the closing reserve for a survivor is $520. Recalculate profit and identify the income error. (Answer: Correct profit is $73.60; income should be $23.20, so profit was understated by $3.20.)

What You'll Learn