Premium Payment Rules in LTC Contracts

Why Premium Rules Depend on the State Model

A long-term care contract may continue to charge premiums after care begins, stop charging while the policyholder is receiving care, or resume charging after recovery. The premium rule is therefore part of the contract’s state definition: it specifies which states generate premium payments. Two policies with the same benefits and transition assumptions can have different expected premium values simply because their premium-paying states differ.

We use a continuous-time model with living states $H$, $C$, and $R$, and an absorbing death state $D$. Here $H$ denotes healthy and not currently receiving care, $C$ denotes receiving care, and $R$ denotes a contract-defined recovery state after care. In this model, $R$ is kept distinct from $H$; the contract’s premium wording determines whether premiums are payable in it. A transition intensity is a rate per year, and the model assumes the stated intensities apply while the policyholder occupies their corresponding states.

Definition: A healthy-only premium rule charges premiums in $H$ and not in $C$ or $R$. A healthy-and-recovery rule charges in $H$ and $R$, but not in $C$. A throughout-survival rule charges in every living state, here $H$, $C$, and $R$, and stops at death. The rule “payable during recovery” is interpreted here as continuing or resuming premiums in $R$, as well as in $H$.

For continuous premiums, let $c$ be the premium rate in dollars per year, payable continuously whenever the policyholder occupies a premium-paying state. Let $\delta$ be the force of interest, and let $p_j(t)$ be the probability that the policyholder is in living state $j$ at time $t$, given the initial state. The expected present value is found by integrating the discounted premium rate times the probability of being in a paying state. This approach allows for entry to care, recovery, relapse, and death to the extent that these are included in the specified state model.

\operatorname{EPV}(\text{premiums}) =c\int_0^T e^{-\delta t}\sum_{j\in\mathcal{P}}p_j(t)\,dt

Here $\mathcal{P}$ is the set of premium-paying states and $T$ is the end of the premium term. For lifetime premiums, $T$ may be taken as infinity when the integral converges. The initial state matters: if the policy is issued in $H$, then the initial occupancy probability of $H$ is one, while the initial probabilities of other states are zero. If issue is in a mixture of states, the occupancy probabilities must reflect that initial distribution.

Why the Occupancy Integral Gives the Expected Value

Let $I_j(t)$ equal one when the policyholder is in state $j$ at time $t$, and zero otherwise. The instantaneous premium amount over a short interval of length $dt$ is $c\sum_{j\in\mathcal{P}}I_j(t)\,dt$. Discounting to time zero and taking its expectation replaces each state indicator by its probability of being in that state. Integrating over the premium term gives the formula.

\begin{aligned} \operatorname{EPV} &=\mathbb{E}\left[\int_0^T e^{-\delta t}c \sum_{j\in\mathcal{P}}I_j(t)\,dt\right]\\ &=c\int_0^T e^{-\delta t} \sum_{j\in\mathcal{P}}\mathbb{E}[I_j(t)]\,dt\\ &=c\int_0^T e^{-\delta t} \sum_{j\in\mathcal{P}}p_j(t)\,dt. \end{aligned}

The key identity is $\mathbb{E}[I_j(t)]=p_j(t)$, since the indicator is one precisely when the policyholder occupies state $j$. This derivation assumes the premium rate is continuously payable in the specified states and that the discounting convention is consistent with force $\delta$. It does not directly value premiums payable only at discrete dates; those require the probability of occupying a paying state at each payment date.

Result: Under a continuous premium rule, sum the occupancy probabilities of exactly the states in which premiums are due, discount that sum, and integrate over the premium term. Do not substitute the probability of being alive unless the rule charges in every living state.

Worked Premium Calculations

Worked Example: Healthy-Only Premiums With a Finite Term

A hypothetical contract starts in $H$. The policyholder leaves $H$ for care $C$ at constant intensity $0.10$ per year and dies from $H$ at intensity $0.02$ per year. Once in $C$, the policyholder cannot return to $H$ in this simplified model. A continuous premium of $1,200 per year is payable only in $H$, for at most five years. Use force of interest $\delta=0.08$. Find the expected present value of premiums.

Method. The probability of still being in $H$ at time $t$ is the probability that neither care entry nor death has occurred. The two competing exit intensities add, so the total exit intensity from $H$ is $0.10+0.02=0.12$. Thus $p_H(t)=e^{-0.12t}$.

Only $H$ pays premiums, so discounting adds the force of interest to the exponent:

\begin{aligned} \operatorname{EPV} &=1200\int_0^5 e^{-0.08t}e^{-0.12t}\,dt\\ &=1200\int_0^5 e^{-0.20t}\,dt\\ &=1200\left(\frac{1-e^{-0.20(5)}}{0.20}\right)\\ &=1200\left(\frac{1-e^{-1}}{0.20}\right)\\ &=1200(3.1606028)\\ &=\$3,792.72. \end{aligned}

The answer is rounded to the nearest cent. As a check, the undiscounted expected time in $H$ over the five-year term is $(1-e^{-0.6})/0.12$, approximately 3.759 years. Discounting reduces the premium-weighted time to 3.161 years, consistent with the smaller present value. The calculation avoids treating care entry as if the policyholder remained eligible to pay premiums in $H$.

Worked Example: Premiums in Healthy and Recovery States

A hypothetical policy begins in $H$ and has no finite premium term. From $H$, care entry $H\to C$ has intensity $0.10$ per year. From $C$, recovery $C\to R$ has intensity $0.25$ per year. Death has intensity $0.02$ in each living state, and there are no other transitions. A continuous premium of $900 per year is payable in $H$ and $R$, but not in $C$. The force of interest is $\delta=0.03$. Calculate the expected present value.

Method. First find the occupancy probability of $H$. Its total exit intensity is $0.10+0.02=0.12$, so $p_H(t)=e^{-0.12t}$. The care-state probability satisfies an inflow-outflow equation: inflow is $0.10p_H(t)$, while the total exit intensity from $C$ is $0.25+0.02=0.27$.

\begin{aligned} p_C'(t)&=0.10e^{-0.12t}-0.27p_C(t),\qquad p_C(0)=0,\\ p_C(t)&=\frac{0.10}{0.27-0.12} \left(e^{-0.12t}-e^{-0.27t}\right)\\ &=\frac{2}{3}\left(e^{-0.12t}-e^{-0.27t}\right). \end{aligned}

Because all living states have the same death intensity, the probability of being alive at time $t$ is $e^{-0.02t}$. The paying states $H$ and $R$ together comprise all living states except $C$, so their combined occupancy probability is $e^{-0.02t}-p_C(t)$. Therefore:

\begin{aligned} \int_0^\infty e^{-0.03t}p_C(t)\,dt &=\frac{2}{3}\left(\frac{1}{0.15}-\frac{1}{0.30}\right) =\frac{20}{9},\\ \int_0^\infty e^{-0.03t}e^{-0.02t}\,dt &=\frac{1}{0.05}=20,\\ \int_0^\infty e^{-0.03t}\left[p_H(t)+p_R(t)\right]dt &=20-\frac{20}{9}=\frac{160}{9}. \end{aligned}

Multiplying by the continuous premium rate gives $900(160/9)=\$16,000.00$. This is the expected present value under the healthy-and-recovery rule. The result is lower than charging in every living state, because premiums are not paid during care. It is higher than the healthy-only value, since recovery state $R$ also generates premiums.

Worked Example: Annual Premiums Throughout Survival

A hypothetical contract charges $1,000 at the start of each policy year for five years, at times $0,1,2,3,4$. Each payment is due if the policyholder is alive on that date, regardless of whether the person is healthy, in care, or recovering. Annual survival probabilities are ${}_kp=0.96^k$, and the effective annual interest rate is $i=0.05$. Find the expected present value.

Method. A payment at time $k$ is due with probability $0.96^k$, and its discount factor is $v^k=(1.05)^{-k}$. The payment at time zero is certain because the policyholder is alive at issue. The expected present value is:

\begin{aligned} \operatorname{EPV} &=1000\sum_{k=0}^{4}(1.05)^{-k}(0.96)^k\\ &=1000\sum_{k=0}^{4}\left(\frac{0.96}{1.05}\right)^k\\ &=1000\left(\frac{1-(32/35)^5}{1-32/35}\right)\\ &=1000(4.2132318)\\ &=\$4,213.23. \end{aligned}

The answer is rounded to the nearest cent. This is a throughout-survival rule, so care-state transitions do not change the probability of paying at a date; only survival matters. The example also shows why annual premiums due in advance are summed at their actual payment dates rather than valued as a continuous stream.

Choosing the Right Probability and Avoiding Traps

For a healthy-only rule, the relevant probability is $p_H(t)$. For the healthy-and-recovery rule in the state definitions above, it is $p_H(t)+p_R(t)$. For throughout-survival premiums, it is the sum of the probabilities of all living states. These quantities are generally not interchangeable. In particular, healthy occupancy is less than or equal to alive occupancy whenever care is a possible living state.

Translate the contract wording.
List the states in which a premium is payable. Clarify whether recovery is a distinct state or is included in the contract’s definition of healthy.

Identify the payment timing.
For continuous premiums, use an integral. For annual or other discrete premiums, sum discounted payment probabilities at the specified dates.

Use the matching occupancy probabilities.
Account for all transitions that can remove the policyholder from a paying state, including care entry and death.

Discount and check the result.
Use the stated interest convention and confirm that adding a premium-paying state cannot reduce the expected present value, all else equal.

Common exam trap: using survival probability for a healthy-only premium. In the first example, an incorrect calculation would use only the death intensity and write $1200\int_0^5 e^{-(0.08+0.02)t}\,dt$, giving $1200(1-e^{-0.5})/0.10=\$4,723.04$ (rounded to cents). This treats a policyholder who enters care as still paying in $H$. The correction is to include care entry in the rate of leaving $H$: use $0.10+0.02$, giving the correct value of $3,792.72. Conversely, under a throughout-survival rule, omitting care-state occupancy would wrongly stop premiums at care entry.

Key takeaway: Premium value depends on both payment timing and contract eligibility. Define the paying states first; then use their occupancy probabilities, not a generic alive probability, unless the contract charges in every living state.

Check Your Understanding

For each question, identify the premium-paying set before calculating. Unless a question specifies otherwise, premiums are continuous and the policy starts in $H$.

A policy has only $H$ and death states. The death intensity is $0.03$, the force of interest is $0.04$, and a continuous premium of $500 is payable for life while in $H$. Find its expected present value. (Answer: $7,142.86)
A contract charges a continuous premium of $800 throughout survival. The death intensity is $0.03$ in every living state, the force of interest is $0.02$, and the premium term is lifetime. What is the expected present value, regardless of transitions among living states? (Answer: $16,000.00)
From $H$, care entry has intensity $0.10$; from $C$, recovery to $R$ has intensity $0.20$. Death has intensity $0.01$ in each living state. Premiums of $600 per year are payable in $H$ and $R$, but not $C$, for life. With force of interest $0.04$, find the expected present value. (Answer: $10,400.00)
Annual premiums of $700 are due at times $0,1,2,3$ if alive, with no premium after death. Annual survival probabilities are ${}_kp=0.97^k$, and the effective annual interest rate is $0.04$. Find the expected present value. (Answer: $2,529.78)

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