Per-Policy Versus Aggregate Asset Shares

Why Aggregate and Per-Policy Asset Shares Must Be Kept Distinct

An asset-share projection can be stated as a total dollar amount for a group of policies or as an amount per policy. Those figures describe the same fund only when the policy count and the basis of the average are clear. If policies leave the group, the aggregate may fall while the average per remaining policy rises, falls, or stays unchanged. If policies have different asset shares, the average can also conceal those differences.

This distinction matters whenever a calculation moves between a cohort-level projection and a per-policy value. A total asset amount is not itself a per-policy asset share, and dividing it by a policy count does not establish that every policy has that amount. The calculation produces an average for the specified group at the specified time.

Definition: Let $A_t$ be the aggregate assets attributed to a specified policy group at time $t$, and let $N_t$ be the number of policies in that group at that time. The average per-policy asset share is $A_t/N_t$, provided $N_t>0$. If all $N_t$ policies have the same asset share, this average is also the asset share of each policy. Otherwise it is only a group average.

Relating Aggregate Assets, Counts, and Cash Flows

First specify which policies are included at each valuation date. The count may change because of deaths, lapses, maturities, withdrawals, or new entrants. A calculation should use the count at the date of the asset amount being converted—not automatically the count at the start of the projection period.

For one year, suppose $A_t$ is the opening aggregate asset amount. Let $C_t$ be the aggregate net cash flow received at the start of the year, with receipts positive and outgo negative. Assume that this amount earns an effective annual investment return $i$, and that $B_t$ is the aggregate benefit or exit payment made at year-end. Under these timing assumptions:

A_{t+1}=(A_t+C_t)(1+i)-B_t.

If $N_{t+1}$ policies remain in the group after year-end exits, the average asset share among those policies is $A_{t+1}/N_{t+1}$. This conversion is valid only if $A_{t+1}$ and $N_{t+1}$ refer to the same group at the same date. The equation also depends on the stated cash-flow timing; a payment made at the start of the year would not earn a full year of investment return under this convention.

Key relationship: If a group contains distinct subgroups $g$, with $N_{g,t}$ policies and average asset share $a_{g,t}$ in each subgroup, then $$ A_t=\sum_g N_{g,t}a_{g,t}, \qquad \frac{A_t}{N_t} = \frac{\sum_g N_{g,t}a_{g,t}}{\sum_g N_{g,t}}, $$ where $N_t=\sum_g N_{g,t}$. The overall per-policy figure is a count-weighted average, not generally the asset share of any one subgroup or policy.

Worked Conversions When the Count Changes

In the examples below, policy counts and payments are specified directly. They are hypothetical realized values, not estimates from a mortality or lapse model. Investment returns are effective annual rates, and each example states the timing needed for its calculation.

Worked Example: Recalculating the Average After Claims and Exits

A hypothetical cohort has 120 policies at the start of a year, each with an asset share of $850. During the year, each policy pays a $100 premium and incurs a $10 expense at the start of the year. The net cash flow then earns a 5% effective annual return. At year-end, three policies make claims of $8,000 each and leave the cohort. Find the aggregate assets and average asset share for the 117 remaining policies.

Method. Calculate the aggregate opening assets, apply the start-of-year net cash flow and investment return, deduct year-end claims, and divide by the remaining count. The claims are year-end payments, so they do not earn investment return in this calculation.

1. Opening assets. The opening amount is $120\times \$850=\$102{,}000$.

2. Start-of-year net cash flow. Aggregate premiums are $120\times \$100=\$12{,}000$, and aggregate expenses are $120\times \$10=\$1{,}200$. Thus $C_t=\$12{,}000-\$1{,}200=\$10{,}800$, giving $\$102{,}000+\$10{,}800=\$112{,}800$ to invest.

3. Investment accumulation. At 5%, the year-end value before claims is $\$112{,}800\times1.05=\$118{,}440$.

4. Year-end claims and remaining assets. The total claim outgo is $3\times \$8{,}000=\$24{,}000$. Therefore $A_{t+1}=\$118{,}440-\$24{,}000=\$94{,}440$. The remaining count is $120-3=117$.

5. Average among remaining policies. $\$94{,}440/117=\$807.179487\ldots$, or $807.18 per remaining policy, rounded to the nearest cent. Using the unrounded average, $117\times \$807.179487\ldots=\$94{,}440$, which checks against the aggregate.

Common mistake avoided: Dividing by the opening count gives $\$94{,}440/120=\$787.00$. That is not the average for the 117 policies remaining at year-end; it uses a count that includes the three policies that exited.

Worked Example: An Unchanged Average Can Hide Different Group Values

A hypothetical portfolio has 40 policies in group A with an average asset share of $1,200, and 60 policies in group B with an average asset share of $700. Ten group A policies then lapse and receive a year-end payment of $900 each. Assume there are no premiums, expenses, or investment returns in this period, and the lapse payments are deducted from group A assets. Find the portfolio average before and after the lapses, and the post-lapse average for each group.

Method. Convert each group average to aggregate assets, add the group amounts, deduct the stated exit payments, and use the remaining counts to calculate the new averages.

1. Opening aggregate assets. Group A has $40\times \$1{,}200=\$48{,}000$. Group B has $60\times \$700=\$42{,}000$. The total is $\$48{,}000+\$42{,}000=\$90{,}000$, across 100 policies, so the initial portfolio average is $\$90{,}000/100=\$900$.

2. Deduct group A lapse payments. The ten payments total $10\times \$900=\$9{,}000$. Group A is left with $\$48{,}000-\$9{,}000=\$39{,}000$ for 30 policies. Group B still has $42,000 for 60 policies.

3. Calculate the post-lapse total and averages. Total assets are $\$39{,}000+\$42{,}000=\$81{,}000$, and 90 policies remain. The portfolio average is $\$81{,}000/90=\$900$. Group A's average is $\$39{,}000/30=\$1{,}300$, while group B's remains $\$42{,}000/60=\$700$.

The unchanged portfolio average does not mean the groups have identical asset shares: the post-lapse values are $1,300 and $700. The weighted-average check is $(30\times \$1{,}300+60\times \$700)/90=(\$39{,}000+\$42{,}000)/90=\$900$.

Common mistake avoided: Reporting $900 as the asset share of every remaining policy would erase the group distinction. It is the portfolio average only; the specified group-level amounts are $1,300 and $700.

Worked Example: Investment Growth Before a Partial Exit

At the start of a year, a hypothetical cohort has 20 policies in group A, each with an asset share of $500, and 30 policies in group B, each with an asset share of $900. All assets earn a 4% effective annual return. At year-end, five group B policies exit, each receiving a payment equal to its year-end asset share. Assume no other cash flows. Find the aggregate assets and average per-policy asset share after the exits, and compare that average with the two remaining group values.

Method. Accumulate each group's opening assets at the common effective return, deduct the exit payments from group B, and divide the remaining aggregate by the remaining policy count.

1. Opening amounts. Group A begins with $20\times \$500=\$10{,}000$; group B begins with $30\times \$900=\$27{,}000$. Together, these are $37,000.

2. Accumulate for one year. Group A's assets become $\$10{,}000\times1.04=\$10{,}400$, equivalent to $520 per policy. Group B's assets become $\$27{,}000\times1.04=\$28{,}080$, equivalent to $936 per policy.

3. Apply the exit payments. Each of the five exiting group B policies receives $936, so total payments are $5\times \$936=\$4{,}680$. Group B retains $\$28{,}080-\$4{,}680=\$23{,}400$, for 25 policies. Group A retains $10,400 for 20 policies.

4. Convert the remaining total to an average. Aggregate assets are $\$10{,}400+\$23{,}400=\$33{,}800$, and 45 policies remain. The portfolio average is $\$33{,}800/45=\$751.111\ldots$, or $751.11 to the nearest cent. The remaining group values are $520 in A and $936 in B. The check $20\times \$520+25\times \$936=\$10{,}400+\$23{,}400=\$33{,}800$ reconciles to the aggregate.

Common mistake avoided: Dividing by 50 would use the starting count, including five policies whose assets have been paid out. Also, the $751.11 portfolio average is not the asset share of either remaining group.

Why the Weighted Average Formula Holds

The aggregate for each subgroup is its number of policies multiplied by its average asset share. Adding those subgroup amounts gives the total assets. Dividing that total by the total number of policies therefore gives a weighted average, with each subgroup's policy count determining its weight.

A_t=\sum_g N_{g,t}a_{g,t} \quad\Longrightarrow\quad \frac{A_t}{N_t} = \frac{\sum_g N_{g,t}a_{g,t}}{\sum_g N_{g,t}}, \qquad N_t=\sum_g N_{g,t}>0.

The weights are the proportions $N_{g,t}/N_t$, which sum to one. Thus a subgroup with more policies contributes more to the overall average. For example, if one subgroup has 30 policies and another has 60, their asset shares receive weights $30/90$ and $60/90$, not equal weights. This also explains why the aggregate divided by the count cannot identify the subgroup values: many different combinations of group asset shares can produce the same total and average.

Common Checks and Interpretation

Before reporting a per-policy figure, check the date, the group, the cash-flow timing, and the denominator. A useful reconciliation is to multiply each reported group average by its corresponding count and confirm that the group totals add to the aggregate. If only an overall average is reported, multiply it by the overall count and confirm that it reproduces the total, allowing for rounding.

Use the matching count. An end-of-year asset amount for continuing policies must be divided by the end-of-year continuing count, not the opening count.
Do not average subgroup averages equally by default. The correct portfolio average weights each subgroup by its policy count.
Do not interpret an average as a universal policy value. That interpretation requires the policies to have the same asset share, or a stated convention that assigns the average to each policy.
Check timing before applying interest. A cash flow received at the start of the year accumulates differently from a payment made at year-end.

The aggregate view is useful for reconciling a fund and its cash flows. The per-policy view is useful for expressing an amount relative to the policies still in the group. Keeping the two views connected—but not interchangeable—helps prevent count errors and avoids presenting a portfolio average as though it describes every policy.

Check Your Understanding

For each problem, use the stated counts and cash-flow timing. Answers are shown at the end of each item for checking.

A cohort has 80 policies, each with an opening asset share of $625. There are no cash flows or exits, and the assets earn a 3% effective annual return. Find the year-end aggregate and average per-policy asset share. (Answer: aggregate $51,500; average $643.75.)
A group has 50 policies with total assets of $40,000. At year-end, five policies withdraw and receive $700 each. There are no other cash flows or investment returns. Find the average asset share for those remaining. (Answer: $811.11 per policy, rounded to the nearest cent.)
Group A has 12 policies at $1,000 each, and group B has 8 policies at $500 each. Four group A policies exit and receive $800 each; there are no other cash flows. Find the remaining aggregate, the overall average, and each group's remaining average. (Answer: aggregate $12,800; overall average $800; group A $1,100; group B $500.)
A cohort has 90 policies and aggregate assets of $72,000. Ten policies exit at year-end and receive payments totaling $8,000. There are no other cash flows. Find the average among continuing policies, and state whether that result establishes that all continuing policies have the same asset share. (Answer: $800 per continuing policy; no, it is only the average unless the policies are known to have equal asset shares.)
Group A has 25 policies with an asset share of $640 each; group B has 15 policies with an asset share of $1,200 each. Five group B policies exit and each receives $1,000. There are no investment returns or other cash flows. Find the remaining portfolio average and the two remaining group averages. (Answer: portfolio average $828.57; group A $640; group B $1,300.)

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