Mortality Pooling and Asset Shares for Survivors

Why Mortality Pooling Changes Survivor Asset Shares

A cohort’s asset share is affected not only by the death benefits paid during a year, but also by the number of policies that remain to share the assets left after those payments. If the assets released when a policy ends exceed its death claim, the surviving policies receive a mortality gain. If the claim exceeds those assets, they bear a mortality loss. The size of either effect per survivor depends on the number of actual survivors.

This tutorial treats a homogeneous cohort: every policy begins the claim period with the same asset share, and each death claim is the same amount. Cash flows and investment returns can also be included, provided their timing is specified. The focus is the allocation across survivors, not a full projection of all possible policy exits.

Definition: Let $N$ be the number of policies immediately before a claim period, $F$ the cohort’s aggregate assets immediately before death claims, $D$ the number of deaths during the period, and $B$ the death benefit paid for each death. If claims are paid at the end of the period and there are no other cash flows before the asset share is measured, the aggregate assets remaining are $F-DB$, and the number of surviving policies is $N-D$. When $N-D>0$, the actual asset share per survivor is $$ \frac{F-DB}{N-D}. $$

If the pre-claim asset share per policy is $a$, then $F=Na$. Substituting gives a useful comparison with the pre-claim share:

\frac{Na-DB}{N-D} = a+\frac{D(a-B)}{N-D}.

Thus the direction of the mortality effect depends on $a-B$. When $a>B$, the assets associated with each policy that dies exceed its claim, and the difference is allocated across survivors. When $a<B$, the claim exceeds those assets, reducing the survivor share. This comparison uses the asset share at the point just before claims; if premiums, expenses, or investment income occur first, they must be reflected in that pre-claim amount.

Actual Experience and Expected-Count Projections

For a projection based on a common annual death probability $q$, the expected number of deaths is $Nq$, and the expected number of survivors is $N(1-q)$. A commonly used expected-count projection allocates the projected aggregate assets over the projected survivor count:

\frac{Na-(Nq)B}{N(1-q)} =\frac{a-qB}{1-q}, \qquad 0\le q<1.

This is a ratio formed from expected cohort counts and corresponding projected cash flows. It is not generally the expected value of the actual random ratio $(Na-DB)/(N-D)$. The actual ratio depends nonlinearly on the random number of survivors, so a ratio of expected amounts and an expectation of a ratio are different quantities. State which basis a question asks for: actual experience uses the observed $D$; an expected-count projection uses $Nq$.

Establish the pre-claim fund.
Apply premiums, expenses, and investment returns in their stated order to find the aggregate assets $F$ just before claims.

Deduct actual or projected claims.
For $D$ deaths, subtract $DB$ from the fund. If projecting by expected counts, use $Nq$ deaths consistently in the projected fund.

Use the matching survivor count.
Divide the remaining fund by $N-D$ for actual experience or by $N(1-q)$ for an expected-count projection.

Check the allocation.
Multiply the per-survivor result by the number of survivors and confirm that it reproduces the aggregate fund after claims.

Worked Applications

Worked Example: Actual Deaths Versus an Expected-Count Projection

A hypothetical cohort has 100 policies, each with a pre-claim asset share of $1,000. The annual death probability is $q=0.06$, and each death benefit is $900. Four deaths actually occur. Find the actual asset share per survivor and the expected-count projection. Assume no other cash flows before claims.

Method. Calculate the actual and projected cases separately, using the matching claim count and survivor count in each case.

Actual experience. The pre-claim fund is $$ 100(1{,}000)=100{,}000. $$ The four claims total $4(900)=3{,}600$, so the fund after claims is $$ 100{,}000-3{,}600=96{,}400. $$ There are $100-4=96$ survivors. Therefore the actual per-survivor share is $$ \frac{96{,}400}{96}=1{,}004.1666\ldots, $$ or $1,004.17 to the nearest cent.

The gain can also be checked against the original $1,000 share. Each death releases $1,000 of pre-claim assets but costs $900, a difference of $100. The four deaths release a total net gain of $4(100)=400$. Allocated over 96 survivors, the increase is $400/96=4.1666\ldots$, which gives the same $1,004.17.

Expected-count projection. The expected number of deaths is $100(0.06)=6$, leaving 94 projected survivors. Projected claims are $6(900)=5{,}400$, leaving $$ 100{,}000-5{,}400=94{,}600. $$ The expected-count share is $94{,}600/94=1{,}006.3829\ldots$, or $1,006.38 to the nearest cent.

Check. The actual calculation allocates $96,400 across 96 people; the projection allocates $94,600 across 94. The results differ because the actual and projected death counts differ. A common mistake is to subtract the actual claims but divide by the projected number of survivors, or vice versa. Each numerator must be paired with its own survivor count.

Worked Example: Carrying Mortality Pooling Into a Second Year

A hypothetical cohort has 30 policies, each with $1,000 of assets immediately before year 1 claims. The year 1 death probability is $q_1=0.10$; four deaths actually occur, and each claim is $1,050. At the start of year 2, each of the 26 survivors pays a $50 premium and incurs a $5 expense. The resulting assets earn an effective annual return of 4%. During year 2, two of the 26 policies die and each receives a $900 claim at year-end. Find the aggregate assets and per-policy asset share for the 24 policies surviving year 2. Use actual death counts in both years.

Method. First allocate the year 1 claim over the 26 actual survivors. Then apply year 2 start-of-year cash flows, the annual return, and year-end claims in that order.

The year 1 fund before claims is $30(1{,}000)=30{,}000$. Four claims total $4(1{,}050)=4{,}200$, leaving $$ 30{,}000-4{,}200=25{,}800. $$ The 26 survivors’ year 1 closing share is $$ \frac{25{,}800}{26}=992.307692\ldots $$ per policy.

At the start of year 2, each survivor adds a net amount of $50-5=45$. The aggregate fund becomes $$ 25{,}800+26(45)=25{,}800+1{,}170=26{,}970. $$ After the 4% return, it is $$ 26{,}970(1.04)=28{,}048.80. $$ The two year 2 claims total $2(900)=1{,}800$, so assets remaining at year-end are $$ 28{,}048.80-1{,}800=26{,}248.80. $$ There are $26-2=24$ survivors, giving a per-survivor share of $$ \frac{26{,}248.80}{24}=1{,}093.70. $$ Thus the aggregate assets are $26,248.80, and the year 2 closing share is $1,093.70 per survivor.

Check. $24(1{,}093.70)=26{,}248.80$. The common timing error is to apply the investment return after subtracting year-end claims, or to apply the start-of-year premium to all 30 original policies. Here only the 26 year 1 survivors pay the year 2 premium and incur the expense, and the return is applied before the year 2 claims.

Worked Example: Including Start-of-Year Cash Flows and Investment Return

A hypothetical cohort has 80 policies with an opening asset share of $1,150 each. At the start of the year, every policy pays a $40 premium and incurs a $10 expense. The assets then earn an effective annual return of 5%. At year-end, three deaths occur, each with a $1,200 claim. The annual death probability used for a projected calculation is $q=0.025$. Find the actual per-survivor share and the expected-count projection.

Method. Build the pre-claim asset share by applying the net start-of-year cash flow and investment return. Then calculate actual and expected-count allocations separately.

The net start-of-year cash flow per policy is $40-10=30$, so the amount invested per policy is $$ 1{,}150+30=1{,}180. $$ After the 5% return, the pre-claim share is $1{,}180(1.05)=1{,}239$. The aggregate pre-claim fund is $80(1{,}239)=99{,}120$.

Actual experience. Three claims total $3(1{,}200)=3{,}600$, leaving $99{,}120-3{,}600=95{,}520$. The survivor count is $80-3=77$, so the actual share is $$ \frac{95{,}520}{77}=1{,}240.51948\ldots, $$ or $1,240.52 to the nearest cent.

As a check, the pre-claim assets associated with each death are $1,239, compared with a $1,200 claim, a gain of $39 per death. The total gain is $3(39)=117$; dividing by 77 gives $117/77=1.51948\ldots$, which raises the share from $1,239 to $1,240.52.

Expected-count projection. Expected deaths are $80(0.025)=2$, and expected survivors are 78. Projected claims total $2(1{,}200)=2{,}400$. Thus the projected share is $$ \frac{99{,}120-2{,}400}{78} =\frac{96{,}720}{78} =1{,}240. $$ The expected-count projection is $1,240.00 per survivor. It differs from actual experience because three deaths occurred rather than the projected count of two.

Check. The actual survivor total is $77(1{,}240.51948\ldots)=95{,}520$, the actual post-claim fund. A common mistake is to calculate the mortality gain using the opening $1,150 rather than the $1,239 pre-claim share: start-of-year cash flows and investment income have already changed the assets exposed to the claims.

Why the Allocation Formula Holds

The formula follows directly from the cohort ledger. Immediately before claims, the cohort owns $F$. Each of $D$ deaths generates a payment of $B$, so the claim outgo is $DB$, leaving $F-DB$. Because those claims are paid on behalf of the policies that die, only $N-D$ policies remain to receive the residual asset share. If that residual is allocated equally, each survivor receives $(F-DB)/(N-D)$. With $F=Na$, subtracting the starting share $a$ gives:

\frac{Na-DB}{N-D}-a =\frac{Na-DB-a(N-D)}{N-D} =\frac{D(a-B)}{N-D}.

This derivation also shows why the denominator matters. The aggregate mortality gain or loss is $D(a-B)$, but its per-survivor effect is that amount divided by $N-D$, not by $N$. The deceased policies are no longer in the group receiving the remaining assets.

Assumptions, Checks, and Exam Traps

The simple formulas assume that the cohort is homogeneous for the quantities being pooled, that the specified death claims are paid from the cohort fund, and that the survivor asset share is allocated equally. They also require at least one survivor for a per-survivor share. If $D=N$, the aggregate fund after claims may still be computed, but there is no survivor share to calculate. If policies have different pre-claim assets or different claim amounts, the simple expression using one $a$ and one $B$ does not describe every policy’s allocation without a further allocation rule.

Use the actual survivor count for actual experience. With $D$ observed deaths, divide by $N-D$, not $N$.
Keep the numerator and denominator on the same basis. Do not combine observed claims with an expected survivor count.
Respect cash-flow timing. Include start-of-year premiums and expenses before investment returns when specified, and subtract year-end claims after the return when specified.
Do not equate expected counts with a guaranteed outcome. The expected-count projection is a cohort projection, not a prediction that the actual number of deaths will equal $Nq$.
Check aggregate conservation. The calculated survivor share multiplied by the survivor count must equal the fund after claims, subject only to stated rounding.

Mortality pooling explains how claim experience can change the assets allocated to continuing policies even when their own benefits and premiums do not change. It is therefore important when interpreting cohort asset shares: a per-survivor change can reflect both the net amount released or consumed by deaths and the number of policies sharing the resulting fund. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Key takeaway: Find the fund immediately before claims, subtract the claims on the appropriate actual or projected basis, and divide by the matching number of survivors. A death creates a gain when pre-claim assets exceed the claim, and a loss when the claim exceeds those assets.

Check Your Understanding

For each problem, use the stated timing and distinguish actual experience from an expected-count projection where requested.

A cohort has 50 policies with $800 each immediately before claims. The annual death probability is $0.04$, and each death benefit is $760. Three deaths actually occur. Find the actual share per survivor and the expected-count projection, both to the nearest cent. (Answer: actual $802.55; expected-count projection $801.67)
A cohort has 40 policies with $950 each immediately before claims. Three deaths occur, and each receives $1,000. Find the asset share for each survivor, to the nearest cent. (Answer: $945.95 per survivor)
A cohort of 60 policies begins with $700 per policy. At the start of the year, each pays a $30 premium and incurs a $5 expense. The fund earns 3% for the year, and two year-end deaths each generate a $750 claim. Find the closing share per survivor to the nearest cent. (Answer: $746.64 per survivor)
A cohort has 100 policies with $1,500 each immediately before claims. The death probability is $0.02$, and the benefit is $1,600. Four deaths actually occur. Calculate the actual share and expected-count projection to the nearest cent, and state which is lower. (Answer: actual $1,495.83; expected-count projection $1,497.96; actual is lower)
A cohort has 25 policies with $2,000 each immediately before claims. Five deaths each receive the same benefit $B$, and the 20 survivors each have a closing share of $1,980. Solve for $B$. (Answer: $2,080 per death benefit)

What You'll Learn