Why a Reserve Pattern Tells a Story
A prospective reserve is the present value, at a specified duration, of future benefits minus the present value of future premiums, conditional on the insured being alive at that duration. Its value is therefore not just a number attached to a policy: it summarizes the insurer’s expected future obligation after allowing for the premiums still due. Reading how that value changes over time helps distinguish the reserve patterns of whole-life, term, and endowment insurance.
The patterns are shaped by two questions: what benefits remain payable, and what premiums remain collectible? A term contract has no insurance benefit after its coverage period. A whole-life contract continues to cover death for life. An endowment contract covers death during its term and also promises a maturity benefit if the insured survives. Those differences influence reserve development, but they do not by themselves guarantee a particular curve or monotonic pattern.
Unless stated otherwise, assume premiums are level and payable annually in advance while the insured is alive, benefits are paid at the end of the year of death, and there are no expenses, lapses, or other cash flows. A maturity benefit is paid at the end of the contract term if the insured survives. These are hypothetical contracts. The valuation assumptions need not describe actual experience.
The One-Year Change in Reserve
A reserve change is easiest to interpret by looking one policy year ahead. Let \(P\) be the annual premium, \(B\) the death benefit payable at the end of the year, \(q_{x+k}\) the probability of death during the year, and \(p_{x+k}=1-q_{x+k}\) the probability of surviving it. If the contract is still in force at the start of that year, the insurer has reserve \({}_kV_x\) and receives premium \(P\). Together they accumulate at rate \(i\). At year-end, the insurer either pays the death benefit or, if the insured survives, holds the next reserve.
This is a balance of expected year-end values. The left side accumulates the reserve and premium. On the right, the death payment occurs with probability \(q_{x+k}\), and the next reserve is needed with probability \(p_{x+k}\). Rearranging gives the reserve at the next duration:
This relation explains why a reserve does not simply rise by interest or fall by the premium. The premium and existing reserve accumulate, but expected claims are also paid during the year. The remaining survivors then need a reserve sufficient for their future contract obligations.
Include only payments that can still occur after the valuation duration, and use the contract’s benefit timing.
Include premiums payable while the insured is alive, beginning at the valuation time under the stated convention.
Subtract the expected present value of future premiums from that of future benefits. A larger reserve can reflect more costly remaining benefits, fewer premiums, or both.
Typical Patterns by Contract Type
| Contract | What remains after duration \(k\) | Typical interpretation |
|---|---|---|
| Whole life | Death coverage continues for life; premiums may continue while alive. | Reserves often build as the insured ages and future mortality costs rise relative to level premiums. |
| Term insurance | Death coverage remains only until the end of the term; no benefit is payable afterward. | A reserve can build during the term and later decline as the coverage period and remaining liability shrink. |
| Endowment insurance | Death coverage remains during the term, and a maturity benefit is due on survival to the end. | The maturity promise adds a future liability, so the reserve may build more strongly than for comparable term coverage. |
These are useful interpretations, not universal rules. The actual pattern depends on mortality, interest, benefits, and premium design. A change in mortality assumptions, for example, may change the relative cost of a future death benefit. The prospective definition, rather than a sketch of a “typical” curve, determines the answer for a specified contract.
Worked Example: A Whole-Life Reserve Under Changing Mortality
A hypothetical whole-life policy pays $100 at the end of the year of death. Annual premiums are payable in advance while the insured is alive. The annual effective interest rate is \(i=0\). Assume death probabilities in successive years are \(q_{x}=0.10\), \(q_{x+1}=0.20\), and \(q_{x+2}=1\). Find the net premium and the reserves at durations 1 and 2.
Method. First calculate the expected present value of benefits and premiums at issue. Then value the remaining benefits and premiums conditional on survival at each requested duration. With \(i=0\), discounting does not change the amounts.
The probabilities of being alive at the starts of years 1, 2, and 3 are \(1\), \(0.90\), and \(0.90(0.80)=0.72\). The probabilities of death in the three years are \(0.10\), \(0.90(0.20)=0.18\), and \(0.72(1)=0.72\). They sum to \(1.00\), so death is certain under this hypothetical mortality model.
The expected present value of the death benefit is therefore \(100(0.10+0.18+0.72)=\$100\). The expected number of premiums is \(1+0.90+0.72=2.62\), because a premium is paid at the start of each year only if the insured is alive then. Equivalence gives:
At duration 1, survival to that time is given. The remaining death benefit has expected value $100 because death is certain within the remaining two years. Conditional on being alive at duration 1, the expected number of future premiums is \(1+0.80=1.80\): one is due now, and the next is due only if the insured survives the next year. Thus:
At duration 2, death in the remaining year is certain. The benefit has value $100 and one premium of \(P\) is due now:
Rounded to the nearest cent, the net premium is $38.17, \({}_1V_x=\)$31.30, and \({}_2V_x=\)$61.83. The reserve rises across these durations in this example: the remaining death benefit is certain, while the number of remaining premiums falls. The one-year balance also checks the duration-1 value: \((0+P)-0.10(100)=38.1679-10=28.1679\) is the expected amount carried by the surviving 0.90 of policies, and \(28.1679/0.90=\$31.2977\). The calculation avoids treating the premium as payable after death.
Worked Example: A Two-Year Term Reserve Can Fall Near Expiry
A hypothetical two-year term policy pays $100 at the end of the year of death during its term. Premiums are payable annually in advance while the insured is alive. Let \(i=0.05\), \(q_x=0.10\), and \(q_{x+1}=0.20\). Calculate the net premium and the reserve at duration 1.
Method. Use the prospective values at issue to determine the level premium. Then, at duration 1, value only the coverage and premium remaining for the second year. Let \(v=1/(1.05)=20/21\). The probability of surviving the first year is \(0.90\).
The expected present value of the death benefit at issue is:
The expected present value of premiums per dollar of annual premium is \(1+0.90v=1+6/7=13/7\). Therefore:
At duration 1, the only remaining death benefit is $100 at the end of the second year if death occurs in that year. Its value is \(100(0.20)v=\$19.0476\ldots\). One premium is due at duration 1, so:
Rounded to the nearest cent, \(P=\)$13.92 and \({}_1V_x=\)$0.51. As a check, the one-year recursion from issue gives \((0+P)(1.05)=\$14.6154\ldots\); subtracting the expected first-year claim \(0.10(100)=\$10\) leaves \(\$4.6154\ldots\) for the 0.90 who survive. Dividing by \(0.90\) gives \(\$5.1282\ldots\), not the prospective value computed above. This discrepancy flags a timing error in using the recursion: its year-end surviving reserve is measured after the first policy year, but the issue reserve equation must include the first-year benefit and premium consistently. Directly, the value at duration 1 should be \(100(0.20)v-P\), which is \(\$5.13\): indeed \(40/21-3800/273=(520-3800)/273\) would be wrong because \(40/21=520/273\); recomputing the premium, \(100(38/147)=3800/147\), divided by \(13/7\) is \(3800/273\). Thus \(40/21-3800/273=(520-3800)/273\), negative. The correct benefit value is \(100(0.20)v=20(20/21)=400/21\), not \(40/21\). Hence:
The corrected reserve is $5.13. The recursion check agrees: \(\{(13.9194)(1.05)-10\}/0.90=\$5.1282\ldots\). The explicit check is important because a missing factor of the benefit amount can change the result by a factor of ten.
Worked Example: The Maturity Promise in an Endowment Reserve
A hypothetical two-year endowment policy pays $100 at the end of the year of death during the term, or $100 at the end of year 2 if the insured survives. Annual premiums are payable in advance while the insured is alive. Let \(i=0.05\), \(q_x=0.10\), and \(q_{x+1}=0.20\). Find the net premium and the reserve at duration 1.
Method. At issue, distinguish death in year 1 from payments in year 2. The latter includes either death in year 2 or survival to maturity. At duration 1, both possible outcomes in year 2 produce the same $100 payment, so their conditional probabilities sum to 1.
With \(v=20/21\), the expected present value of benefits at issue is:
The expected present value of premiums per dollar of annual premium is still \(1+0.90v=13/7\). Thus:
At duration 1, the end-of-year-2 payment is certain under the stated contract: it is a death benefit if death occurs during year 2, and otherwise a maturity benefit. Its present value is \(100v=2000/21=\$95.2381\ldots\). One premium is due at duration 1, giving:
Rounded to the nearest cent, \(P=\)$49.08 and \({}_1V_x=\)$46.15. The year-2 payment is not just a death benefit weighted by the death probability: omitting the survival maturity benefit would understate the liability. Conversely, adding a separate maturity payment to the full year-2 death payment would count mutually exclusive outcomes twice.
Interpreting the Differences
The whole-life example had a reserve that rose from duration 1 to duration 2 because fewer premiums remained to fund a death benefit that was still certain. For many level-premium whole-life models, increasing mortality at older ages contributes to a rising reserve: the premium is level, but the expected cost of near-term coverage tends to increase. This is an interpretation of common assumptions, not a theorem that every whole-life reserve must increase.
For term insurance, the reserve reflects only the remaining term coverage and premiums. Its pattern can build as future claim costs become more concentrated in the remaining period, then fall as the coverage period runs down. The term example also shows that “near the end” does not mean “large”: the actual prospective value depends on the remaining death probability, discounting, and premium due. An endowment reserve includes a further source of liability, the promised maturity payment on survival. That promise can materially change the reserve relative to term insurance with the same death benefit.
A useful exam habit is to write the remaining cash flows at each duration before calculating. Check whether a premium is due at the valuation time, whether the benefit is paid at the start or end of a period, and whether survival creates a maturity payment. The prospective reserve is always a conditional future-value comparison; a familiar pattern cannot replace that calculation. This treatment of reserve interpretation is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, state the remaining cash flows before calculating.
- A one-year hypothetical term policy has one year of coverage remaining, \(i=0\), death benefit $200, death probability 0.25, and a premium of $30 due now. Find the reserve immediately before that premium. (Answer: $20)
- A hypothetical whole-life policy has \(i=0\), death benefit $100, and remaining death probabilities of 0.20 in the next year and 1.00 in the following year. Premiums are $25 annually in advance while alive. Find the reserve now, immediately before the next premium. (Answer: $55)
- A two-year hypothetical endowment pays $500 on death during the term or on survival to maturity. Let \(i=0\), \(q_x=0.20\), and \(q_{x+1}=0.25\). Find the net annual premium and the reserve at duration 1. (Answer: Premium \(=\$277.78\); duration-1 reserve \(=\$222.22\), rounded to cents)
- For a hypothetical term policy, the reserve immediately before a premium is $40, the premium is $12, \(i=0.05\), the annual death probability is 0.10, and the death benefit is $100. Using the one-year recursion, find the reserve at the next duration conditional on survival. (Answer: $49.56)