Asset-Share Sensitivity to Mortality Experience

Why Mortality Experience Changes Asset Shares

An asset share for a group of policies is affected by more than the amount of each death claim. Mortality experience determines how many claims are paid, how much is paid in total, and how many policies remain to share the accumulated fund. Consequently, an adverse mortality outcome can reduce the cohort fund and also reduce the number of policies over which that fund is divided.

A useful comparison changes mortality experience while holding other assumptions fixed. This isolates the mortality effect from differences in premiums, expenses, or investment returns. The calculations below use annual periods and explicitly state when premiums, expenses, interest, and claims are applied.

Definition: At time $t$, let $N_t$ be the number of in-force policies and $F_t$ the aggregate fund allocated to that cohort, immediately before the period-$t$ premium and expense cash flows. The per-policy asset share is $A_t=F_t/N_t$, provided $N_t>0$. If $D_t$ policies die during the period, then $N_{t+1}=N_t-D_t$. The aggregate death claims paid during the period are denoted by $C_t$.

For the annual model used here, each in-force policy pays premium $P$ and incurs expense $e$ at the start of the year. The net amount added per in-force policy is $P-e$. The fund, including these cash flows, earns effective annual interest $i$, and death claims are paid at the end of the year. There are no other cash flows in the period.

F_{t+1}=\bigl(F_t+N_t(P-e)\bigr)(1+i)-C_t, \qquad N_{t+1}=N_t-D_t, \qquad A_{t+1}=\frac{F_{t+1}}{N_{t+1}}.

If all death claims are a fixed amount $B$, then $C_t=D_tB$. If benefits vary by policy, the aggregate claim is the sum of the amounts payable on the actual deaths: $C_t=\sum_{j=1}^{D_t}B_{t,j}$. The number of deaths and the claim amount per death are therefore distinct inputs to the fund calculation.

Why the Recurrence Works

The recurrence follows directly from the specified cash-flow timing. The fund at the start of the year is $F_t$. Adding $N_t$ premiums and subtracting $N_t$ expenses gives $F_t+N_t(P-e)$. That amount earns interest for the full year, so its end-year value before claims is multiplied by $1+i$. Claims are paid at year-end, so they are subtracted after accumulation and do not themselves earn interest in this model. The resulting fund is then allocated over the $N_t-D_t$ survivors.

\underbrace{F_t+N_tP-N_te}_{\text{fund after start-year cash flows}} \ \xrightarrow{\text{earn }i}\ \bigl(F_t+N_t(P-e)\bigr)(1+i) \ \xrightarrow{\text{pay claims}}\ F_{t+1}.

Changing the timing changes the formula. For example, a claim paid at the beginning rather than the end of the year would not be included in the amount earning a full year of interest. The recurrence should therefore be derived from the stated timing rather than applied mechanically.

Comparing Mortality Outcomes

For a homogeneous group with annual death probability $q$, a deterministic projection commonly uses expected deaths $N_tq$, expected survivors $N_t(1-q)$, and, for a fixed death benefit $B$, expected claims $N_tqB$. An actual experience comparison instead uses the observed death count and actual claim amounts. These are different calculations: expected counts need not be whole numbers, while actual counts are whole numbers.

For an experience comparison, it is helpful to show both the change in the aggregate fund and the change in the survivor count. The per-survivor asset share is their ratio, so it can move in a different proportion from the aggregate fund. With heterogeneous benefits, two groups with the same number of deaths can have different aggregate claims and hence different funds.

Worked Example: More Deaths Than Expected

A hypothetical cohort has 100 policies and an opening aggregate fund of $20,000. Each policy pays a premium of $500 and incurs an expense of $50 at the start of the year. The effective annual interest rate is 5%, and each death claim of $10,000 is paid at year-end. Compare a projection with 2 deaths to actual experience with 4 deaths. Assume no other cash flows.

Method. Apply the annual aggregate recurrence separately to each mortality outcome, then divide each ending fund by its own survivor count.

1. Start-year cash flows. The net amount per in-force policy is $500-50=\$450$. For 100 policies, net cash inflow is $100(450)=\$45{,}000$. The fund before interest is $20{,}000+45{,}000=\$65{,}000$.

2. Fund before claims. Accumulating for one year gives $65{,}000(1.05)=\$68{,}250$.

3. Projection with 2 deaths. Claims are $2(10{,}000)=\$20{,}000$, leaving $68{,}250-20{,}000=\$48{,}250$. There are $100-2=98$ survivors, so the projected asset share is $48{,}250/98=\$492.3469\ldots$, or $492.35 per survivor to the nearest cent.

4. Actual experience with 4 deaths. Claims are $4(10{,}000)=\$40{,}000$, leaving $68{,}250-40{,}000=\$28{,}250$. There are $100-4=96$ survivors, so the actual asset share is $28{,}250/96=\$294.2708\ldots$, or $294.27 per survivor to the nearest cent.

The actual fund is $20,000 lower, exactly the additional two claims. The number of survivors is also two lower. The per-survivor asset share falls by $\$492.35-\$294.27=\$198.08$, using the rounded asset shares. The calculation avoids treating the reduced survivor count as if it offset the extra claim outgo.

Worked Example: Same Death Count, Different Claim Amounts

A hypothetical cohort has 40 policies and an opening aggregate fund of $12,000. Each policy pays a premium of $600 and incurs a start-year expense of $100. The effective annual interest rate is 4%. Two policies die, and their actual benefits are $8,000 and $14,000. Compare this experience with a benchmark that has the same two deaths but assumes a $9,000 benefit for each death. Claims are paid at year-end, and no other cash flows occur.

Method. Hold the death count, premiums, expenses, and interest fixed. Compare aggregate claims first, then compute the asset share using the same survivor count.

1. Accumulated fund before claims. The net amount per policy is $600-100=\$500$, so start-year net cash flow is $40(500)=\$20{,}000$. The pre-interest fund is $12{,}000+20{,}000=\$32{,}000$. After interest it is $32{,}000(1.04)=\$33{,}280$.

2. Benchmark claims and asset share. The benchmark claims total $2(9{,}000)=\$18{,}000$. The benchmark fund is $33{,}280-18{,}000=\$15{,}280$. With $40-2=38$ survivors, its asset share is $15{,}280/38=\$402.1052\ldots$, or $402.11 per survivor.

3. Actual claims and asset share. Actual claims total $8{,}000+14{,}000=\$22{,}000$. The actual fund is $33{,}280-22{,}000=\$11{,}280$. The survivor count remains 38, so the actual asset share is $11{,}280/38=\$296.8421\ldots$, or $296.84 per survivor.

The actual claim total exceeds the benchmark by $4,000, and the asset share is lower by $\$402.11-\$296.84=\$105.27$ using rounded values. The difference is due to claim amounts, not claim frequency or survivor count. Replacing the two actual benefits by their average, $11,000, would give the same aggregate claims here, but that shortcut is valid only when the average is calculated from all actual claims in the group.

Worked Example: Mortality Differences Over Two Years

A hypothetical cohort begins with 20 policies and no aggregate fund. Each in-force policy pays a premium of $1,000 and incurs an expense of $100 at the start of each year. The effective annual interest rate is 10%. Claims are paid at year-end. Compare two mortality paths: Path A has 2 deaths in year 1, each with a $5,000 claim, followed by 3 deaths in year 2 with benefits of $4,000, $5,000, and $6,000. Path B has 1 death in year 1 with a $5,000 claim, followed by 2 deaths in year 2, each with a $5,000 claim. No other cash flows occur.

Method. Roll forward the aggregate fund and policy count at the end of each year. Use the year-start survivor count to calculate that year’s premiums and expenses.

Path A, year 1. Net cash flow is $20(1{,}000-100)=\$18{,}000$. After interest the fund is $18{,}000(1.10)=\$19{,}800$. Claims total $2(5{,}000)=\$10{,}000$, leaving $F_1=\$9{,}800$. The survivor count is $N_1=20-2=18$.

Path A, year 2. Net cash flow is $18(900)=\$16{,}200$. The fund before claims is $(9{,}800+16{,}200)(1.10)=\$28{,}600$. The three claims total $4{,}000+5{,}000+6{,}000=\$15{,}000$, leaving $F_2=\$13{,}600$. There are $18-3=15$ survivors, so the year-2 asset share is $13{,}600/15=\$906.6666\ldots$, or $906.67 to the nearest cent.

Path B, year 1. Net cash flow is again $20(900)=\$18{,}000$, so the fund before claims is $19,800. Claims are $5,000, leaving $F_1=\$14{,}800$, and $N_1=19$.

Path B, year 2. Net cash flow is $19(900)=\$17{,}100$. The fund before claims is $(14{,}800+17{,}100)(1.10)=\$35{,}090$. Claims total $2(5{,}000)=\$10{,}000$, leaving $F_2=\$25{,}090$. There are $19-2=17$ survivors, so the year-2 asset share is $25{,}090/17=\$1{,}475.8823\ldots$, or $1,475.88 to the nearest cent.

Path A has more deaths and higher total claims over the two years: $25,000 compared with $15,000. It also has fewer survivors, and its year-2 fund is smaller. Each feature contributes to the lower year-2 asset share. This projection also shows why year-2 premiums must be based on the year-2 opening count, not the original cohort size.

Interpreting Expected and Actual Asset Shares

An expected mortality projection is useful as a benchmark, but it is not automatically the expected per-policy asset share under random mortality. In general, the ratio of expected values is not the expectation of a ratio. If $F_{t+1}$ and $N_{t+1}$ are random, then

\frac{E[F_{t+1}]}{E[N_{t+1}]} \ne E\!\left[\frac{F_{t+1}}{N_{t+1}}\right]

in general. The left-hand side is the fund expectation divided by the expected survivor count; the right-hand side averages the per-policy asset share across possible outcomes. They answer different questions. Also, the per-policy ratio requires a positive survivor count; if no policies survive, it is undefined.

For a practical comparison, report the assumptions and the components that drive the result. At minimum, identify the opening fund and cohort, premium and expense timing, interest rate, death count, total claims, survivor count, and ending asset share. If benefits vary, state the actual aggregate claim amount rather than relying on a claim count alone.

Common Mortality-Experience Traps

Using the original cohort count for later premiums. Premiums in a later year are based on the policies in force at that year’s start. After deaths, using $N_0$ instead of the current $N_t$ overstates premium income.
Subtracting claims at the wrong point. Under the stated end-year claim timing, claims are subtracted after the fund earns interest. Accumulating the fund after first subtracting claims incorrectly removes interest on money that was available during the year.
Tracking claims but not survivors. The ending asset share is the remaining fund divided by the remaining policy count. Reporting only the aggregate fund does not answer the per-survivor question.
Assuming a death count fully describes mortality cost. When benefits differ, the same count can produce different aggregate claims. Sum the benefits actually paid.
Calling an expected ratio an expected per-policy result. Dividing expected fund by expected survivors gives a ratio of expectations, not generally the expected value of the random per-policy ratio.

Key takeaway: Mortality sensitivity has two linked components: death claims change the aggregate fund, while deaths also change the number of policies sharing that fund. Calculate both explicitly, using the correct cash-flow timing and actual claim amounts.

Check Your Understanding

For each question, use the stated annual timing: premiums and expenses at the start of each year, interest on the resulting fund, and death claims at year-end. Round dollar asset shares to the nearest cent.

A cohort has 60 policies, an opening fund of $9,000, premium $400 per policy, expense $40 per policy, and effective annual interest rate 3%. Three policies die, each with a $6,000 claim. Find the ending fund and asset share per survivor. (Answer: ending fund $13,518; asset share $237.16.)
A cohort has 25 policies, opening fund $5,000, premium $800, expense $100, and effective annual interest rate 5%. Two policies die with claims of $7,000 and $9,000. Find the aggregate claims and ending asset share. (Answer: aggregate claims $16,000; ending asset share $331.52.)
A cohort has 30 policies and no opening fund. Annual premium is $500, expense is $50, and effective annual interest is 4%. In year 1, one policy dies with a $6,000 claim. In year 2, two policies die, each with a $6,000 claim. Find the year-2 ending fund and asset share. (Answer: ending fund $9,933.60; asset share $367.91.)
A cohort has 50 policies, opening fund $30,000, premium $300, expense $30, and effective annual interest rate 4%. Compare actual experience of four deaths with benefits $5,000, $6,000, $7,000, and $8,000 against a benchmark of two deaths with benefits of $6,500 each. Find each ending fund and asset share, and state which has the larger asset share. (Answer: actual fund $19,240 and asset share $418.26; benchmark fund $32,240 and asset share $671.67; the benchmark has the larger asset share.)

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