Why Begin with a State Map?
Integrated benefit cases often combine provisions that respond to different events: a pension may depend on retirement, disability income on being disabled, and a care supplement on entering or remaining in long-term care. Before valuing any of them, translate the contract language into a model that makes the insured person’s possible conditions and movements explicit. A clear state map is a compact way to do this.
The map is not a substitute for selecting transition probabilities or valuing cash flows. Its purpose is to prevent a more basic error: applying the right benefit to the wrong condition, transition, or time. The discipline is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Specify the Map Before the Cash Flows
For the examples below, use a discrete-time model with annual steps. Let \(p_{ij}\) be the probability of being in state \(j\) at the end of a year, conditional on being in state \(i\) at the beginning of that year. Each row of probabilities must sum to 1, and a transition not allowed by the map has probability 0. This convention describes annual transitions; it does not specify when within the year a transition occurs.
A useful map names states, draws or lists the permitted arrows, and records the initial condition. Then write each provision beside the event that triggers it. An annual pension paid to someone in care is generally attached to occupancy of the care state. A lump sum paid only on becoming disabled is attached to the active-to-disabled arrow. Those triggers are not interchangeable.
| Model element | Question to answer | Typical notation |
|---|---|---|
| State | What condition is the person in at the valuation date? | \(A\): active; \(D\): disabled |
| Allowed transition | Which movements can occur over one model step? | \(A\to D\) |
| Initial condition | Where does the model start? | In state \(A\) with probability 1 |
| State cash flow | What is paid because the person occupies a state? | Annual income while in \(D\) |
| Transition cash flow | What is paid because a particular movement occurs? | Lump sum on \(A\to D\) |
The state definitions must be sufficiently precise to make them mutually exclusive. For example, “retired” and “in care” overlap if a retired person can also be receiving care. A map can avoid this ambiguity by using separate states such as “retired, not in care” and “retired, in care,” or by defining the care state to include the base pension and care supplement. Choose a representation that makes every benefit trigger unambiguous.
Probability Propagation and Cash-Flow Placement
Write \(\pi_i(t)\) for the probability of occupying state \(i\) at time \(t\). Given the state probabilities at time \(t\), the probability of occupying state \(j\) one year later is
This follows from the law of total probability: the person must have been in exactly one state \(i\) at time \(t\), and then move from that state to \(j\). The mutually exclusive starting states partition all possibilities, so their contributions add. In particular, probabilities at each time should sum to 1, including the probability of already being dead if death is an absorbing state.
Suppose a payment \(b_j\) is made at year-end for occupying state \(j\), and a separate payment \(c_{ij}\) is made at that year-end for a transition from \(i\) to \(j\). If those are distinct benefits, their expected total payment is
The first sum uses end-of-year state probabilities; the second uses the probability of each transition during the year. When a payment is made only on a new transition, do not count people who were already in the destination state at the start of the year. For a year-end payment and effective annual interest rate \(i\), its present value at the start of the year is the expected payment multiplied by \(v=1/(1+i)\). A payment at another time requires a discount factor appropriate to that time.
Worked Example: Disability Income and a Death Benefit
A hypothetical one-year disability contract begins with a person active. The states are active \(A\), disabled \(I\), and dead \(D\). Over one year, the transition probabilities are \(p_{AA}=0.90\), \(p_{AI}=0.06\), \(p_{AD}=0.04\); \(p_{IA}=0.25\), \(p_{II}=0.70\), \(p_{ID}=0.05\); and \(p_{DD}=1\). All other transitions are prohibited. A $1,000 premium is received at time 0 while the person is active. At year-end, $12,000 is paid if the person is disabled, and $50,000 is paid if death occurred during the year. The effective annual interest rate is 4%. Find the expected present value of the insurer’s net cash flow, treating premiums as positive and benefits as negative.
Method: map the benefit triggers, then use the corresponding probabilities. The initial condition is \(\pi_A(0)=1\), with \(\pi_I(0)=\pi_D(0)=0\). Thus the end-of-year state probabilities are \(0.90\) active, \(0.06\) disabled, and \(0.04\) dead. The disability income is state-contingent on occupying \(I\) at year-end. The death benefit is transition-contingent on entering \(D\) during the year; because the person starts active, its probability is \(0.04\).
Expected disability income is \(0.06(12{,}000)=\$720\). Expected death benefit is \(0.04(50{,}000)=\$2{,}000\). The expected year-end benefit total is therefore \(\$720+\$2{,}000=\$2{,}720\), with present value \(\$2{,}720/1.04=\$2{,}615.38\). The premium is received at time 0, so no discounting applies to it. The expected present value of net insurer cash flow is
Check: The end-of-year probabilities sum to \(0.90+0.06+0.04=1\), and the calculation discounts the benefits once, not the time-0 premium. A common mistake is to multiply the death benefit by the probability of being dead at year-end in a multi-year case; that would wrongly include deaths from earlier years. Here there are no earlier deaths, so the end-of-year death probability and the probability of a new death transition coincide.
Integrated Care and Pension Provisions
A state map for long-term care should distinguish care occupancy from entry into care if the contract treats them differently. For example, a monthly or annual care supplement may be payable throughout care, while a one-time assessment benefit may be payable only on entering care. A pension may continue in care, stop in care, or be supplemented there; the map itself does not decide which treatment applies. The wording of the hypothetical provision determines the cash flow attached to each state or arrow.
Worked Example: Retirement, Care Entry, and a Continuing Pension
A hypothetical member begins the year retired, in state \(R\). The other states are in care \(C\) and dead \(D\). Permitted transitions are \(R\to R,C,D\), \(C\to C,R,D\), and \(D\to D\). From \(R\), the one-year probabilities are \(0.75\) to \(R\), \(0.15\) to \(C\), and \(0.10\) to \(D\). From \(C\), they are \(0.70\) to \(C\), \(0.20\) to \(R\), and \(0.10\) to \(D\). The member receives a $18,000 year-end pension in either \(R\) or \(C\), an additional $7,000 at year-end if in \(C\), a one-time $3,000 payment on transition \(R\to C\), and a $9,000 death benefit on transition into \(D\). The effective annual interest rate is 5%. Find the present value of the expected one-year benefits.
Method: separate occupancy benefits from arrow-triggered benefits. The initial condition is \(R\) with probability 1. At year-end, the probabilities of \(R,C,D\) are \(0.75,0.15,0.10\). The pension is paid in \(R\) and \(C\), so its expected amount is \((0.75+0.15)(18{,}000)=\$16{,}200\). The care supplement is paid only in \(C\), giving \(0.15(7{,}000)=\$1{,}050\).
The one-time care-entry payment is attached to \(R\to C\), which has probability \(0.15\), so its expected amount is \(0.15(3{,}000)=\$450\). The death benefit is attached to transition into \(D\), with probability \(0.10\), so its expected amount is \(0.10(9{,}000)=\$900\). The expected year-end total is \(\$16{,}200+\$1{,}050+\$450+\$900=\$18{,}600\). Discounting gives
Check: A person who moves from \(R\) to \(C\) receives both the base pension and care supplement under the stated terms, as well as the one-time entry payment. No care-entry payment is assigned to someone already in \(C\), since the initial state is \(R\) and the only care entrants in this one-year model use the \(R\to C\) arrow. A common mistake is to apply the care supplement to the transition probability and omit the base pension that continues in care.
Multi-Year Maps: Track the Path and the Initial Condition
In a multi-year case, the cash flow at time \(t\) can depend on more than the state occupied then. A death benefit is a typical example: the state \(D\) at time \(t\) includes people who died earlier, but a death benefit is usually paid only once, on the transition into \(D\). Use the state probabilities to value occupancy payments and transition probabilities to value payments triggered by new movements. An absorbing dead state is convenient for probability bookkeeping; it does not imply repeated death payments.
Worked Example: A Deferred Pension with Care and a New-Death Benefit
A hypothetical pension arrangement has states active \(A\), vested but not retired \(V\), retired \(R\), in care \(C\), and dead \(D\). The allowed transitions and annual probabilities are: from \(A\), \(0.80\) to \(A\), \(0.10\) to \(V\), \(0.05\) to \(R\), and \(0.05\) to \(D\); from \(V\), \(0.60\) to \(V\), \(0.30\) to \(R\), and \(0.10\) to \(D\); from \(R\), \(0.80\) to \(R\), \(0.15\) to \(C\), and \(0.05\) to \(D\); from \(C\), \(0.70\) to \(C\), \(0.20\) to \(R\), and \(0.10\) to \(D\); and \(D\to D\) with probability 1. Initially the member is in \(A\). At the end of year 2, a $10,000 pension is paid in \(R\), a total of $15,000 is paid in \(C\), and a $20,000 death benefit is paid for a death transition during year 2. Find the present value at time 0 of the expected year-2 payment using an effective annual interest rate of 4%.
Method: propagate the state probabilities one year at a time. At the end of year 1, the probabilities in \(A,V,R,C,D\) are \(0.80,0.10,0.05,0,0.05\). The year-2 probability of \(A\) is \(0.80(0.80)=0.64\). The probability of \(V\) is \(0.80(0.10)+0.10(0.60)=0.14\). The probability of \(R\) is \(0.80(0.05)+0.10(0.30)+0.05(0.80)=0.11\). The probability of \(C\) is \(0.05(0.15)=0.0075\).
The year-2 probability of being dead is the \(0.05\) already dead at the end of year 1, plus new deaths during year 2: \(0.80(0.05)+0.10(0.10)+0.05(0.05)=0.0525\). Thus the year-2 dead-state probability is \(0.1025\). The full distribution checks: \(0.64+0.14+0.11+0.0075+0.1025=1\).
The expected state-contingent pension payment is \(0.11(10{,}000)+0.0075(15{,}000)=\$1{,}212.50\). The death benefit uses only the new-death transition probability, so its expected amount is \(0.0525(20{,}000)=\$1{,}050\). The expected year-2 payment is \(\$1{,}212.50+\$1{,}050=\$2{,}262.50\). It is paid two years from time 0, so
Check and trap: The total year-2 probability of being dead is \(0.1025\), but using \(0.1025(20{,}000)\) would pay the death benefit again for the \(0.05\) already dead after year 1. That incorrect calculation would overstate the expected year-2 payment by \(0.05(20{,}000)=\$1{,}000\). The correct transition probability for new deaths is \(0.0525\).
Common State-Mapping Errors
A state map is useful only if its states, transitions, and cash-flow triggers match the benefit provisions. Before valuing an integrated case, check the following points:
- Overlapping states: “retired” and “in care” may both apply. Decide whether to create combined states or define a care state that includes the pension.
- Missing or impossible arrows: Write down the permitted movements. Do not silently allow recovery, retirement, or re-entry if the model excludes it.
- State versus transition trigger: A payment while disabled is not the same as a lump sum on becoming disabled.
- Initial condition: Starting in a care state is different from entering care during the projection. A one-time entry benefit should not be assigned merely because the person occupies care.
- Payment timing: A year-end amount is discounted for the full year; a time-0 premium is not.
- Absorbing states: Probability can remain in an absorbing dead state, but a transition benefit is normally paid only when the transition into that state occurs.
Check Your Understanding
For each problem, map the relevant trigger before calculating. Assume annual steps and the stated payment timing.
- A hypothetical one-year contract begins in the active state. The year-end probabilities are 0.82 active, 0.12 disabled, and 0.06 dead. A $9,000 benefit is paid if disabled at year-end, and $30,000 is paid on death during the year. At an effective annual rate of 3%, find the present value of expected benefits. (Answer: $2,796.12)
- A hypothetical retired member has year-end probabilities 0.70 retired, 0.20 in care, and 0.10 dead. A $15,000 pension is paid in retirement or care, a $4,000 supplement is paid in care, and $5,000 is paid on a death transition. At an effective annual rate of 4%, find the present value of expected one-year benefits. (Answer: $14,230.77)
- A member starts vested \(V\). In year 1, the probabilities are \(0.50\) to \(V\), \(0.40\) to retired \(R\), and \(0.10\) to dead \(D\). From \(V\), the year-2 probabilities are \(0.50\) to \(V\), \(0.40\) to \(R\), and \(0.10\) to \(D\). From \(R\), they are \(0.80\) to \(R\), \(0.10\) to care \(C\), and \(0.10\) to \(D\). State \(D\) is absorbing. At the end of year 2, $8,000 is paid in \(R\), $12,000 in \(C\), and $10,000 on a new death transition. Find the expected year-2 payment, excluding any discounting. (Answer: $5,540)
- A member begins active. During the year, the probabilities are 0.70 for active-to-active, 0.20 for active-to-retired, and 0.10 for active-to-dead. A $3,000 lump sum is paid only on active-to-retired transition, and a separate $5,000 pension is paid at year-end if retired. Find the expected total year-end payment. (Answer: $1,600)