Waiver of Premium During Disability

Introduction

A waiver-of-premium provision changes which scheduled premiums are collected; it does not create a payment to the policyholder. While the insured is disabled, a premium that would otherwise be due is waived. If the insured recovers, premiums resume, provided the policy remains in force and premiums are still scheduled.

The actuarial value of this provision is therefore the expected present value of the premiums not collected during disability. The central quantity is the probability that the insured is disabled at each time a premium would be due. That is different from the probability that disability has begun by that time: recovery can return the insured to the active state.

Definition: Let $A$ denote the active state, $D$ the disabled state, and $0$ death. A waiver-of-premium benefit is a state-contingent benefit whose amount at a scheduled premium time is the premium otherwise due if the insured is in state $D$, and zero otherwise. In the usual model, no premium is due after death, and premiums resume after recovery to $A$.

Valuing Waived Premiums

Assume the policy starts active at time 0. Let $p_D(t)=P(X_t=D\mid X_0=A)$ be the probability of being alive and disabled at time $t$. This is a state-occupancy probability: it includes all possible paths that lead to state $D$, including a disability, recovery, and a later recurrence if the model allows it.

If premiums are payable continuously at rate $P$ per year until time $T$, and the force of interest is $\delta$, the expected present value (EPV) of premiums waived is

\operatorname{EPV}(\text{waived premiums}) =\int_0^T P e^{-\delta t}p_D(t)\,dt.

For premiums $P_k$ due at times $t_k$, with effective annual interest rate $i$ and $v=1/(1+i)$, the corresponding expression is

\operatorname{EPV}(\text{waived premiums}) =\sum_k P_k v^{t_k}p_D(t_k).

These formulas assume that a premium is waived exactly when the insured is disabled at the relevant time. A contract with a waiting period, a different rule for premiums due on the date of recovery, or a maximum waiver period requires its own benefit indicator. The formulas also assume premiums cease at death; if the premium schedule or waiver rule differs, the cash flow must be represented accordingly.

Constant-Intensity Model and a Useful Derivation

For a basic disability model, suppose transitions from active to disabled occur at constant intensity $\alpha$, and transitions from disabled to active occur at constant intensity $\beta$. First assume no mortality. With initial state $A$, the probability of being disabled at time $t$ satisfies

\frac{dp_D(t)}{dt}=\alpha(1-p_D(t))-\beta p_D(t), \qquad p_D(0)=0.

The first term is probability flow into disability: the insured is active with probability $1-p_D(t)$, and the transition intensity is $\alpha$. The second term is probability flow out of disability through recovery. Rearranging gives $p_D'(t)=\alpha-(\alpha+\beta)p_D(t)$. Multiplying by the integrating factor $e^{(\alpha+\beta)t}$ and integrating from 0 to $t$ yields

\begin{aligned} \frac{d}{dt}\left(e^{(\alpha+\beta)t}p_D(t)\right) &=\alpha e^{(\alpha+\beta)t},\\ e^{(\alpha+\beta)t}p_D(t) &=\frac{\alpha}{\alpha+\beta}\left(e^{(\alpha+\beta)t}-1\right),\\ p_D(t) &=\frac{\alpha}{\alpha+\beta}\left(1-e^{-(\alpha+\beta)t}\right). \end{aligned}

The solution reflects both incidence and recovery. It starts at zero, increases toward the long-run disabled-state probability $\alpha/(\alpha+\beta)$, and does not generally equal the probability of ever having become disabled.

Mortality can be included by treating death as an absorbing state. In general, the probability $p_D(t)$ is obtained from the transient-state transition probabilities of the assumed multi-state model. A useful special case occurs when mortality has the same constant intensity $\mu$ in both $A$ and $D$, and is independent of disability transitions. Then survival to time $t$ contributes a factor $e^{-\mu t}$, so

p_D(t)=e^{-\mu t}\frac{\alpha}{\alpha+\beta} \left(1-e^{-(\alpha+\beta)t}\right).

This factorization does not apply when mortality intensities differ by state or depend on the disability history. In those cases, use the transition probabilities from the full model rather than multiplying by a single common survival probability.

Worked Example: Continuous Premiums With Recovery

A hypothetical policy has a continuous premium rate of $180 per year for two years. The insured starts active. Assume no mortality, constant disability intensity $\alpha=0.25$, constant recovery intensity $\beta=0.25$, and force of interest $\delta=0.03$. Calculate the EPV of premiums waived. Use the exact disability-state occupancy probability and round the final amount to the nearest cent.

Method. The premium is waived only while disabled, so first find the probability of being disabled at each time. Here $\alpha+\beta=0.50$, and $\alpha/(\alpha+\beta)=0.50$. Therefore

p_D(t)=0.5(1-e^{-0.5t}).

Substitute this occupancy probability into the continuous-premium EPV formula:

\begin{aligned} \operatorname{EPV} &=\int_0^2 180e^{-0.03t}\,0.5(1-e^{-0.5t})\,dt\\ &=90\int_0^2(e^{-0.03t}-e^{-0.53t})\,dt\\ &=90\left[ \frac{1-e^{-0.06}}{0.03} -\frac{1-e^{-1.06}}{0.53} \right]. \end{aligned}

The two integral terms are

\frac{1-e^{-0.06}}{0.03}\approx1.941183, \qquad \frac{1-e^{-1.06}}{0.53}\approx1.233105.

Thus the EPV is $90(1.941183-1.233105)\approx \$63.73$. The result is positive and less than the value of paying the full premium continuously for two years, as expected. The common mistake this method avoids is valuing premiums waived using only the probability of first becoming disabled, which ignores recovery.

Scheduled Premiums and State at the Due Date

For premiums payable at discrete times, the relevant question is the insured’s state at each due date. If a premium is waived when the insured is disabled at time $t_k$, its expected waived amount is $P_k p_D(t_k)$. Discounting that expected amount to time 0 and summing across due dates gives the discrete EPV formula.

This calculation is not the same as valuing a continuous stream between due dates. If a policy only has annual premiums, disability during the year does not by itself mean the full annual premium is waived: the contract’s rule at the actual due date determines whether that premium is waived. Conversely, a continuous premium stream requires the disability occupancy probability throughout the period.

Worked Example: Annual Premiums With Recovery

A hypothetical policy charges an annual premium of $1,000 at times $t=1,2,3$. The insured starts active. Assume no mortality, constant disability and recovery intensities $\alpha=\beta=0.20$, and an effective annual interest rate $i=0.05$. A premium is waived if the insured is disabled at its due date. Calculate the EPV of waived premiums, to the nearest cent.

Method. With no mortality, $p_D(t)=0.5(1-e^{-0.4t})$. At the three due dates,

\begin{aligned} p_D(1)&=0.5(1-e^{-0.4})\approx0.164840,\\ p_D(2)&=0.5(1-e^{-0.8})\approx0.275336,\\ p_D(3)&=0.5(1-e^{-1.2})\approx0.349403. \end{aligned}

The discount factor is $v=1/1.05\approx0.952381$. The three present values are

\begin{aligned} 1000v^1p_D(1)&\approx \$156.99,\\ 1000v^2p_D(2)&\approx \$249.74,\\ 1000v^3p_D(3)&\approx \$301.83. \end{aligned}

Adding the unrounded terms gives an EPV of approximately $708.56. The value uses the probability of being disabled at each due date, not the probability of having had any disability by then. Recovery can reduce the probability of being disabled at a later premium date.

Including Mortality

A premium is generally no longer payable after death, so the probability used for waiver valuation must be the probability of being both alive and disabled. In a full model with state-dependent mortality, calculate the disabled-state probability from the multi-state transition model, then use it in the same continuous or discrete EPV expression. Do not treat a dead insured as disabled simply because disability occurred before death.

Worked Example: Common Mortality in Both Living States

A hypothetical contract has a continuous premium rate of $240 per year for three years. The insured starts active. Disability intensity is $\alpha=0.20$, recovery intensity is $\beta=0.30$, and mortality intensity is $\mu=0.10$ in both active and disabled states. Assume constant intensities, independent mortality, and force of interest $\delta=0.04$. Calculate the EPV of waived premiums to the nearest cent.

Method. Since mortality is common to both living states, the probability of being alive and disabled is

p_D(t)=e^{-0.10t}\frac{0.20}{0.20+0.30} (1-e^{-0.50t}) =0.4e^{-0.10t}(1-e^{-0.50t}).

Substitute into the continuous-premium formula. The discount factor and survival factor combine to give an exponent of $0.04+0.10=0.14$:

\begin{aligned} \operatorname{EPV} &=240\int_0^3e^{-0.04t}\,0.4e^{-0.10t}(1-e^{-0.50t})\,dt\\ &=96\left[ \frac{1-e^{-0.14(3)}}{0.14} -\frac{1-e^{-(0.14+0.50)(3)}}{0.64} \right]\\ &=96\left[ \frac{1-e^{-0.42}}{0.14} -\frac{1-e^{-1.92}}{0.64} \right]. \end{aligned}

The two terms in brackets are approximately $2.449666$ and $1.333427$. Hence the EPV is $96(2.449666-1.333427)\approx \$107.16$. This is less than the no-mortality value under the same disability assumptions because premiums are no longer due after death.

Common Mistakes and Practical Checks

The waiver is a benefit contingent on being disabled when a premium would otherwise be payable. Its value is not the value of the entire premium stream, nor is it necessarily based on the probability of a first disability onset. Before calculating, identify the premium schedule, the rule for the state at each due time, the states in which premiums remain payable, and the discount basis.

Ignoring recovery. The probability of ever becoming disabled is not the probability of being disabled at a later time. Use state occupancy, which allows for recovery.
Applying the wrong payment timing. Use an integral for a continuous premium stream and a sum at the actual due dates for scheduled premiums.
Omitting mortality. If premiums stop at death, use the probability of being alive and disabled, not a disability probability conditional on survival, unless it is multiplied by the probability of survival.
Discounting inconsistently. For continuous cash flows use $e^{-\delta t}$; for annual payments under effective rate $i$, use $v^t$.
Counting premiums after the contract ends. Stop the valuation at the end of the premium term or at any earlier contractual termination time.

A useful check is that the EPV of waived premiums cannot exceed the EPV of all scheduled premiums under the same termination assumptions: the waiver applies to only a subset of those premium cash flows. Also, if disability intensity is zero, the waiver value is zero; if recovery is ignored when it should be allowed, later disabled-state probabilities will generally be overstated.

Key takeaway: Value a waiver-of-premium provision by multiplying each scheduled premium by the probability that the insured is alive and disabled when it would be due, discounting to the valuation date, and summing or integrating over the premium term. Recovery matters because premiums resume when the insured returns to the active state.

Check Your Understanding

For each problem, calculate the EPV of waived premiums using the state occupancy probability appropriate to the stated payment timing. Round dollar answers to the nearest cent.

A continuous premium of $120 per year is payable for one year. The insured starts active, there is no mortality, $\alpha=0.10$, $\beta=0.10$, and $\delta=0.02$. Find the EPV of waived premiums. (Answer: $5.55)
Annual premiums of $500 are due at times 1 and 2. The insured starts active, there is no mortality, $\alpha=\beta=0.50$, and $i=0$. A premium is waived if the insured is disabled at its due date. Find the EPV. (Answer: $374.20)
A continuous premium of $300 per year is payable for two years. The insured starts active; $\alpha=0.10$, $\beta=0.30$, and mortality intensity is $0.04$ in both living states. Assume independent constant intensities and $\delta=0.06$. Find the EPV. (Answer: $41.13)
Annual premiums of $1,000 are due at times 1, 2, and 3. There is no mortality; $\alpha=0.10$, $\beta=0.20$, and $i=0.04$. The insured starts active, and premiums are waived only if disabled at the due date. Find the EPV. (Answer: $397.97)

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