Deriving the Prospective Reserve Identity

Why the Prospective Reserve Is Conditional

A multi-state contract can have different future cash flows depending on whether the policy is active, disabled, or in another contract state at the valuation date. Even for the same policy and valuation time, the reserve can therefore differ by current state. The prospective reserve gives a precise value for each case: it is the expected present value of future outgo less the expected present value of future premium income, conditional on the state occupied now.

This tutorial derives that identity and applies it to discrete-time models. It uses a deterministic effective annual interest rate and cash flows at specified future valuation dates. Transition probabilities may come from any appropriate state model; the identity does not require a particular transition structure.

Definition: Let $Z_t$ denote the contract state at valuation time $t$. For a contract currently in state $j$, the prospective reserve $V_j(t)$ is the conditional expected present value, at time $t$, of future benefits and expenses minus future premiums. Benefits and expenses are positive outgo; premiums received by the insurer are subtracted. Cash flows at time $t$ itself are excluded unless the valuation convention explicitly includes them.

For a discrete model with future cash flows at times $t+1,\ldots,t+n$, let $B_{t+k}$, $E_{t+k}$, and $P_{t+k}$ denote, respectively, the benefit, expense, and premium at time $t+k$. These amounts can depend on the path of states, including transitions during the interval. Write the net outgo at that date as $C_{t+k}=B_{t+k}+E_{t+k}-P_{t+k}$. If the effective annual rate $i$ is fixed, then $v=1/(1+i)$, and the prospective reserve is

V_j(t)=\sum_{k=1}^{n}v^k\,\mathbb{E}\!\left[C_{t+k}\mid Z_t=j\right] =\sum_{k=1}^{n}v^k\,\mathbb{E}\!\left[B_{t+k}+E_{t+k}-P_{t+k}\mid Z_t=j\right].

The conditioning event matters: the probabilities used to value future cash flows are the probabilities of future paths given that the contract is in state $j$ now. A state-specific reserve is not multiplied by the probability that the policy is in state $j$. That probability is relevant when calculating an average reserve over a population, not when valuing the contract conditional on its observed state.

Why the Identity Holds

The reserve definition can be written as the conditional expected present value of a single random total. On each possible future path, discount each future net outgo to time $t$, add the discounted amounts, and then take the conditional expectation given the current state. For a finite set of payment dates, linearity of conditional expectation allows the expectation of the sum to be written as a sum of conditional expectations. Since the interest rate is deterministic, the discount factor $v^k$ is fixed and can be taken outside the expectation.

Form the pathwise net cash flow.
At each future date $t+k$, set $C_{t+k}=B_{t+k}+E_{t+k}-P_{t+k}$. This sign convention treats benefits and expenses as insurer outgo and premiums as insurer income.

Discount along each path.
With a deterministic effective annual rate $i$, the time-$t$ present value of the total future net outgo is $\sum_{k=1}^{n}v^k C_{t+k}$.

Condition on the current state.
For a policy known to be in state $j$, take $\mathbb{E}[\sum_{k=1}^{n}v^k C_{t+k}\mid Z_t=j]$.

Separate the dates.
Conditional linearity and deterministic discounting give $\sum_{k=1}^{n}v^k\mathbb{E}[C_{t+k}\mid Z_t=j]$, which is the stated prospective reserve identity.

This argument requires finite expected present values so that the expectations and sums are well-defined. It also relies on deterministic discount factors. If interest rates are stochastic, the appropriate expression is the conditional expectation of the discounted cash flows, with the random discount factor inside the expectation; it is generally not valid to replace it by a fixed $v^k$. Likewise, the identity does not prescribe the cash-flow timing: the model must first specify when each benefit, expense, and premium is paid.

Worked Applications

Worked Example: Active and Disabled States with Termination

A hypothetical two-year contract is valued at time 0 in state $A$, active. The annual effective interest rate is $i=5\%$, so $v=1/1.05$. The possible states are active $A$, disabled $D$, and terminated $X$. There is no separate mortality decrement; termination $X$ is an absorbing contract state. From $A$, the one-year transition probabilities are $P(A\to A)=0.80$ and $P(A\to D)=0.20$. From $D$, they are $P(D\to D)=0.60$ and $P(D\to X)=0.40$.

At the end of year 1, an $A\to A$ transition has expense $20$ dollars and premium income $150$ dollars. An $A\to D$ transition pays a benefit of $300$ dollars and incurs an expense of $35$ dollars, with no premium. At the end of year 2, $A\to A$ has expense $20$ and premium income $150$; $A\to D$ pays a benefit of $500$ and incurs expense $35$; $D\to D$ has expense $35$; and $D\to X$ pays a termination benefit of $1{,}000$ and incurs expense $10$. Find $V_A(0)$.

Method. Find the conditional expected net outgo at each future date, then discount each date’s expected amount. This method keeps transition benefits attached to the transition that triggers them.

Year 1. The net outgo on $A\to A$ is $20-150=-130$ dollars. On $A\to D$, it is $300+35=335$ dollars. Therefore,

\mathbb{E}[C_1\mid Z_0=A] =0.80(-130)+0.20(335) =-104+67=-37.

Year 2. The path probabilities conditional on $Z_0=A$ are $P(AA)=0.80(0.75)=0.60$, $P(AD)=0.80(0.25)=0.20$, $P(DD)=0.20(0.60)=0.12$, and $P(DX)=0.20(0.40)=0.08$, where the probabilities from $A$ in year 2 are $0.75$ to $A$ and $0.25$ to $D$. The year-2 net outgo on these paths is, respectively, $-130$, $500+35=535$, $35$, and $1{,}000+10=1{,}010$ dollars. Hence,

\begin{aligned} \mathbb{E}[C_2\mid Z_0=A] &=0.60(-130)+0.20(535)+0.12(35)+0.08(1{,}010)\\ &=-78+107+4.2+80.8=114. \end{aligned}

Discounting the two conditional expected net outgo amounts gives

\begin{aligned} V_A(0) &=\frac{-37}{1.05}+\frac{114}{1.05^2}\\ &=-35.2380952+103.4013605\\ &=68.1632653. \end{aligned}

The reserve is $\$68.16$ to the nearest cent. As a check, the year-1 expectation is an expected net inflow of $37$ dollars, while the larger expected outgo in year 2 more than offsets it after discounting. The positive reserve is therefore consistent with the calculated discounted net outgo.

Worked Example: A Disabled Contract with State-Dependent Premiums

A hypothetical two-year income-protection contract is currently disabled, state $D$. The annual effective interest rate is $4\%$. The states are $D$ and active $A$; there is no separate mortality or termination transition in the two-year model. From $D$, the next-year probabilities are $0.70$ to $D$ and $0.30$ to $A$. From $A$, they are $0.20$ to $D$ and $0.80$ to $A$. At each year-end, a contract ending the year in $D$ pays a benefit of $400$ dollars; either state incurs an expense of $20$ dollars; and a contract ending in $A$ receives premium income of $100$ dollars. Find $V_D(0)$.

Method. First calculate the net outgo associated with each year-end state, then determine the state probabilities for each future year conditional on starting in $D$. The net outgo is $400+20=420$ dollars in $D$ and $20-100=-80$ dollars in $A$.

At the end of year 1, the state probabilities are $P(D)=0.70$ and $P(A)=0.30$. Thus,

\mathbb{E}[C_1\mid Z_0=D] =0.70(420)+0.30(-80) =294-24=270.

At the end of year 2, the probability of being in $D$ is the probability of reaching $D$ through either year-1 state:

P(Z_2=D\mid Z_0=D) =0.70(0.60)+0.30(0.20) =0.42+0.06=0.48.

The remaining probability is $P(Z_2=A\mid Z_0=D)=1-0.48=0.52$. Therefore,

\mathbb{E}[C_2\mid Z_0=D] =0.48(420)+0.52(-80) =201.6-41.6=160.

Using $v=1/1.04$, the reserve is

\begin{aligned} V_D(0) &=\frac{270}{1.04}+\frac{160}{1.04^2}\\ &=259.6153846+147.9289941\\ &=407.5443787. \end{aligned}

Thus $V_D(0)=$$\$407.54$ to the nearest cent. The year-2 calculation uses the probability of each state at year 2, not the year-1 probabilities carried forward unchanged.

Worked Example: A Maturity Payment Conditional on the Final State

A hypothetical contract is valued at time 0 in state $A$, active. Its annual effective interest rate is $5\%$. The states are $A$ and disabled $D$, with no mortality or termination in the two-year model. In year 1, the transition probabilities from $A$ are $0.90$ to $A$ and $0.10$ to $D$. In year 2, the probabilities are $0.80$ from $A$ to $A$, $0.20$ from $A$ to $D$, $0.30$ from $D$ to $A$, and $0.70$ from $D$ to $D$.

At the end of year 1, every state incurs an expense of $5$ dollars, an $A$-state contract receives a premium of $70$ dollars, and a $D$-state contract receives a benefit of $120$ dollars. At the end of year 2, there is no premium. Every state incurs an expense of $5$ dollars, an $A$-state contract receives a maturity benefit of $500$ dollars, and a $D$-state contract receives a benefit of $200$ dollars. Find the reserve.

Method. Calculate year-end net outgo by state, find the conditional state distribution at each payment date, and discount. In year 1 the net outgo is $5-70=-65$ dollars in $A$ and $120+5=125$ dollars in $D$. Thus,

\mathbb{E}[C_1\mid Z_0=A] =0.90(-65)+0.10(125) =-58.5+12.5=-46.

At year 2, the probability of being in $A$ is $0.90(0.80)+0.10(0.30)=0.72+0.03=0.75$. The probability of being in $D$ is $0.90(0.20)+0.10(0.70)=0.18+0.07=0.25$. The year-2 net outgo is $500+5=505$ dollars in $A$ and $200+5=205$ dollars in $D$. Consequently,

\mathbb{E}[C_2\mid Z_0=A] =0.75(505)+0.25(205) =378.75+51.25=430.

The maturity benefit is included at its actual payment date, year 2. The reserve is

\begin{aligned} V_A(0) &=\frac{-46}{1.05}+\frac{430}{1.05^2}\\ &=-43.8095238+390.0226757\\ &=346.2131519. \end{aligned}

The reserve is $\$346.21$ to the nearest cent. The state probabilities at year 2 sum to $0.75+0.25=1$, and the maturity payment is included once, in the year-2 cash flow.

Common Exam Trap: Mixing Up Signs or Conditioning

A frequent error is to add premiums to benefits and expenses instead of subtracting them. In the first example, this error would make the year-1 expected amount $0.80(20+150)+0.20(300+35)=203$ dollars. That is not the insurer’s expected net outgo: the $150$-dollar premium is income, so the correct amount is $0.80(20-150)+0.20(300+35)=-37$ dollars. A sign error can change both the size and direction of the reserve.

A second error is to confuse a conditional reserve with a population average. If a fraction of policies are disabled, the average reserve across all policies may weight state-specific reserves by state probabilities. But $V_D(t)$ is already conditional on the policy being disabled at time $t$; multiplying it by the probability of being disabled again would answer a different question.

A third check is the time attached to each discount factor. A payment at $t+2$ is discounted by $v^2$, even when its amount is triggered by a transition in year 2. Applying $v$ to every future payment understates the value of later cash flows.

When the Identity Is Useful

The prospective reserve identity is the valuation basis for a state-specific liability: it translates the contract’s future guarantees, expenses, and premium income into a value at the current date and state. Its form remains the same whether a benefit is paid on entering a state, while remaining in a state, or on termination, provided the cash-flow definition and timing correctly capture the contractual payment.

Before calculating, identify the current state, list the possible future payment dates, state the cash-flow sign convention, and determine the conditional state probabilities at each date. Then take the conditional expected net outgo and discount it from its payment date. This sequence keeps the reserve tied to the contract’s actual state and prevents transition-contingent payments from being omitted or counted at the wrong time.

Check Your Understanding

For each problem, use the prospective reserve identity and the stated cash-flow timing. Amounts are in dollars.

A contract is currently disabled. At the end of one year, it remains disabled with probability $0.60$ or recovers to active with probability $0.40$. The annual effective interest rate is $5\%$. The disabled state pays a benefit of $300$ and incurs an expense of $10$; the active state incurs an expense of $10$ and receives premium income of $80$. Find the current state-specific reserve. (Answer: $\$150.48$)
A contract is currently active. At each year-end, net outgo is $-50$ in active and $140$ in disabled. In year 1, the transition probabilities from active are $0.80$ to active and $0.20$ to disabled. In year 2, the transition probabilities are $0.75$ from active to active, $0.25$ from active to disabled, $0.50$ from disabled to disabled, and $0.50$ from disabled to active. At $i=5\%$, find the two-year prospective reserve. (Answer: $-\$5.08$)
A contract is currently disabled. At the end of year 1 it is disabled with probability $0.50$ and active with probability $0.50$. For year 2, the transition probabilities are $0.40$ from disabled to disabled, $0.60$ from disabled to active, $0.10$ from active to disabled, and $0.90$ from active to active. At each year-end, a disabled contract receives a benefit of $100$, every contract incurs an expense of $5$, and an active contract receives premium income of $30$. At the end of year 2, every contract also receives a maturity payment of $200$. With $i=4\%$, find the reserve. (Answer: $\$230.31$)
In one sentence, state why $V_j(t)$ is not generally the same as the reserve averaged over all policies in a population containing several contract states. (Answer: $V_j(t)$ is conditional on the contract being in state $j$, whereas a population average weights reserves across the states represented in that population.)
Suppose a two-year contract has deterministic annual interest and a benefit payment at the end of year 2. State the discount factor for that payment and explain why. (Answer: $v^2=1/(1+i)^2$, because the payment is two years after the valuation date.)

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