Death and Other Absorbing-State Reserves

Introduction

A death benefit may be payable precisely when a policyholder enters the death state, while the reserve after entry into that state may be zero. These statements are consistent: the claim is a payment on the transition, whereas a reserve in the resulting state represents obligations still due in the future. The distinction also matters for other absorbing states. A state representing permanent care, for example, may have a substantial reserve if payments continue after entry.

This tutorial develops that distinction using prospective reserves: expected present values of future benefits and expenses less future premiums, conditional on the current state and valuation time. The key questions are when a cash flow is due and whether it remains due after the transition.

Definition: A state is absorbing if, once entered, the process cannot leave it. In a discrete-time model, state $z$ is absorbing when $p_{zz}(t)=1$ at each subsequent step. Its prospective reserve $V_z(t)$ is the expected present value, at time $t$, of future benefits and expenses less future premiums, conditional on being in state $z$ at time $t$. A benefit payable on entry into $z$ is a transition benefit; it is not automatically part of $V_z(t)$.

Two Different Values at an Absorbing Transition

Suppose a contract is in a transient state $i$ just before an instantaneous transition into an absorbing state $z$. Let $B_{iz}(t)$ be the net amount payable because of that transition, expressed as an insurer outgo less any premium received at the transition. Let $V_z(t+)$ be the reserve immediately after the transition and its associated payment. At that instant, with no time for interest to accrue,

\text{value immediately before the transition} = B_{iz}(t)+V_z(t+).

If a death claim is paid at death and no further cash flows are due after death, then $V_z(t+)=0$, but the value immediately before the transition is still the death benefit. If payments continue in the absorbing state, the pre-transition value includes both the transition amount and the reserve for those later payments.

A state being absorbing only says that the process cannot leave it. It does not say that the contract has no future cash flows there. A model must specify whether benefits, expenses, or premiums continue after absorption. For example, a hypothetical permanent-care benefit might continue for a fixed number of years; its reserve is generally positive even though recovery is impossible under the model.

One-Year Reserve Recursion

For a discrete-time model, take $V_i(t)$ to be the reserve at time $t$, just after any cash flows scheduled at that time. Suppose the next transition occurs at time $t+1$, at effective annual interest rate $i$, with $v=1/(1+i)$. Let $p_{ij}(t)$ be the probability of being in state $j$ at time $t+1$, conditional on state $i$ at time $t$. Let $C_{ij}(t+1)$ be the net outgo at time $t+1$ on the transition from $i$ to $j$: benefits and expenses less premiums due on that transition. All cash flows in this recursion are assumed to occur at the end of the year.

V_i(t)=v\sum_j p_{ij}(t)\left[C_{ij}(t+1)+V_j(t+1)\right].

This form assumes there are no other cash flows during the year. A premium due at time $t$, for example, is not included: the reserve is defined just after that cash flow. If cash flows occur at different times, they must be placed at their actual payment times and discounted accordingly.

Justification: Condition on the state $j$ reached at time $t+1$. On that outcome, the insurer has net outgo $C_{ij}(t+1)$ and then faces future expected net outgo with value $V_j(t+1)$. Weight each outcome by $p_{ij}(t)$, sum the conditional values, and discount one year by $v$. This gives the recursion. The transition payment and the successor-state reserve are separate terms because they refer to different cash flows and valuation positions.

For an absorbing death state $z$ with no cash flows after entry, $V_z(t)=0$. That zero boundary condition does not remove a death benefit due at the moment of death: the benefit belongs in $C_{iz}$. Conversely, if a periodic benefit continues after entry, its expected present value belongs in $V_z$. A terminal payment due only at the end of a specified contract term is another distinct cash flow: it is included in the state-contingent amount due at that term, not assumed to be a payment on entry into every absorbing state.

Worked Example: A Death Claim and a Zero Post-Death Reserve

A hypothetical two-year term policy has conditional annual death probability $0.20$ in each year, conditional on being alive at its start. It pays $5,000 at the end of the year of death and nothing on survival to the end of year 2. A $400 premium is due at time 1 if the insured is alive then. The effective annual interest rate is $i=0\%$. Find the reserve at time 0 just after any time-0 cash flows, and the reserve at time 1 just after the premium if the insured is alive.

Method. Enumerate the mutually exclusive death and survival outcomes, then value future net cash flows. There are no payments after entry into the death state, so its post-death reserve is zero.

At time 0. Death in year 1 has probability $0.20$, so its expected benefit is $0.20(5{,}000)=\$1{,}000$. Survival through year 1 followed by death in year 2 has probability $0.80(0.20)=0.16$, so the expected benefit for that outcome is $0.16(5{,}000)=\$800$. The premium at time 1 is received with probability $0.80$, giving expected premium income $0.80(400)=\$320$. With $i=0\%$, no discounting is required. Thus

V_{\text{alive}}(0)=1{,}000+800-320=\$1{,}480.00.

At time 1, given survival and immediately after the premium: only the second year remains. The death benefit has expected value $0.20(5{,}000)=\$1{,}000$, and there is no further premium. Hence $V_{\text{alive}}(1)=\$1{,}000.00$. At the moment of death at the end of year 2, the $5,000 claim is due; immediately after that claim is paid, $V_{\text{death}}=0$.

Check. From time 1, the expected benefit is $1,000. At time 0, the year-1 death benefit plus the discounted expected time-1 value for survivors less the expected premium is $1{,}000+0.80(1{,}000)-0.80(400)=1{,}480$, agreeing with the outcome calculation.

Reserves When Obligations Continue After Absorption

An absorbing state can carry future obligations. Suppose a hypothetical contract pays a fixed amount at each of the next $n$ year-ends after entry into a permanent state, with no uncertainty about payment once the state has been entered. Its reserve at entry is the present value of those scheduled payments. This is a state-specific prospective reserve: it is not the transition benefit that may also be due at entry.

Worked Example: A Continuing Benefit in a Permanent State

A hypothetical contract pays $1,000 at each of the next three year-ends after entry into a permanent care state. The state is absorbing, the payments are certain once the state is entered, and there are no premiums or expenses in that state. The effective annual interest rate is $i=5\%$. Find the reserve immediately after entry.

Method. Discount each future payment from its own due date. Here $v=1/1.05=0.95238095$, so the time-0 reserve at entry is

V_{\text{care}} =1{,}000(v+v^2+v^3) =1{,}000(0.95238095+0.90702948+0.86383760) =\$2{,}723.25.

The payments have present values $1{,}000/1.05=\$952.38095$, $1{,}000/1.05^2=\$907.02948$, and $1{,}000/1.05^3=\$863.83760$. Their unrounded sum is $2,723.24803, which rounds to $2,723.25. The reserve is positive because the care-state benefit continues after entry. If instead the contract paid a single amount only upon entry and nothing afterward, that amount would be a transition benefit and the post-entry reserve would be zero.

Check. The reserve exceeds the first $1,000 payment because it also includes two later payments. It is below the undiscounted total of $3,000 because all three payments are discounted.

Valuing Immediately Before Entry

When entry into an absorbing state can occur during a future year, the reserve before that year’s transition is a probability-weighted value, not simply the amount due on entry. Under the one-year assumptions above, if the only possible outcomes are entry into $z$ or remaining in a state with no subsequent obligations, then

V_i(t)=v\,p_{iz}(t)\left[B_{iz}(t+1)+V_z(t+1)\right].

This expression discounts both the transition payment and the successor-state reserve because both are valued at time $t+1$. If the contract has obligations on other outcomes, their net cash flows and successor reserves must also be included. If the transition occurs at a known instant and the valuation is immediately before that instant, there is no one-year discount or transition probability to apply: the conditional value is the transition amount plus the reserve immediately after entry.

Worked Example: A Transition Payment Plus a Care-State Reserve

A hypothetical one-year contract starts in an active state. At the end of the year, there is a $0.10$ probability of entering an absorbing permanent-care state and a $0.90$ probability of remaining active. If care begins, the contract pays $4,000 on entry and then pays $2,000 at each of the next two year-ends. If the insured remains active, the contract expires with no payment. There are no premiums or expenses, and $i=5\%$. Find the reserve at time 0.

Method. First calculate the reserve after entry into care; then include the transition payment and weight and discount the combined amount.

At the care-state entry time, $v=1/1.05=0.95238095$. The post-entry reserve is

V_{\text{care}}(1) =2{,}000(v+v^2) =2{,}000(0.95238095+0.90702948) =\$3{,}718.82.

On entry, the $4,000 transition payment is also due, so the total conditional value at the end of year 1 is $4{,}000+3{,}718.82086=\$7{,}718.82086$. The active-survival outcome has no payment or successor reserve. Therefore

V_{\text{active}}(0) =0.10(7{,}718.82086)v+0.90(0) =\$735.13.

Check. The conditional value on entry is greater than the $4,000 transition payment because two later care payments remain. The time-0 reserve is much smaller than that conditional amount because entry has probability $0.10$ and the amount is discounted for one year.

Common Timing Errors

The most frequent error is to count a transition payment twice: once in the transition cash flow and again in the reserve of the absorbing state. A different error is to omit continuing payments merely because the state is absorbing. State the reserve position and cash-flow timing before calculating.

Wrong: For the preceding example, use $V_{\text{care}}(1)=\$7{,}718.82$ by adding the $4,000 entry payment to the two-year care reserve and calling the result the post-entry reserve.
Correction: The post-entry reserve is only $V_{\text{care}}(1)=\$3{,}718.82$. The $4,000 is due on the transition, so it is added when valuing the transition outcome. Then multiply by the $0.10$ transition probability and discount one year to obtain $735.13 at time 0.
Wrong: Set every absorbing-state reserve to zero.
Correction: Set it to zero only when no future net cash flows remain after entry. Continuing benefits or expenses, and any future premiums, must be included according to the contract.
Wrong: Include a premium that was already paid at the reserve valuation time.
Correction: If the reserve is defined just after that premium, it is not a future cash flow. Include only premiums due after the valuation time and on outcomes where they are payable.

Key takeaway: Identify the valuation instant, the cash flow due on the transition, and the obligations remaining afterward. The transition amount belongs to the transition; the successor-state reserve belongs to the future. Apply transition probabilities and discounting only when valuing the outcome from an earlier time.

Check Your Understanding

For each problem, distinguish any payment on entry from the reserve after entry. Unless stated otherwise, payments occur at year-end and reserves are measured just after cash flows at the stated time.

A one-year policy pays $8,000 at the end of the year if death occurs, with probability $0.15$, and nothing otherwise. There are no payments after death. The effective annual interest rate is $4\%$. Find the time-0 reserve. (Answer: $1,153.85)
An absorbing state has no premiums or expenses and pays $900 at each of the next two year-ends. At $i=5\%$, find its reserve immediately after entry. (Answer: $1,673.47)
At the end of one year, a contract enters an absorbing state with probability $0.08$. It pays $3,000 on entry and has a post-entry reserve of $5,000 at that time. On the other outcome there are no future obligations. At $i=0\%$, find the time-0 reserve. (Answer: $640.00)
A two-year term policy has conditional annual death probability $0.10$ in each year, a $10,000 benefit at the end of the year of death, and a $600 premium due at time 1 if alive then. There are no payments after death, and $i=0\%$. Find the time-0 reserve just after any time-0 cash flows. (Answer: $1,360.00)
A hypothetical one-year contract has a $0.20$ probability of entering an absorbing state at the end of the year. Entry triggers a $12,000 payment, and the post-entry state pays $2,500 at each of the next three year-ends. There are no obligations on survival, and $i=5\%$. Find the time-0 reserve. (Answer: $3,582.50)

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