Why Boundary Conditions Need a Timing Convention
A reserve boundary condition is a value assigned at a contract’s starting point or at an endpoint such as maturity or termination. The amount depends not only on the benefit named in the contract, but also on which cash flows have already occurred at the instant the reserve is measured. A premium due at issue, a maturity benefit, and an expense paid at termination can each change the answer.
The discussion below uses annual discrete cash flows. Unless stated otherwise, premiums are received at the start of a policy year, death benefits and death-related expenses are paid at the end of the year of death, and maturity benefits are paid at the maturity date. All calculations use the stated modified valuation basis, not necessarily the assumptions used to price the contract.
For a valuation basis with effective annual interest rate \(i\), future cash flows are discounted using \(v=1/(1+i)\). Probabilities in a prospective reserve are conditional on the policy being in force at the valuation time. Thus, for example, a benefit payable at the end of the next year is weighted by the probability of the relevant event during that year.
Issue: Zero Is a Result, Not a Default
At issue, the reserve is found by valuing the contract’s future cash flows on the modified basis. If the initial premium is due at issue and the reserve is measured just before that premium, it is included as a future premium and subtracted. Any issue expense payable at the same instant is included as a future expense and added. These cash flows must not be omitted merely because they occur at time zero.
A zero issue reserve follows only if the present values of future outgo and premium income are equal on the modified valuation basis, with consistent timing and included cash flows. A premium established on a different pricing basis does not generally produce that equality. If the initial premium and issue expense are exchanged at issue, then the reserve just after those cash flows is \(V_0^+=V_0^-+P_0-e_0\), where \(P_0\) is the premium received and \(e_0\) is the expense paid.
Worked Example: Issue Reserve With a Modified Expense Basis
A hypothetical two-year endowment contract pays $1,000 at the end of the year of death, if death occurs during the term, or $1,000 at the end of year 2 if the insured survives to maturity. Annual premiums of $500 are due at issue and at time 1 if the policy is then in force. On the modified valuation basis, \(i=10\%\), the first-year mortality probability is \(q_0=0.10\), and the conditional second-year mortality probability is \(q_1=0.10\). There is an issue expense of $100 and a renewal expense of $20 at time 1 if the policy is in force. Find the issue reserve immediately before the issue premium and expense.
Method. Calculate the prospective value of benefits, expenses, and premiums separately, weighting each future cash flow by the probability of being payable and discounting it at 10%. The issue premium and issue expense are both time-zero cash flows.
The probability of death in year 1 is \(0.10\), so the year-1 death-benefit value is
The probability of surviving the first year is \(0.90\). Conditional on that survival, the policy pays $1,000 at time 2 whether death occurs in year 2 or the insured survives to maturity. The unconditional probability of being in force at time 2 is \(0.90\), so the time-2 benefit value is
Using the unrounded components, the total benefit value is
The issue expense is $100 at time 0. The renewal expense is payable at time 1 only if the policy survives the first year, giving total expense value
The issue premium is $500 at time 0. The time-1 premium is received only if the policy survives the first year. Thus the total premium value is
The issue reserve immediately before the time-zero cash flows is therefore
As a timing check, immediately after receiving the $500 premium and paying the $100 issue expense, the reserve is \(41.9834711+500-100=441.9834711\), or $441.98. This equals the present value of the remaining benefits and renewal expense less the remaining time-1 premium. The issue reserve is not zero because the stated premium does not exactly balance the future cash flows on this modified basis.
Maturity and Other Contract Endpoints
At maturity, first specify whether the reserve is measured just before or just after the final payment. Immediately before a maturity benefit is paid, that benefit is an immediate future outgo. If a maturity expense is also payable, it belongs in the same boundary value. Immediately after all final cash flows, no contractual future cash flows remain, so the reserve for the terminated contract is zero.
when those are the only cash flows at maturity and there are no simultaneous premiums or other receipts. The formula is conditional on the policy reaching maturity. A policy that instead terminates earlier through death or surrender has a different endpoint cash flow.
Worked Example: Maturity and Death Boundaries on a Modified Basis
A hypothetical two-year contract has a $500 death benefit payable at the end of the year of death and a $800 maturity benefit payable at time 2 if the insured survives. A death expense of $5 is paid with the death benefit, and a maturity expense of $10 is paid with the maturity benefit. On the modified basis, \(i=5\%\), and the conditional probability of death during year 2 is \(q_1=0.25\). At time 1, an annual premium of $200 is due and a renewal expense of $8 is payable. Find the endpoint values at time 2 and the reserve at time 1 immediately before the premium and renewal expense.
Method. Set each endpoint value equal to the cash outgo due at that endpoint. Then use those endpoint amounts to calculate the prospective reserve at time 1, conditional on the policy being in force at that time.
If death occurs at time 2, the immediate outgo is the $500 death benefit plus the $5 death expense. Thus \(V_2^-=500+5=\$505\) on the death-termination branch. Once that settlement is made, the contract has ended and \(V_2^+=0\).
If the insured survives to maturity, the immediate outgo is the $800 maturity benefit plus the $10 maturity expense. Thus \(V_2^-=\$810\) on the survival-to-maturity branch, and \(V_2^+=0\) after payment. These are alternative endpoints, not amounts payable together.
At time 1, conditional on the policy being in force, the expected time-2 outgo is
Discount this amount one year at 5%, then include the time-1 premium and expense, since the reserve is measured just before both cash flows. The reserve is
Immediately after receiving the premium and paying the renewal expense, the reserve is \(506.8095238+200-8=\$698.8095238\), exactly the discounted value of the expected time-2 outgo. The boundary values and the time-1 calculation are consistent with the cash-flow timing.
Termination: Separate the Settlement From the Active-Policy Reserve
“Termination” can refer to different events, including death, surrender, or another contractually specified exit. At the termination instant, identify the cash flow that settles the contract and state whether the reserve is measured just before or just after settlement. If the contract has no remaining liability after that settlement, the reserve for the terminated contract is zero just after payment.
An important qualification is that an active-policy prospective reserve is not automatically equal to a surrender benefit. The active-policy reserve values the future cash flows of a policy that remains in force under the valuation model. A surrender payment is a cash flow on the surrender branch. Equating the active-policy reserve to the surrender amount without a stated rule or definition confuses two different quantities.
Worked Example: A Surrender Settlement With a Termination Expense
A hypothetical modified-basis contract is surrendered at time 3. The contract pays a surrender benefit of $300 at that instant, and the insurer incurs a $15 termination expense at the same time. No premium is due at time 3, and no benefits, premiums, or expenses arise after surrender. Find the reserve immediately before settlement and immediately after settlement.
Method. At the termination boundary, list the cash flows still to occur from the valuation instant. Because there are no future contract cash flows after settlement, only the immediate surrender benefit and expense enter the pre-settlement liability.
Immediately before settlement, the insurer’s outgo is \(300+15=\$315\). There are no future premiums to subtract and no later cash flows to value, so the reserve for the terminating contract is \(V_3^-=\$315\). Immediately after paying the surrender benefit and incurring the expense, the contract has ended and \(V_3^+=\$0\).
The check is direct: the decrease from $315 just before settlement to zero just after settlement equals the $315 total outgo. The $300 surrender benefit alone would understate the pre-settlement amount by the $15 termination expense. This boundary calculation does not assert that the reserve for a policy continuing in force immediately before time 3 is necessarily $315.
Why the Boundary Formula Holds
A prospective reserve is the discounted value of cash flows that remain after the valuation instant. The endpoint result follows directly by applying that definition at a contract endpoint: after the final settlement, there are no future benefits, expenses, or premiums for the terminated contract. Just before settlement, the settlement cash flow has not yet occurred and must be included. This gives a general timing check.
Include every benefit, expense, premium, or receipt payable at the boundary instant, and identify which party pays or receives it.
For a reserve immediately before the cash flows, include them in the prospective value. For a reserve immediately after them, remove them from future cash flows.
If the contract has ended and no further amounts are due, the post-settlement reserve is zero. If the contract continues, value its remaining cash flows instead.
Equivalently, when the only cash flows at a terminal time \(T\) are an immediate benefit \(B_T\) and expense \(e_T\), with no premium or other receipt, the prospective value immediately before settlement is \(B_T+e_T\). After settlement, there is no remaining liability. If a premium or other inflow is also due at \(T\), it must be included with its correct sign rather than silently omitted.
Common Boundary-Condition Traps
- Assuming every issue reserve is zero. Zero requires equivalence on the modified basis, including the correct premium timing and expenses. A premium set on another basis need not satisfy that condition.
- Mixing before- and after-payment values. A reserve just before maturity includes the final payment; a reserve just after it does not.
- Using a survival probability twice. A boundary value at maturity is conditional on reaching maturity. Use survival probabilities when valuing that endpoint from an earlier date, not again in the conditional boundary amount.
- Omitting endpoint expenses. If an expense is payable with a death, maturity, or surrender settlement, it is part of the relevant cash flow.
- Setting the active reserve equal to surrender value by default. A surrender benefit is the cash flow on a termination branch; it does not automatically determine the value of a policy that remains active.
Check Your Understanding
For each problem, state the requested boundary value and use the stated cash-flow timing.
- A hypothetical one-year contract pays $1,000 with certainty at time 1. At issue, the modified basis has \(i=5\%\), an issue expense of $30, and a premium of $900 received at time 0. Find the reserve immediately before the issue cash flows. (Answer: $82.38)
- A contract pays a $650 maturity benefit and incurs a $12 maturity expense at time \(T\), with no other cash flows at that instant. What are the reserve immediately before settlement and immediately after settlement? (Answer: $662 before settlement; $0 after settlement)
- A policy is surrendered at time 4 for $240, and a $10 termination expense is payable at the same time. No premium is due then and no cash flows remain after surrender. Find the reserve immediately before and after settlement. (Answer: $250 before settlement; $0 after settlement)
- At time 1, a policy is in force and has one year remaining. On the modified basis, \(i=10\%\); the conditional probability of death during the year is \(0.20\), with a $400 death benefit, and survival leads to a $900 maturity benefit. A $100 premium is due at time 1. There are no expenses. Find the reserve immediately before the premium. (Answer: $627.27)
- A one-year contract pays $700 at maturity if the insured survives and $300 at the end of the year if death occurs during the year. At time 1, the insured is alive and maturity is reached, so the $700 benefit and an $8 maturity expense are payable. What are the reserves immediately before and after these final cash flows? (Answer: $708 before settlement; $0 after settlement)