Per-Policy and Portfolio Reserve Movements

Why the Number of Policies Matters

A reserve is stated per policy, but a portfolio reserve is the sum of the amounts held for policies still in force. Those two quantities can move differently. A reserve may rise for each continuing contract while the total portfolio reserve falls because enough contracts ended during the period. Conversely, a substantial per-policy reserve release may be outweighed by growth in the number of policies.

The key is to keep two questions separate: how did the reserve change for a continuing policy? and how did the number of policies carrying reserves change? This distinction is important when interpreting a reserve-adjusted profit result. The material on reserve-adjusted profit and survival weighting provides the setting; here the focus is on the arithmetic that connects individual policies to the portfolio.

Definition: Let $N_t$ be the number of policies in force at the start of year $t$, and let $V_t$ be the opening reserve per policy. Let $N_{t+1}$ be the number still in force at the end of the year, and $V_{t+1}$ their closing reserve per policy. For a homogeneous group, opening aggregate reserves are $N_tV_t$, closing aggregate reserves are $N_{t+1}V_{t+1}$, and the aggregate reserve movement is $\(N_{t+1}V_{t+1}-N_tV_t$\).

For the basic formulas below, assume a closed cohort: there are no new policies entering during the year. Assume also that each policy still in force at the end has reserve $V_{t+1}$, while a policy that dies, lapses, or matures has no reserve after its exit is settled. Any claim, maturity benefit, or other settlement is included in the period’s cash flows, not in the closing reserve. If reserves vary by policy, the portfolio amounts are sums of the individual reserves rather than a common reserve multiplied by a policy count.

Separating Survivor Development from Exits

Write $E_t=N_t-N_{t+1}$ for the number of policies that exit during the year. Add and subtract $N_{t+1}V_t$ to the aggregate reserve movement:

\begin{aligned} N_{t+1}V_{t+1}-N_tV_t &=N_{t+1}(V_{t+1}-V_t) +(N_{t+1}-N_t)V_t\\ &=N_{t+1}(V_{t+1}-V_t)-E_tV_t. \end{aligned}

The first term is the per-policy reserve movement for the policies that remain, multiplied by the number of those policies. The second term is the release of opening reserves on policies that exit. Under the stated assumption that their reserve is extinguished at exit, this is a reduction in aggregate reserves.

Theorem: For a homogeneous closed cohort whose exiting policies have zero closing reserve after settlement, $\(\text{aggregate reserve movement}=\text{survivors}\times\text{per-survivor reserve movement}-\text{exits}\times\text{opening reserve}$\). The per-survivor movement is $V_{t+1}-V_t$; it is not the aggregate movement per original policy.

This is an accounting identity, not a claim that exiting policies all experience the same event. If deaths, lapses, and maturities have different cash settlements, those differences belong in cash profit. Their reserves are still removed from the closing aggregate when their contracts end. In a model where an exit attracts a continuing reserve or a separate settlement reserve, that amount must be represented explicitly rather than assumed to be zero.

Find the in-force count.
Separate continuing policies from policies ending through death, lapse, maturity, or another specified exit.

Calculate survivor development.
Multiply $V_{t+1}-V_t$ by the number of policies still in force.

Allow for reserve release on exits.
Subtract the opening reserves on policies whose contracts have ended, if their closing reserves are zero.

Reconcile the portfolio totals.
Check that the result equals closing aggregate reserves minus opening aggregate reserves.

Worked Portfolio Calculations

Worked Example: Expected Reserve Movement with Deaths and Lapses

A hypothetical portfolio has 1,000 identical policies at the start of a year. The opening reserve is $400 per policy, and the closing reserve for each continuing policy is $460. Assume an effective annual interest rate of $i=4\%$; the two reserve amounts are supplied on this valuation basis. During the year, each policy has a 0.04 probability of death, a 0.06 probability of lapse, and a 0.90 probability of remaining in force. Outcomes follow a multinomial model with those probabilities for 1,000 policies. Find the expected aggregate reserve movement.

Method. Use expected policy counts in the aggregate identity. The expected number continuing is $1{,}000(0.90)=900$. The expected number exiting is $1{,}000(0.04+0.06)=100$.

The expected opening aggregate reserve is $1{,}000(\$400)=\$400{,}000$. The expected closing aggregate reserve is $900(\$460)=\$414{,}000$. Thus the expected aggregate reserve movement is $\$414{,}000-\$400{,}000=\$14{,}000$.

The decomposition gives the same result. Survivor development is $900(\$460-\$400)=900(\$60)=\$54{,}000$. The reserve released on exits is $100(\$400)=\$40{,}000$. Therefore the net movement is $\$54{,}000-\$40{,}000=\$14{,}000$, an increase.

Check. The opening and closing portfolio totals differ by $14,000, matching the decomposition. The common error is to multiply the $60 per-survivor increase by all 1,000 starting policies: that gives $60,000 and omits the $40,000 reserve release on exits.

Worked Example: Different Types of Exit

A hypothetical closed block has 500 policies with opening reserve $1,200 each. The closing reserve is $1,280 for each policy still in force. Assume $i=5\%$, with these reserves supplied on that basis. The end-of-year outcomes follow a multinomial model: 0.88 remain in force, 0.04 die, 0.06 lapse, and 0.02 mature. Find the expected aggregate reserve movement and show how the exits contribute.

The expected number continuing is $500(0.88)=440$. The expected deaths, lapses, and maturities are $500(0.04)=20$, $500(0.06)=30$, and $500(0.02)=10$, respectively. Thus there are 60 expected exits in total.

Opening aggregate reserves are $500(\$1{,}200)=\$600{,}000$. Closing aggregate reserves are $440(\$1{,}280)=\$563{,}200$. The expected aggregate movement is $\$563{,}200-\$600{,}000=-\$36{,}800$.

For the decomposition, survivor development is $440(\$1{,}280-\$1{,}200)=440(\$80)=\$35{,}200$. The opening reserve released on all exits is $60(\$1{,}200)=\$72{,}000$. The net is $\$35{,}200-\$72{,}000=-\$36{,}800$. The cause-specific exit counts sum to $20+30+10=60$, so all three causes are included exactly once.

Check. The negative aggregate movement is consistent with the closing total being $36,800 below the opening total, even though the per-policy reserve for continuers increased by $80. This calculation does not imply that the cash settlements for death, lapse, and maturity are identical; it accounts only for the removal of their reserves.

Connecting Aggregate Reserves to Profit

Under a standard one-year reserve-adjusted profit convention, period profit equals cash profit plus closing aggregate reserves minus opening aggregate reserves. Cash profit collects the period’s premiums, expenses, benefits, and other cash items under the specified timing convention. The reserve movement then recognizes the change in the liability held for the portfolio. If opening and closing reserves are measured consistently and exit settlements are included in cash flows, the identity reconciles the period’s cash experience and the reserve balance.

The identity can also be applied by policy group. A continuing policy contributes its cash profit plus its individual reserve change. An exiting policy contributes its cash profit and the release of its opening reserve, subject to the assumption that its closing reserve is zero. Summing those contributions gives portfolio profit. This is why the number of policies associated with a reserve movement must match the policies to which that movement applies.

Worked Example: Reserve Movement in a Profit Test

A hypothetical portfolio begins with 200 policies. During the year, each policy has probability 0.80 of remaining in force, probability 0.10 of death, and probability 0.10 of lapse. Outcomes follow a multinomial model. Assume $i=0\%$, so no investment accumulation is needed in this one-year illustration. Every policy pays a $1,000 premium and incurs a $100 expense; each death claim is $5,000. Opening reserve is $900 per policy and closing reserve is $950 per continuing policy. Find expected profit, using cash profit plus the aggregate reserve movement.

The expected counts are 160 continuing policies, 20 deaths, and 20 lapses. Total premiums are $200(\$1{,}000)=\$200{,}000$; total expenses are $200(\$100)=\$20{,}000$; and expected death claims are $20(\$5{,}000)=\$100{,}000$. Cash profit is therefore $\$200{,}000-\$20{,}000-\$100{,}000=\$80{,}000$.

The expected closing aggregate reserve is $160(\$950)=\$152{,}000$, while the opening aggregate reserve is $200(\$900)=\$180{,}000$. The aggregate reserve movement is $\$152{,}000-\$180{,}000=-\$28{,}000$. Expected profit is consequently $\$80{,}000-\$28{,}000=\$52{,}000$.

A group-by-group check gives the same result. Each continuing policy has cash profit $\$1{,}000-\$100=\$900$ and reserve increase $\$950-\$900=\$50$, for total contribution $160(\$950)=\$152{,}000$. Each lapse has cash profit $900 and releases a $900 opening reserve, contributing zero; the 20 lapses contribute $0. Each death has cash profit $\$900-\$5{,}000=-\$4{,}100$ and releases a $900 reserve, contributing $-\$5{,}000$; 20 deaths contribute $-\$100{,}000$. The sum is $\$152{,}000+\$0-\$100{,}000=\$52{,}000$.

Check. The group calculation agrees with the portfolio identity. A mistake would be to use $200(\$950-\$900)=\$10{,}000$ as the reserve movement; that incorrectly applies the continuing-policy reserve change to policies that exited.

Expected Counts, Actual Counts, and Interpretation

When outcomes are random, multiplying a starting count by a probability gives an expected count, which need not be a whole number. The portfolio identity remains valid for actual counts. For expectations, it can be applied to expected counts when the reserve amounts are fixed for each relevant status, as in the examples. If reserves depend on policy characteristics or the event that occurred, use the corresponding expected sum of individual reserve amounts rather than multiplying one average reserve by an unrelated count.

Keep the following distinctions clear:

Per-survivor reserve movement is $V_{t+1}-V_t$, conditional on a policy remaining in force.
Survivor contribution to the aggregate movement is $N_{t+1}(V_{t+1}-V_t)$.
Exit contribution is $-E_tV_t$ when each exiting policy’s reserve is extinguished and its settlement is included in cash flows.
Aggregate reserve movement is closing aggregate reserves minus opening aggregate reserves; it is not generally the per-policy movement multiplied by the starting count.

This distinction matters in profit testing because a portfolio result combines reserve development and changes in the in-force population. A per-survivor reserve increase is not, by itself, evidence that total reserves rose. Always reconcile the calculated movement to the opening and closing portfolio balances, and keep cash settlements separate from reserve releases.

Key takeaway: First identify which policies carry closing reserves. Multiply per-policy reserve development by that continuing count, then account for reserves removed when contracts exit. The result must equal closing aggregate reserves minus opening aggregate reserves.

Check Your Understanding

For each question, calculate the reserve movement using the stated exit and reserve assumptions.

A hypothetical group has 300 policies, opening reserve $700, and closing reserve $740 for each continuer. The annual probabilities are 0.85 continuing, 0.05 death, and 0.10 lapse, with multinomial outcomes. Assume $i=3\%$ and that the supplied reserves use this basis. Find the expected aggregate reserve movement. (Answer: $-\$21{,}300$)
A hypothetical block starts with 120 policies and opening reserve $2,000 each. The closing reserve is $2,100 for each continuer. Assume $i=4\%$, supplied reserves on that basis, and multinomial probabilities 0.90 continuing, 0.05 death, and 0.05 lapse. Find the expected aggregate movement. (Answer: $-\$13{,}200$)
A hypothetical portfolio has 400 policies, opening reserve $500, and closing reserve $520 for each continuer. The probabilities are 0.75 continuing, 0.15 death, and 0.10 lapse. With $i=2\%$ and supplied reserves on that basis, calculate the survivor-development term, exit-release term, and net movement. (Answer: $+\$6{,}000$, $-\$50{,}000$, and $-\$44{,}000$, respectively)
A hypothetical group of 100 policies has probabilities 0.80 continuing, 0.10 death, and 0.10 lapse. Each policy pays a $600 premium and incurs a $50 expense; each death claim is $3,000. Opening reserve is $400 and closing reserve is $450 per continuer. Assume $i=0\%$, all cash items occur within the year as stated, and outcomes follow a multinomial model. Find expected profit using cash profit plus reserve movement. (Answer: cash profit $=\$25{,}000$, reserve movement $=-\$4{,}000$, expected profit $=\$21{,}000$)
A hypothetical cohort has 100 policies with opening reserve $100 each. In year 1, 90% continue and the closing reserve is $120 per continuer. In year 2, 80% of those year-1 continuers remain, and the year-2 closing reserve is $150 per continuer. Assume no new business, no other exits, $i=0\%$, and the stated continuation probabilities apply independently by policy-year. Find the expected aggregate reserve movement over both years relative to the initial portfolio reserve. (Answer: $+\$800$)

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