Cohort Asset Shares With Multiple Entry Years

Why Entry Year Matters

A portfolio may contain policies issued in different calendar years. Even when the policies have the same product design, their asset shares at a given valuation date need not be the same: one cohort may have paid premiums for longer, incurred a different sequence of expenses, or experienced a different sequence of investment returns since issue. A valid projection keeps those histories attached to the cohort that experienced them.

The key distinction is between calendar time and policy duration. Calendar time identifies the year’s investment return and the date cash flows occur. Duration counts how many policy years have elapsed since a particular cohort entered. At a common calendar valuation date, cohorts can therefore be at different durations.

Definition: An entry-year cohort is a group of policies that enter a portfolio in the same calendar year and are projected using the same specified assumptions and experience allocation. A cohort asset share is the accumulated value, on the stated per-policy or aggregate basis, of that cohort’s allocated cash flows and investment experience from entry to the valuation date. It does not include cash flows or investment returns from before that cohort entered.

Set the Timeline and Cash-Flow Convention

Before calculating, state when premiums, expenses, benefits, and investment returns occur. The recurrence and accumulation factors depend on this timing. In the examples below, policies enter at the beginning of their entry year. Premiums and expenses are paid at the beginning of each covered year; the investment return for that calendar year is then applied; and any specified benefit outgo is paid at year-end. Thus a beginning-of-year net cash flow earns that year’s return, while a year-end benefit does not.

Let $A_{c,t}$ be the asset share at the end of calendar year $t$ for cohort $c$. Let $e_c$ be its entry year, $P_{c,t}$ its beginning-of-year premium, $E_{c,t}$ its beginning-of-year expenses, $B_{c,t}$ its year-end benefit outgo, and $j_{c,t}$ its investment return for the year. The return may be common to all cohorts, in which case write $j_t$, or cohort-specific if the model allocates different investment experience to different cohorts.

Recurrence: With no opening asset share before entry, set $A_{c,e_c-1}=0$. Under the timing convention above, $$ A_{c,t}=(A_{c,t-1}+P_{c,t}-E_{c,t})(1+j_{c,t})-B_{c,t}, \qquad t\ge e_c. $$ The recurrence applies only to years from the cohort’s entry onward. A cash flow paid at a different time must be accumulated for the corresponding fraction or sequence of investment periods, rather than inserted unchanged into this formula.

This is a retrospective projection: begin with the cohort’s opening asset share, add the cash flows received at the start of the year, accumulate that balance at the cohort’s return, and subtract any outgo paid at year-end. If an acquisition expense is paid at entry, include it in $E_{c,e_c}$ when it follows this timing convention. Do not apply returns from years before entry to that expense or to any later contribution.

Worked Example: Two Entry Years at One Valuation Date

Two hypothetical cohorts are valued at the end of calendar year 3. Returns are 5% in year 1, −2% in year 2, and 4% in year 3. Cohort A enters at the beginning of year 1 and receives a premium of $1,000 with an expense of $100 at the beginning of each year. Cohort B enters at the beginning of year 2 and receives a premium of $800 with an expense of $80 at the beginning of each covered year. There are no benefits or decrements. Find each per-policy asset share at the valuation date.

Method. Apply the recurrence separately from each cohort’s entry year. The annual net cash flows are $900 for A and $720 for B. The negative return in year 2 is −0.02, so its accumulation factor is $1-0.02=0.98$.

Cohort A. At the end of year 1, $A_{A,1}=(0+900)(1.05)=945$. At the end of year 2, $A_{A,2}=(945+900)(0.98)=1,845(0.98)=1,808.10$. At the end of year 3, $A_{A,3}=(1,808.10+900)(1.04)=2,708.10(1.04)=2,816.424$.

Cohort B. B has no asset share from year 1 because it has not entered. At the end of year 2, $A_{B,2}=(0+720)(0.98)=705.60$. At the end of year 3, $A_{B,3}=(705.60+720)(1.04)=1,425.60(1.04)=1,482.624$.

Rounded to the nearest cent, A’s asset share is $2,816.42 and B’s is $1,482.62. The result is checked against the timelines: A has three beginning-of-year net contributions and experiences all three returns; B has two net contributions and experiences only the year 2 and year 3 returns. The common mistake avoided is giving B a year 1 investment return or premium history.

Why the Cohort Recurrence Works

Repeated application of the recurrence shows exactly how the cohort’s own cash flows reach a later valuation date. For a cohort entering in year $e_c$, each beginning-of-year net inflow is accumulated from the start of its payment year through that year’s return and all subsequent returns. A year-end benefit is accumulated only through returns after its payment year.

To see this, write $C_{c,t}=P_{c,t}-E_{c,t}$. For one year, the recurrence is $A_{c,t}=C_{c,t}(1+j_{c,t})-B_{c,t}$, since the opening value is zero. For the next year, substitute the first year’s ending value into the recurrence:

\begin{aligned} A_{c,e_c+1} &=\bigl(C_{c,e_c}(1+j_{c,e_c})-B_{c,e_c}+C_{c,e_c+1}\bigr) (1+j_{c,e_c+1})-B_{c,e_c+1}. \end{aligned}

The first-year net inflow earns both year $e_c$’s return and year $e_c+1$’s return. The first-year benefit outgo, paid at the end of year $e_c$, earns only the return for the following year. Continuing this substitution to a valuation at the end of year $T$ gives:

A_{c,T} = \sum_{k=e_c}^{T} C_{c,k}\prod_{s=k}^{T}(1+j_{c,s}) - \sum_{k=e_c}^{T} B_{c,k}\prod_{s=k+1}^{T}(1+j_{c,s}).

An empty product, such as the benefit accumulation factor when $k=T$, equals 1. Thus a benefit paid at the valuation date is subtracted without further accumulation. The formula also makes the entry-year boundary explicit: both sums begin at $e_c$, not at the portfolio’s first calendar year. It is valid under the stated timing convention; end-of-year premiums, mid-year expenses, or other payment timings require adjusted accumulation factors.

Worked Example: Different Premium Histories and Year-End Outgo

Two hypothetical cohorts are valued at the end of year 3. Calendar-year returns are 5% in year 1, 0% in year 2, and 10% in year 3. Cohort C enters in year 1 and has beginning-of-year premiums of $400, $500, and $600 in years 1, 2, and 3, respectively, with a $50 expense in each year. C also pays a $100 benefit at the end of year 2. Cohort D enters in year 2, has a $700 premium and $100 expense in year 2, and a $650 premium and $90 expense in year 3. D pays a $300 benefit at the end of year 3. Find both asset shares.

Method. Track each cohort from its own entry. Apply each beginning-of-year premium less expense before the return, then subtract the specified year-end benefit. The year 2 benefit for C is not exposed to the year 2 return under this timing convention.

Cohort C. Its net beginning-of-year cash flows are $400-50=350$, $500-50=450$, and $600-50=550$. At the end of year 1, $A_{C,1}=350(1.05)=367.50$. At the end of year 2, first accumulate the opening share and year 2 net inflow: $(367.50+450)(1.00)=817.50$. Then subtract the year-end benefit, giving $A_{C,2}=817.50-100=717.50$. At the end of year 3, $A_{C,3}=(717.50+550)(1.10)=1,267.50(1.10)=1,394.25$.

Cohort D. D’s year 2 net cash flow is $700-100=600$, so $A_{D,2}=600(1.00)=600$. Its year 3 net cash flow is $650-90=560$. Before the year-end benefit, the share is $(600+560)(1.10)=1,160(1.10)=1,276$. Subtracting the $300 benefit gives $A_{D,3}=1,276-300=976.00$.

At the end of year 3, C’s asset share is $1,394.25 and D’s is $976.00. The year 2 cash flow for C includes a benefit outgo, while D has no year 1 history. These checks confirm that neither the cash-flow timing nor the entry dates have been ignored. The common mistake avoided is accumulating a year-end benefit through a return earned before that benefit is paid.

Investment Experience Belongs to the Years Experienced

When cohorts share the same investment pool, they may use the same return for a calendar year in which both are invested. They still do not have the same sequence of returns from entry to valuation. A cohort entering later has no exposure to earlier years, and its later contributions have not been invested for as long as its opening balance. A single cumulative return factor applied to all cash flows therefore generally gives the wrong result.

Worked Example: Avoiding a Portfolio-Wide Return Shortcut

Cohort A enters at the beginning of year 1 and cohort B at the beginning of year 2. Each cohort receives a net beginning-of-year cash flow of $1,000 in every year from entry through year 3. There are no benefits. The common investment returns are 20% in year 1, −10% in year 2, and 10% in year 3. Find both end-of-year-3 asset shares and assess the shortcut of multiplying B’s total contributions by the portfolio’s full three-year accumulation factor.

Cohort A. At the end of year 1, $A_{A,1}=1,000(1.20)=1,200$. At the end of year 2, $A_{A,2}=(1,200+1,000)(0.90)=2,200(0.90)=1,980$. At the end of year 3, $A_{A,3}=(1,980+1,000)(1.10)=2,980(1.10)=3,278$.

Cohort B. B’s year 2 contribution earns the year 2 return, so $A_{B,2}=1,000(0.90)=900$. At the end of year 3, $A_{B,3}=(900+1,000)(1.10)=1,900(1.10)=2,090$.

The full three-year accumulation factor is $1.20(0.90)(1.10)=1.188$. The incorrect shortcut would give B $2,000(1.188)=2,376$, as if both of its contributions had been invested from the beginning of year 1. But B did not enter until year 2; its correct share is $2,090. A’s correct share is $3,278. The discrepancy is not a rounding issue: the shortcut assigns returns for a period in which B had no invested asset share and overstates how long B’s later contribution was invested.

A Reliable Cohort Projection Process

A disciplined projection makes it easier to detect errors before results are combined or compared. For each cohort, maintain its own entry year, opening value, duration, cash-flow history, and applicable return sequence. If a cohort has policies with different terms or experience allocation, it may need to be split into more homogeneous groups for the calculation.

Set the valuation date and entry year.
For each cohort, record the first calendar year in which its policies are invested and count duration only from that entry year.

Specify timing and cash flows.
Identify whether premiums, expenses, and benefits occur at the beginning, middle, or end of each year. Use an opening asset share of zero at entry unless the model explicitly assigns an initial amount.

Use the cohort’s own return history.
Apply only returns earned after entry, and use a cohort-specific return where the investment allocation requires one.

Reconcile the ending value.
Check the recurrence year by year and confirm that each contribution has the right accumulation period and each outgo is deducted at the stated time.

Common Mistakes and Relevance

Cohort projections are useful whenever a portfolio grows over time and results are examined at a common date. They ensure that an older cohort is not treated as if it had the same duration as a newer one, and that a later entrant is not credited with returns from before issue. The same discipline applies whether the calculation is per policy or on another explicitly stated basis.

Starting every cohort at the portfolio’s first year. A cohort’s opening asset share is established at its own entry, and pre-entry premiums and returns do not belong to it.
Confusing duration with calendar year. A duration-one policy year can occur in different calendar years for different cohorts, so the applicable investment return must be matched to the calendar year actually experienced.
Using a single return factor for cash flows paid at different dates. Each inflow or outgo has its own accumulation period, determined by payment timing and the valuation date.
Changing the timing convention mid-calculation. If a benefit is paid at year-end, do not deduct it before that year’s return unless the model specifies a different payment time.

A cohort asset share is only as meaningful as its underlying allocation rules. The projection must state whether it represents a policy or a group, how cash flows are assigned, and what returns are credited. With those assumptions explicit, separate cohort recurrences preserve the distinct histories needed for a defensible valuation. This topic is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

Use the beginning-of-year inflow and year-end outgo convention stated above. Round money to the nearest cent where needed.

A cohort enters at the beginning of year 2. It receives net cash flows of $500 at the beginnings of years 2 and 3. Returns are −5% in year 2 and 10% in year 3, with no outgo. Find its end-of-year-3 asset share. (Answer: $1,072.50)
A cohort enters in year 1 with net beginning-of-year cash flows of $300, $400, and $200 in years 1, 2, and 3. Returns are 0%, 5%, and −10%, respectively. It pays a $50 benefit at the end of year 2. Find its end-of-year-3 asset share. (Answer: $796.50)
Cohort A enters in year 1 and receives net cash flows of $1,000 at the beginnings of years 1 and 2. Cohort B enters in year 2 and receives $1,000 at the beginning of year 2 only. Returns are 4% in year 1 and 6% in year 2. At the end of year 2, how much larger is A’s asset share than B’s? (Answer: $1,102.40)
Six identical policies enter a cohort in year 2. Each has net beginning-of-year cash flows of $300 in year 2 and $400 in year 3. Returns are −10% in year 2 and 5% in year 3, with no benefits. Find the cohort’s aggregate asset share at the end of year 3. (Answer: $4,221.00)

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