Reinsurance Cash Flows in Asset-Share Projections

Why Reinsurance Changes an Asset Share

An asset share accumulates the premiums and other amounts available for a policy, less expenses and benefits, with investment income applied according to the timing of each cash flow. Reinsurance changes that ledger in two distinct ways: the insurer may pay a reinsurance premium, and it may receive a recovery when a covered claim occurs. The premium is an outgo when due; the recovery is an inflow only when the contract’s claim conditions are met.

The key is to match each cash flow to the risk ceded and the event that triggers payment. A recovery is not automatically a fixed proportion of every policy cash flow. It depends on the reinsurance terms: the covered benefit, the ceded share or retention, any attachment point and limit, and any eligibility conditions. The examples below use hypothetical contracts and data.

Definition: In an asset-share projection, the reinsurance premium is the amount paid by the insurer for cover, recorded as an outgo when contractually due. A reinsurance recovery is an amount received from the reinsurer for a qualifying covered claim, recorded as an inflow when payable under the contract. The net claim outgo is the gross benefit paid to the policyholder less the recovery for that claim.

Timing and the Basic Ledger

Let $A_t$ be the asset share at the start of year $t$, $P_t$ the premium collected at the start of the year, $E_t$ the expenses, $T_t$ any applicable taxes, and $Q_t$ the reinsurance premium. Suppose these beginning-of-year amounts are paid or received immediately, and the fund earns effective annual investment return $i_t$. Immediately before any year-end claim, the accumulated fund is

\widetilde{A}_{t+1}=(A_t+P_t-E_t-T_t-Q_t)(1+i_t).

If a claim of amount $B_t$ occurs at year-end and triggers a recovery $R_t$, the claim reduces the fund by $B_t-R_t$. If the policy survives and has no claim, this claim adjustment does not apply. If the contract makes the reinsurance premium due at a different time, the premium must instead be placed at that time and accumulated only for the period it remains in the fund.

The recurrence is a cash-flow ledger, not a rule that every reinsurance contract must be charged at the start of a year. Its timing assumptions must agree with the projection being made. In particular, a recovery paid after the year-end would not be available to earn the year’s investment return under this setup.

Worked Example: A Surviving Policy With a Quota-Share Premium

A hypothetical policy has an asset share of $1,000 at the start of a year. The insurer collects a $600 premium at the start of the year, pays $50 of expenses, and pays a $30 reinsurance premium at that time. There are no taxes or claims during the year. The effective annual investment return is 4%. Find the asset share at year-end.

Method. Apply the annual asset-share recurrence, treating the premium, expense, and reinsurance premium as beginning-of-year cash flows. The reinsurance premium is an outgo; it is not a claim recovery.

First calculate the amount invested:

$$ 1{,}000+600-50-30=1{,}520. $$

Accumulate this amount for one year at 4%:

$$ 1{,}520(1.04)=1{,}580.80. $$

The year-end asset share is $1,580.80. As a check, the investment income is $1{,}520(0.04)=\$60.80$, so the result is also $1{,}520+60.80=\$1{,}580.80$. A common mistake is to add the $30 reinsurance premium as if it were a recovery; doing so would overstate the invested amount by $60 and the year-end share by $62.40.

Expected Claims and Asset Shares for Survivors

For a cohort, claim cash flows are often projected in expectation. Consider a simple one-year model with $N_t$ identical policies in force at the start of the year. Let $q_t$ be the probability that a policy has the covered claim during the year. Suppose every such claim pays benefit $B_t$, and the reinsurance contract pays recovery $R_t$ for each qualifying claim. Assume there are no lapses, all policies have the same starting asset share and cash flows, and claims and recoveries occur at year-end. Then expected net claim outgo per starting policy is $q_t(B_t-R_t)$.

The reason is that each starting policy has probability $q_t$ of producing a claim, and the net outgo conditional on that claim is $B_t-R_t$. The expected outgo is therefore probability times conditional amount. It would not be correct to subtract the recovery from every policy’s cash flows: only policies with qualifying claims generate recoveries.

E[\text{net claim outgo per starting policy}] =q_t(B_t-R_t).

Under these assumptions, the expected asset amount remaining for the cohort after year-end claims is divided among the expected survivors, $N_t(1-q_t)$. This gives the following expected asset share per survivor:

A_{t+1}^{\text{surv}} = \frac{ N_t(A_t+P_t-E_t-T_t-Q_t)(1+i_t) - N_tq_t(B_t-R_t) }{ N_t(1-q_t) }.

This formula assumes the reinsurance premium is payable by every policy in force at the start of the year, while the recovery is payable only for a qualifying claim. It also assumes the claim and recovery are both at year-end, so neither earns investment income in the year. It is not an individual surviving policy’s realized cash-flow path: it is a deterministic expected cohort projection divided by the expected number of survivors.

Calculate beginning-of-year funds.
Add premiums and subtract expenses, taxes, and any reinsurance premium due at that time.

Apply investment return over the correct period.
Accumulate only the amounts held in the fund during that period.

Apply covered claim cash flows.
For each qualifying claim, subtract the gross benefit and add the recovery, or subtract their net amount.

Use the correct policy count.
For an expected share per survivor, divide remaining cohort assets by expected survivors, not by the original cohort size.

Worked Example: Quota-Share Recovery in a Mortality Pool

A hypothetical cohort has 100 policies in force at the start of a year. Each policy has a starting asset share of $2,000, pays a $600 premium, incurs $100 of expenses, and is charged a $50 reinsurance premium at the start of the year. The effective annual investment return is 4%. The probability of a year-end death claim is 5%. Each death benefit is $20,000, and a quota-share agreement reimburses 50% of that benefit for each qualifying death claim. Find the expected asset share per survivor at year-end.

Method. Accumulate the beginning-of-year funds, subtract the expected net claim outgo for the cohort, and divide by the expected number of survivors. The recovery is conditional on a death claim.

For each starting policy, the amount invested is

$$ 2{,}000+600-100-50=2{,}450. $$

After investment income, this becomes

$$ 2{,}450(1.04)=2{,}548. $$

The recovery per death claim is $0.50(20{,}000)=\$10{,}000$, so the net claim outgo per death is $20{,}000-10{,}000=\$10{,}000$. Expected net claim outgo per starting policy is

0.05(10{,}000)=\$500.

For 100 starting policies, expected assets before claims are $100(2{,}548)=\$254{,}800$. Expected net claims are $100(500)=\$50{,}000$, leaving

254{,}800-50{,}000=\$204{,}800.

The expected number of survivors is $100(1-0.05)=95$. Therefore,

\frac{204{,}800}{95}=\$2{,}155.789\ldots

Rounded to the nearest cent, the expected asset share per survivor is $2,155.79. As a check, five expected claims each cost the cohort a net $10,000, for total expected outgo of $50,000. Subtracting the gross $20,000 benefit without allowing for the matching recovery would understate the survivor share.

Recoveries Depend on the Contract’s Covered Risk

A quota-share agreement commonly specifies a proportion of a defined covered amount, but the contract must define that amount. It may be a specified death benefit or another agreed measure of risk; it should not be assumed to be the policy’s entire premium, asset share, or every benefit payment. An excess-of-loss agreement works differently: the insurer retains losses up to an attachment point, and the reinsurer covers a specified layer above it, subject to the contract limit.

For an excess-of-loss layer with attachment $a$, maximum recovery $L$, and covered claim amount $C$, a simplified recovery formula is

R(C)=\min\{\max(C-a,0),L\}.

This formula applies when the contract covers the stated claim amount on this basis, with no additional exclusions, aggregations, or adjustments. Below the attachment, the recovery is zero. Above the attachment, the recovery increases with the claim until it reaches the limit.

Worked Example: Excess-of-Loss Recovery on a Covered Claim

A hypothetical insurer has a policy with a $15,000 asset share at the start of the year. It receives a $1,000 premium and pays $100 of expenses and a $300 reinsurance premium at the start of the year. There are no taxes. The effective annual return is 3%. The policy then has a covered year-end claim of $18,000. The excess-of-loss contract attaches at $10,000 and has a maximum recovery of $15,000. Find the asset share after the claim.

Method. First apply the attachment and limit to the covered claim. Then accumulate the beginning-of-year net cash flows and subtract the year-end net claim. The $18,000 claim is within the stated covered basis.

The recovery before applying the limit is $18{,}000-10{,}000=\$8{,}000$. Since $8,000 is below the $15,000 maximum, the payable recovery is $8,000. Net claim outgo is

18{,}000-8{,}000=\$10{,}000.

The beginning-of-year fund before investment is

15{,}000+1{,}000-100-300=\$15{,}600.

After earning 3%, the amount immediately before the claim is

15{,}600(1.03)=\$16{,}068.

The claim and recovery occur at year-end, so the year-end asset share is

16{,}068-10{,}000=\$6{,}068.

The result is $6,068. As a check, the $8,000 recovery plus the insurer’s $10,000 net outgo equals the $18,000 gross claim. Applying a fixed percentage to the claim would ignore the attachment and could produce the wrong recovery.

Common Mistakes and Checks

A useful audit is to reconcile every claim to the contract: identify the gross covered amount, calculate the recovery under the stated terms, and confirm that gross benefit less recovery equals net claim outgo. Then check that the recovery is assigned to the right claim period and that the reinsurance premium is charged when due.

Reversing the premium sign. A reinsurance premium paid by the insurer reduces the asset share; it is not an inflow.
Crediting recoveries without a qualifying claim. A claim-contingent recovery is not received on a policy that has no qualifying claim.
Applying the recovery to the wrong amount. A percentage or excess layer applies only to the amount defined by the agreement, not automatically to all premiums or benefits.
Double-counting claim outgo. Either subtract the gross claim and add the recovery, or subtract the net claim. Do not subtract the gross claim and then subtract the recovery again.
Ignoring timing. A beginning-of-year reinsurance premium affects the amount earning interest; a year-end recovery does not earn a full year’s interest under the timing used here.

For the cohort calculation, a common sign error would be to subtract the $100,000 expected gross claims in the quota-share example and then subtract the $50,000 of recoveries as well. That would leave only $104,800 before division by 95 survivors, rather than the correct $204,800. The recovery is an inflow: subtracting net claims of $50,000 is equivalent to subtracting $100,000 of gross claims and adding $50,000 of recoveries.

Key takeaway: Record the reinsurance premium as an outgo when payable, and record a recovery only when the covered claim conditions are met. Calculate the recovery from the contract’s defined risk amount, preserve the correct cash-flow timing, and use net claim outgo exactly once.

Check Your Understanding

For each problem, track the reinsurance cash flow separately from the policyholder premium and benefit. Monetary answers are in dollars.

A policy starts the year with an asset share of $800. At the start of the year it receives a $500 premium and pays $40 of expenses and a $20 reinsurance premium. It survives without a claim, and the effective annual return is 5%. Find its year-end asset share. (Answer: $1,302.00)
A cohort has a 4% probability of a covered claim. The benefit is $25,000, and the reinsurance contract pays $15,000 for each qualifying claim. Find expected net claim outgo per starting policy. (Answer: $400.00)
A cohort of 200 policies has $3,000 of assets per starting policy after investment but before year-end claims. Each policy has a 2% probability of a qualifying claim, and the net claim outgo per claim is $12,000. Find the expected asset share per survivor. (Answer: $2,816.33)
An excess-of-loss contract has an attachment of $5,000 and a maximum recovery of $12,000. A covered claim is $20,000. Find the recovery and the insurer’s net claim outgo. (Answer: Recovery $12,000; net claim outgo $8,000)
A covered claim has probability 3% per starting policy. The gross benefit is $40,000, and the recovery is $25,000 whenever the claim qualifies. Find expected net claim outgo per starting policy. (Answer: $450.00)

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