Retrospective Recurrence for Annual Asset Shares

Introduction

An asset share is built by carrying past policy cash flows forward to a common date. In an annual projection, the essential operation is a recurrence: begin with the opening asset share, add premiums, deduct expenses, credit investment earnings, and deduct claims. The arithmetic is straightforward only after the timing and per-policy basis of every amount have been made explicit.

This tutorial uses a specific annual convention to derive and apply that recurrence. It then expands the recurrence into a cumulative cash-flow calculation, which provides an independent check. The convention is not universal: a premium, expense, or claim paid at another point in the year must be accumulated for the appropriate period.

Definition: Let $A_t$ be the asset share at the end of year $t$, on a stated policy or cohort basis. In year $t$, let $P_t$ be premium income and $E_t$ be expenses, both received or paid at the start of the year; let $i_t$ be the effective investment return for that year; and let $C_t$ be claim outgo paid at the end of the year. Under this timing convention, $$ A_t=(A_{t-1}+P_t-E_t)(1+i_t)-C_t. $$

Reading the Recurrence

The opening balance $A_{t-1}$ and the net beginning-of-year cash flow $P_t-E_t$ are invested together for the year. Consequently, both are multiplied by $1+i_t$. The end-of-year claim $C_t$ is then subtracted without investment accumulation during that year. A positive $C_t$ is an outgo; it is not a benefit added to the asset share.

All quantities must use a consistent basis. For example, if the opening asset share is stated per original policy in a cohort, premiums, expenses, and claims must also be stated per original policy. If amounts are instead expressed per policy in force at the start of each year, the basis must be maintained consistently when moving between years. The examples state their basis explicitly.

The recurrence assumes the stated effective return applies to the invested balance for the whole year, and that the specified premium and expense amounts are available or paid at the beginning of the year. It does not imply that all actual policy cash flows occur at those times. If a cash flow occurs at a different time, its investment accumulation must be adjusted rather than mechanically using this formula.

Worked Example: Rolling a Two-Year Asset Share Forward

A hypothetical policy has an opening asset share of $100. At the start of each year it receives a premium of $80 and incurs expenses of $10. Claims are paid at each year-end: $20 in year 1 and $30 in year 2. The effective investment returns are 5% in year 1 and 4% in year 2. Treat all amounts and returns as deterministic; no mortality model is used, and the claim outgoes are given cash-flow amounts. Calculate the end-of-year asset shares.

Year 1. The amount invested at the start of the year is $100+80-10=170$. After investment earnings, it is $170(1.05)=178.50$. Deduct the year-end claim: $A_1=178.50-20=\$158.50$.

Year 2. The amount invested at the start of the year is $158.50+80-10=228.50$. After investment earnings, it is $228.50(1.04)=237.64$. Deduct the year-end claim: $A_2=237.64-30=\$207.64$.

Check. The year 2 starting balance of $158.50 agrees with the year 1 ending balance. The final asset share is $207.64. This method applies the annual recurrence directly and avoids treating the year-end claim as if it had earned a full year's investment return.

Why the Recurrence Holds

The recurrence is a balance-sheet roll-forward. Immediately after the beginning-of-year premium and expense transactions, the assets available for investment are $A_{t-1}+P_t-E_t$. A return $i_t$ on those assets gives an end-of-year amount of $(A_{t-1}+P_t-E_t)(1+i_t)$. Paying the claim at year-end reduces that amount by $C_t$. Combining these three ledger steps gives the formula.

Start with the opening balance.
Carry forward $A_{t-1}$, the asset share at the end of the preceding year.

Record beginning-of-year flows.
Add $P_t$ and subtract $E_t$, giving $A_{t-1}+P_t-E_t$ available for investment.

Credit investment earnings.
Multiply the invested amount by $1+i_t$, since it earns the stated effective return for the year.

Record the year-end claim.
Subtract $C_t$, which is paid after the year's investment accumulation under this convention.

The same reasoning also gives a cumulative formula. Repeatedly substitute the recurrence for each prior year. Each beginning-of-year net flow earns the return for its own year and every later year up to $n$; each year-end claim earns returns only in the years after it is paid. With an empty product defined to equal 1, the result is

A_n=A_0\prod_{j=1}^{n}(1+i_j) +\sum_{t=1}^{n}(P_t-E_t)\prod_{j=t}^{n}(1+i_j) -\sum_{t=1}^{n}C_t\prod_{j=t+1}^{n}(1+i_j).

For example, the $t=n$ claim is multiplied by an empty product, so it is deducted at its stated year-end amount. The $t=n$ beginning-of-year net flow is multiplied by $1+i_n$, because it is invested during year $n$. These different accumulation factors are a useful check on timing.

Worked Example: Expected Claims for a Surviving Cohort

A hypothetical cohort begins with an asset share of $50 per original policy. Premiums and expenses are paid at the start of each year. The annual effective returns are 0% and 4%. In year 1, each original policy is charged a $100 premium and $10 expense. Each policy has a 2% probability of death during the year, with a $500 claim paid at year-end. For policies surviving year 1, year 2 has a $80 premium and $5 expense per policy in force at the start of that year; the conditional probability of death in year 2 is 5%, with the same $500 claim. Calculate the expected year 2 asset share per original policy. Assume the stated death probabilities define Bernoulli death outcomes, and use expected cash flows on the original-cohort basis.

Year 1 claim outgo. Per original policy, the expected year 1 claim is $0.02(500)=\$10$. The year 1 asset share per original policy is therefore $(50+100-10)(1.00)-10=\$130$.

Year 2 exposure. The probability of surviving year 1 is $1-0.02=0.98$. Thus, per original policy, expected year 2 premium income is $0.98(80)=\$78.40$, and expected year 2 expenses are $0.98(5)=\$4.90$. The expected year 2 claim outgo per original policy is $0.98(0.05)(500)=\$24.50$.

Year 2 asset share. Apply the recurrence to the cohort amounts: $(130+78.40-4.90)(1.04)-24.50$. The amount before investment earnings is $203.50$, and after earnings it is $203.50(1.04)=\$211.64$. Therefore the expected year 2 asset share is $211.64-24.50=\$187.14$ per original policy.

Check. The survival factor is applied to each year 2 cash flow because those amounts arise only for policies surviving year 1. The year 2 expected claim is deducted after the 4% return, consistent with the specified year-end claim timing. This method avoids mixing per-survivor amounts with per-original-policy asset shares.

Checking by Accumulating Each Cash Flow

An independent check is to take each cash flow separately and accumulate it to the end of the projection. This is especially useful when returns vary by year: use the product of the actual annual accumulation factors, not one average rate. The opening balance is accumulated through every year; beginning-of-year net flows are accumulated from their year to the end; and year-end claims are accumulated only after their payment dates.

Worked Example: Verifying a Three-Year Projection Cumulatively

For a hypothetical deterministic ledger, $A_0=\$50$. At the start of each year, premiums and expenses are respectively $100 and $15 in year 1, $80 and $20 in year 2, and $90 and $10 in year 3. Year-end claims are $11, $12, and $5. The effective returns are 10%, 0%, and 5%, respectively. No mortality distribution is modeled; these are fixed cash flows. Verify the year 3 asset share using both methods.

Recurrence calculation. In year 1, $A_1=(50+100-15)(1.10)-11=135(1.10)-11=\$137.50$. In year 2, $A_2=(137.50+80-20)(1.00)-12=197.50-12=\$185.50$. In year 3, $A_3=(185.50+90-10)(1.05)-5=265.50(1.05)-5=\$273.775$.

Cumulative calculation. Accumulate each item to the end of year 3:

\begin{aligned} A_3={}&50(1.10)(1.00)(1.05) +85(1.10)(1.00)(1.05)-11(1.00)(1.05)\\ &+60(1.00)(1.05)-12(1.05) +80(1.05)-5\\ ={}&57.75+98.175-11.55+63.00-12.60+84.00-5\\ ={}&273.775. \end{aligned}

The two methods agree. Reporting money to the nearest cent, the year 3 asset share is $273.78. Rounding is performed only at the final step.

Common Timing Mistakes

A frequent error is to subtract a year-end claim before applying the year's return. In the first example, the incorrect calculation $(100+80-10-20)(1.05)=\$157.50$ treats the $20 claim as though it were paid at the start of the year and therefore removes it from the amount earning interest. The correct year-end claim calculation is $(100+80-10)(1.05)-20=\$158.50$. The difference is the return that the incorrectly early deduction would have lost: $20(0.05)=\$1.00$.

Other checks that help prevent errors include:

Confirm whether the stated asset share is at the start or end of a year before applying a return.
Keep the policy or cohort basis consistent for the opening balance and every cash flow.
Do not multiply a year-end claim by the return for the year in which it is paid.
Use each year's own return when rates vary, and retain full precision until the final reported amount.
If a cash flow is not at the assumed time, change its accumulation factor; do not force it into the recurrence unchanged.

Key takeaway: Under the stated convention, begin with the opening asset share, add premiums and subtract expenses at the start of the year, accumulate that net amount at the annual effective return, and subtract end-of-year claims. A cumulative cash-flow expansion should reproduce the same result when every amount is carried to the same date.

Check Your Understanding

Use the stated beginning-of-year premium and expense, end-of-year claim convention in each question. Amounts are hypothetical.

An asset share opens at $200. The start-year premium is $90, expenses are $10, the effective return is 3%, and the year-end claim is $12. Find the closing asset share. (Answer: $276.40)
A two-year ledger starts at zero. In each year, the premium is $100 and expenses are $20. Returns are 4% and 5%; year-end claims are $10 and $8. Find the year 2 closing asset share. (Answer: $152.86)
A cohort has an opening share of $50 per original policy. In year 1, premium is $100, expense is $10, return is 0%, death probability is 2%, and the death benefit is $500. In year 2, among year 1 survivors, premium is $80, expense is $5, and conditional death probability is 5%; the same benefit is paid at year-end. The year 2 return is 4%. Using expected cohort cash flows per original policy, find the year 2 asset share. (Answer: $187.14)
A ledger opens at $40. Year 1 has premium $85, expense $15, return 5%, and year-end claim $10. Year 2 has premium $60, expense $10, return 2%, and year-end claim $5. Find the year 2 closing share. (Answer: $153.61)
A one-year asset share is to close at $200. The opening share is unknown; the beginning-year premium is $160, expense is $20, return is 5%, and year-end claim is $10. Find the required opening share. (Answer: $60.00)

What You'll Learn