Prospective Reserves by Contract State

What a Reserve by Contract State Measures

A contract can provide different benefits, collect different premiums, or incur different expenses depending on its current state. An active policy, for example, may require a premium and provide no immediate disability benefit, while a disabled policy may provide an income benefit and require no premium. A prospective reserve must therefore reflect not only the policy’s terms, but also the state the contract occupies at the valuation time.

The reserve is calculated on a stated valuation basis: the interest rate, assumptions about future transitions, and the future cash flows included in the valuation. In this tutorial, benefits and expenses are positive outgo, while premiums are income that reduces outgo. The examples use annual cash flows paid at each year-end. The valuation time is immediately before the future periods shown, so there is no cash flow at that instant included in the reserve.

Definition: Let $J_t$ denote the contract state at valuation time $t$. The prospective reserve in state $r$, denoted here by $V_r(t)$, is the expected present value at time $t$ of future benefits and expenses less future premiums, conditional on $J_t=r$, using the valuation assumptions. If future cash flows occur at integer durations $k=1,\ldots,n$, and $Z_{t+k}$ is benefits plus expenses less premiums at time $t+k$, then $$ V_r(t)=E\left[\left.\sum_{k=1}^{n}v^k Z_{t+k}\,\right|\,J_t=r\right], \qquad v=\frac{1}{1+i}. $$

The conditional expectation uses the future probabilities appropriate to starting in state $r$. A cash flow may depend on the state reached, and the state probabilities at later durations depend on the transitions allowed by the model. If the contract has no further possible cash flows after a specified time, the sum ends there. A negative reserve is possible under this definition: it means the expected present value of future premiums exceeds that of future benefits and expenses on the stated basis.

The formula is a valuation definition, not a statement about the amount of money actually held by an insurer. It is also not a reserve for every policy in a group: it is the value conditional on the particular contract state specified. Changing the valuation interest rate, transition assumptions, or cash-flow basis can change the result even when the contract state is unchanged.

Why Conditioning on State Works

For a fixed future history of the contract, discount each future net outgo to time $t$ and add the discounted amounts. That produces one present value for that history. The prospective reserve averages these present values over the possible future histories, assigning each history its probability conditional on the state at time $t$.

This follows directly from the definition of conditional expectation and the linearity of expectation. For a finite set of possible histories $h$, let $p(h\mid J_t=r)$ be the probability of history $h$ given the current state, and let $z_k(h)$ be its net outgo at duration $k$. Then

V_r(t) =\sum_h p(h\mid J_t=r)\sum_{k=1}^{n}v^kz_k(h) =\sum_{k=1}^{n}v^k\sum_h p(h\mid J_t=r)z_k(h).

The first expression averages each history’s present value. The second averages the cash flow at each duration and then discounts it. Thus either calculation order is valid if the same conditional probabilities, cash flows, and discount periods are used. The current state matters because it changes which histories are possible and their probabilities.

Worked Calculations

Worked Example: One-Year Reserve in Active and Disabled States

Consider a hypothetical income-protection contract with one year remaining. The annual effective valuation interest rate is $i=4\%$, so $v=1/1.04$. At year-end, an active policy has a benefit of $0, an expense of $8, and a premium of $40. A disabled policy has a benefit of $70, an expense of $8, and no premium. During the year, an active contract remains active with probability 0.90 and becomes disabled with probability 0.10. A disabled contract becomes active with probability 0.30 and remains disabled with probability 0.70. Find the reserve at the start of the year in each state.

Method. First calculate net outgo in each year-end state as benefit plus expense less premium. Then take the expectation using transition probabilities conditional on the starting state, and discount the resulting expected outgo one year.

For an active policy, year-end net outgo is $0+8-40=-\$32$ if active and $0+8-0=\$8$ would not apply to the disabled state; in the disabled state it is $70+8-0=\$78$. Therefore, conditional on starting active, expected year-end net outgo is

0.90(-32)+0.10(78)=-28.80+7.80=-\$21.00.

Discounting one year gives $V_A(0)=-21/1.04=-\$20.1923$, or −$20.19 to the nearest cent.

For a disabled policy, the probabilities of ending active and disabled are 0.30 and 0.70. Its expected year-end net outgo is

0.30(-32)+0.70(78)=-9.60+54.60=\$45.00.

Thus $V_D(0)=45/1.04=\$43.2692$, or $43.27 to the nearest cent. The state-specific values differ because starting states imply different transition probabilities. The calculation also shows why premiums must be subtracted rather than added to outgo.

Check. Both results use one year of discounting. The active-state result is negative because the expected premium income exceeds expected benefit and expense outgo on this one-year basis.

Worked Example: Two-Year Reserve Starting in the Disabled State

A hypothetical contract has active state $A$ and disabled state $D$. At each year-end, net outgo is −$20 in state $A$ and $80 in state $D$; these amounts already equal benefits plus expenses less premiums. The annual effective interest rate is $i=5\%$. Each year, a contract in $A$ moves to $A$ with probability 0.80, while a contract in $D$ moves to $A$ with probability 0.40. The remaining probability in each state is a transition to $D$. There are two year-end cash flows and no later payments. Find the reserve conditional on starting in $D$.

Method. Calculate expected net outgo for each future year using the state probabilities for that year, discount each expectation by its own duration, and add the present values.

At the end of year 1, starting in $D$, the probabilities are 0.40 in $A$ and 0.60 in $D$. Expected net outgo is

0.40(-20)+0.60(80)=-8+48=\$40.

For year 2, the probability of being in $A$ is the sum of the probabilities of the two ways to reach it: starting in $D$, reaching $A$ in year 1, then remaining in $A$; or reaching $D$ in year 1, then moving to $A$. Hence

P(J_2=A\mid J_0=D)=0.40(0.80)+0.60(0.40)=0.32+0.24=0.56.

The probability of being in $D$ at year 2 is $1-0.56=0.44$. The year-2 expected net outgo is therefore

0.56(-20)+0.44(80)=-11.20+35.20=\$24.

The reserve is

V_D(0)=\frac{40}{1.05}+\frac{24}{1.05^2} =38.095238\ldots+21.768707\ldots = \$59.86.

The result is rounded to the nearest cent only at the end. Common mistake avoided: the year-2 state probabilities must be obtained from the year-1 distribution; using the one-year probabilities again would ignore the second transition.

Worked Example: Separating Cash-Flow Components Before Valuation

A hypothetical two-year contract has states $A$ and $D$, with annual effective interest $i=4\%$. At each year-end, the active state has a benefit of $0, an expense of $5, and a premium of $30. The disabled state has a benefit of $60, an expense of $5, and no premium. From $A$, the next state is $A$ with probability 0.80 and $D$ with probability 0.20. From $D$, the next state is $A$ with probability 0.25 and $D$ with probability 0.75. Calculate the reserve at time 0 conditional on starting in $A$.

Method. Convert the stated benefits, expenses, and premiums to net outgo in each state before applying the transition probabilities. The active-state net outgo is $0+5-30=-\$25$; the disabled-state net outgo is $60+5-0=\$65$.

At the end of year 1, the probabilities are 0.80 in $A$ and 0.20 in $D$, so expected net outgo is

0.80(-25)+0.20(65)=-20+13=-\$7.

At the end of year 2, the probability of being in $A$ is

$$ 0.80(0.80)+0.20(0.25)=0.64+0.05=0.69, $$

so the probability of being in $D$ is $1-0.69=0.31$. Expected year-2 net outgo is

0.69(-25)+0.31(65)=-17.25+20.15=\$2.90.

Discount the year-1 amount once and the year-2 amount twice:

V_A(0)=\frac{-7}{1.04}+\frac{2.90}{1.04^2} =-6.730769\ldots+2.681953\ldots =-\$4.05.

The reserve is rounded to the nearest cent. Common mistake avoided: discounting the second-year expected amount only once would value it at the wrong time. The negative result is consistent with the sign convention: over the two years, the discounted expected premium income is slightly greater than discounted expected benefits and expenses.

Checks That Keep the Valuation Consistent

A state-specific prospective reserve is only meaningful when the conditioning event, cash-flow signs, and timing agree. Before finalizing a calculation, check the following:

State at the valuation time: identify the state on which the conditional value is based. Do not substitute probabilities associated with a different starting state.
Net-outgo sign: use benefits plus expenses less premiums. A premium is income and reduces the reserve; an expense is outgo and increases it.
Future periods: include only cash flows after the valuation time. Do not include a payment already made or an immediate amount unless the stated valuation convention requires it.
Discount duration: a payment one year ahead is discounted by $v$, a payment two years ahead by $v^2$, and so on. Match the exponent to the time from valuation to payment.
Conditional state probabilities: for later periods, account for every route by which the contract can reach a state. The probabilities for a later duration are generally not the same as those for the first duration.
Basis consistency: use one specified interest and transition basis throughout the calculation. A change in assumptions defines a different prospective valuation.

These definitions and calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In exam work, a brief statement of the starting state, the net-outgo convention, and the timing of payments often makes the calculation easier to audit and helps reveal errors before rounding.

Key takeaway: $V_r(t)$ values future net outgo conditional on the contract being in state $r$ at time $t$. Use the transition probabilities from that state, subtract premiums from benefits and expenses, discount each future payment to the valuation time, and add the resulting expected present values.

Check Your Understanding

For each problem, use benefits plus expenses less premiums as net outgo and round dollar answers to the nearest cent.

A one-year contract has states $A$ and $D$. At year-end, net outgo is −$10 in $A$ and $50 in $D$. Starting in $A$, the probabilities of ending in $A$ and $D$ are 0.70 and 0.30. If $i=3\%$, find the reserve. (Answer: $7.77)
A one-year contract has net outgo −$24 in $A$ and $56 in $D$. Starting in $D$, the probabilities of ending in $A$ and $D$ are 0.25 and 0.75. If $i=4\%$, find the reserve. (Answer: $34.62)
A two-year contract has net outgo −$12 in $A$ and $48 in $D$ at each year-end. From $A$, the next state is $A$ with probability 0.75; from $D$, it is $A$ with probability 0.50. Starting in $A$, find the reserve if $i=0\%$. (Answer: $9.75)
A two-year contract has net outgo −$15 in $A$ and $45 in $D$. From $A$, the next state is $A$ with probability 0.80; from $D$, it is $A$ with probability 0.40. Starting in $D$, find the reserve if $i=5\%$. (Answer: $30.34)
A two-year contract has the following year-end cash flows: in $A$, benefit $20, expense $5, and premium $35; in $D$, benefit $90, expense $5, and no premium. From $A$, the next state is $A$ with probability 0.60; from $D$, it is $A$ with probability 0.20. Starting in $A$, find the reserve if $i=4\%$. (Answer: $75.89)

What You'll Learn