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Exam ALTAM · Long-Term Care Models · Tutorial 196 of 999

Integrated Long-Term Care Pricing Case

Build and check a net premium by projecting care-state cash flows, applying contract limits, and discounting premiums and benefits at their actual payment times.

Intermediate 12 min read

What You'll Learn

  • Set up an annual state model with active, waiting, low-care, high-care, and death states
  • Apply an elimination period before determining when benefits become payable
  • Calculate expected capped benefits using state probabilities and care-level amounts
  • Value premiums when they are payable only while the insured remains active
  • Use the equivalence principle to calculate and check a level net premium
  • Track a maximum benefit-payment period without paying beyond the cap

Why an Integrated Pricing Case Needs Careful Timing

A long-term care (LTC) premium depends on more than the amount payable in each care level. The expected cost also depends on when care begins, whether recovery is possible, how mortality varies by state, when an elimination period is satisfied, and how long benefits can be paid. A pricing calculation is reliable only when each of these assumptions is tied to the correct cash-flow date.

This tutorial develops a hypothetical annual model and applies the equivalence principle: the net premium is chosen so that the expected present value of premiums equals the expected present value of benefits. The setup is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Model definition: At each integer time, the insured occupies exactly one state. A transition probability is the probability of moving from the state at the start of a year to a state at its end. Death is absorbing. Unless otherwise stated, benefits are paid at the end of a year when the contract conditions for payment have been met. Premiums are paid at the start of the stated year.

Worked Example: A Two-Year Net Premium With an Elimination Period

Worked Example: A Two-Year Net Premium With an Elimination Period

Scenario and assumptions. Consider a hypothetical policy issued to a person in the active state at time 0. The annual effective interest rate is \(i=4\%\), so \(v=1/1.04\). The states are active \(A\), waiting-period care \(W\), eligible low care \(L\), eligible high care \(H\), and dead \(D\). The elimination period is one full year in care: an insured entering \(W\) must remain in care for that year to move to an eligible state. On transition from \(W\), recovery to \(A\) is possible. The annual transition probabilities are:

$$ \begin{array}{c|ccccc} & A & W & L & H & D\\ A & 0.85 & 0.10 & 0 & 0 & 0.05\\ W & 0.10 & 0.10 & 0.50 & 0.20 & 0.10\\ L & 0.20 & 0 & 0.55 & 0.15 & 0.10\\ H & 0.05 & 0 & 0.10 & 0.65 & 0.20\\ D & 0 & 0 & 0 & 0 & 1 \end{array} $$

The probabilities of death during a year are state-specific: 0.05 from \(A\), 0.10 from \(W\) or \(L\), and 0.20 from \(H\). A benefit of $10,000 is payable at year-end in state \(L\); in state \(H\), the stated benefit of $22,000 is subject to a maximum annual benefit of $18,000. Thus the payable high-care amount is $18,000. Benefits are payable for at most two years, and premiums are waived after leaving \(A\). A level premium is due at time 0 and, if the insured is in \(A\) at time 1, at time 1.

Step 1: Project the state probabilities. Starting in \(A\), the time-1 probabilities are \(0.85\) in \(A\), \(0.10\) in \(W\), and \(0.05\) in \(D\). There is no eligible-care state at time 1, so no benefit is paid at the end of year 1.

For year 2, multiply the time-1 probabilities by the transition probabilities. For example, the probability of being in \(L\) at time 2 is \(0.10(0.50)=0.05\), because only those in \(W\) at time 1 can enter \(L\) in year 2. The probability of being in \(H\) at time 2 is \(0.10(0.20)=0.02\). The complete time-2 vector, in the order \(A,W,L,H,D\), is:

$$ \begin{aligned} \boldsymbol{p}_2 &=(0.85,0.10,0,0,0.05) \begin{pmatrix} \text{transition probabilities} \end{pmatrix}\\ &=(0.7325,\ 0.095,\ 0.050,\ 0.020,\ 0.1025). \end{aligned} $$

For clarity, the death probability at time 2 includes the 0.05 already dead at time 1, as well as \(0.85(0.05)+0.10(0.10)=0.0525\) newly dead during year 2. Thus \(P(D\text{ at time }2)=0.05+0.0525=0.1025\). The components sum to 1.0000.

Step 2: Calculate expected benefits. At the end of year 1, benefits are zero. At the end of year 2, the probabilities of being in eligible states \(L\) and \(H\) are 0.05 and 0.02. Therefore:

$$ E[B_2]=0.05(10{,}000)+0.02(18{,}000) =500+360=\$860. $$

The benefit cap is applied before the expected value: using $22,000 rather than the capped $18,000 would overstate the high-care component. The expected present value of benefits is:

$$ \operatorname{EPV}(B) =\frac{860}{1.04^2} =\$795.1183\ldots \approx \$795.12. $$

Step 3: Calculate the premium value. A premium is paid at time 0 with certainty. The time-1 premium is payable only if the insured is active at time 1, an event with probability 0.85. Hence the expected present value per dollar of level premium is:

$$ 1+0.85v =1+\frac{0.85}{1.04} =\frac{189}{104} =1.8173077\ldots. $$

Equating premium and benefit present values gives:

$$ P =\frac{795.118343\ldots}{1.8173077\ldots} =\frac{1{,}075{,}000}{2{,}457} =\$437.5254\ldots \approx \boxed{\$437.53}. $$

Check. The expected present value of premiums at the unrounded premium is \(437.5254\ldots(1.8173077\ldots)=\$795.1183\ldots\), matching the benefit value. This is a net premium: expenses, profit margins, and risk margins have not been added.

Why the Equivalence Calculation Works

Let \(B_t\) be the benefit paid at time \(t\), and let \(I_t\) indicate whether a premium is payable then. If the level premium is \(P\), the present value of premiums is \(P\sum_t v^t I_t\), while the present value of benefits is \(\sum_t v^t B_t\). The net premium equivalence principle equates their expectations.

$$ P =\frac{\displaystyle\sum_t v^t E[B_t]} {\displaystyle\sum_t v^t P(I_t=1)}. $$

To justify the formula, discount each cash flow to time 0 and add the discounted amounts. By linearity of expectation, the expected value of the sum is the sum of the expected values; no independence assumption between different years is needed. Since the premium amount \(P\) is fixed, it factors out of the expected premium present value. Solving the resulting equation for \(P\) gives the displayed expression. The formula assumes the stated interest rate and transition model, and it applies to a net premium only; additional loadings require additional terms or a different pricing basis.

Worked Example: The Waiting Year Determines the Payment Date

Worked Example: The Waiting Year Determines the Payment Date

Scenario and assumptions. A hypothetical insured is already in state \(W\) at time 0, with one full year of the elimination period still to complete. The annual effective interest rate is 4%. At the end of the year, the probabilities from \(W\) are 0.10 to active \(A\), 0.10 to remain in \(W\), 0.50 to eligible low care \(L\), 0.20 to eligible high care \(H\), and 0.10 to death \(D\). The low-care benefit is $9,000. The stated high-care benefit is $21,000, capped at $15,000. A benefit is paid at year-end upon entering \(L\) or \(H\).

Method. Apply the elimination-period rule to the end-of-year state, then apply the benefit cap, and finally discount the expected payment one year.

The payable benefit amounts are $9,000 in \(L\) and $15,000 in \(H\). The expected year-end benefit is:

$$ E[B_1] =0.50(9{,}000)+0.20(15{,}000) =4{,}500+3{,}000 =\$7{,}500. $$

The present value is \(7{,}500/1.04=\$7{,}211.5384\ldots\), or $7,211.54 to the nearest cent. The probability of remaining in \(W\) is not itself a benefit probability: the benefit begins only for the 0.50 and 0.20 probabilities that reach eligible care states. A common error is to discount this payment for two years on the grounds that there is an elimination period. The insured in this scenario is already in the waiting state at time 0, and the contract pays at the end of the one year specified; discounting by \(1.04^2\) would use the wrong payment date.

Check. The probability of a year-end benefit is \(0.50+0.20=0.70\), and the expected amount conditional on a benefit is \(7{,}500/0.70=\$10{,}714.29\). Their product is $7,500, consistent with the direct calculation.

Worked Example: A Two-Payment Maximum With Recovery

Worked Example: A Two-Payment Maximum With Recovery

Scenario and assumptions. A hypothetical insured starts in eligible low care \(L\); the elimination period has already been satisfied. The annual effective interest rate is 4%. Benefits are paid at year-end if the insured is in \(L\) or \(H\) then, but no more than two annual payments are allowed. A low-care benefit of $10,000 is payable; the stated high-care benefit of $22,000 is capped at $18,000. Recovery ends eligibility for this claim, and there is no recurrent claim after recovery. The transition probabilities are:

$$ \begin{array}{c|cccc} & R & L & H & D\\ L & 0.20 & 0.50 & 0.20 & 0.10\\ H & 0.10 & 0.10 & 0.60 & 0.20\\ R & 1 & 0 & 0 & 0\\ D & 0 & 0 & 0 & 1 \end{array} $$

Here \(R\) is recovered and no longer eligible for this claim. Year 1: From \(L\), the expected benefit is \(0.50(10{,}000)+0.20(18{,}000)=\$8{,}600\). The \(0.20\) probability of recovery and \(0.10\) probability of death produce no year-end benefit.

Year 2: At time 1, the insured is in \(L\) with probability 0.50 and in \(H\) with probability 0.20. Conditional on being in \(L\), the expected year-2 benefit is \(0.50(10{,}000)+0.20(18{,}000)=\$8{,}600\), contributing \(0.50(8{,}600)=\$4{,}300\). Conditional on being in \(H\), it is \(0.10(10{,}000)+0.60(18{,}000)=\$11{,}800\), contributing \(0.20(11{,}800)=\$2{,}360\). Thus \(E[B_2]=4{,}300+2{,}360=\$6{,}660\).

Year 3: Any insured receiving payments in both years 1 and 2 has reached the two-payment maximum. Those who recover or die do not become eligible again under the stated assumptions. Therefore \(B_3=0\); projecting another year of care benefits without tracking the payment maximum would be incorrect.

The expected present value is calculated from the unrounded amounts:

$$ \begin{aligned} \operatorname{EPV}(B) &=\frac{8{,}600}{1.04}+\frac{6{,}660}{1.04^2}\\ &=8{,}269.230769\ldots+6{,}157.544378\ldots\\ &=\$14{,}426.775147\ldots \approx \boxed{\$14{,}426.78}. \end{aligned} $$

Check. The year-2 expected benefit includes only transitions from the care states occupied at time 1. The two-year maximum is respected, the cap is applied to the high-care amount, and there is no year-3 payment. Rounding is performed only after adding the unrounded discounted values.

Common Traps in an Integrated LTC Calculation

An integrated calculation is especially vulnerable to errors that are locally plausible but inconsistent with the contract timeline. Before calculating a premium, write down the state at each valuation time, the event that makes a benefit payable, and the exact payment date.

  • Counting the waiting state as eligible care. A person in \(W\) has not yet satisfied the elimination period in the first example; \(W\) has no immediate benefit.
  • Discounting to the wrong date. A benefit payable at the end of year 2 is multiplied by \(v^2\), not \(v\). A premium due at time 1 is discounted once, even if its payment depends on being active then.
  • Ignoring death already accumulated. In a full state-probability vector with absorbing death, the death probability at a later time includes those already dead as well as new deaths. Check that all state probabilities sum to 1.
  • Applying the cap after averaging. If the high-care benefit is capped at $18,000, use $18,000 in the expected value, not the uncapped $22,000.
  • Paying beyond a maximum duration. A care-state probability alone does not establish that another benefit is payable; the number of prior payments may also matter.
  • Mixing rounded terms. Keep full precision through the final sum, then round the final dollar amount to cents.
Key takeaway: Price benefits from eligible state probabilities and actual payment dates; price premiums from the probability and timing of premium payment. The equivalence calculation is only as sound as those contract conditions and state projections.

Check Your Understanding

For each problem, keep the stated payment date and eligibility condition explicit. Amounts are hypothetical.

  1. A person is in the waiting state at time 0. At the end of year 1, transition probabilities are 0.40 to low care, 0.30 to high care, 0.10 to active, 0.10 to remain waiting, and 0.10 to death. The annual effective interest rate is 3%. Low care pays $8,000; high care pays $17,000 subject to a $14,000 cap. A benefit is payable at the end of year 1 upon entering eligible care. Find its expected present value. (Answer: $7,184.47)
  2. A level net premium is paid at time 0 with certainty and at time 1 only if the insured is active, an event with probability 0.90. The annual effective interest rate is 5%, and the expected present value of benefits is $950. Find the net premium to the nearest cent. (Answer: $511.54)
  3. At time 1, the probability of being in the waiting state is 0.12. From that state during year 2, the probabilities of entering low care and high care are 0.40 and 0.25. The annual effective interest rate is 4%. Low care pays $9,000; high care pays $20,000 but is capped at $16,000. Find the expected present value of the year-2 benefit. (Answer: $843.20)
  4. A policy starts active. During each year, active transitions with probabilities 0.80 to active, 0.15 to waiting, and 0.05 to death. From waiting, probabilities are 0.10 to active, 0.20 to waiting, 0.40 to eligible low care, 0.20 to eligible high care, and 0.10 to death. Death is absorbing. Find the probability of being dead at time 2 and the time-2 expected benefit if low care pays $10,000 and high care pays $15,000. (Answer: Probability dead = 0.1050; expected benefit = $1,050)
  5. An insured starts in low care with the elimination period satisfied. Benefits are paid at each year-end in low care or high care, subject to a two-payment maximum. From low care, probabilities to recovered, low care, high care, and death are 0.20, 0.50, 0.20, and 0.10; from high care they are 0.10, 0.10, 0.60, and 0.20. Recovery is permanent. Low care pays $10,000; high care pays $22,000 capped at $18,000. At 4% annual effective interest, find the expected present value of benefits over the maximum payment period. (Answer: $14,426.78)

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