Premium Waivers After a State Change

Introduction

A premium waiver changes the future premium cash flows after a specified event or state change. The key valuation question is not simply whether the policyholder has ever been disabled, but which premiums remain payable under the contract’s precise waiver rule. A waiver may be permanent after first entry into a trigger state, or it may apply only while the policyholder is currently in that state.

These provisions can produce different expected present values even when they use the same state model. The method is to identify, at each premium due date, the event that makes the premium payable; calculate the probability of that event; and discount the premium to the valuation date. All products and figures below are hypothetical.

Definition: A permanent waiver triggered by entry into a set of states $W$ means premiums are waived from the first entry into $W$ onward, even if the policyholder later leaves $W$. A current-state waiver means premiums are waived at a due date if the policyholder is then in a state in $W$, but may become payable again after leaving $W$. A premium is eligible if the contract requires it to be paid at its due date, subject to any separate conditions such as policy termination.

Premium Eligibility and Expected Present Value

Suppose premiums $C_k$ are due at times $t_k$, for $k=1,\ldots,n$. Let $I_k$ equal 1 if the premium due at $t_k$ is payable and 0 otherwise. Under an effective annual interest rate $i$, the discount factor to time 0 is $v^{t_k}$, where $v=1/(1+i)$. For fractional years, this formula uses annual effective accumulation over the corresponding fraction of a year. Linearity of expectation gives

\operatorname{EPV}(\text{premiums}) =\operatorname{E}\left[\sum_{k=1}^{n} C_k v^{t_k} I_k\right] =\sum_{k=1}^{n} C_k v^{t_k}P(I_k=1).

This result does not require the eligibility events at different due dates to be independent. Each premium’s expected present value depends on its own probability of being payable. It assumes the stated premium amount is deterministic and that the interest rate is fixed. If premiums vary with the policyholder’s state, the amount and eligibility must be considered together; the formula above applies directly when $C_k$ is the amount payable whenever eligible.

For a current-state waiver, if premiums are payable in the states outside $W$, then $P(I_k=1)$ is the probability that the policyholder is in one of those states at time $t_k$, provided the policy remains in force. For a permanent waiver, eligibility instead requires that the policyholder has not entered $W$ by $t_k$. That is a first-entry condition, not merely a condition on the state occupied at $t_k$.

Why the Eligibility Probability Depends on the Trigger

Consider a time-homogeneous continuous-time Markov chain with states Healthy $H$ and Disabled $D$. If the transition intensities are $H\to D=\alpha$ and $D\to H=\beta$, the transition probability $p_{HH}(t)$ is the probability of being in $H$ at time $t$, starting in $H$. The forward equation for this two-state model is

\frac{d}{dt}p_{HH}(t) =-\alpha p_{HH}(t)+\beta\bigl(1-p_{HH}(t)\bigr), \qquad p_{HH}(0)=1.

The two terms represent, respectively, probability flowing out of $H$ and probability flowing into $H$. Solving this linear differential equation gives

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

Thus, if the waiver applies only while currently disabled, the premium is payable with probability $p_{HH}(t)$. If instead the first entry into $D$ permanently waives future premiums, the premium is payable only if no transition $H\to D$ has occurred by time $t$. With no other exit from $H$, that probability is $e^{-\alpha t}$. The distinction matters because recovery can restore current-state eligibility but cannot undo a permanent waiver.

Worked Example: A Permanent Waiver with Competing Disability and Death

A hypothetical policyholder starts in $H$. From $H$, disability $D$ occurs at constant intensity $0.3$ per year and death occurs at constant intensity $0.2$ per year. Both are competing transitions, and there are no exits from either state. A premium of $900 is due at time 2 years only if the policyholder is still healthy; entry into $D$ permanently waives later premiums. The effective annual interest rate is 5%. Find the expected present value of this premium.

Method. Eligibility requires remaining in $H$ throughout the two-year period. The total intensity of leaving $H$ is $0.3+0.2=0.5$ per year. Therefore, the probability of remaining in $H$ to time 2 is

P(I_1=1)=e^{-(0.3+0.2)2}=e^{-1}=0.3678794.

The discount factor to time 2 is $v^2=1/(1.05)^2=1/1.1025$. Hence

\operatorname{EPV} =900\left(\frac{0.3678794}{1.1025}\right) =900(0.3336775) =300.3098.

To the nearest cent, the expected present value is $300.31. The calculation includes death as a competing exit from the premium-paying state: a policyholder who dies before time 2 cannot pay the premium. A common mistake is to use $e^{-0.3(2)}$, which ignores death and gives the probability of avoiding disability, rather than the probability of being alive and healthy at the due date.

Current-State Waivers and Permanent Waivers Compared

When recovery is possible, a current-state waiver can allow premiums to resume. A permanent waiver cannot. For the symmetric two-state model with $\alpha=\beta=0.5$, the preceding transition-probability formula simplifies to $p_{HH}(t)=0.5+0.5e^{-t}$. In contrast, the probability of no disability entry by time $t$ is $e^{-0.5t}$. The next example values both provisions using the same transition model and premium schedule.

Worked Example: Comparing Reversible and Permanent Waivers

A hypothetical policyholder starts in $H$. There is no death. The transition intensities $H\to D$ and $D\to H$ are both $0.50$ per year. Premiums of $800 are due at times 1 and 2 years. Under the first provision, premiums are waived only while the policyholder is currently in $D$. Under the second provision, first entry into $D$ permanently waives all later premiums. The effective annual interest rate is 5%. Calculate and compare the two expected present values.

Current-state waiver. The premium is payable at time $t$ if the policyholder is in $H$ then. Starting in $H$,

p_{HH}(t)=0.5+0.5e^{-t}.

At time 1, $p_{HH}(1)=0.5+0.5e^{-1}=0.6839397$. Its discounted expected premium is $800(0.6839397)/1.05=521.10$. At time 2, $p_{HH}(2)=0.5+0.5e^{-2}=0.5676676$. Its discounted expected premium is $800(0.5676676)/1.1025=411.91$. Therefore,

\operatorname{EPV}_{\text{current}} =521.10+411.91 =\$933.01.

Permanent waiver. A premium is payable only if the policyholder has never entered $D$. The probability of that event is $e^{-0.5t}$. At time 1, the probability is $e^{-0.5}=0.6065307$, giving $800(0.6065307)/1.05=462.12$. At time 2, it is $e^{-1}=0.3678794$, giving $800(0.3678794)/1.1025=266.94$. Thus,

\operatorname{EPV}_{\text{permanent}} =462.12+266.94 =\$729.06.

The permanent waiver reduces the expected present value by $933.01-729.06=\$203.95$. The common trap is to treat $p_{HH}(t)$ as the probability of never having been disabled. It is only the probability of being healthy at time $t$; it includes policyholders who were disabled and recovered.

Waivers Triggered by a Later State

A contract may waive premiums only after entry into a more severe state, while continuing to collect them during an earlier state. This requires tracking the probability that the trigger state has not yet been reached. The probability is not necessarily the probability of remaining in the initial state.

Worked Example: Waiver Triggered by Long-Term Care Entry

A hypothetical policyholder starts in $H$, then may transition from $H$ to Disabled $D$ at intensity $0.50$ per year and from $D$ to Long-Term Care $L$ at intensity $0.50$ per year. There are no other transitions. Entry into $L$ permanently waives premiums, but premiums remain payable in both $H$ and $D$. A $1,000 premium is due at time 2 years. The effective annual interest rate is 4%. Find its expected present value.

Method. At time 2, the premium is payable if the policyholder is in $H$ or $D$, equivalently if entry into $L$ has not occurred. The probability of still being in $H$ is

P(H\text{ at }2)=e^{-0.5(2)}=e^{-1}=0.3678794.

To be in $D$ at time 2, the transition to $D$ must occur at some time $s$ between 0 and 2, followed by no transition from $D$ to $L$ before time 2. The probability is

P(D\text{ at }2) =\int_0^2 0.5e^{-0.5s}e^{-0.5(2-s)}\,ds =\int_0^2 0.5e^{-1}\,ds =e^{-1} =0.3678794.

The probability of being in a premium-paying state at the due date is therefore $0.3678794+0.3678794=0.7357589$. Discounting the $1,000 premium gives

\operatorname{EPV} =1000\left(\frac{0.7357589}{1.04^2}\right) =1000\left(\frac{0.7357589}{1.0816}\right) =\$680.25.

The result is checked against the state interpretation: the policyholder may be in either $H$ or $D$ and still owe the premium, so using only the probability of remaining in $H$ would understate the expected value. Entry into $L$, not entry into $D$, is the trigger.

Practical Checks and Common Traps

Before valuing premiums under a waiver provision, translate the contract wording into an eligibility event at each due date. In particular, determine whether the trigger acts permanently, whether premiums can resume after recovery, and whether death or another termination state ends premium payments. The model must also match the assumed transition structure: a probability of being in a state at a due date cannot generally substitute for a probability of never having entered a state.

Confusing present state with state history. A recovered policyholder may owe a premium under a current-state waiver but not under a permanent waiver.
Ignoring competing exits. If disability and death both end the premium-paying period, the probability of remaining healthy uses the sum of their outgoing intensities.
Using the wrong trigger set. If the waiver begins only in long-term care, premiums may still be due while the policyholder is disabled.
Discounting at the wrong time. A premium due at $t_k$ is discounted by $v^{t_k}$, not by a factor for the number of transitions or the duration spent in a state.
Assuming a state probability captures a permanent waiver. When recovery is possible, permanent waiver valuation needs a first-entry probability or an augmented model that records whether the trigger has ever occurred.

These calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The central discipline is to specify the trigger and due-date eligibility before applying transition probabilities or discounting.

Check Your Understanding

For each problem, identify the relevant eligibility event before calculating the premium value.

A policyholder starts healthy. Disability occurs at intensity 0.20 per year and death at intensity 0.10 per year, with no recovery. Disability permanently waives premiums. A $600 premium is due at year 1, and the effective annual interest rate is 5%. Find its expected present value. (Answer: $423.32)
There is no death. A policyholder starts in $H$, with transition intensities $H\to D=0.40$ and $D\to H=0.40$ per year. A $500 premium is due at year 1 and is waived only if the policyholder is currently in $D$. The effective annual interest rate is 4%. Find its expected present value. (Answer: $348.40)
A policyholder starts in $H$, then transitions $H\to D\to L$, with each transition intensity $0.25$ per year. There are no other transitions. Entry into $L$ permanently waives premiums, but premiums remain payable in $H$ and $D$. A $1,000 premium is due at year 2, and the interest rate is 0%. Find its expected present value. (Answer: $909.80)
A policyholder starts healthy. Disability occurs at intensity $0.40$ per year and death at intensity $0.10$ per year; both are absorbing. Disability permanently waives premiums. Premiums of $400 are due at years 1 and 2 only while the policyholder remains healthy. The effective annual interest rate is 3%. Find the expected present value of both premiums. (Answer: $374.25)

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