Introduction
A long-term care contract can pay different amounts depending on the policyholder’s current health state. A healthy policyholder may receive no benefit, a person recovering from an impairment may receive a partial benefit, and a person in a more severe care state may receive a larger amount. The calculation begins by translating the contract wording into a payment amount for each state.
This tutorial focuses on the amount payable at one assessment, such as a monthly benefit assessment. It does not calculate the present value of a stream of future payments or determine a premium. Assume that any required eligibility conditions have already been tested; a benefit is assigned only where the contract says it is payable. No interest rate is needed for a single assessment because there is no discounting.
Translate the Contract Into State Payments
The state labels describe the policyholder’s condition under the model. The contract then specifies what amount, if any, is associated with each state. The schedule can be written as a table before calculating an expected payment. This makes it easier to separate two tasks: first determine the payment conditional on each state, then weight those payments by the probabilities of the states.
For a fixed-indemnity benefit, the payment is specified directly by state. For a reimbursement benefit, the payment usually depends both on the state and on eligible expense. If a contract reimburses a proportion \(r_j\) of eligible expense \(e_j\), subject to an insurer-payment cap \(c_j\), the payment in state \(j\) is
This formula applies when the cap is a maximum on the insurer’s payment, as stipulated in the contract. If the cap instead applies to eligible expense before the reimbursement percentage is applied, the formula would be different. Read carefully which quantity is capped.
| Contract feature | Question to answer | Typical payment rule |
|---|---|---|
| State trigger | Which state or states qualify? | Set the payment to zero in nonqualifying states |
| Fixed benefit | What amount is specified for the state? | Use the stated amount \(b_j\) |
| Reimbursement | What expense is eligible, and what proportion is reimbursed? | Calculate \(r_j e_j\) |
| Payment cap | Does the cap limit the insurer payment? | Use the smaller of the uncapped payment and the cap |
The probabilities must describe the distribution of states at the assessment being valued. They must be mutually exclusive and collectively exhaustive for the modeled population, so they sum to one. If death is included in that distribution and the contract pays no benefit at death, its probability still belongs in the state distribution; its payment is simply zero.
Expected Payment From State Probabilities
Let \(p_j=P(J=j)\) be the probability that the policyholder is in state \(j\) at the assessment. When each state has a specified payment \(b_j\), the expected payment is the probability-weighted average:
This is an expectation for one assessment under the stated state distribution. It is not a probability of entering care and does not by itself describe a future payment stream. If the state probabilities are conditional on a policyholder being alive, they must not be combined as though they also included a death probability; use a distribution whose conditioning matches the question.
Worked Example: Fixed Benefits by State
A hypothetical contract makes one monthly assessment and pays $180 in recovery, $600 in moderate care, and $1,100 in severe care. It pays nothing in the healthy or death states. At the assessment, the state probabilities are \(P(H)=0.25\), \(P(R)=0.15\), \(P(C_1)=0.30\), \(P(C_2)=0.20\), and \(P(D)=0.10\). Find the expected payment. There is no discounting or interest-rate adjustment for this single assessment.
Method. List the payment for each state, including zero amounts, and multiply each payment by its state probability. The probabilities sum to \(0.25+0.15+0.30+0.20+0.10=1.00\), as required.
The state payments are \(b_H=0\), \(b_R=\$180\), \(b_{C_1}=\$600\), \(b_{C_2}=\$1{,}100\), and \(b_D=0\). Therefore,
The expected payment is $427 per monthly assessment. As a check, only the three benefit-eligible states contribute to the sum; including the healthy and death probabilities as zero-payment terms does not change it. A common mistake is to omit one state’s weighted benefit or to multiply a benefit by the wrong state probability. Writing the five terms explicitly prevents that error.
Reimbursement Amounts and Payment Caps
For a reimbursement contract, do not substitute the expense itself for the insurer’s payment. First multiply eligible expense by the reimbursement percentage; then apply any cap on the insurer payment. Only after the state-specific amounts have been determined should they be weighted by state probabilities.
Worked Example: State-Specific Reimbursement With Caps
A hypothetical policy makes one monthly assessment. In recovery it reimburses 60% of eligible expense, up to a maximum insurer payment of $300. In moderate care it reimburses 70%, capped at $650. In severe care it reimburses 80%, capped at $1,200. There is no benefit in the healthy or death states. Eligible expenses are $400 in recovery, $1,000 in moderate care, and $1,800 in severe care. The state probabilities are \(0.25,0.20,0.25,0.15,0.15\) for \(H,R,C_1,C_2,D\), respectively. Find the expected payment. No interest or mortality intensity is needed; the stated probabilities already give the assessment-time distribution, including death.
Method. For each eligible state, calculate the percentage of expense and compare it with that state’s insurer-payment cap. The payment is the smaller amount.
In recovery, 60% of $400 is \(0.60(400)=\$240\). Since $240 is below the $300 cap, the payment is $240. In moderate care, 70% of $1,000 is \(0.70(1{,}000)=\$700\). The $650 cap reduces the payment to $650. In severe care, 80% of $1,800 is \(0.80(1{,}800)=\$1{,}440\). The $1,200 cap reduces the payment to $1,200.
The probabilities sum to \(0.25+0.20+0.25+0.15+0.15=1.00\). Now weight each payment by its state probability:
The expected payment is $390.50 per assessment. The common mistake is to treat the caps as limits on eligible expense and apply the reimbursement percentage afterward. Here they are explicitly caps on insurer payments, so the moderate-care amount is \(\min(\$700,\$650)=\$650\), not 70% of $650.
When Expense Varies Within a State
A state may not determine a unique payment if eligible expenses differ among policyholders in that state. In that case, use the conditional expected payment within the state. If \(B\) is the payment and \(J=j\), the contribution from state \(j\) to the overall expectation is \(P(J=j)E[B\mid J=j]\). Thus
For a reimbursement benefit with a cap, apply the percentage and cap to each expense outcome before averaging. In general, the average of capped payments is not the same as applying the cap to the average expense. The cap creates a nonlinear payment rule.
Worked Example: Averaging Capped Payments Within Severe Care
A hypothetical monthly benefit reimburses 75% of eligible expense in severe care, subject to a maximum insurer payment of $900. Among policyholders in severe care, 40% have $800 of eligible expense and 60% have $1,600. The probability of severe care at the assessment is 0.30; all other states have zero payment. Find the expected payment. No interest adjustment is made.
Method. Calculate the payment for each expense group after applying the cap, find the conditional mean payment in severe care, and multiply by the probability of severe care.
For the $800 expense group, 75% of expense is \(0.75(800)=\$600\), below the $900 cap, so the payment is $600. For the $1,600 group, 75% of expense is \(0.75(1{,}600)=\$1{,}200\), so the payment is reduced to $900. The mean payment conditional on severe care is
The overall expected payment is therefore
The answer is $234 per assessment. To check the calculation directly, the two expense groups contribute \(0.30(0.40)(600)=\$72\) and \(0.30(0.60)(900)=\$162\), for a total of $234.
A tempting but incorrect method is to average expenses first: \(0.40(800)+0.60(1{,}600)=\$1{,}280\), then calculate 75% of $1,280 and cap it at $900. That gives \(0.30(900)=\$270\), which is wrong. The correct calculation gives \(0.30[0.40(600)+0.60(900)]=\$234\). The cap applies to each person’s payment before averaging.
Common Traps and a Reliable Workflow
State-dependent benefit calculations are usually short, but errors often come from mixing up the contract rule and the probability calculation. Keep the two stages separate: determine what the contract pays in each relevant state, then calculate the probability-weighted average. In a reimbursement design with variable expense, the first stage may itself require an average over expense outcomes conditional on the state.
Determine which states qualify at this assessment and whether the supplied probabilities are unconditional or conditional on survival or another event.
Record a payment for every possible state, including zero payments for states not covered by the stated benefit.
For reimbursement, apply the percentage to eligible expense and then apply a cap to the quantity it is defined to limit. If expense varies within a state, calculate each outcome before averaging.
Use the probability of each state for the same assessment and confirm that the state probabilities sum to one.
Report the expected amount per assessment, distinguish it from eligible expense, and verify the result by summing the state contributions.
- Do not pay benefits in every nonhealthy state automatically. The contract determines whether recovery or a particular care level qualifies.
- Do not confuse benefit amount with expense. A reimbursement percentage, cap, or both may make the insurer’s payment different from the expense incurred.
- Do not apply one state’s schedule to another state. Benefit amounts and caps may differ by state.
- Do not average capped payments incorrectly. When expenses vary, apply the contract rule to each expense outcome before calculating the conditional average.
- Do not discard the death probability without checking the distribution. If death is one of the exhaustive states, its zero payment is part of the expectation calculation.
Check Your Understanding
For each question, calculate the payment schedule first, then use the stated probabilities. Amounts are per assessment, with no discounting.
- A hypothetical fixed-benefit policy pays $120 in recovery, $500 in moderate care, and $900 in severe care, and nothing in healthy or death. The probabilities for \(H,R,C_1,C_2,D\) are \(0.30,0.10,0.25,0.20,0.15\). Find the expected payment. (Answer: $317)
- A contract reimburses 60% of moderate-care expense, capped at $500, and 80% of severe-care expense, capped at $1,000. Expenses are $700 in moderate care and $1,500 in severe care. The probabilities of moderate and severe care are 0.30 and 0.20; all other states pay zero. Find the expected payment. (Answer: $326)
- A hypothetical fixed-indemnity contract pays $80 in recovery, $300 in moderate care, and $700 in severe care. The probabilities for \(H,R,C_1,C_2,D\) are \(0.40,0.10,0.20,0.20,0.10\). Calculate the expected payment. (Answer: $208)
- In severe care, half of policyholders have $800 in eligible expense and half have $1,600. The contract reimburses 75% of expense, capped at an insurer payment of $900. Severe care has probability 0.30, and all other states pay zero. Find the expected payment, applying the cap separately to each expense group. (Answer: $225)