Asset Shares Versus Prospective Reserves

Two Values at the Same Valuation Date

An asset share and a prospective reserve answer different questions. An asset share looks backward: what funds have accumulated from past premiums and other cash flows, after allowing for claims, expenses, and investment returns? A prospective reserve looks forward: what is the present value of future benefits and expenses, less the present value of future premiums? Comparing them is useful only when both refer to the same policy status, valuation time, and cash-flow conventions.

For example, a premium due at the valuation date may already be included in an asset share calculated immediately after the premium. It must also be treated as paid when calculating the reserve at that same instant. Comparing an after-premium asset share with a before-premium reserve can create an apparent difference that is only a timing mismatch.

Definition: The asset share at time $t$, denoted $AS_t$, is the accumulated value per relevant in-force policy of the cash flows assigned to that policy or cohort from issue to time $t$. These cash flows can include premiums, investment income, expenses, taxes, reinsurance cash flows, claims, and other specified outgo. The prospective reserve at time $t$, denoted $V_t$, is the expected present value at $t$ of future benefits and expenses less future premiums, conditional on the policy being in force at $t$, under a stated valuation basis.

Neither quantity is fully specified by its name alone. An asset share depends on the cash flows allocated and the experience or projection basis used to accumulate them. A reserve depends on the future cash flows included and the assumptions used to value them. A meaningful comparison therefore states the valuation instant, whether amounts are before or after premiums due at that instant, and the assumptions for interest, mortality, expenses, and other decrements.

First Align the Timing

The examples below use annual periods. Unless stated otherwise, premiums are paid at the start of a policy year, benefits are paid at the end of the year in which the insured dies, and valuation is immediately after any premium due at that time has been paid. Let $i$ be the annual effective interest rate and $v=1/(1+i)$. When a reserve is calculated just before a premium due at the valuation time, that premium is a future premium from that instant; just after payment, it is not.

Worked Example: Matching Values for a Pure Endowment

A hypothetical two-year savings contract has no mortality, expenses, or other decrements. The insurer receives premiums of $100 at times 0 and 1, each immediately after the valuation instant at that time. The maturity benefit is $215.25 at time 2. The annual effective interest rate is 5%. Calculate the asset share and prospective reserve at time 1 immediately after the second premium, and check whether they agree.

Method. Accumulate past premiums to find the asset share. Discount future benefits, with no future premiums remaining, to find the reserve. The assumptions and timing are identical for both values.

The asset share immediately after the time-0 premium is $100. By time 1, it has earned 5% interest:

$$ AS_1=100(1.05)+100=105+100=205. $$

At time 1 immediately after the premium, the only future cash flow is the $215.25 maturity benefit one year later. Thus:

V_1=215.25v=\frac{215.25}{1.05}=205.

Therefore $AS_1=V_1=205$. As a check on the date convention, immediately before the time-1 premium the asset share is $105. The reserve at that instant is also $105, since the premium of $100 is then still due: $215.25/1.05-100=105$. Both measures agree at either instant when treated consistently.

Why the Recurrences Can Agree

The relationship can be seen in a one-year mortality model. Suppose $AS_t$ is measured just after the premium at time $t$. Over the following year, the fund earns interest at rate $i$. A proportion $q_t$ of the cohort is expected to die and receive benefit $B_t$ at year-end; the remaining proportion is $p_t=1-q_t$. At time $t+1$, surviving policies pay premium $P_{t+1}$, and the asset share is measured just after that premium. Assume there are no other cash flows in this derivation.

For each policy in the starting cohort, the fund before the year-end claims is $AS_t(1+i)$. Expected claims per starting policy are $q_tB_t$. The remaining fund is shared among the expected survivors, whose proportion is $p_t$; consequently the amount per survivor before the next premium is $[AS_t(1+i)-q_tB_t]/p_t$. Adding the next premium gives the cohort asset-share recurrence:

AS_{t+1}=\frac{AS_t(1+i)-q_tB_t}{p_t}+P_{t+1}.

Rearranging this equation gives:

AS_t=v\left[q_tB_t+p_t(AS_{t+1}-P_{t+1})\right].

The same expression follows from prospective valuation. At time $t$, the year-end benefit is paid with probability $q_t$. With probability $p_t$, the policy survives; at time $t+1$, its reserve after the premium is $AS_{t+1}$ in the equality case, while the premium $P_{t+1}$ is received at that time. Discounting these expected future net amounts one year gives $V_t=v[q_tB_t+p_t(V_{t+1}-P_{t+1})]$. Thus, if $AS_{t+1}=V_{t+1}$, the recurrences imply $AS_t=V_t$. Applying the same argument backward across the term establishes equality throughout when the terminal values and assumptions also agree.

This is a conditional statement, not a claim that every asset share must equal every reserve. It relies on matching cash flows, assumptions, and timing. Additional expenses, lapse payments, taxes, reinsurance, or different investment and mortality assumptions must be included consistently before applying the comparison.

Worked Example: Mortality Pooling With a Two-Year Term Benefit

A hypothetical two-year term-insurance cohort has 1,000 policies in force at time 0. Each policy pays a premium of $10 at the start of each year it is in force. The death benefit is $100, payable at the end of the year of death. In each year, the conditional death probability is $q=0.10$ and the survival probability is $p=0.90$. The annual effective interest rate is 0%; there are no expenses or other decrements. Use expected cohort counts. Compare the survivor asset share and prospective reserve at time 1, first immediately before and then immediately after the premium due then.

Method. Deduct expected claims from the cohort fund, divide by expected survivors to obtain the amount per survivor, and compare with the reserve at the same instant. This avoids treating a pooled cohort asset share as though every original policy survived.

At time 0, the cohort receives $1{,}000(10)=\$10{,}000$. During the first year, expected deaths are $1{,}000(0.10)=100$, so expected claims are $100(100)=\$10{,}000$. Expected survivors are $1{,}000(0.90)=900$. The remaining fund per survivor just before the time-1 premium is therefore:

AS_{1,\text{before}}=\frac{10{,}000-10{,}000}{900}=\$0.

At time 1 before the premium, each survivor owes $10 immediately and has one remaining year of coverage. Expected claims in that year are $0.10(100)=\$10$ per survivor. With zero interest, the prospective reserve before the premium is $10-10=\$0$. The two values agree at this instant.

After the premium is paid, the survivor asset share is $0+10=\$10$. The premium is no longer future income in the reserve calculation; the reserve after payment is the expected remaining claim, $0.10(100)=\$10$. They agree again. At the end of the second year, the term coverage has expired, so the reserve for any surviving policy is zero. The result checks at both time-1 instants and at the contract’s end.

When the Values Differ

A difference is informative only after checking that the comparison is valid. Common reasons for a genuine difference include using actual or projected experience to accumulate an asset share but a prudent valuation basis for the reserve, using different expense assumptions, or including cash flows in one measure but not the other. Differences may also arise because the asset share is an average for a cohort, while the reserve is calculated conditionally for an in-force policy under a specified set of assumptions.

The next example isolates the effect of using different interest assumptions. The contract has no mortality, so there is no need to interpret an expected survivor value.

Worked Example: Different Accumulation and Valuation Interest Rates

A hypothetical three-year savings contract receives premiums of $100 at times 0, 1, and 2. There are no expenses or decrements, and the maturity benefit is $330 at time 3. The asset share uses a 4% annual effective net investment return. The prospective reserve at time 2 uses a 3% annual effective valuation rate. Compare the asset share and reserve at time 2 immediately after the third premium. Round money to the nearest cent.

Method. Accumulate the premiums already received at the asset-share return. Separately discount the remaining benefit at the reserve valuation rate. The difference is expected because the two measures use different interest assumptions.

The time-0 premium accumulates for two years to $100(1.04)^2=\$108.16$. The time-1 premium accumulates for one year to $100(1.04)=\$104.00$. The time-2 premium has no time to earn interest at the valuation instant and remains $100. Thus:

AS_2=108.16+104.00+100.00=\$312.16.

There are no future premiums. The $330 benefit is due one year after time 2, so the reserve is:

V_2=\frac{330}{1.03}=\$320.388349\ldots=\$320.39.

The reserve exceeds the asset share by $320.39-312.16=\$8.23$. This is not an inconsistency: past premiums were accumulated at 4%, whereas the future benefit was discounted at 3%. If both calculations used the same rate and matching cash-flow assumptions, the comparison would be different. The arithmetic also checks directly: $312.16(1.03)=\$321.5248$, which is less than the $330 benefit, so $312.16 accumulated at 3% would not fund the stated maturity amount.

Common Timing and Basis Traps

A frequent error is to compare values on opposite sides of a premium date. In the first worked example, the time-1 asset share immediately after the $100 premium is $205. A wrong comparison takes the reserve just before that premium, $105, and reports a $100 difference. The correction is to compare like with like: the reserve immediately after the premium is $205, matching the after-premium asset share. Alternatively, compare the before-premium values, which are both $105.

Check policy status. A reserve for an in-force policy is not automatically comparable with an asset share averaged over a cohort that includes policies that have already terminated.
Check future premiums. Include premiums due after the valuation instant as future income in the prospective reserve; do not subtract a premium already paid.
Check mortality pooling. If claims are deducted from a cohort fund and the remainder is allocated to survivors, divide by the expected number of survivors when calculating an amount per survivor.
Check assumptions and scope. Record which expenses, decrements, taxes, investment returns, and benefits are included in each calculation. A difference may reflect those choices rather than an arithmetic error.

The comparison is useful because it places accumulated experience alongside the forward-looking value of contractual obligations. It does not, by itself, explain the source of a difference; that requires identifying the cash flows and assumptions responsible for it.

Key takeaway: Compare an asset share and a prospective reserve at the same instant, for the same policy status, with premiums and other cash flows treated consistently. Agreement is expected only when the cash-flow definitions and assumptions are aligned; a difference can be meaningful when they are not.

Check Your Understanding

For each problem, state the valuation timing before comparing or calculating the two values.

A hypothetical savings policy has premiums of $50 at times 0 and 1, no decrements or expenses, and a maturity benefit of $115.50 at time 2. The annual effective interest rate is 10%. Find the asset share and reserve at time 1 immediately after the premium. (Answer: $AS_1=V_1=\$105.00$.)
A one-year term cohort has 1,000 policies, each paying a $10 premium at time 0. The death benefit is $50 at year-end; $q=0.20$, $p=0.80$, and $i=0$. There are no expenses. Find the survivor asset share and reserve immediately before the premium due at time 1, assuming a second year of identical coverage and a $10 premium at time 1. (Answer: Both are $\$0.00$ before the time-1 premium.)
A hypothetical two-premium savings contract receives $80 at times 0 and 1. At time 1 immediately after the second premium, the asset-share accumulation basis is 5%. The benefit of $174 is payable at time 2, and the reserve valuation rate is 4%. Find both values and the amount by which the reserve exceeds the asset share. (Answer: $AS_1=\$164.00$, $V_1=\$167.31$, and the reserve exceeds the asset share by $\$3.31$.)
At time 1 immediately before a premium of $20 is due, a policy has an asset share of $95. It has no decrements or expenses, earns 0% interest, and pays a benefit of $120 at time 2 if in force. Find the prospective reserve at time 1 before the premium and compare it with the asset share. (Answer: $V_1=\$100.00$, which exceeds the asset share by $\$5.00$.)

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