Comparing Asset Shares With Guaranteed Policy Benefits

Purpose and Comparison Basis

An accumulated asset share describes the assets attributed to a policy or group of policies after allowing for specified cash flows and investment returns. A guaranteed benefit is a contractual amount payable when its stated conditions are met. Comparing the two at the same duration can show whether the accumulated amount is above or below that guarantee on the chosen basis.

The comparison is only meaningful when the amounts are measured at the same time and on a consistent population basis. A per-policy asset share for a survivor must be compared with the guarantee for that survivor. An aggregate asset share must be compared with the aggregate guarantee for the policies entitled to payment. A benefit payable only if the policy remains in force should not be multiplied by the original cohort size if some policies have died or otherwise exited.

Definition: At duration $t$, let $A_t$ be the accumulated asset share on a stated policy or cohort basis, and let $G_t$ be the guaranteed benefit due at that duration on the same basis. Define the comparison margin as $M_t=A_t-G_t$. If $M_t>0$, there is an asset-share excess over the guarantee; if $M_t<0$, there is a shortfall of $G_t-A_t$. The comparison depends on the cash-flow, investment, and population assumptions used to calculate $A_t$.

This is a direct funding comparison under stated assumptions. It does not, by itself, calculate a prospective reserve, a regulatory capital requirement, or the full financial position of an insurer. Those require additional definitions and assumptions. For this topic, the central task is to calculate the asset share correctly, identify exactly which guaranteed benefits are due, and compare like with like. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Accumulating an Asset Share Before Comparing

First specify when each cash flow occurs. In the examples below, a premium and expense occur at the beginning of each year, and the net amount is invested for the full year. The opening asset share is also invested for the year. If the annual effective investment return is $i$, the end-of-year asset share is therefore the opening asset share plus the beginning-of-year net cash flow, all multiplied by $1+i$. Other timing conventions require a different accumulation.

A_{t+1}=(A_t+P_t-E_t)(1+i)

Here $P_t$ is the premium allocated to the policy during year $t+1$, and $E_t$ is the expense allocated at the same time. The formula assumes no intervening claims, lapses, or other cash flows during the year. If these occur, they must be incorporated at their actual specified times or in an explicitly stated aggregate approximation.

Worked Example: Comparing a Single-Policy Asset Share at Maturity

A hypothetical policy has an opening asset share of $300 immediately before its first annual premium. At the beginning of each of the next three years, it receives a premium of $700 and incurs an expense of $100. The annual effective investment return is 5%. Assume the policy remains in force, there are no claims or other cash flows, and a guaranteed maturity benefit of $2,400 is payable at the end of year 3. Calculate the asset share and its margin against the guarantee.

Method. Use the stated beginning-of-year timing and apply the recurrence for each year. This avoids treating the premiums as end-of-year deposits.

The net beginning-of-year cash flow is $700-100=\$600$. Starting from $A_0=\$300$:

End of year 1: $A_1=(300+600)(1.05)=900(1.05)=\$945.00$.

End of year 2: $A_2=(945+600)(1.05)=1,545(1.05)=\$1,622.25$.

End of year 3: $A_3=(1,622.25+600)(1.05)=2,222.25(1.05)=\$2,333.3625$, or $2,333.36 to the nearest cent.

The margin is $M_3=2,333.3625-2,400=-\$66.6375$. Thus the asset share is below the guaranteed maturity benefit by $66.64 per policy, rounded to cents. As a check, the accumulated amount is less than $2,400, so the sign must indicate a shortfall. The calculation does not imply that the contract's guarantee changes; it reports the comparison on the stated asset-share basis.

Common mistake avoided: adding the premium to the asset share after applying interest would give the premium no investment return, contrary to the stated beginning-of-year timing.

Mortality Pooling and the Survivor Basis

When a cohort is subject to mortality, claims paid on deaths reduce the assets remaining for survivors. A survivor asset share is not generally the remaining aggregate assets divided by the original number of policies. It is the remaining aggregate assets divided by the number of policies still in force at the comparison date.

A simple one-year formula makes the distinction clear. Suppose there are $N$ identical policies, each with opening asset share $A$. The annual effective return is $i$, a fraction $q$ die during the year, and each death claim $B$ is paid at year-end. Assume no other cash flows and that $qN$ is the expected number of deaths. Expected assets after investment and claims are $NA(1+i)-qNB$; expected survivors are $N(1-q)$. Dividing the former by the latter gives the expected-basis asset share per survivor:

A_{\text{surv}}=\frac{NA(1+i)-qNB}{N(1-q)} =\frac{A(1+i)-qB}{1-q}

This derivation follows from the cohort balance: investment applies to the opening assets, death claims are then deducted, and the remaining fund is allocated across survivors. The formula assumes deaths and claims are represented by the stated expected fraction, all policies have the same opening asset share, and the claims are paid at year-end. It does not directly apply to a different claim-timing convention or to heterogeneous policies without appropriate aggregation.

Worked Example: Comparing a Survivor Asset Share With a Guarantee

A hypothetical cohort has 1,200 policies, each with an opening asset share of $1,100. The annual effective return is 2%. During the year, 5% of the policies are expected to die, each generating a $1,600 claim at year-end. Assume no premiums, expenses, lapses, or other cash flows. At year-end, each survivor is entitled to a guaranteed benefit of $1,100. Find the expected survivor asset share and compare it with the guarantee.

Method. Calculate the aggregate fund after investment and death claims, then divide by the expected number of survivors. This avoids dividing by the original cohort size when the comparison concerns survivors.

Expected deaths are $1,200(0.05)=60$, so expected survivors are $1,200-60=1,140$. The expected aggregate claims are $60(1,600)=\$96,000$.

The opening aggregate asset share is $1,200(1,100)=\$1,320,000$. After the 2% return, it is $1,320,000(1.02)=\$1,346,400$. After claims, the remaining fund is $1,346,400-96,000=\$1,250,400$.

The survivor asset share is $1,250,400/1,140=\$1,096.842105\ldots$, or $1,096.84 per survivor to the nearest cent. The margin against the $1,100 guarantee is $1,096.842105\ldots-1,100=-\$3.157895\ldots$. Therefore, the shortfall is $3.16 per survivor.

As a second check, calculate on a per-starting-policy basis before dividing by the survivor fraction: $1,100(1.02)-0.05(1,600)=1,122-80=\$1,042$. Then $1,042/0.95=\$1,096.842105\ldots$, agreeing with the aggregate calculation. The expected aggregate shortfall is approximately $1,140(3.157895)=\$3,600$, consistent with $1,250,400-1,140(1,100)=-\$3,600$.

Common mistake and correction: dividing the remaining fund by 1,200 gives $1,250,400/1,200=\$1,042$, but that is the remaining amount per original policy, not per survivor. The required survivor denominator is 1,140. Thus the correct comparison is a $3.16 shortfall per survivor, not a comparison of $1,042 directly with the survivor guarantee.

Comparisons at More Than One Duration

A policy may have different guaranteed amounts at different durations. Recalculate or carry forward the asset share to each comparison date, and use the guarantee payable at that date. A surplus at one duration does not establish an excess at every other duration: premiums, expenses, investment returns, claims, and the guaranteed amount may all differ over time.

Worked Example: Comparing Guarantees at Two Durations

A hypothetical policy has an opening asset share of $200. At the beginning of each year it receives a premium of $500 and incurs an expense of $50. The annual effective return is 2%, with no claims, lapses, or other cash flows. The guaranteed benefit is $1,150 at the end of year 2 and $1,700 at the end of year 3, if the policy remains in force. Compare the asset share with the guarantee at both durations.

Method. Project the asset share recursively using the net beginning-of-year cash flow, then compare each year-end amount with the guarantee for that exact duration.

The net cash flow at the beginning of each year is $500-50=\$450$. At the end of year 1, $A_1=(200+450)(1.02)=650(1.02)=\$663$.

At the end of year 2, $A_2=(663+450)(1.02)=1,113(1.02)=\$1,135.26$. The margin against the year-2 guarantee is $1,135.26-1,150=-\$14.74$, a shortfall of $14.74.

At the end of year 3, $A_3=(1,135.26+450)(1.02)=1,585.26(1.02)=\$1,616.9652$, or $1,616.97 to the nearest cent. The margin against the year-3 guarantee is $1,616.9652-1,700=-\$83.0348$, a shortfall of $83.03.

The year-3 shortfall is larger even though the asset share has grown, because the guarantee has also increased. Each comparison uses the benefit attached to that duration. As a check, both projected asset shares are below the corresponding guarantees, so both margins are negative.

Common mistake avoided: carrying the year-2 shortfall forward as if it were the year-3 comparison ignores both the year-3 cash flows and the changed guaranteed amount.

Interpreting the Margin

The comparison margin is useful because it puts the accumulated assets and the guaranteed amount in a common unit, usually dollars at the specified duration. At a per-policy level, it shows the difference for the selected policy basis. At a cohort level, it indicates the aggregate difference for the policies entitled to the benefits included in the calculation. State the basis alongside the number: for example, “$3.16 shortfall per expected survivor at the end of the year,” rather than simply “$3.16 shortfall.”

A positive margin indicates that the calculated asset share exceeds the stated guarantee on the assumptions used. A negative margin indicates that it falls below the guarantee. Neither sign alone proves that a company is solvent or insolvent. The asset share may use a particular allocation of premiums, expenses, taxes, reinsurance, or investment return, while a formal liability or solvency measure may use different assumptions and definitions. The comparison is a funding indicator for the specified basis, not a substitute for those measures.

Fix the comparison date and benefit.
Identify the exact duration and the guaranteed amount payable then, including any condition such as survival or remaining in force.

Project the asset share with stated timing.
Accumulate premiums and opening assets, deduct expenses and claims at their specified times, and account for the relevant population changes.

Align the basis.
Compare per-policy amounts with per-policy amounts, or aggregate assets with the aggregate benefits due to the applicable policies.

Calculate and interpret the margin.
Subtract the guarantee from the asset share; report a positive result as an excess and a negative result as a shortfall.

A frequent exam trap is to use a mathematically correct amount on the wrong basis: for instance, dividing a remaining cohort fund by the initial cohort size when the question asks for the amount per survivor. Other common errors are applying investment return to cash flows at the wrong time and comparing an asset share at one duration with a guarantee payable at another. Writing the timing, population, and guarantee explicitly before calculating helps prevent all three.

Key takeaway: A sound comparison requires the asset share and guaranteed benefit to refer to the same duration, the same eligible policies, and the same per-policy or aggregate basis. Report the difference with its sign and basis, and interpret it only under the assumptions used to produce the asset share.

Check Your Understanding

For each problem, calculate the asset share on the stated basis and compare it with the guarantee. Amounts are hypothetical.

A policy has an opening asset share of $800. At the beginning of the year, a net premium after expenses of $300 is invested for a year at an annual effective return of 5%. The year-end guaranteed benefit is $1,200. (Answer: asset share $1,155.00; shortfall $45.00.)
A policy has an opening asset share of $1,000. At the beginning of the year, a net cash inflow of $250 is received and invested for the year at 4%. The year-end guarantee is $1,275. (Answer: asset share $1,300.00; excess $25.00.)
A cohort of 500 policies has an opening asset share of $900 each. The annual effective return is 3%; 4% are expected to die and each death claim of $1,500 is paid at year-end. There are no other cash flows. Compare the expected survivor asset share with a $920 guarantee per survivor. (Answer: 480 expected survivors; aggregate claims $30,000; survivor asset share $903.13; shortfall $16.88 per survivor.)
A policy starts with an asset share of $200. At the beginning of each of two years, it receives a net cash inflow of $450, invested at an annual effective return of 2%. The year-2 guaranteed benefit is $1,100. (Answer: year-1 asset share $663.00; year-2 asset share $1,135.26; excess $35.26.)
A cohort of 800 policies has an opening asset share of $1,250 each. The annual effective return is 4%; 2% are expected to die, each generating a year-end claim of $3,000. There are no other cash flows. Compare the expected survivor asset share with a $1,280 guarantee per survivor. (Answer: 784 expected survivors; aggregate claims $48,000; survivor asset share $1,265.31; shortfall $14.69 per survivor.)

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