Reserves in Multi-State Insurance Models

Why Reserves Depend on the Current State

For a long-term contract, the same policy can have different future values depending on whether the insured is active, disabled, or receiving long-term care. A prospective reserve must therefore be conditional on the policy’s state at the valuation time. It must also reflect that the state can change before future premiums and benefits are due.

The calculation combines two probability questions: which state will the policy occupy at a future time, and which transition will occur during each period? It then values the cash flows attached to those states and transitions. All examples below use hypothetical contracts and data.

Definition: Let the contract have states $0,1,\ldots,m$, including any termination state, and let $R_i(t)$ be the prospective reserve at time $t$, conditional on the policy being in state $i$ at that time. The reserve is the expected present value of future benefits and expenses minus the expected present value of future premiums, conditional on that state and the contract terms then in force.

A reserve is conditional on a state, not on the policy having followed one particular path to reach it. If two policies are both in the disabled state at time $t$, their reserves under the same contract assumptions are the same, even if they entered that state by different routes, unless the contract explicitly makes benefits depend on that history.

Set the Model and Cash-Flow Timing

For a discrete-time model, use the following convention throughout a valuation: at each integer time, the premium for the coming year is paid at the start of that year if the contract is in a state that requires a premium. Benefits and expenses for that year are paid at its end. Let $p_{jk}(s)$ be the probability that a policy in state $j$ at time $s$ is in state $k$ at time $s+1$. The probabilities leaving each state sum to one, including any probability of termination.

Let $P_j(s)$ be the premium paid at time $s$ in state $j$, and let $b_{jk}(s)$ be the benefit and expense outgo paid at time $s+1$ when the policy moves from $j$ to $k$. Either amount may be zero. A state-based end-of-year payment can be represented by setting $b_{jk}(s)$ equal to the same amount for all destinations $k$ from the relevant starting state $j$. A transition benefit, such as a payment on recovery or death, can have a different amount for each transition.

Assume a constant effective annual interest rate $i$ for the examples, with $v=1/(1+i)$. If assumptions vary by year, each future payment must instead be discounted using the applicable sequence of interest rates. The formulas here also presume that the transition probabilities and cash flows are known for each period, conditional on the state and transition. They do not cover contracts whose future terms depend on unrecorded individual history.

Definition: Write $q_{ij}(t,n)$ for the probability that a policy in state $i$ at time $t$ occupies state $j$ at time $t+n$. In particular, $q_{ij}(t,0)$ is 1 if $i=j$, and 0 otherwise. These are occupancy probabilities; they do not specify the precise path used to arrive at the future state.

In a finite-state model, occupancy probabilities can be calculated by considering the possible states one period earlier. For $n\geq 0$,

q_{ij}(t,n+1)=\sum_{\ell=0}^{m}q_{i\ell}(t,n)p_{\ell j}(t+n).

This is a scalar probability calculation: add the probabilities of reaching each possible intermediate state and then moving from it to $j$. It does not require the future cash flows to be the same in every state.

Prospective Valuation Across States and Transitions

Suppose the contract ends at time $N$, and there are no future cash flows after that time. From state $i$ at time $t$, the expected premium paid at time $t+n$ is the sum over states of the probability of occupying each state multiplied by its premium. The expected end-of-period outgo is calculated by first considering the state at the period’s start and then each possible destination state.

R_i(t)= -\sum_{n=0}^{N-t-1}v^n \sum_{j=0}^{m}q_{ij}(t,n)P_j(t+n) +\sum_{n=0}^{N-t-1}v^{n+1} \sum_{j=0}^{m}q_{ij}(t,n) \sum_{k=0}^{m}p_{jk}(t+n)b_{jk}(t+n).

The premium at time $t+n$ is discounted $n$ years because it is paid at the start of that year. The benefit or expense at the end of that period is discounted $n+1$ years. A termination state can be included in the sums: if it has no subsequent premiums or benefits, its future cash flows are zero.

Justification: For period $n$, split the possible future paths into disjoint cases according to the state $j$ occupied at time $t+n$. The probability of each case, conditional on state $i$ at time $t$, is $q_{ij}(t,n)$. Given that state $j$, split again according to the destination $k$, with conditional probability $p_{jk}(t+n)$. Thus the expected period-end outgo is $\sum_j q_{ij}(t,n)\sum_k p_{jk}(t+n)b_{jk}(t+n)$. The expected start-of-period premium is $\sum_j q_{ij}(t,n)P_j(t+n)$. Discount each expected amount to time $t$, subtract premiums from outgo, and add over the remaining periods to obtain the reserve formula.

Worked Calculations

Worked Example: A One-Year Reserve with a Death Benefit

A hypothetical one-year contract is valued immediately before the premium due at time 0. Its states are active, disabled, and dead. The annual effective interest rate is $i=5\%$, so $v=1/1.05=20/21$. From the active state, the probabilities of being active, disabled, or dead at year-end are 0.65, 0.20, and 0.15, respectively. The start-of-year premium is $100. End-of-year benefits are $0 if active, $300 if disabled, and $2,000 if dead. Calculate the reserve conditional on being active at time 0.

Method. Since there is one year remaining, value the premium at time 0 without discounting, then calculate the expected year-end benefit and discount it one year. The expected benefit is

$$ 0.65(0)+0.20(300)+0.15(2{,}000)=0+60+300=360. $$

The present value of the expected benefit is $360(20/21)=342.8571$. Subtract the $100 premium:

R_{\text{active}}(0)=-100+342.8571=242.8571.

Rounded to the nearest cent, the reserve is $242.86. The calculation is checked by noting that the year-end benefit expectation includes all three mutually exclusive outcomes, and the premium is paid at time 0 rather than year-end.

Common mistake. Omitting the death outcome gives an expected benefit of only $0.20(300)=60$, and a reserve of $-100+60(20/21)=-42.86$. That calculation wrongly treats the 0.15 death probability as if it produced no payment; the stated death benefit must be included.

Worked Example: A Two-Year Reserve with State-Dependent Premiums

A hypothetical two-year contract has active, disabled, care, and dead states. The annual effective interest rate is 5%, so $v=20/21$. The transition probabilities are constant in both years:

State at start of year	Active at year-end	Disabled at year-end	Care at year-end	Dead at year-end
Active	0.60	0.20	0.10	0.10
Disabled	0.20	0.50	0.20	0.10
Care	0.00	0.10	0.70	0.20
Dead	0.00	0.00	0.00	1.00

The year-end benefit is $0 in the active state, $200 in the disabled state, $500 in the care state, and $1,000 in the dead state, regardless of the transition that led there. A $120 premium is due at the start of each year when active; no premium is due in the other states. Calculate the reserve at time 0 conditional on being active.

Method. First calculate the one-year remaining reserve at time 1 for each possible living state. From active, the expected year-2 benefit is $0.60(0)+0.20(200)+0.10(500)+0.10(1{,}000)=190$, so $R_{\text{active}}(1)=-120+190(20/21)=60.9524$. From disabled, the expected benefit is $0.20(0)+0.50(200)+0.20(500)+0.10(1{,}000)=300$, giving $R_{\text{disabled}}(1)=300(20/21)=285.7143$. From care, the expected benefit is $0.00(0)+0.10(200)+0.70(500)+0.20(1{,}000)=570$, giving $R_{\text{care}}(1)=570(20/21)=542.8571$. The reserve in the dead state is zero because the contract has no later cash flows there.

In the first year, the expected benefit from active is also $190. The time-1 state probabilities are 0.60 active, 0.20 disabled, 0.10 care, and 0.10 dead. Therefore the expected time-1 reserve is

$$ 0.60(60.9524)+0.20(285.7143)+0.10(542.8571)+0.10(0)=148.0000. $$

The time-0 reserve includes the $120 premium due now, the first year’s benefit discounted one year, and the expected time-1 reserve discounted one year:

R_{\text{active}}(0)=-120+\frac{190}{1.05}+\frac{148}{1.05} =-120+180.9524+140.9524=201.9048.

Rounded to the nearest cent, the reserve is $201.90. As a check, the first-year transition probabilities sum to one, and the second-year premium is included only in the active-state reserve at time 1.

Common mistake. Discounting the time-1 reserve twice would give $-120+190v+148v^2=195.19$. The reserve of $148 is already valued at time 1, so it is discounted once, by $v$, to time 0.

Worked Example: Transition-Specific Payments from the Care State

A hypothetical one-year contract is valued immediately before a $50 premium due while the insured is in care. The annual effective interest rate is 5%, so $v=20/21$. During the year, the probabilities from care are 0.10 to active, 0.20 to disabled, 0.50 to remain in care, and 0.20 to dead. The corresponding end-of-year transition benefits are $400, $700, $1,200, and $5,000. An additional $100 expense is paid at year-end for every outcome. Find the reserve conditional on being in care.

Method. Add the benefit and expense for each transition, then weight by that transition’s probability. The expected end-of-year outgo is

$$ 0.10(400+100)+0.20(700+100)+0.50(1{,}200+100)+0.20(5{,}000+100) =50+160+650+1{,}020=1{,}880. $$

Discount the expected outgo for one year and subtract the premium paid at time 0:

R_{\text{care}}(0)=-50+1{,}880(20/21) =-50+1{,}790.4762=1{,}740.4762.

Rounded to the nearest cent, the reserve is $1,740.48. The weighted probabilities sum to 1.00, so the fixed $100 expense contributes exactly $100 to expected outgo.

Common mistake. Applying the $1,200 payment for remaining in care to every outcome would produce an expected benefit of $1,200, ignoring recovery, disability, and death payments. The payment depends on the transition, so each stated amount must be multiplied by its own probability.

Interpreting and Checking State Reserves

State-specific reserves need not increase in a simple order from active to disabled to care. A care state might have larger immediate benefit outgo but no future premiums; an active state might have lower immediate benefit outgo but more premiums. The result depends on all future transition probabilities, cash flows, and the remaining term. A negative prospective reserve is possible when the expected present value of future premiums exceeds that of future outgo.

For a population distributed across states, an average reserve can be calculated only after the conditional reserves are determined. If the probabilities of occupying the states at a valuation time are $w_i$, where $\sum_i w_i=1$, then the expected reserve across that population is $\sum_i w_iR_i(t)$. This average is not a substitute for the reserve conditional on a particular policy state.

Key checks: State probabilities leaving each state must sum to one; each transition payment must be attached to the correct event; premiums and end-of-period outgo must use their correct discount times; and a state with no remaining contract cash flows must have reserve zero.

Check Your Understanding

For each problem, use the stated timing convention: premiums are paid at the start of a year, and benefits are paid at its end.

A one-year contract has $i=4\%$. From active, the year-end probabilities are 0.80 active, 0.10 disabled, and 0.10 dead. The respective benefits are $0, $500, and $1,500; the premium at time 0 is $80. Find the active-state reserve. (Answer: $112.31)
For a one-year contract with $i=5\%$, a disabled insured has probabilities 0.25 of recovery to active, 0.60 of remaining disabled, and 0.15 of death. The respective transition benefits are $200, $600, and $1,000, with no premium due while disabled. Find the disabled-state reserve. (Answer: $533.33)
A two-year contract has active, disabled, and dead states. In each year, active transitions with probabilities 0.50 to active, 0.30 to disabled, and 0.20 to dead; disabled transitions with probabilities 0.20 to active, 0.60 to disabled, and 0.20 to dead. The year-end benefits are $0, $300, and $900 according to destination state. A $100 premium is paid at the start of each year when active and none otherwise. With $i=5\%$, find the reserve at time 0 given active status. (Answer: $329.93)
At time 0, a one-year contract has $i=0$. From active, transition probabilities to active, disabled, and dead are 0.70, 0.20, and 0.10; from disabled they are 0.10, 0.70, and 0.20. End-of-year benefits by destination are $0, $400, and $1,000. The active premium is $100 and the disabled premium is zero. Find each living-state reserve, then the expected reserve for a population that is 75% active and 25% disabled. (Answer: Active reserve $80; disabled reserve $480; population average $180)
A contract has two living states, active and disabled, and a dead termination state. At time 0, the state occupancy probabilities are 0.60 active, 0.30 disabled, and 0.10 dead. The corresponding conditional reserves are $250, $700, and $0. Calculate the expected reserve across the population. (Answer: $360)

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