Why the Asset-Share Recursion Needs a Cash-Flow Timeline
An asset share is a measure of the accumulated assets allocated to a policy or group of similar policies under a specified projection basis. In a period-by-period projection, it is not enough to add premiums and apply an investment return. The calculation must also allow for expenses, benefits paid on exit, payments to continuing policies, and tax. Just as important, each amount must be assigned to the correct point in the policy year and divided by the correct number of policies.
This tutorial derives a one-year recursion from a cohort cash-flow balance, then applies it to hypothetical participating policies. The projection is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
All cash flows in this expression are on a consistent per-opening-policy basis until the final division by \(p_t\). In particular, \(B_t\) is expressed per survivor, so its cost per opening policy is \(p_tB_t\). The formula assumes \(p_t>0\), that premiums and \(E_t^0\) are paid at the beginning of the year, and that the remaining listed costs are paid at year-end. If a product uses different payment times, the relevant cash flows must be accumulated to a common valuation time before applying the balance.
Deriving the Recursion from the Cohort Balance
The denominator in the recursion is not a technical adjustment: it converts the remaining assets for the whole starting cohort into assets per policy still in force. The following derivation makes that distinction explicit.
Take \(N\) policies in force at the beginning of the year. Their opening asset shares total \(NA_t\), and beginning premiums less beginning expenses add \(N(P_t-E_t^0)\).
The amount invested is \(N(A_t+P_t-E_t^0)\). At an effective annual return \(i_t\), its year-end value is \(N(A_t+P_t-E_t^0)(1+i_t)\).
There are \(Nq_t\) expected exits, costing \(Nq_tC_t\). There are \(Np_t\) expected survivors, costing \(Np_tB_t\) for a benefit of \(B_t\) each. Opening-policy expenses and tax cost \(NE_t^1\) and \(NT_t\).
The cohort has \(Np_t\) expected policies remaining. Divide its total remaining assets by \(Np_t\) to obtain the asset share per survivor.
Algebraically, the remaining cohort assets are
Dividing this expression by \(Np_t\) cancels \(N\) and gives the stated recursion. The derivation uses expected cohort counts; it does not assert that an actual small cohort will have exactly \(Nq_t\) exits. For a projection with several exit causes, replace \(q_tC_t\) by the sum of each cause’s exit probability multiplied by its corresponding benefit, and use the probability of remaining in force in the denominator.
Worked Applications
Worked Example: Correctly Allocating a Survivor Bonus
A hypothetical participating block has an opening asset share of $300 per policy. Each policy pays a beginning-of-year premium of $120, and the beginning expense is $20. The annual effective investment return is 3%. During the year, 5% of policies die and each death generates a $2,000 claim. Each survivor receives a year-end bonus of $15. There are no other expenses or taxes. Find the remaining assets per opening policy and the asset share per survivor.
Method. First accumulate the opening asset share and premium after the beginning expense. Then deduct each year-end cost on its proper basis, and divide the remaining cohort assets per opening policy by the 95% survival probability.
The amount invested per opening policy is
After investment return, this becomes
The expected death-claim cost per opening policy is \(0.05(2{,}000)=100\). The bonus is $15 per survivor, so its expected cost per opening policy is \(0.95(15)=14.25\). Thus remaining assets per opening policy are
The survivor probability is \(0.95\), so the asset share per survivor is
Rounded to the nearest cent, the results are $297.75 per opening policy and $313.42 per survivor. The check is that \(0.95(313.4210526\ldots)=297.75\), recovering the remaining cohort assets per opening policy.
Worked Example: Including Tax as a Year-End Cash Flow
A hypothetical block begins the year with an asset share of $700 per policy. The beginning premium is $300 and the beginning expense is $40. The effective annual investment return is 5%. At year-end, maintenance expense is $15 per opening policy, 1% of policies exit with a $20,000 claim each, and each survivor receives a $25 bonus. Tax is specified as $20 per opening policy, paid at year-end. Calculate the asset share per survivor.
Method. Use the recursion with all year-end items stated per opening policy except the survivor bonus, whose per-opening cost is \(p_tB_t\).
The beginning amount invested per policy is \(700+300-40=960\), and its year-end value is
Here \(q_t=0.01\) and \(p_t=0.99\). The expected claim cost is \(0.01(20{,}000)=200\), and the expected bonus cost per opening policy is \(0.99(25)=24.75\). After maintenance expense and tax, the remaining amount per opening policy is
Therefore,
The asset share is $755.81 per survivor, rounded to the nearest cent. As a check, \(0.99(755.8080808\ldots)=748.25\). The tax amount here is an input specified by the problem; it should not be replaced by a different tax convention unless one is given.
Worked Example: A Two-Year Participating Projection
A hypothetical participating policy has an opening asset share of $0. In each of two years, the beginning premium is $1,000, the effective annual investment return is 4%, the probability of death is 1%, and the death claim is $10,000. At each year-end, maintenance expense is $20 per opening policy and every survivor receives a declared bonus. The year-one bonus is $10 per survivor and the year-two bonus is $20 per survivor. Beginning expenses are $100 in year one and $40 in year two. Tax is 20% of the investment income earned on the amount invested at the start of each year, and is paid at year-end. Find the asset share per survivor at the end of year two.
Method. Apply the recursion sequentially. For each year, calculate tax from the stated investment-income base, deduct it at year-end, and carry forward the resulting per-survivor asset share.
In year one, the amount invested is \(0+1{,}000-100=900\). Investment income is \(900(0.04)=36\), so tax is \(0.20(36)=7.20\). The invested amount accumulates to \(900(1.04)=936\). With \(p_0=0.99\), the claim cost per opening policy is \(0.01(10{,}000)=100\), and the bonus cost is \(0.99(10)=9.90\). The remaining amount per opening policy is
Thus the year-one asset share per survivor is
For year two, the opening asset share is \(807.9797979\ldots\). After the $1,000 premium and $40 beginning expense, the amount invested is \(1{,}767.9797979\ldots\). Its investment income is
so year-two tax is \(0.20(70.7191919\ldots)=14.1438384\ldots\). The accumulated amount is \(1{,}767.9797979\ldots(1.04)=1{,}838.6989899\ldots\). The year-two claim cost per opening policy is again \(100\), and the bonus cost per opening policy is \(0.99(20)=19.80\). The remaining amount is
Dividing by the survivor probability gives
The year-two asset share is $1,701.77 per survivor, rounded to the nearest cent. Multiplying this amount by \(0.99\) recovers approximately $1,684.76 per opening policy, consistent with the unrounded cohort balance.
Common Exam Trap: Mixing the Per-Survivor and Per-Opening Bases
In the first example, an incorrect calculation might deduct the full $15 bonus per opening policy, giving \(412-100-15=297\). That treats every opening policy as receiving a bonus, including those who died. The correction is to charge \(0.95(15)=14.25\) per opening policy, leaving $297.75. A second error is to stop at $297.75 and call it the survivor asset share. That figure is the remaining cohort amount per opening policy; the survivor asset share is \(297.75/0.95=\$313.42\).
Before using the recursion, label each amount as one of the following: per opening policy, per survivor, or per exiting policy. Then convert it to a common per-opening-policy basis in the numerator. Only after that balance is complete should it be divided by the probability of survival. If a benefit is paid at a different time from the assumed year-end, it must also be adjusted for investment return over the appropriate part of the year.
Interpretation and Limits of the Projection
The recursion is a cohort accounting identity under its stated timing and decrement assumptions. It does not, by itself, determine whether the asset share equals a reserve, a policy value, or a benefit payable under the contract. Those quantities can be defined on different bases and for different purposes. In a participating projection, a declared bonus can be included as a cost to the asset share when allocated, but the recursion does not decide how that bonus should be set.
The formula also depends on how tax is defined. If tax is based on taxable profit rather than the investment income specified in the two-year example, the tax calculation must follow that stated basis. Similarly, if expenses are incurred only when a policy survives, their expected per-opening cost is \(p_t\) times the per-survivor expense. A projection should not silently switch between expected cash flows and actual realized experience.
Check Your Understanding
For each problem, use the stated timing and report the asset share to the nearest cent where appropriate.
- A block has an opening asset share of $200, a beginning premium of $400, a beginning expense of $50, and a 5% effective annual return. The exit probability is 2%, the exit claim is $5,000, year-end expense is $10 per opening policy, and each survivor receives a $12 bonus. There is no tax. Find the asset share per survivor. (Answer: $465.04)
- A block has an opening asset share of $1,000, a beginning premium of $250, a beginning expense of $50, and a 4% effective annual return. The exit probability is 1%, the exit claim is $8,000, year-end expense is $20 per opening policy, and tax is $10 per opening policy. No survivor benefit is paid. Find the asset share per survivor. (Answer: $1,149.49)
- A block has an opening asset share of $500, a beginning premium of $300, a beginning expense of $20, and a 6% effective annual return. The exit probability is 2%, the exit claim is $10,000, year-end expense is $15 per opening policy, and each survivor receives $5. Tax is 25% of the investment income on the beginning invested amount and is paid at year-end. Find the asset share per survivor. (Answer: $607.35)
- A block has an opening asset share of $0, a beginning premium of $600, a beginning expense of $60, and a 5% effective annual return. The exit probability is 10%, year-end expense is $10 per opening policy, and each survivor receives $20. There is no tax. What exit claim per exiting policy would produce a next-year asset share of $400 per survivor? (Answer: $1,790)