Waiver of Premium After Care Entry

Premium Waiver as a State-Dependent Cash Flow

A long-term care contract may waive its premium when the insured enters a state that satisfies the contract’s care trigger. The waiver changes the premium cash flow, not necessarily the policy’s other benefits. If the insured later recovers, the contract may require premiums to resume. The correct cash flow therefore depends on the insured’s state on each premium due date, rather than only on whether care was ever entered.

This tutorial is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. All products and values below are hypothetical. We focus on premiums due at specified dates; benefit valuation is not needed to determine whether a premium is payable.

Definition: Let $Z_t$ denote the insured’s state at time $t$. Let $\mathcal{W}$ be the set of states that trigger premium waiver, and let $\mathcal{P}$ be the set of living states in which a premium is payable. Under a state-dependent waiver, a premium due at time $t_k$ is payable only if $Z_{t_k}\in\mathcal{P}$. If recovery moves the insured from a waiver state into a payable state, premiums resume at the next applicable due date. Unless the contract says otherwise, premiums missed during a waiver are not accumulated as arrears.

For example, suppose $H$ denotes a state in which the insured is healthy or otherwise not receiving eligible care, $C$ denotes an eligible care state, and $D$ denotes death. If the contract waives premiums in $C$, a simple model has $\mathcal{P}=\{H\}$ and $\mathcal{W}=\{C\}$. Death is neither a payable state nor a waiver state: no further premium is due after death. If several care levels qualify, $\mathcal{W}$ can contain several states. If some care level does not qualify, it should not be included merely because it represents care.

A waiver that ends upon recovery is different from a permanent waiver triggered by the first care entry. Under the first design, a transition from $H$ to $C$ suspends premiums, and a subsequent transition from $C$ to $H$ restores them. Under the second design, care entry changes a separate contract status permanently, even if the insured later recovers. The model must reflect the contract wording; treating these designs as equivalent changes the expected premium cash flows.

From State Occupancy to Premium Cash Flows

Suppose premiums are due at times $t_0,t_1,\ldots,t_n$, with premium amount $G_k$ due at $t_k$. Define an indicator $I_k$ equal to 1 if the premium is payable at that date and 0 otherwise. The actual premium cash flow is $G_k I_k$. If $\pi_j(t_k)=P(Z_{t_k}=j)$ is the probability of occupying state $j$ at that date, then

E[G_k I_k] = G_k\sum_{j\in\mathcal{P}}\pi_j(t_k).

This formula assumes the premium is due only while the insured is alive and in a payable state, and that the stated premium amount is due in full at the date. The probabilities in the sum must be unconditional state occupancy probabilities, so they already reflect any chance of having entered care, recovered, or died before the due date. If premiums are payable in advance, evaluate the state at the due date immediately before payment. If a contract has a grace period or a different timing rule, use that rule instead.

With effective annual interest rate $i$, let $v=1/(1+i)$. When all payment dates are integer durations, the expected present value of premiums is

E[\operatorname{PV}(\text{premiums})] = \sum_{k=0}^{n} v^{t_k}G_k \sum_{j\in\mathcal{P}}\pi_j(t_k).

This follows by writing the present value on each path as $\sum_k v^{t_k}G_k I_k$, then taking expectations and using the linearity of expectation. It does not require the indicators at different dates to be independent. Indeed, state occupancy at different dates is generally dependent; the formula needs only the probability of being in a payable state at each individual due date.

Worked Example: A Waiver With No Recovery

A hypothetical policy charges an annual premium of $1,200 at times 0, 1, and 2. The insured is in state $H$ at issue. At the end of each year, an insured still in $H$ enters eligible care state $C$ with probability 0.5; otherwise the insured remains in $H$. Care is absorbing for this example, and death is ignored. Premiums are waived in $C$. The effective annual interest rate is 5%. Find the expected present value of premiums.

Method. Find the probability of being in the payable state $H$ at each due date, multiply by the premium, and discount each expected payment.

At time 0, the insured is certainly in $H$, so the probability of paying the premium is 1. At time 1, the probability of remaining in $H$ is $0.5$. At time 2, the insured must remain in $H$ in both years, so the probability is $0.5^2=0.25$. Thus the expected premium cash flows at times 0, 1, and 2 are respectively $1{,}200$, $1{,}200(0.5)=600$, and $1{,}200(0.25)=300$.

The discount factor is $v=1/1.05$. Therefore,

E[\operatorname{PV}] = 1{,}200+\frac{600}{1.05}+\frac{300}{(1.05)^2} = 1{,}200+571.4286+272.1088 = 2{,}043.5374.

Rounded to the nearest cent, the expected present value is $2,043.54. The result is checked by confirming that the time-1 and time-2 payments are weighted by their respective probabilities of still being in $H$, and that both are discounted for the correct number of years. A common mistake is to calculate the probability of care entry by time 2 and use it as the probability of paying at time 2; payment instead depends on the probability of still being in $H$, which is 0.25.

Recovery and Resumption of Premiums

When recovery is possible, the payable-state probability need not decrease over time. A person in care at one due date may have recovered by the next one. For a discrete-time model, use a transition matrix whose rows are the current states and whose columns are the states one period later. The occupancy distribution at each due date is obtained by multiplying the distribution from the preceding date by that matrix. The premium at each date is then weighted by the total probability in payable states.

For a simple two-state illustration with no death, write $H$ for a payable state and $C$ for a waiver state. If the one-year transition probabilities are $P(H\to C)=a$ and $P(C\to H)=b$, then

\pi_H(k+1)=\pi_H(k)(1-a)+\pi_C(k)b.

The first term represents insureds who were payable and remain payable; the second represents insureds who were in care and recover. Omitting the second term treats care entry as a permanent waiver, even when the policy resumes premiums after recovery.

Worked Example: Premiums Resume After Recovery

A hypothetical contract charges $1,000 annually at times 0, 1, 2, and 3. It waives premiums in care state $C$ and resumes them in state $H$ after recovery. There is no death in this illustration. For each year, an insured in $H$ remains in $H$ with probability 0.8 and enters $C$ with probability 0.2; an insured in $C$ recovers to $H$ with probability 0.5 and remains in $C$ with probability 0.5. The insured starts in $H$, and $i=0.04$. Find the expected present value of premiums.

Method. Recursively calculate the probability of being in $H$ at each payment date, then discount the expected payment. The one-year transition matrix, with rows and columns ordered $H,C$, is

\mathbf{P} = \begin{pmatrix} 0.8 & 0.2\\ 0.5 & 0.5 \end{pmatrix}.

At time 0, $(\pi_H,\pi_C)=(1,0)$. At time 1, $\pi_H(1)=1(0.8)+0(0.5)=0.8$, so the expected payment is $800. At time 2, $\pi_C(1)=0.2$, and therefore

\pi_H(2)=0.8(0.8)+0.2(0.5)=0.64+0.10=0.74.

At time 3, $\pi_C(2)=1-0.74=0.26$, so

\pi_H(3)=0.74(0.8)+0.26(0.5)=0.592+0.130=0.722.

The expected payments at times 0 through 3 are thus $1,000, $800, $740, and $722. Discounting at 4% gives

E[\operatorname{PV}] = 1{,}000+\frac{800}{1.04} +\frac{740}{(1.04)^2} +\frac{722}{(1.04)^3} = 1{,}000+769.2308+684.1716+641.8554 = 3{,}095.2578.

To the nearest cent, the expected present value is $3,095.26. The state probabilities at each date sum to 1, and the expected payments equal $1,000 times the probability of being in $H$, providing checks on the recursion and cash flows.

Exam trap. Treating the first care entry as a permanent waiver gives the incorrect payable-state probabilities $1,0.8,0.64,0.512$, because that calculation discards all recoveries. The transition matrix shows that recovery adds $0.2(0.5)$ to $\pi_H(2)$, and $0.26(0.5)$ to $\pi_H(3)$. Since premiums resume after recovery, those terms must be included.

Continuous-Time Transitions Between Due Dates

Care entry and recovery may occur at any time, even when premiums are payable only annually. In that case, calculate the occupancy probability at each premium date from the continuous-time model. Consider a hypothetical two-state process with no death, constant transition intensity $\alpha$ from $H$ to $C$, and constant recovery intensity $\beta$ from $C$ to $H$. These intensities are measured per year. Starting in $H$, let $p_H(t)=P(Z_t=H)$.

Over a short interval of length $dt$, probability flows out of $H$ at rate $\alpha p_H(t)$ and into $H$ at rate $\beta(1-p_H(t))$. Hence

\frac{dp_H(t)}{dt} = -\alpha p_H(t)+\beta(1-p_H(t)) = \beta-(\alpha+\beta)p_H(t).

The equilibrium value is $\beta/(\alpha+\beta)$. Subtracting this value from $p_H(t)$ gives a differential equation with solution proportional to $e^{-(\alpha+\beta)t}$. Applying $p_H(0)=1$ yields

p_H(t) = \frac{\beta}{\alpha+\beta} + \frac{\alpha}{\alpha+\beta}e^{-(\alpha+\beta)t}.

The formula applies under the stated assumptions: constant transition intensities, only the two living states, and no death or other exits. If death or additional states are present, the occupancy probabilities must come from the corresponding larger transition model; this two-state expression should not be used unchanged.

Worked Example: Annual Premium Dates With Continuous Recovery

A hypothetical policy has a $500 premium due at times 0, 1, and 2. Premiums are waived in care state $C$ and payable in state $H$. The insured starts in $H$. Assume a continuous-time model with constant intensity $\alpha=0.2$ per year from $H$ to $C$, recovery intensity $\beta=0.3$ per year from $C$ to $H$, and no death during the two-year illustration. The effective annual interest rate is 4%. Find the expected present value of premiums.

Method. Use the two-state occupancy solution at each annual due date. Here $\alpha+\beta=0.5$, and $\beta/(\alpha+\beta)=0.6$, so

p_H(t)=0.6+0.4e^{-0.5t}.

At time 0, $p_H(0)=1$. At time 1, $e^{-0.5}=0.6065307$, giving $p_H(1)=0.6+0.4(0.6065307)=0.8426123$. At time 2, $e^{-1}=0.3678794$, giving $p_H(2)=0.6+0.4(0.3678794)=0.7471518$. Thus the expected payments at the three dates are $500, $421.3061, and $373.5759.

Discounting at 4%,

E[\operatorname{PV}] = 500+\frac{500(0.8426123)}{1.04} +\frac{500(0.7471518)}{(1.04)^2} = 500+405.1020+345.3919 = 1{,}250.4939.

The expected present value is $1,250.49 to the nearest cent. As a check, the occupancy probabilities are between 0 and 1, and the probability of being payable has decreased from 1 at issue while remaining positive because recovery can return the insured to $H$. Care entry is not treated as an irreversible event.

Contract Interpretation and Common Traps

Before calculating premium cash flows, translate the contract wording into a rule that can be applied at each due date. In particular, distinguish the state that triggers waiver from a state that merely precedes eligibility. If the waiver requires a qualifying period, the relevant model must record whether that condition has been met; being in a care state on a due date may not by itself be sufficient. Likewise, if the waiver applies only to some care levels, include only those levels in $\mathcal{W}$.

Keep premium timing separate from state transitions. In an annual model, a transition made at the end of a year affects the state at the next specified due date according to the problem’s timing convention. In a continuous-time model, use the occupancy probability at the due date; do not substitute the probability of having entered care by that date. The probability of ever entering care and the probability of currently occupying a payable state are different quantities, particularly when recovery is possible.

Waiver is not automatically permanent. If premiums resume on recovery, add the recovered insureds to the payable-state probability.
Entry probability is not occupancy probability. A premium is determined by the state at its due date under the stated contract rule, not merely by whether care entry occurred at some earlier time.
Death stops future premiums. Do not include a premium after death unless the contract explicitly specifies a different payment arrangement.
Use the correct discount time. A payment at time $t_k$ is discounted by $v^{t_k}$, not by a factor for one period regardless of its date.
Do not create arrears without contract wording. A waived premium is generally omitted from that date’s cash flow; it is not automatically collected after recovery.

Key takeaway: A premium waiver is a rule on premium cash flows indexed by policy state and payment date. Identify the payable states, obtain their occupancy probabilities at each due date, multiply by the applicable premium, and discount each expected payment. If recovery ends the waiver, recovered policyholders must re-enter the payable-state calculation.

Check Your Understanding

For each hypothetical contract, determine the premium cash flow using the stated waiver and transition assumptions.

A $800 premium is due at times 0, 1, and 2. The insured starts in $H$, care state $C$ is absorbing, and an insured in $H$ enters $C$ during each year with probability 0.25. Premiums are waived in $C$, and $i=0.04$. What is the expected present value of premiums, to the nearest cent? (Answer: $1,792.97)
A $600 premium is due at times 0, 1, and 2. The insured starts in $H$; over each year, $P(H\to H)=0.9$, $P(H\to C)=0.1$, $P(C\to H)=0.4$, and $P(C\to C)=0.6$. Premiums are waived only in $C$. What is the expected undiscounted total of the three premiums? (Answer: $1,650)
At a due date, a hypothetical policy has occupancy probabilities 0.65 in payable state $H$, 0.20 in waiver state $C_1$, and 0.15 in care state $C_2$. The contract waives premiums only in $C_1$, and the due premium is $1,000. What is the expected premium cash flow at that date? (Answer: $800)
A continuous-time two-state model has intensity 0.3 per year from $H$ to $C$, recovery intensity 0.2 per year from $C$ to $H$, and starts in $H$. A $700 premium is due at time 1 and is waived in $C$. There is no death. What is the expected premium due at time 1, to the nearest cent? (Answer: $534.74)
A $900 premium is due at time 1. At that date, the insured has probability 0.72 of being in payable state $H$, 0.18 of being in waiver state $C$, and 0.10 of being dead. The waiver applies only in $C$, and premiums stop at death. At $i=0.05$, what is the expected present value of this premium, to the nearest cent? (Answer: $617.14)

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