Prospective Reserves in Annual Profit Tests

Introduction

In an annual profit test, a reserve at a policy-year boundary summarizes the expected value, at that boundary, of the contract’s future cash flows. It is not the profit earned during the year just completed. Instead, it looks forward: future benefits and expenses increase the insurer’s expected outgo, while future premiums reduce it.

A correct prospective reserve depends on more than the benefit amount and interest rate. The calculation must also specify exactly when the reserve is measured relative to cash flows due at the boundary, and which future outcomes remain possible given that the policy is in force. The examples below use a consistent before-payment convention: include cash flows due at the valuation boundary in the reserve.

Definition: The prospective reserve at time $t$, conditional on the policy being in force immediately before time-$t$ cash flows, is the expected present value at time $t$ of future benefits and expenses less future premiums. A positive reserve means expected future outgo exceeds expected future premium income. A negative reserve means the reverse.

Cash-Flow Sign and Timing

Let $B_s$ denote benefits paid at time $s$, $E_s$ expenses paid at time $s$, and $P_s$ premiums received at time $s$. From the insurer’s perspective, define net outgo at time $s$ as $B_s+E_s-P_s$. Under the before-payment convention, cash flows at time $t$ have discount factor 1, while a cash flow at time $t+j$ is discounted by $v^j$, where $v=1/(1+i)$.

The amounts in this expression may depend on future events. For example, a death benefit is paid only on death, and a renewal premium may be received only if the policyholder is still in force when it falls due. Therefore, future amounts must be averaged over the possible outcomes conditional on the policy being in force at time $t$.

Theorem: For a fixed annual effective interest rate $i$ and a finite projection ending at time $n$, the prospective reserve under the before-payment convention is

{}_tV = \operatorname{E}\!\left[ \sum_{j=0}^{n-t}v^j\bigl(B_{t+j}+E_{t+j}-P_{t+j}\bigr) \;\middle|\; \text{policy in force immediately before time }t \right].

Here cash flows not due in a particular outcome are zero. If no benefits, expenses, or other obligations remain after time $n$, the terminal reserve is zero. The formula assumes that the stated interest basis applies to all discounted cash flows; if interest rates vary by year, each cash flow must instead use the appropriate accumulated discount factors.

If cash flows are known at time $t$, they appear in the expectation without uncertainty and are not discounted. For a death benefit paid at the end of a year, the relevant probability is the probability of death during that year conditional on being alive and in force at its start. When considering a later year, the probability of reaching that year must also be included. This conditioning is what prevents an unconditional death probability from being applied to a policy already known to be in force at the valuation date.

Why the Prospective Formula Holds

The formula follows from valuing each future cash flow at the reserve date and then taking its conditional expected value. For one future payment $C$ due at time $t+j$, its value at time $t$, if it occurs, is $v^jC$. If it occurs only in certain outcomes, its expected value is $v^j\operatorname{E}[C\mid\text{in force at }t]$. Adding these values over all future dates gives the expected present value of total outgo.

Because expectation is additive for a finite sum, the expected present value of benefits plus expenses less premiums is the expected present value of each benefit and expense minus the expected present value of premiums. This yields the reserve formula. The argument does not require independence between future cash flows; it requires that each expectation correctly reflect the conditional probabilities and cash-flow rules of the contract.

Fix the boundary convention.
Decide whether time-$t$ premiums and expenses are included. Here, the reserve is measured before those payments, so include them.

List cash flows by date and outcome.
Record benefits, expenses, and premiums separately, and specify whether each is certain or conditional on death, survival, or remaining in force.

Apply conditional probabilities.
Use probabilities conditional on the policy being in force at the reserve date; for later years, include the probability of surviving or remaining in force until then.

Discount and reconcile signs.
Discount each expected net outgo to time $t$, add the amounts, and check that premiums have negative signs and benefits and expenses have positive signs.

Worked Calculations at Policy-Year Boundaries

Worked Example: A Final-Year Term Benefit

A hypothetical two-year term policy is valued at time 1, immediately before the premium and maintenance expense due then. The policy is in force at time 1. A premium of $150 and expense of $20 are due at time 1. During the final year, the conditional probability of death is $0.04$, and a death benefit of $5,000 is paid at time 2 if death occurs. There are no further premiums or expenses. The effective annual interest rate is $4\%$. Find the reserve at time 1.

Method. List the time-1 net outgo first, then add the discounted expected death benefit. The boundary cash flow is an expense less a premium:

$$ E_1-P_1=20-150=-130. $$

The expected death benefit at time 2, conditional on being in force at time 1, is $0.04(5{,}000)=200$. Its present value at time 1 is:

v(200)=\frac{200}{1.04}=192.3077.

Therefore,

{}_1V=-130+192.3077=62.3077.

Rounded to cents, the reserve is $62.31. The positive amount is consistent with expected discounted future benefit outgo exceeding the net premium income due at the boundary. The common mistake is to omit the time-1 expense or premium, or to use $150-20$ as the insurer’s net outgo; the correct boundary amount is $20-150=-130$.

Worked Example: A Benefit Certain at the End of the Remaining Term

A hypothetical two-year endowment policy is valued at time 1 before payments due at that time. A premium of $800 and maintenance expense of $40 are due at time 1. At time 2, the policy pays $1,200 if the insured dies during year 2 or survives to maturity. These outcomes exhaust the possibilities for a policy in force at time 1. The effective annual interest rate is $5\%$. Find the reserve at time 1.

Method. First evaluate the time-1 cash flow: expense less premium is $40-800=-760$. The time-2 payment is certain conditional on being in force at time 1: either death in year 2 triggers the payment, or survival to maturity triggers it. Thus its expected amount at time 2 is $1,200, not a probability-weighted fraction of $1,200.

{}_1V=(40-800)+\frac{1{,}200}{1.05} =-760+1{,}142.8571 =382.8571.

Rounded to cents, the reserve is $382.86. The common mistake is to multiply the maturity payment by the survival probability and omit the death outcome. The payment is certain because death in year 2 and survival to maturity are mutually exclusive and exhaustive outcomes, and both result in the same amount at time 2.

Worked Example: Survival-Weighted Premiums and Benefits

A hypothetical three-year term policy is valued at time 1, just before the premium and expense for the year beginning then. A premium of $500 and expense of $50 are due at the start of each remaining year while the policy is in force. A $2,000 death benefit is paid at the end of year 2 if death occurs in that year, and a $3,000 benefit is paid at the end of year 3 if death occurs in that year. The conditional death probabilities are $0.10$ in year 2 and $0.20$ in year 3, given survival to the start of the corresponding year. The effective annual interest rate is $0\%$. Find the reserve at time 1.

Method. The net outgo at each premium date is $50-500=-450$. At time 1, this amount is due with certainty. At time 2, it is due only if the insured survives year 2, which has probability $1-0.10=0.90$. The year-2 death benefit is paid with probability $0.10$. A year-3 death benefit requires survival through year 2 and death in year 3, so its unconditional probability given in-force status at time 1 is $0.90(0.20)=0.18$.

At zero interest, no discounting is needed. The expected present value contributions are:

\begin{aligned} \text{time-1 premium and expense} &=-450,\\ \text{time-2 premium and expense} &=0.90(-450)=-405,\\ \text{year-2 death benefit} &=0.10(2{,}000)=200,\\ \text{year-3 death benefit} &=0.90(0.20)(3{,}000)=540. \end{aligned}

Adding the contributions gives:

{}_1V=-450-405+200+540=-115.

The reserve is -$115.00. The negative value is possible: on this valuation basis, expected future premium income exceeds expected future benefits and expenses. A common mistake is to count the year-3 death benefit with probability $0.20$ alone. That probability is conditional on reaching year 3, so it must be multiplied by the $0.90$ probability of surviving year 2.

Worked Example: A Time-Zero Endowment Reserve

A hypothetical two-year endowment is valued at time 0 before the first premium and expense. A premium of $300 and expense of $20 are due at the start of each year in force. A $1,000 death benefit is paid at the end of year 1 if death occurs in that year. At time 2, $1,000 is paid whether death occurs in year 2 or the insured survives to maturity. The conditional death probabilities are $0.10$ in year 1 and $0.20$ in year 2, given survival to the start of that year. The effective annual interest rate is $10\%$. Find the reserve at time 0.

Method. At time 0, the net cash flow is $20-300=-280$. At time 1, the premium and expense are due only if the insured survives year 1, so their expected amount is $0.90(20-300)=-252$. The year-1 death benefit has expected amount $0.10(1{,}000)=100$ at time 1. Discount both time-1 amounts by $1.10$.

At time 2, the $1,000 endowment payment is certain conditional on survival to time 1: death during year 2 and survival to maturity both trigger it. The probability of reaching time 1 is $0.90$, so its expected time-0 present value is $0.90(1{,}000)/(1.10)^2$. Thus:

\begin{aligned} {}_0V &=-280+\frac{-252+100}{1.10} +\frac{0.90(1{,}000)}{(1.10)^2}\\ &=-280-138.1818+743.8017\\ &=325.6199. \end{aligned}

Using unrounded values, the result is approximately $325.6198; rounded to cents, the reserve is $325.62. The common mistake is to treat the time-2 payment as contingent only on survival to maturity. Once the policy survives year 1, the same payment is made at time 2 whether death occurs in year 2 or maturity is reached.

Interpreting and Checking the Reserve

A prospective reserve is a conditional valuation, not a realized cash balance and not a forecast of a single policy’s actual outcome. Its value depends on the valuation basis: interest, mortality or other decrements, contract benefits, expenses, and premium rules. Changing any of these can change the reserve even when the policy terms remain unchanged.

The reserve’s sign is informative but should not be interpreted as a claim that the insurer will necessarily have a surplus or deficit on an individual policy. For example, the negative reserve in the third example reflects expected premium income exceeding expected outgo under the stated assumptions. It does not imply that future claims are impossible or that no assets are needed for the portfolio.

In annual profit tests, reserves provide a valuation of obligations at policy-year boundaries and are used alongside the period’s cash flows to analyze profit emergence. Keep the reserve calculation distinct from the profit calculation: first establish the cash-flow timing and conditional future values correctly. The reserve-balance relationships that connect reserves across successive years are a separate step.

Boundary timing: Under the convention used here, include premiums and expenses due at the valuation time. If a problem instead defines the reserve after those payments, remove those boundary cash flows consistently.
Conditional probability: A death probability for a future year is conditional on reaching that year. Weight it by the probability of surviving or remaining in force until then.
Exhaustive outcomes: If different outcomes produce the same payment at the same date, check whether the payment is certain conditional on the current state before applying a probability.
Sign: Use benefits plus expenses minus premiums. Reversing this convention reverses the interpretation of a positive or negative reserve.

Key takeaway: A prospective reserve at an annual boundary is the conditional expected present value of future benefits and expenses less future premiums. State the boundary convention, condition on in-force status, and account for every possible outcome before discounting.

Check Your Understanding

For each problem, use the before-payment convention and report the reserve to the nearest cent. Assume no cash flows other than those stated.

A one-year term policy is valued at time 1. A premium of $70 and expense of $10 are due at time 1. The conditional probability of death by time 2 is $0.05$, and the $1,000 benefit is paid at time 2. At $5\%$ effective annual interest, find the reserve. (Answer: -$12.38)
A hypothetical endowment is valued at time 1. A premium of $500 and expense of $25 are due then. At time 2, $900 is paid whether death occurs during year 2 or the insured survives to maturity. At $4\%$ effective annual interest, find the reserve. (Answer: $390.38)
A two-year term policy is valued at time 0. A premium of $120 and expense of $30 are due at the start of each year in force. The year-1 death probability is $0.10$, with a $1,000 benefit at time 1. Conditional on surviving year 1, the year-2 death probability is $0.20$, with a $1,500 benefit at time 2. At zero interest, find the reserve. (Answer: $199.00)
A two-year endowment is valued at time 0. A premium of $250 and expense of $20 are due at the start of each year in force. A $600 death benefit is paid at time 1 if death occurs in year 1. At time 2, $1,000 is paid whether death occurs in year 2 or survival to maturity occurs. The year-1 conditional death probability is $0.10$; the year-2 conditional death probability is $0.20$. At $5\%$ effective annual interest, find the reserve. (Answer: $446.33)

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