Premiums Conditional on the Current State

Why Premium Eligibility Depends on State

A policy may require premiums only while the policyholder occupies certain states. For example, a hypothetical contract might collect a premium while a person is active, but not while the person is in a specified inactive state or after death. The key valuation question is not simply how many premiums are scheduled: it is what is the probability that the policyholder is in a premium-paying state when each premium is due?

The calculation begins by specifying the premium-paying states and the timing of payments. A premium due at the end of a year is conditional on the state at that payment time. A continuous premium, in contrast, accrues only over time spent in an eligible state. These are different cash-flow patterns and use different valuation formulas.

Definition: Let $\mathcal{C}$ be the set of premium-paying states. A premium is payable at time $t$ under a state-at-payment condition if the policyholder occupies a state in $\mathcal{C}$ at time $t$. For a continuous premium rate $c_j(t)$ in state $j$, the premium rate at time $t$ is $c_j(t)$ when the current state is $j$, and zero in states outside $\mathcal{C}$.

Unless stated otherwise, assume the state process is modeled as specified, interest is deterministic, and premiums are paid exactly at the stated times. At a discrete premium date, an eligible state is assessed at that date. For continuous premiums, a transition instant has zero duration and therefore contributes no separate premium amount.

Premiums Due at Discrete Times

Suppose premiums of amount $P_k$ are due at times $t_k$, and let $p_j(t_k)$ denote the probability that the policyholder occupies state $j$ at time $t_k$. If the initial state is known, the state occupancy probabilities follow from the transition model. With annual effective interest rate $i$, the expected present value (EPV) is

\operatorname{EPV} = \sum_k v^{t_k} P_k \sum_{j\in\mathcal{C}}p_j(t_k), \qquad v=\frac{1}{1+i}.

The inner sum is the probability of occupying any premium-paying state on the payment date. If the premium amount differs by state, replace $P_k$ by the applicable state-specific amount inside the sum. A premium due at time zero is included without discounting, and is payable only if the initial state is eligible.

Worked Example: Annual Premiums Only in State A

A hypothetical policyholder starts in state $A$. At each year-end, the state transitions according to the following probabilities, which are the same in each year: from $A$, the probabilities of moving to $A$, $B$, and $D$ are $0.50$, $0.30$, and $0.20$; from $B$, they are $0.20$, $0.60$, and $0.20$. State $D$ is absorbing. A premium of $600 is due at times 1, 2, and 3 only if the policyholder is in $A$ at the payment time. The annual effective interest rate is $i=0.10$. Find the EPV.

Method. Find the probability of occupying $A$ at each payment time, then multiply each probability-weighted premium by its discount factor. The premium condition is assessed at year-end, not at the start of the year.

At time 1, the probability of being in $A$ is $p_A(1)=0.50$. The time-1 probability of being in $B$ is $0.30$.

At time 2, a policyholder can be in $A$ either by moving from $A$ to $A$, or by moving from $B$ to $A$. Thus

$$ p_A(2)=0.50(0.50)+0.30(0.20)=0.25+0.06=0.31. $$

The time-2 probability of being in $B$ is $p_B(2)=0.50(0.30)+0.30(0.60)=0.15+0.18=0.33$. At time 3,

$$ p_A(3)=p_A(2)(0.50)+p_B(2)(0.20) =0.31(0.50)+0.33(0.20) =0.155+0.066=0.221. $$

The discount factors are $v=1/1.10$, $v^2=1/1.21$, and $v^3=1/1.331$. Therefore,

\begin{aligned} \operatorname{EPV} &=600\left(\frac{0.50}{1.10} +\frac{0.31}{1.21} +\frac{0.221}{1.331}\right)\\ &=600(0.4545455+0.2561983+0.1660406)\\ &=600(0.8767844)=\$526.0706. \end{aligned}

Rounded to the nearest cent, the EPV is $526.07. The occupancy check is consistent: at each time the probabilities of $A$, $B$, and $D$ sum to 1. The common mistake avoided here is using only the probability of staying in $A$ continuously; the policyholder can leave $A$ and return before a later premium date.

Continuous Premiums During State Occupancy

For a continuous premium rate $c_j(t)$ payable in state $j$, let $p_j(t)$ be the probability of occupying that state at time $t$. Under force of interest $\delta$, the EPV is the integral of the discounted expected premium rate. The expression applies when the expected discounted premium stream is integrable; for a finite term with bounded rates this condition is satisfied.

\operatorname{EPV} = \int_0^T e^{-\delta t} \sum_{j\in\mathcal{C}}p_j(t)c_j(t)\,dt.

This follows by considering a small interval of length $dt$. The probability of being in state $j$ at time $t$ is $p_j(t)$, so the expected premium paid during that interval is $p_j(t)c_j(t)\,dt$. Its time-$0$ value is approximately $e^{-\delta t}p_j(t)c_j(t)\,dt$. Summing over states and intervals and taking the limit gives the integral. The result uses state occupancy, not merely the probability of never having left the initial state.

Worked Example: A Continuous Premium with Possible Return to A

A hypothetical policyholder starts in $A$ in a two-state continuous-time Markov model. The transition intensity from $A$ to $B$ is $0.30$ per year, and from $B$ to $A$ it is $0.10$ per year. There are no other transitions. A continuous premium of $250 per year is payable while the policyholder is in $A$, with no term limit. The force of interest is $\delta=0.10$. Find the EPV.

Method. Use the two-state occupancy probability and integrate its discounted value. Let $p_A(t)$ be the probability of being in $A$ at time $t$, given the initial state $A$. The stationary probability of $A$ is $0.10/(0.30+0.10)=0.25$, and the deviation from this value decays at rate $0.30+0.10=0.40$. Hence

p_A(t)=0.25+0.75e^{-0.40t}.

This expression has the required initial value $p_A(0)=1$ and tends to $0.25$ as $t$ increases. Applying the continuous-premium formula gives

\begin{aligned} \operatorname{EPV} &=250\int_0^\infty e^{-0.10t} \left(0.25+0.75e^{-0.40t}\right)\,dt\\ &=250\left( 0.25\int_0^\infty e^{-0.10t}\,dt +0.75\int_0^\infty e^{-0.50t}\,dt \right)\\ &=250\left(\frac{0.25}{0.10}+\frac{0.75}{0.50}\right)\\ &=250(2.5+1.5)=\$1{,}000. \end{aligned}

The EPV is $1,000.00. The return transition from $B$ to $A$ matters: premiums are payable again during later periods in $A$. Treating the premium as stopping permanently at the first $A$-to-$B$ transition would value a different cash-flow rule.

State Eligibility and Stopping on a Transition

A state-based condition and a permanent stopping condition are not automatically the same. If premiums are payable whenever the current state is $A$, then a later return to $A$ makes the policyholder eligible again. By contrast, if premiums cease permanently upon a specified transition, eligibility depends on the path taken, not only the current state.

A permanent stop can be represented by distinguishing states that have different premium histories. For instance, a state entered after premiums have permanently stopped must remain distinct from an otherwise similar state in which premiums could still be due. If the model already has an absorbing state representing termination and that transition ends premiums, no additional distinction is needed. The valuation must match the stated contract condition; it should not silently treat a recoverable state as absorbing.

Worked Example: State-Dependent Continuous Premiums Before Termination

A hypothetical contract starts in state $A$. State $D$ is absorbing and has no premiums. The transition intensities per year are $A\to B:0.20$, $A\to D:0.10$, $B\to A:0.10$, and $B\to D:0.20$. Continuous premiums are $100 per year in $A$ and $200 per year in $B$. The force of interest is $\delta=0.10$. Find the EPV from initial state $A$.

Method. Define $V_A$ and $V_B$ as the EPVs of future premiums conditional on starting in $A$ and $B$, respectively. In either transient state the total rate of leaving is $0.30$. Over a small interval $dt$, discounting contributes $1-\delta\,dt$, the probability of leaving contributes the relevant intensity times $dt$, and the premium earned is the state premium rate times $dt$. Keeping terms of order $dt$, dividing by $dt$, and letting $dt$ tend to zero gives

(\delta+0.30)V_A=100+0.20V_B, \qquad (\delta+0.30)V_B=200+0.10V_A.

Substitute $\delta=0.10$, so the equations become

0.40V_A=100+0.20V_B, \qquad 0.40V_B=200+0.10V_A.

Divide the first equation by $0.40$ to obtain $V_A=250+0.50V_B$. Divide the second to obtain $V_B=500+0.25V_A$. Substituting the second expression into the first gives

V_A=250+0.50(500+0.25V_A) =500+0.125V_A, \qquad 0.875V_A=500, \qquad V_A=\frac{4000}{7}=571.4286.

Thus the EPV from $A$, rounded to the nearest cent, is $571.43. As a check, $V_B=500+0.25(4000/7)=4500/7$, and $0.40(4500/7)=200+0.10(4000/7)$, so both equations are satisfied. The common mistake avoided is to omit the $B$-to-$A$ return premium stream or to treat $B$ as absorbing.

A Reliable Valuation Procedure

For a state-contingent premium problem, separate the contract rule from the probability calculation. The following sequence keeps timing, eligibility, and discounting distinct.

Specify the premium rule.
List the states in which a premium is payable, the amount or rate in each state, and whether the condition is assessed at a payment date or throughout an interval.

Identify the state probabilities or values needed.
For discrete premiums, calculate occupancy probabilities at due dates. For continuous premiums, use occupancy probabilities over time or solve conditional EPV equations.

Apply the correct discounting.
Use $v^{t}$ for a payment at time $t$ under an effective annual rate, or $e^{-\delta t}$ under a constant force of interest.

Check the contract interpretation.
Confirm whether a return to an eligible state restarts premiums, and whether a stated stopping transition is permanent in the model.

Common Exam Traps

Using the wrong state time. For a premium due at year-end, use the state at year-end. Do not use the probability of occupying the state at the beginning of the year unless the problem says the premium is due then.
Ignoring return transitions. A policyholder may leave and later re-enter a premium-paying state. State-based eligibility counts that later occupancy unless the contract specifies permanent cessation.
Confusing rates with amounts. A continuous premium rate of $100 per year is multiplied by time spent in the state; it is not a single $100 payment at each integer time.
Using the wrong discount convention. With an effective annual rate $i$, a time-$t$ payment is discounted by $(1+i)^{-t}$. With force $\delta$, it is discounted by $e^{-\delta t}$.
Dropping an exit intensity. In a continuous-time model, the rate governing decay of a two-state occupancy deviation is the sum of the two transition intensities, not just the intensity out of the initial state.

Key takeaway: The EPV of conditional premiums is determined by the expected premium amount at each time: premium amount or rate multiplied by the probability of occupying an eligible state. Correct state timing and the treatment of return transitions are essential.

Check Your Understanding

For each problem, identify the eligible state condition before calculating.

A policy starts in $A$. One year later it is in $A$ with probability $0.80$. A $100 premium is due at time 1 only if the state is $A$. The annual effective interest rate is $0.05$. Find the EPV. (Answer: $76.19)
A policy starts in $A$. At each year-end, $A\to A$ has probability $0.70$, $A\to B$ has probability $0.30$, $B\to A$ has probability $0.40$, and $B\to B$ has probability $0.60$. A $200 premium is due at times 1 and 2 only in $A$. If $i=0.10$, find the EPV. (Answer: $228.10)
A continuous-time policy starts in $B$, with intensities $A\to B:0.20$ and $B\to A:0.10$ per year and no other transitions. A continuous premium of $1,200 per year is payable in $A$ indefinitely. If $\delta=0.10$, find the EPV. (Answer: $3,000.00)
A policy starts in $A$; $D$ is absorbing. The intensities are $A\to B:0.10$, $A\to D:0.10$, $B\to A:0.20$, and $B\to D:0.10$ per year. Continuous premiums are $80 per year in $A$ and $120 per year in $B$. If $\delta=0.10$, find the EPV from $A$. (Answer: $440.00)
A discrete-time policy starts in $A$, with transitions $A\to A:0.60$, $A\to B:0.40$, $B\to A:0.25$, and $B\to B:0.75$. A $100 premium is due at times 0, 1, and 2 only if the state is $A$. If $i=0.04$, find the EPV. (Answer: $200.22)

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