Boundary Conditions at Issue, Expiry, and Maturity

Why Reserve Boundaries Matter

A reserve recursion relates a policy value at one duration to the value one year later. To use it correctly, the calculation needs a starting value, a terminal value, or both. These boundary values come from the contract’s cash flows and the exact time at which the reserve is defined. A zero at issue or expiry is not a universal rule: it is correct only under the appropriate contract and reserve convention.

This tutorial uses an annual, discrete model. Premiums are due at times $0,1,\ldots,n-1$, conditional on the policy being in force then. Death benefits are paid at the end of the year of death. Any maturity benefit is paid at time $n$, provided the insured survives to that time. Reserves are measured immediately before a premium due at that time; at maturity they are measured immediately before any maturity benefit is paid. This convention makes the issue and terminal boundaries explicit.

Definition: Under the convention used here, ${}_kV_x$ is the prospective net reserve at time $k$, conditional on the insured aged $x+k$ being alive and the policy being in force, immediately before any premium due at time $k$. It is the present value at time $k$ of future benefits minus the present value of future premiums, calculated on the stated interest and mortality basis. A premium due at time $k$ is included among those future premiums.

The Recursion and Its Boundaries

Suppose the annual premium at time $k$ is $P_k$, the death benefit paid at the end of year $k+1$ is $B_{k+1}$, and the policy has no other benefit during that year. Write $i$ for the effective annual interest rate, and let ${}_1q_{x+k}$ and ${}_1p_{x+k}$ be the one-year death and survival probabilities. They satisfy ${}_1p_{x+k}+{}_1q_{x+k}=1$. The one-year reserve recursion is

({}_kV_x+P_k)(1+i) = {}_1q_{x+k}B_{k+1} + {}_1p_{x+k}{}_{k+1}V_x.

The premium is added to the reserve because it is received at time $k$. The resulting amount accumulates at interest over the year. At the year’s end, the death outcome requires the death benefit; the survival outcome requires the next reserve. The next reserve is conditional on survival and is not multiplied by a probability inside its own definition, so the survival probability appears in the recursion.

Issue boundary under net premiums: If the net premium is determined by the equivalence principle using the same mortality, interest, benefit, and payment timing as the reserve, then ${}_0V_x=0$. At issue, the present value of future net benefits equals the present value of premiums, including the premium due immediately at time 0. This boundary need not hold for a gross reserve with expenses, a reserve using a different basis, or a convention that measures the reserve after the issue premium.

At the other end of a finite contract, the boundary is set by what remains payable at that time. For an $n$-year term policy with no survival benefit, there is no liability after the term expires, so ${}_nV_x=0$. For an $n$-year endowment with a maturity benefit $E$, the reserve immediately before the payment at time $n$ is ${}_nV_x=E$. Immediately after that benefit is paid, the policy has ended and the reserve is zero. These are different time points, not competing answers.

Contract boundary	Reserve value under the stated convention	Reason
Issue, net premium equivalence	${}_0V_x=0$	Present values of future benefits and premiums balance
Term expiry, no survival benefit	${}_nV_x=0$	No future benefit or premium remains
Endowment maturity, benefit $E$	${}_nV_x=E$	The maturity benefit is due at time $n$
Immediately after final benefit payment	Zero	The contract has terminated

Why the One-Year Relation Holds

The recursion can be justified directly from the prospective definition. At time $k$, the premium $P_k$ is a future inflow, so the reserve after allowing for that inflow is ${}_kV_x+P_k$. One year later, conditional on death during the year, the benefit $B_{k+1}$ is paid. Conditional on survival, the continuing policy has reserve ${}_{k+1}V_x$. Therefore the time-$k$ reserve, after including the current premium, is the discounted expected value of those year-end amounts:

{}_kV_x+P_k = v\left( {}_1q_{x+k}B_{k+1} + {}_1p_{x+k}{}_{k+1}V_x \right), \qquad v=\frac{1}{1+i}.

Multiplying both sides by $1+i$ gives the stated recursion. The argument assumes the annual premium is received at the beginning of the year, the death benefit is paid at its end, and the next reserve is measured at that same year-end time before any premium due there. If premiums or benefits are paid at different times, their values must be adjusted to match the actual contract timing.

Set the reserve convention.
State whether the value is before or after a premium or benefit at each boundary.

Translate the contract ending into a terminal value.
Use zero for a term contract with no survival benefit, or the amount payable for a maturity benefit immediately before payment.

Apply the recursion in the required direction.
Use the issue boundary to move forward, the terminal boundary to move backward, or both to check a proposed premium and reserve path.

Worked Examples

Worked Example: Net Premium and Issue-to-Expiry Check for Term Insurance

A hypothetical two-year term policy pays \$1,000 at the end of the year of death and nothing on survival to the end of the term. The effective annual interest rate is $i=0.10$, and the one-year mortality probabilities are ${}_1q_{x}= {}_1q_{x+1}=0.10$, with survival probabilities $0.90$ in both years. Level net premiums are due at times 0 and 1. Find the premium and the reserve at time 1. Check the issue and expiry boundaries.

Method. Work backward from the term-expiry value, then use the net premium issue condition ${}_0V_x=0$. The expiry boundary is ${}_2V_x=0$, since no benefit is payable on survival and no premiums are due after time 1.

For year 2, the recursion gives

({}_1V_x+P)(1.10) = 0.10(1{,}000)+0.90(0)=100, \qquad {}_1V_x=\frac{100}{1.10}-P.

For year 1, use the issue boundary:

(0+P)(1.10) = 0.10(1{,}000)+0.90{}_1V_x.

Substitute ${}_1V_x=100/1.10-P$:

1.10P = 100+0.90\left(\frac{100}{1.10}-P\right) = 181.81818\ldots-0.90P,

so $2P=181.81818\ldots$, and $P=90.90909\ldots$. Rounded to the nearest cent, the annual net premium is \$90.91. Then

{}_1V_x = \frac{100}{1.10}-90.90909\ldots=0.

Check. The forward recursion from ${}_0V_x=0$ gives ${}_1V_x=0$, and the next year gives ${}_2V_x=0$. Both boundary values are satisfied. The common mistake avoided here is setting the expiry reserve to the death benefit: that benefit is paid only if death occurs during the final year, not if the insured survives to expiry.

Worked Example: Terminal Value for a One-Year Endowment

A hypothetical one-year endowment pays \$200 at the end of the year if death occurs during the year, or \$50 at time 1 if the insured survives. The effective annual interest rate is $i=0.10$, with ${}_1q_x=0.20$ and ${}_1p_x=0.80$. A single net premium is due at time 0. Find the premium and state the reserve immediately before and immediately after the maturity payment.

Method. The terminal reserve is the maturity benefit immediately before it is paid: ${}_1V_x=50$. At issue, net premium equivalence gives ${}_0V_x=0$. Apply the recursion for the one policy year:

$$ (0+P)(1.10) = 0.20(200)+0.80(50) = 40+40=80. $$

Thus $P=80/1.10=72.72727\ldots$, or \$72.73 to the nearest cent. The terminal reserve immediately before payment is ${}_1V_x=\$50$; immediately after the payment, the contract has ended and the reserve is zero.

Check. The expected end-of-year benefit is \$80, and its present value is $80/1.10=\$72.72727\ldots$, equal to the premium. The common mistake avoided is assigning zero at time 1 merely because the contract ends then; the maturity benefit is still due immediately before termination.

Worked Example: Forward and Backward Validation of a Two-Year Term Reserve

A hypothetical two-year term policy has level annual premiums of \$27.50 due at times 0 and 1. The effective annual interest rate is $i=0.25$. In each year, the death probability is $0.25$ and the survival probability is $0.75$. The death benefit is \$100 in year 1 and \$200 in year 2; there is no survival benefit. Verify the reserve path using both boundaries and both directions of recursion.

The issue and expiry boundaries are ${}_0V_x=0$ under net premium equivalence and ${}_2V_x=0$ for term insurance without a maturity benefit. First move forward from issue. For year 1, solve the recursion for ${}_1V_x$:

{}_1V_x = \frac{(0+27.50)(1.25)-0.25(100)}{0.75} = \frac{34.375-25}{0.75} = 12.50.

For year 2:

{}_2V_x = \frac{(12.50+27.50)(1.25)-0.25(200)}{0.75} = \frac{50-50}{0.75} = 0.

Now check backward from expiry. The year-2 recursion gives

{}_1V_x = \frac{0.25(200)+0.75(0)}{1.25}-27.50 = 40-27.50 = 12.50.

Then the year-1 recursion gives

{}_0V_x = \frac{0.25(100)+0.75(12.50)}{1.25}-27.50 = \frac{34.375}{1.25}-27.50 = 0.

Check. Forward and backward calculations agree at both intermediate and boundary durations. The common mistake avoided is applying the year-1 benefit to the year-2 recursion; each recursion uses the benefit payable at the end of its own policy year.

Interpreting Boundary Conditions Carefully

A boundary is a statement about a specific time and a specific cash-flow convention. In particular, “the reserve at maturity is zero” is incomplete unless it says whether the value is taken before or after the maturity benefit. For an endowment, the reserve just before payment includes the amount due; the reserve after payment is zero. The same distinction applies to any contract with a benefit payable at the final boundary.

The issue value also depends on the premium basis. Under net premium equivalence, a level premium balances the present value of benefits and premiums at issue, so the reserve immediately before the issue premium is zero. If expenses are included, or if the premium was not set using the same assumptions as the reserve, the issue value need not be zero. Do not force a boundary condition merely because it is common in a simpler model.

A reliable reserve calculation should therefore state the policy duration, cash-flow timing, mortality and interest assumptions, and whether a boundary value is measured before or after a payment. Once those are fixed, the terminal value and recursion provide a strong consistency check: a proposed premium and reserve path must satisfy the recursion at every year and the relevant boundary values.

Key takeaway: For a finite policy, assign the issue value from the premium principle and the terminal value from the benefit still due at that instant. A term policy with no survival benefit ends with zero reserve; an endowment has a reserve equal to its maturity benefit immediately before payment. Use the recursion to verify that these boundaries are consistent throughout the contract.

Check Your Understanding

For each problem, use the reserve convention and annual recursion stated above. Monetary answers are rounded to the nearest cent where needed.

A one-year endowment has $i=0.05$, ${}_1q_x=0.10$, and ${}_1p_x=0.90$. It pays \$500 on death during the year and \$100 on survival at time 1. Find the single net premium and state the terminal reserve immediately before payment. (Answer: Premium \$133.33; terminal reserve \$100.00 immediately before payment.)
A two-year term policy pays \$100 at the end of either year of death and nothing on survival to expiry. The effective annual interest rate is $i=0$, and each year has death probability $0.20$ and survival probability $0.80$. Find the level net premium and the time-1 reserve. (Answer: Premium \$20.00; ${}_1V_x=\$0.00$.)
A two-year term policy has $i=0.25$, annual death probability $0.25$, and survival probability $0.75$ in each year. Its death benefits are \$80 in year 1 and \$160 in year 2. Find the level net premium and ${}_1V_x$. (Answer: Premium \$22.00; ${}_1V_x=\$10.00$.)
A three-year endowment has $i=0.10$, annual death probability $0.10$, and survival probability $0.90$ in each year. It pays \$100 at the end of a year of death and \$300 at time 3 on survival. Level premiums of \$100 are due at times 0, 1, and 2. Find ${}_2V_x$. (Answer: ${}_2V_x=\$154.55$.)
A five-year policy has $i=0.05$, ${}_1q_{x+4}=0.10$, and ${}_1p_{x+4}=0.90$. Its final-year death benefit is \$1,000, and its premium at time 4 is \$80. Calculate ${}_4V_x$ first for a term policy with no survival benefit, then for an endowment with a \$1,000 maturity benefit. State the respective values of ${}_5V_x$. (Answer: Term: ${}_5V_x=0$, ${}_4V_x=\$15.24$. Endowment: ${}_5V_x=\$1,000$ immediately before payment, ${}_4V_x=\$872.38$.)

Contract boundary	Reserve value under the stated convention	Reason
Issue, net premium equivalence	\({}_0V_x=0\)	Present values of future benefits and premiums balance
Term expiry, no survival benefit	\({}_nV_x=0\)	No future benefit or premium remains
Endowment maturity, benefit \(E\)	\({}_nV_x=E\)	The maturity benefit is due at time \(n\)
Immediately after final benefit payment	Zero	The contract has terminated

What You'll Learn