Numerical Net Premium for Multi-Level Care Benefits

From Care-State Probabilities to a Net Premium

A long-term care (LTC) contract may pay different benefit amounts in different care states. Pricing it requires more than finding the probability of entering care: the valuation must account for how long the insured occupies each benefit-paying state, the interest discounting of payments, and the conditions under which premiums are paid. A multi-state model keeps these pieces consistent.

This tutorial uses a hypothetical contract with four states: healthy, moderate care, severe care, and death. The model has no recovery transitions and no direct transition from healthy to severe care. These are assumptions for the examples, not universal features of LTC contracts. The calculations use continuous benefit and premium payments; changing payment timing requires valuing the actual payment pattern.

Definition: Let $p_j(t)$ be the probability that an insured who is healthy at issue occupies living state $j$ at time $t$. For a force of interest $\delta$, define the discounted expected occupancy of state $j$ by $z_j=\int_0^\infty e^{-\delta t}p_j(t)\,dt$. It is the expected present value of one dollar per year payable continuously while the insured occupies state $j$.

Model and Pricing Equation

Use the following constant transition intensities, all measured per year: healthy to moderate care at rate $\alpha$, healthy to death at rate $\mu_H$, moderate care to severe care at rate $\beta$, moderate care to death at rate $\mu_M$, and severe care to death at rate $\mu_S$. Death is absorbing. The total exit intensities from the three living states are $r_H=\alpha+\mu_H$, $r_M=\beta+\mu_M$, and $r_S=\mu_S$.

Suppose the contract pays continuously at rates $b_M$ and $b_S$ dollars per year while in moderate and severe care, respectively. Suppose also that a level premium $P$ dollars per year is payable continuously in every living state, with no waiver on entry to care. Applying the equivalence principle, the expected present value of premiums equals that of benefits:

P(z_H+z_M+z_S)=b_Mz_M+b_Sz_S, \qquad P=\frac{b_Mz_M+b_Sz_S}{z_H+z_M+z_S}.

The denominator reflects the stated premium condition: premiums continue in healthy, moderate-care, and severe-care states, and stop at death. If premiums stop at care entry or are payable at discrete dates, this denominator must be changed to match that contract.

Why the Discounted Occupancies Have a Simple Form

Starting healthy, the probability of remaining healthy to time $t$ is $p_H(t)=e^{-r_Ht}$. Therefore its discounted expected occupancy is the integral of that probability with interest discounting. For moderate care, entry occurs from healthy at intensity $\alpha$; each such entry is discounted to time zero and then weighted by the expected discounted time in moderate care before its next exit. The same reasoning applies to severe care, entered from moderate care.

z_H=\int_0^\infty e^{-(\delta+r_H)t}\,dt =\frac{1}{\delta+r_H}, \qquad z_M=\frac{\alpha z_H}{\delta+r_M}, \qquad z_S=\frac{\beta z_M}{\delta+r_S}.

For example, the expression for $z_M$ can also be seen by integrating the rate of transition into moderate care over time. Each unit of expected transition flow from healthy contributes the discounted expected occupancy $1/(\delta+r_M)$ after entry. Hence $z_M=\int_0^\infty e^{-\delta t}\alpha p_H(t)/(\delta+r_M)\,dt=\alpha z_H/(\delta+r_M)$. This reasoning relies on constant transition intensities and the stated one-way transitions; recovery or additional routes require the corresponding multi-state equations.

Specify transitions and cash flows.
State which transitions are possible, which states pay benefits, and when premiums are payable.

Calculate discounted occupancies.
Use transition intensities and the force of interest to find $z_H,z_M,z_S$.

Value each side of the equivalence equation.
Multiply each occupancy by its applicable payment rate, then divide expected benefit value by the premium annuity value.

Worked Example: State Probabilities After One Year

A hypothetical insured starts healthy. The transition intensities are $\alpha=0.10$, $\mu_H=0.05$, $\beta=0.20$, $\mu_M=0.10$, and $\mu_S=0.25$, per year. Find the probabilities of occupying each state after one year. Interest does not affect these state probabilities.

Method. Use the constant-intensity transition model. The healthy exit intensity is $r_H=0.10+0.05=0.15$, so $p_H(1)=e^{-0.15}=0.860707976$.

The moderate-care probability is the probability flow from healthy into moderate care, accumulated over all possible entry times: $p_M(1)=\int_0^1 0.10e^{-0.15u}e^{-0.30(1-u)}\,du$. Evaluating the integral gives $p_M(1)=\frac{0.10}{0.30-0.15}(e^{-0.15}-e^{-0.30})=0.079926503$.

For severe care, first use the moderate-care occupancy at time $u$, then allow transition to severe care at rate $0.20$, followed by survival in severe care until time 1:

p_S(1)=\frac{0.20(0.10)}{0.30-0.15} \left[ \frac{e^{-0.15}-e^{-0.25}}{0.25-0.15} - \frac{e^{-0.30}-e^{-0.25}}{0.25-0.30} \right] =0.007922759.

Death is absorbing, so its probability is the complement of the three living-state probabilities: $p_D(1)=1-0.860707976-0.079926503-0.007922759=0.051442762$. To four decimal places, the probabilities are healthy $0.8607$, moderate care $0.0799$, severe care $0.0079$, and death $0.0514$. The unrounded values sum to 1.000000000. A common mistake is to treat the probability of entering a care state as the probability of occupying it at the end of the year; the integral accounts for both entry time and subsequent transition or death.

Worked Example: Level Net Premium for Two Care Benefits

Use the transition intensities from the preceding example. A hypothetical contract pays continuously at $b_M=\$12{,}000$ per year in moderate care and $b_S=\$24{,}000$ per year in severe care. The level premium is payable continuously in all living states and ceases at death. The force of interest is $\delta=0.05$ per year. Calculate the level net premium.

Method. Calculate discounted occupancy for each living state, then apply the equivalence principle. Here $r_H=0.15$, $r_M=0.30$, and $r_S=0.25$. Thus:

z_H=\frac{1}{0.05+0.15}=5, \qquad z_M=\frac{0.10(5)}{0.05+0.30}=\frac{10}{7}=1.428571, \qquad z_S=\frac{0.20(10/7)}{0.05+0.25}=\frac{20}{21}=0.952381.

The expected present value of benefits is $12{,}000(10/7)+24{,}000(20/21)=17{,}142.857+22{,}857.143=\$40{,}000.00$. The premium annuity value is $5+10/7+20/21=155/21=7.380952$. Therefore, $P=40{,}000/(155/21)=8{,}400{,}000/155=\$5{,}419.35$ per year, rounded to the nearest cent.

The result is checked by multiplying the premium by its annuity value: $5{,}419.3548387(155/21)=\$40{,}000.00$ before rounding. A common mistake is to divide by $z_H$ alone, producing $\$8{,}000$. That would value premiums payable only while healthy, contrary to the contract terms. The correct denominator includes all three premium-paying states.

Worked Example: Solving for a Severe-Care Benefit Rate

Keep the transition rates, force of interest, and premium-payment condition from the pricing example. Suppose the continuous premium is fixed at $\$6{,}000$ per year and the moderate-care benefit rate is $\$10{,}000$ per year. Find the severe-care benefit rate $b_S$ that satisfies the net premium equivalence principle.

Method. First calculate the expected present value of premiums: $6{,}000(155/21)=\$44{,}285.714$. The moderate-care benefit value is $10{,}000(10/7)=\$14{,}285.714$. Thus severe-care benefits must provide the remaining value: $\$44{,}285.714-\$14{,}285.714=\$30{,}000.00$.

Since severe-care discounted occupancy is $z_S=20/21$, the required rate is $b_S(20/21)=30{,}000$, giving $b_S=30{,}000(21/20)=\$31{,}500$ per year. Check: the severe-care value is $31{,}500(20/21)=\$30{,}000$, and total benefit value is $30{,}000+14{,}285.714=\$44{,}285.714$, equal to the premium value. The answer is a benefit rate, not a lump sum payable on entry to severe care.

Interpretation and Common Traps

Discounted expected occupancy links the transition model to the contract cash flows. A state can have a modest probability at a particular time yet contribute materially to expected benefits if it is occupied for a long time or has a high payment rate. Conversely, a high benefit rate alone does not determine value: its discounted occupancy matters as well.

Use the stated premium-paying states. Premiums payable while alive use $z_H+z_M+z_S$; premiums payable only while healthy use $z_H$.
Keep interest and transition rates distinct. The force of interest discounts cash flows; it is not an additional cause of death or care entry.
Match payment units. Continuous benefit rates in dollars per year multiply continuous discounted occupancy in years. Do not treat them as lump sums.
Check the state probabilities. At any fixed time, probabilities across healthy, care, and death states must sum to 1, apart from rounding.

This level net premium is based on expected present values under the specified assumptions. It does not include expenses, profit margins, risk loads, or other pricing adjustments. The result is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

Use the stated model and payment conditions in each question. Round money to the nearest cent and probabilities to four decimal places.

A hypothetical insured starts healthy, with healthy-state exit intensity $0.12$ per year. What is the probability of still being healthy after two years? (Answer: $e^{-0.24}=0.7866$)
For $\delta=0.04$, let $\alpha=0.12,\ \mu_H=0.08,\ \beta=0.15,\ \mu_M=0.10,\ \mu_S=0.20$, all per year. Find $z_H,z_M,z_S$. (Answer: $z_H=4.1667,\ z_M=1.7241,\ z_S=1.0776$ years)
Using the rates in question 2, suppose continuous benefits are $\$9{,}000$ per year in moderate care and $\$18{,}000$ per year in severe care. Premiums are continuous in all living states. Find the level net premium. (Answer: $\$5{,}010.31$ per year)
Use the transition rates and force of interest from the main pricing example. If the moderate-care benefit is $\$12{,}000$ per year and the continuous premium is $\$5{,}000$ per year in every living state, what severe-care benefit rate balances expected present values? (Answer: $\$20{,}750$ per year)
Use the main pricing example's transition rates and $\delta=0.05$, but set benefits to $\$8{,}000$ per year in moderate care and $\$20{,}000$ per year in severe care. Premiums remain continuous in every living state. Find the level net premium. (Answer: $\$4{,}129.03$ per year)

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