Premiums Payable Only While Active

Introduction

A premium payable only while an insured is active is contingent on the insured’s state at each premium time. Disability stops the premium stream, but recovery can make later premiums payable again. The relevant quantity is therefore not simply the probability of avoiding disability; it is the probability of being active at each time, including after any recovery.

This tutorial values that premium stream by summing or integrating discounted active-state occupancy probabilities. The result is the expected present value of premiums, not an equivalence premium: determining a premium that balances benefits and expenses is a separate calculation.

Definition: Let $A$, $D$, and $X$ denote the active, disabled, and dead states. For an insured initially active, define $p_A(t)=P(\text{in state }A\text{ at time }t\mid\text{active at time }0)$. A premium is payable only while active if it is due at time $t$ precisely when the insured is in state $A$ at the applicable payment time.

Valuing Continuous Premiums

Suppose premiums are paid continuously at rate $c$ dollars per year while the insured is active, for a term from time 0 to time $T$. Let $\delta$ be the constant force of interest. The expected present value of this premium stream is

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

At each instant, the payment rate is $c$ if the insured is active and zero otherwise. Its expected discounted rate at time $t$ is thus $c e^{-\delta t}p_A(t)$. Integrating those expected rates over the premium-paying term gives the formula. For an unlimited term, use $T=\infty$, provided the integral is finite.

The occupancy probability $p_A(t)$ counts all ways the insured can be active at time $t$. In particular, it includes paths in which the insured became disabled and subsequently recovered. It is not generally equal to the probability of remaining continuously active since issue.

Constant-Intensity Model and a Useful Derivation

Consider a continuous-time Markov model with constant transition intensities: $\alpha$ from active to disabled, $\lambda_A$ from active to dead, $\beta$ from disabled to active, and $\lambda_D$ from disabled to dead. Death is absorbing. The transient-state generator, with state order $A,D$, is

Q= \begin{pmatrix} -(\alpha+\lambda_A)&\alpha\\ \beta&-(\beta+\lambda_D) \end{pmatrix}.

For an unlimited continuous premium stream, define $x_A$ as the expected present value of one dollar per year paid while active, starting active; define $x_D$ in the same way when starting disabled. The active-state payment rates are 1 in state $A$ and 0 in state $D$. A first-step argument over a short interval $dt$ gives, for transient state $i$,

\delta x_i=r_i+\sum_{j\in\{A,D\}}q_{ij}x_j,

where $r_A=1$, $r_D=0$, and $q_{ij}$ are entries of $Q$. To see why, over $dt$ the expected payment is $r_i\,dt+o(dt)$. The probability of a transition from $i$ to $j\ne i$ is $q_{ij}dt+o(dt)$, and the probability of remaining in $i$ is $1+q_{ii}dt+o(dt)$. Discounting the continuation value by $1-\delta dt+o(dt)$, then subtracting $x_i$, dividing by $dt$, and taking the limit gives the stated equation.

Substitution of the generator entries yields two equations. Solving the second for $x_D$ and substituting into the first gives the active-start factor:

\begin{aligned} (\delta+\alpha+\lambda_A)x_A-\alpha x_D&=1,\\ -\beta x_A+(\delta+\beta+\lambda_D)x_D&=0,\\ x_A&=\frac{\delta+\beta+\lambda_D} {(\delta+\alpha+\lambda_A)(\delta+\beta+\lambda_D)-\alpha\beta}. \end{aligned}

This result assumes constant intensities, constant force of interest, and a premium rate paid only in state $A$. It applies to the unlimited term; for a finite term, the occupancy integral is still valid, but this unlimited-term expression does not directly give the finite-term value.

Worked Example: Continuous Premiums With Recovery

A hypothetical contract charges a continuous premium at rate $1,200 per year whenever the insured is active. The insured starts active. Assume constant intensities $\alpha=0.10$, $\lambda_A=0.02$, $\beta=0.20$, and $\lambda_D=0.03$ per year, a force of interest $\delta=0.05$, and an unlimited premium-paying term. Find the expected present value of premiums.

Method. Use the two-state transient model and solve for the unlimited active-state occupancy factor. This accounts for premiums resuming after recovery.

The two diagonal coefficients in the equations are

a=\delta+\alpha+\lambda_A=0.05+0.10+0.02=0.17,\qquad b=\delta+\beta+\lambda_D=0.05+0.20+0.03=0.28.

The determinant is $ab-\alpha\beta$:

$$ (0.17)(0.28)-(0.10)(0.20)=0.0476-0.0200=0.0276. $$

Therefore the active-start factor is

x_A=\frac{0.28}{0.0276}=10.1449275\text{ years}.

Multiplying by the continuous premium rate gives

\operatorname{EPV}(\text{premiums})=1200(10.1449275)=\$12{,}173.91.

Check. The corresponding disabled-start factor is $x_D=\beta/0.0276=0.20/0.0276=7.2463768$. Substitution gives $0.17x_A-0.10x_D=1.0000$ and $-0.20x_A+0.28x_D=0.0000$, as required. The monetary result is rounded to the nearest cent.

Common mistake avoided. Counting only the first active spell would omit premiums after recovery and would understate the value. The equations value every future active period.

Valuing Discrete Premiums

If premiums are due only at specified times, use the probability of being active at each due date, rather than integrating between dates. Suppose an $n$-year contract has annual premiums of amount $C$ in advance at times $k=0,1,\ldots,n-1$, and the insured is initially active. At an effective annual interest rate $i$, let $p_A(k)$ be the probability of being active at time $k$. Then

\operatorname{EPV}(\text{premiums})= C\sum_{k=0}^{n-1}v^k p_A(k), \qquad v=\frac{1}{1+i}.

The time-zero premium is included because the insured is active at issue. If premiums are due in arrears, the dates and the number of payments change; the formula must use the actual due dates. At a future due date, include a premium only if the insured is active at that date under the contract’s payment condition.

For a discrete-time Markov model, let $P$ be the one-year transition matrix. If the starting-state probability vector is $\boldsymbol{\pi}_0$, then $\boldsymbol{\pi}_k=\boldsymbol{\pi}_0P^k$. The active-state probability at time $k$ is the entry of that vector corresponding to $A$. A recovered insured can therefore be active at a later premium date even if disabled at an earlier one.

Worked Example: Annual Premiums With Recurrence

A hypothetical three-year contract charges $500 annually in advance at times 0, 1, and 2, but only if the insured is active on the due date. The insured starts active. For this example, the one-year transition probabilities among active, disabled, and dead are

P= \begin{pmatrix} 0.80&0.15&0.05\\ 0.25&0.65&0.10\\ 0&0&1 \end{pmatrix}.

Assume an effective annual interest rate $i=0.05$. Calculate the expected present value of premiums.

Method. Find the active probability at each due date, then discount each conditional expected payment. At time 0, $p_A(0)=1$. At time 1, the probability of being active is the active-to-active transition probability:

$$ p_A(1)=0.80. $$

At time 2, the insured can be active after being active at time 1, or can have recovered from disability:

$$ p_A(2)=(0.80)(0.80)+(0.15)(0.25)=0.6400+0.0375=0.6775. $$

The time-2 term includes the recurrence path. With $v=1/1.05$, the expected present value is

\begin{aligned} \operatorname{EPV} &=500\left(1+\frac{0.80}{1.05}+\frac{0.6775}{1.05^2}\right)\\ &=500(1+0.7619048+0.6145125)\\ &=500(2.3764173)=\$1{,}188.21. \end{aligned}

Check. The three undiscounted expected payments are $500, $400, and $338.75. Discounting the latter two gives $400/1.05=\$380.95$ and $338.75/1.1025=\$307.26$; the total is $1,188.21 after rounding to the nearest cent.

Common mistake avoided. A premium at time 2 is not ruled out merely because disability occurred earlier. The insured may have recovered, and the transition calculation includes that possibility.

Finite-Term Continuous Premiums

A finite term limits the times over which the continuous premium rate is paid. The same occupancy principle applies, but the integral ends at $T$. In a particularly simple model with no recovery, an initially active insured leaves the active state at total intensity $\gamma$, through disability or death. The probability of still being active at time $t$ is $e^{-\gamma t}$. Hence

\operatorname{EPV}(\text{premiums}) =c\int_0^T e^{-(\delta+\gamma)t}\,dt =c\frac{1-e^{-(\delta+\gamma)T}}{\delta+\gamma}.

This simplification requires that leaving the active state is permanent for purposes of active occupancy. If recovery is possible, $e^{-\gamma t}$ is not generally the active-state probability; use the full multi-state occupancy probabilities instead.

Worked Example: A Four-Year Continuous Premium Term

A hypothetical policy charges $1,500 per year continuously while active, for four years. The insured starts active. Assume constant active-to-disabled intensity $0.06$, active mortality intensity $0.01$, no recovery, and force of interest $\delta=0.03$. Find the expected present value of premiums.

Method. Since disability and death both end active occupancy permanently in this model, the total exit intensity is

\gamma=0.06+0.01=0.07.

Thus $p_A(t)=e^{-0.07t}$, and the discounting and active-occupancy exponents combine:

\begin{aligned} \operatorname{EPV} &=1500\int_0^4 e^{-0.03t}e^{-0.07t}\,dt\\ &=1500\frac{1-e^{-(0.10)(4)}}{0.10}\\ &=1500\frac{1-e^{-0.4}}{0.10}\\ &=1500\frac{1-0.6703200}{0.10}\\ &=1500(3.2967995)=\$4{,}945.20. \end{aligned}

Check. The active probability falls from 1 at issue to $e^{-0.28}=0.7558$ at year 4, so the discounted occupancy factor of 3.2968 years is less than the undiscounted four-year term. The monetary result is rounded to the nearest cent.

Common mistake avoided. Applying the active exit intensity but omitting interest would value undiscounted occupancy. The factor in the integral is $e^{-(\delta+\gamma)t}$, because both discounting and leaving the active state reduce the expected present value.

Common Timing and State-Eligibility Traps

The key modelling decision is to match the premium condition to the contract’s payment timing. For continuous premiums, active occupancy matters at each instant of the term. For discrete premiums, the insured’s state at each due date matters. A premium due in advance at time $k$ is not treated like one due at the end of the year.

Using first-disability probabilities instead of occupancy. First-onset probabilities do not by themselves describe whether the insured is active at a later time when recovery is possible.
Stopping premiums permanently after disability. That is appropriate only if the contract or model makes premiums permanently cease. If premiums resume on recovery, later active occupancy must be counted.
Discounting with the wrong rate convention. Use $e^{-\delta t}$ with a force of interest for continuous time, and $v^k$ with an effective annual rate for annual due dates.
Including a payment at the wrong date. A contract with annual payments in advance over $n$ years has due dates $0$ through $n-1$, not $1$ through $n$.

These valuations are useful for pricing and contract analysis because they isolate the expected value of the premium cash flows under a specified state model. They do not, on their own, determine whether a premium is adequate: that requires valuing the other cash flows and applying the contract’s pricing basis.

Key takeaway: Value premiums payable only while active by multiplying the premium amount or rate by the probability of being active at the relevant payment time, discounting, and summing or integrating over the premium-paying term. Allow for recovery whenever the model permits it.

Check Your Understanding

For each problem, use the stated timing convention and state probabilities.

A hypothetical contract pays a continuous premium rate of $4,000 per year while active, with no recovery. The active-to-disabled intensity is 0.08, active mortality intensity is 0.02, the force of interest is 0.04, and the term is unlimited. Find the expected present value. (Answer: $28,571.43)
An insured starts active and pays $6,000 annually in advance at times 0 and 1, only if active. The probability of being active at time 1 is 0.70, and the effective annual interest rate is 0.04. Find the expected present value. (Answer: $10,038.46)
An insured starts disabled. At annual dates the transition probabilities from active are 0.75 to active, 0.20 to disabled, and 0.05 to dead; from disabled they are 0.30 to active, 0.60 to disabled, and 0.10 to dead. Premiums of $1,000 are due at times 1 and 2 only if active. At $i=0.05$, find the expected present value. (Answer: $653.06)
A four-year contract pays a continuous premium rate of $1,500 per year while active. The insured starts active, cannot recover after disability, and has active-to-disabled and active-mortality intensities of 0.06 and 0.01. If $\delta=0.03$, find the expected present value. (Answer: $4,945.20)
In a constant-intensity model, $\alpha=0.10$, $\lambda_A=0.01$, $\beta=0.20$, and $\lambda_D=0.04$, all per year. For an unlimited continuous premium stream at rate $750 per year while active, with $\delta=0.05$, find the expected present value starting active. (Answer: $8,238.64)

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