Why Reserves Change Annual Profit
A policy can generate a positive operating cash flow during a year and still report a loss if the reserve required at year-end has increased substantially. Conversely, a policy may report profit when it terminates and releases a reserve, even if its current-year cash flow is modest. The reserve adjustment is therefore essential when annual profit is calculated for a long-term policy.
In a multi-state contract, the reserve at the end of the year can depend on the state reached: a continuing policy, a policyholder receiving disability benefits, and a terminated policy can have different ending liabilities. The calculation must pair the beginning reserve with the ending reserve appropriate to each transition. This tutorial develops that annual calculation and shows how to form a transition-weighted profit vector. The treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
The notation \(i_{\!a}\) is used here for the annual effective interest rate to distinguish it from the state label \(i\). In the examples, premiums, benefits, and expenses are combined into \(C_{ij}\), expressed at the end of the year. If a cash flow occurs earlier, it must first be accumulated to year-end under the stated investment assumption. The formula assumes no other asset items, such as taxes or shareholder transfers, beyond those explicitly included in \(C_{ij}\).
The reserve values may be supplied by a valuation basis or calculated elsewhere. This tutorial focuses on how those values enter annual profit, not on deriving the reserves themselves. The formula applies whether the policy remains in force or ends during the year, provided the ending reserve matches the liability, if any, after the transition.
Deriving the Reserve Adjustment
The calculation follows an end-year asset balance. The beginning reserve grows with interest to \((1+i_{\!a})V_i^{B}\). Add the net cash flow accumulated to year-end. The ending reserve must then remain available to support the policy after the year closes. What is left is profit.
Separating the interest earned on the beginning reserve from the change in reserve gives an equivalent form:
Thus, an increase in reserve, \(V_j^{E}-V_i^{B}>0\), reduces profit; a decrease increases profit. This decomposition uses the unaccumulated beginning reserve in the reserve change and adds interest separately. It is equivalent to accumulating the full beginning reserve and subtracting the ending reserve.
Use the reserve for the policy’s state at the start of the period, not a reserve for a possible later state.
Combine premiums and other inflows positively, and benefits and expenses negatively. Accumulate earlier cash flows to year-end if needed.
Subtract the reserve required in the actual or specified ending state. Use zero if no liability remains after termination.
For expected profit conditional on the start state, multiply each transition profit by its transition probability and sum.
Worked Annual Profit Calculations
Worked Example: An Increase in the Reserve for a Continuing Policy
A hypothetical one-year policy remains in force throughout the year. The annual effective interest rate is 5%. The beginning reserve is $800. At year-end, the policy has received a $300 premium and incurred $100 of benefits and $40 of expenses, so its net end-year cash flow is \(C=300-100-40=\$160\). The ending reserve is $900. Calculate the annual profit.
Method. Accumulate the beginning reserve, add the net cash flow, and deduct the reserve required at year-end.
The accumulated beginning reserve is $$ 1.05(800)=\$840. $$
The end-year amount before setting aside the ending reserve is $$ 840+160=\$1{,}000. $$
After retaining the $900 ending reserve, profit is $$ \Pi=1{,}000-900=\boxed{\$100}. $$
As a check, the reserve increased by \(900-800=\$100\), and interest on the beginning reserve is \(0.05(800)=\$40\). Therefore, $$ \Pi=40+160-100=\$100. $$ The two forms agree. The common mistake is to add the reserve increase to profit; it is an additional liability and must reduce profit.
Worked Example: A Profit Vector From an Active Policy
A hypothetical active policy has beginning reserve $500. The annual effective interest rate is 5%. At year-end, it can be active with probability 0.70, disabled with probability 0.20, or terminated by death with probability 0.10. The corresponding ending reserves are $560, $1,200, and $0. The path cash flows, already stated as end-year amounts, are: active, a $400 premium less $50 of expenses; disabled, a $200 benefit and $30 of expenses; death, a $400 premium less $50 of expenses and a $10,000 death benefit. Calculate each path profit and the expected profit conditional on starting active.
Method. First calculate each path cash flow and profit using the reserve for that ending state. Then weight the profits by the stated transition probabilities.
The accumulated beginning reserve is the same for all three paths: $$ 1.05(500)=\$525. $$ For the active path, \(C_{\mathrm{active}}=400-50=\$350\), so $$ \Pi_{\mathrm{active}}=525+350-560=\$315. $$ For the disabled path, \(C_{\mathrm{disabled}}=-200-30=-\$230\), so $$ \Pi_{\mathrm{disabled}}=525-230-1{,}200=-\$905. $$ For the death path, \(C_{\mathrm{death}}=400-50-10{,}000=-\$9{,}650\), and no reserve remains after death: $$ \Pi_{\mathrm{death}}=525-9{,}650-0=-\$9{,}125. $$
The expected profit conditional on starting active is $$ 0.70(315)+0.20(-905)+0.10(-9{,}125) =220.50-181.00-912.50 =\boxed{-\$873.00}. $$
As a check, the expected ending reserve is \(0.70(560)+0.20(1{,}200)+0.10(0)=\$632\). Expected path cash flow is \(0.70(350)+0.20(-230)+0.10(-9{,}650)=-\$766\). Thus \(525-632-766=-\$873\), matching the transition-weighted result. The probabilities sum to one, and the death path correctly uses a zero ending reserve.
Worked Example: Reserve Release on Recovery and Termination
A hypothetical policy starts the year in a disabled state with a beginning reserve of $1,200. The annual effective interest rate is 4%. At year-end, the policyholder may recover to an active state with probability 0.25, remain disabled with probability 0.65, or die with probability 0.10. Ending reserves are $500 after recovery, $1,300 if still disabled, and $0 after death. End-year path cash flows, excluding reserve movements and investment income, are a $20 expense after recovery, a $300 benefit plus $20 expense while disabled, and a $6,000 death benefit plus $20 expense on the death path. Calculate the path profits and expected profit.
Method. Accumulate the disabled-state beginning reserve once, then apply each path’s own cash flow and ending-state reserve.
The accumulated beginning reserve is $$ 1.04(1{,}200)=\$1{,}248. $$ After recovery, \(C_{\mathrm{recovery}}=-\$20\), so the $700 reserve release contributes positively through the reduction in required reserve: $$ \Pi_{\mathrm{recovery}}=1{,}248-20-500=\$728. $$ If the policy remains disabled, \(C_{\mathrm{disabled}}=-300-20=-\$320\), and the reserve increases by $100: $$ \Pi_{\mathrm{disabled}}=1{,}248-320-1{,}300=-\$372. $$ On death, \(C_{\mathrm{death}}=-6{,}000-20=-\$6{,}020\), with no ending reserve: $$ \Pi_{\mathrm{death}}=1{,}248-6{,}020-0=-\$4{,}772. $$
Weighting the results gives $$ 0.25(728)+0.65(-372)+0.10(-4{,}772) =182-241.80-477.20 =\boxed{-\$537.00}. $$
As a check, the expected ending reserve is \(0.25(500)+0.65(1{,}300)=\$970\), and expected cash flow is \(0.25(-20)+0.65(-320)+0.10(-6{,}020)=-\$815\). Then \(1{,}248-970-815=-\$537\). Recovery releases reserve and raises that path’s profit relative to holding a larger liability; remaining disabled increases the reserve and reduces profit.
Common Sign and Weighting Traps
The reserve adjustment is not an extra cash benefit or claim. It represents the amount of the end-year fund that must remain tied to the contract. A reserve increase consumes part of the amount otherwise available as profit; a reserve release makes more available.
- Reversing the sign. In the equivalent form, subtract \(V_j^{E}-V_i^{B}\). Do not add that increase. In Worked Example 1, the incorrect calculation \(40+160+100=\$300\) adds the $100 reserve increase. The correction is \(40+160-100=\$100\).
- Using the beginning-state reserve at year-end. In a multi-state policy, the ending reserve depends on the state reached. For example, using $560 rather than $1,200 for the disabled outcome in Worked Example 2 would overstate that path’s profit by $640.
- Weighting reserves but not cash flows, or vice versa. For an expected profit from a starting state, weight each complete path profit. Equivalently, weight the accumulated beginning reserve, path cash flows, and ending reserves consistently.
- Keeping a reserve after termination without justification. If death or another transition ends all future liability, the ending reserve is zero. If a contract provides a continuing benefit after the event, use the reserve for that remaining liability instead.
Check Your Understanding
For each question, calculate annual profit on the stated end-year basis. Where transitions are given, report expected profit conditional on the specified starting state.
- A policy has a beginning reserve of $1,000, a 3% annual effective interest rate, end-year net cash flow of $75, and ending reserve of $1,050. What is its profit? (Answer: $55)
- A policy starts with reserve $600. The annual effective interest rate is 4%, end-year net cash flow is \(-\$90\), and the ending reserve is $450. What is profit, and is this a reserve increase or release? (Answer: $84 profit; $150 reserve release)
- An active policy has beginning reserve $200 and annual effective interest rate 5%. It remains active with probability 0.60, with cash flow $50 and ending reserve $220; or becomes disabled with probability 0.40, with cash flow \(-\$40\) and ending reserve $700. Find expected profit. (Answer: \(-\$188\))
- A policy starts in a care state with reserve $750, and the annual effective interest rate is 4%. It continues in care with probability 0.80, end-year cash flow $100, and ending reserve $780; or lapses with probability 0.20, end-year cash flow \(-\$20\), and zero ending reserve. Find expected profit. (Answer: $232)
- A policy starts active with a $400 reserve and earns 5% annually. It ends either active, with probability 0.75, end-year cash flow $30 and reserve $420; or dead, with probability 0.25, end-year cash flow \(-\$2{,}000\) and zero reserve. Find each path profit and expected profit. (Answer: active profit $30; death profit \(-\$1{,}580\); expected profit \(-\$372.50\))