State-Contingent Benefit Valuation

Why State-Contingent Benefits Need Occupancy Probabilities

A multi-state contract may pay a benefit only while the policyholder is in a particular state. For example, a hypothetical long-term care contract might pay an income benefit at each year-end if the insured is then in a care state. The value depends on two separate items: the probability that the policy occupies the eligible state when payment is due, and the discount factor from that payment time back to the valuation time.

The essential timing question is whether the benefit depends on the state at a payment time or on a transition between states. A payment due at the end of year 2 only if the insured is disabled then is valued using the probability of being disabled at time 2. It is not valued using only the probability of entering disability during year 2. This tutorial concerns benefits payable because the policy occupies specified states at payment times, or continuously while it occupies those states.

Definition: Let $J(t)$ denote the policy state at time $t$, and let $i$ be the effective annual interest rate with $v=1/(1+i)$. A state-contingent benefit is a payment whose amount or eligibility depends on $J(t)$. If $p_j(t)=P(J(t)=j\mid J(0)=i_0)$, then $p_j(t)$ is the probability of occupying state $j$ at time $t$, conditional on the specified initial state $i_0$. A payment $b_j(t)$ due at time $t$ only in state $j$ has expected present value $v^t p_j(t)b_j(t)$ when $t$ is measured in years and the benefit is paid at that fixed time.

For a discrete model, let $p_{ij}(k)$ be the probability of being in state $j$ at time $k$, given that the policy began in state $i$. If a benefit $b_{j,k}$ is paid at the end of period $k$ when the policy is in state $j$, the expected present value (EPV) is

\operatorname{EPV} = \sum_{k=1}^{n} v^k \sum_{j\in\mathcal{S}} p_{ij}(k)b_{j,k},

where $\mathcal{S}$ is the set of states and a benefit amount is zero in states that are not eligible. A termination state can be included in $\mathcal{S}$; if no benefit is paid after termination, its benefit amount is zero. The probabilities in this expression are unconditional probabilities from the initial state, not probabilities renormalized among policies still in force.

Why the Expected-Present-Value Formula Holds

At a fixed payment time $k$, write the actual payment as a random variable $B_k$. If the payment is $b_{j,k}$ in state $j$, then $B_k$ takes the value $b_{j,k}$ when $J(k)=j$. Since the state events are mutually exclusive, its expectation is the sum of each possible payment weighted by its probability.

E[B_k] = \sum_{j\in\mathcal{S}}P(J(k)=j)b_{j,k} = \sum_{j\in\mathcal{S}}p_{ij}(k)b_{j,k}.

With a deterministic effective annual interest rate, the present value of that payment is $v^kB_k$. Linearity of expectation gives $E[v^kB_k]=v^kE[B_k]$. Summing over payment times gives the discrete EPV formula. This argument requires no independence between payments at different times; it uses the expected value of each payment separately. If interest is stochastic or dependent on the policy state, the discount factor cannot in general be taken outside the expectation in this way.

Worked Example: Two Year-End Benefits in a Three-State Model

A hypothetical two-year contract starts in the active state $A$. At each year-end, it pays $500 if the insured is disabled, state $D$. The states are active, disabled, and terminated, with terminated state $T$ absorbing. The annual transition probabilities are $A\to A=0.75$, $A\to D=0.20$, $A\to T=0.05$, $D\to A=0.30$, $D\to D=0.65$, and $D\to T=0.05$. There is no benefit in $A$ or $T$. At an effective annual interest rate of 5%, find the EPV. Assume the stated transitions apply for each year and the benefit is paid only if the state at that year-end is $D$.

Method. Find the disabled-state probability at each payment date, multiply it by $500, and discount each expected payment to time 0.

At time 1, the policy starts in $A$, so $P(J(1)=D)=0.20$. The expected year-1 benefit is $500(0.20)=\$100$.

At time 2, the policy can be disabled after either being active or disabled at time 1. Thus $P(J(2)=D)=0.75(0.20)+0.20(0.65)=0.15+0.13=0.28$. The terminated state contributes zero because it cannot return to $D$. The expected year-2 benefit is $500(0.28)=\$140$.

The discount factor is $v=1/1.05$, so $v^2=1/1.1025$. Therefore,

\operatorname{EPV} = \frac{100}{1.05} + \frac{140}{1.1025} = 95.238095\ldots+126.984127\ldots = \$222.22.

The result is rounded to the nearest cent. A check is that the undiscounted expected payments total $240, so a positive-interest EPV of $222.22 is plausible. The method avoids treating the probability of being disabled at year 2 as merely the probability of remaining disabled from year 1; entry into disability from the active state also contributes.

Continuous Benefits While in an Eligible State

Some benefits accrue continuously rather than as a lump sum at a fixed date. Suppose a contract pays at a rate of $c_j(t)$ dollars per year while the policy occupies state $j$, and the force of interest is a constant $\delta$. If payments stop at time $n$, the expected present value of that state-specific income is

\operatorname{EPV} = \int_0^n e^{-\delta t}p_j(t)c_j(t)\,dt.

The factor $p_j(t)c_j(t)$ is the expected payment rate at time $t$; discounting it and integrating adds the expected present values over time. More generally, state-specific rates can be summed over eligible states. The formula assumes the stated rate is paid whenever the policy occupies the state and that the interest force is deterministic. It does not value a lump sum paid only on entering the state: that is a transition-triggered benefit and requires transition probabilities over the relevant interval.

Worked Example: Continuous Income During Disability

A hypothetical contract starts in active state $A$ and pays $1,200 per year continuously while the insured is disabled, state $D$, for at most two years. The only transitions are $A\to D$ at constant rate $\alpha=0.20$ per year and $D\to A$ at constant rate $\beta=0.10$ per year; there is no termination in this two-state model. The force of interest is $\delta=0.05$. Find the EPV.

Method. First derive the probability of occupying $D$ at time $t$, then integrate the discounted expected payment rate over the two-year term.

Let $p_D(t)=P(J(t)=D\mid J(0)=A)$. Over a short interval, the probability of moving into $D$ from $A$ contributes $\alpha(1-p_D(t))\,dt$, while the probability of moving out of $D$ contributes $-\beta p_D(t)\,dt$. Therefore $p_D'(t)=\alpha-(\alpha+\beta)p_D(t)$, with $p_D(0)=0$. Solving this differential equation gives

p_D(t) = \frac{\alpha}{\alpha+\beta} \left(1-e^{-(\alpha+\beta)t}\right) = \frac{2}{3}\left(1-e^{-0.30t}\right).

The EPV is consequently

\begin{aligned} \operatorname{EPV} &=1200\int_0^2 e^{-0.05t} \frac{2}{3}\left(1-e^{-0.30t}\right)\,dt\\ &=800\left[ \frac{1-e^{-0.10}}{0.05} - \frac{1-e^{-0.70}}{0.35} \right]\\ &=800\left[ 1.90325164-1.43832770 \right]\\ &=800(0.46492394) =\$371.94. \end{aligned}

The result is rounded to the nearest cent. As a check, $p_D(t)$ begins at zero and approaches $2/3$, so the expected discounted duration in $D$ over two years is positive and less than two years. The answer is $1200$ times that expected discounted duration. The method avoids applying a year-end discount factor to a benefit that accrues continuously.

Checking State Occupancy and Payment Timing

A state-occupancy calculation should match the contract wording exactly. A payment at the end of a year depends on the state at that year-end. A continuous payment rate depends on the time spent in the eligible state. A benefit triggered by a transition, such as a lump sum on entering care, is a different cash-flow definition. In particular, the probability of occupying care at a date is not generally the same as the probability of having entered care by that date.

Specify the valuation time and payment times.
Use the time from which all benefits are discounted and identify whether each payment is at a fixed date or accrues over an interval.

Identify the eligible state or states.
Translate the benefit wording into the policy states that produce a payment. Assign zero benefit to ineligible states.

Calculate occupancy probabilities.
Use the transition model from the stated initial state. Include termination or death in the probability calculation when applicable, even if those states pay no benefit.

Apply the correct discount factor.
For a fixed payment at time $k$, use $v^k$ under an effective annual rate. For continuous payments under constant force $\delta$, use $e^{-\delta t}$ inside the integral.

Sum or integrate and check.
Combine expected discounted payments, then check that probabilities are between 0 and 1 and that discounting reduces positive future values.

Worked Example: Care-State Payments with Re-entry

A hypothetical policy begins in care state $C$. At each year-end it pays $900 if the policy is then in $C$, for the first two years only. The other living state is active, $A$, and $T$ is absorbing termination. Annual transitions are $A\to A=0.6$, $A\to C=0.3$, $A\to T=0.1$, $C\to A=0.2$, $C\to C=0.7$, and $C\to T=0.1$. The effective annual interest rate is 4%. Find the EPV.

Method. Track the probability of occupying $C$ at each payment date, including movement from $C$ to $A$ and back to $C$.

At the end of year 1, the policy is in $C$ with probability $0.7$. The expected year-1 payment is $900(0.7)=\$630$.

At the end of year 2, the policy can be in $C$ by moving from $A$ to $C$, or by remaining in $C$. The year-1 probabilities are $P(A)=0.2$ and $P(C)=0.7$, so $P(J(2)=C)=0.2(0.3)+0.7(0.7)=0.06+0.49=0.55$. The terminated state has probability 0.1 at year 1 and is absorbing, so it contributes nothing to year-2 care occupancy. The expected year-2 payment is $900(0.55)=\$495$.

With $v=1/1.04$ and $v^2=1/1.0816$,

\begin{aligned} \operatorname{EPV} &=\frac{630}{1.04}+\frac{495}{1.0816}\\ &=605.769231\ldots+457.655325\ldots\\ &=\$1,063.42. \end{aligned}

The result is rounded to the nearest cent. The undiscounted expected payments are $630+495=\$1,125$, which is greater than the discounted EPV, as expected.

Common mistake and correction. A calculation that counts only policies remaining in $C$ gives a year-2 care probability of $0.7(0.7)=0.49$, omitting policies that move from $C$ to $A$ and then back to $C$. That incorrect calculation gives $630/1.04+441/1.0816=\$1,013.50$. The correction is to include both possible routes into $C$ at year 2: $0.2(0.3)+0.7(0.7)=0.55$, giving the correct EPV of $1,063.42.

Common Exam Traps

Using the wrong event. “Pay if in care at year-end” calls for the probability of occupying care at that year-end, not the probability of entering care during the year.
Ignoring entry into an eligible state. A policy can leave and later re-enter the state. Include every route that can lead to occupancy at the payment date.
Renormalizing after termination. If the benefit is valued from the initial policy state, use unconditional occupancy probabilities. Do not divide by the probability of remaining in force unless the question explicitly asks for a value conditional on survival or continued coverage.
Discounting at the wrong time. A benefit paid at the end of year $k$ receives $v^k$. A continuously paid benefit uses discounting inside an integral.
Mixing interest conventions. Under an effective annual rate $i$, use $v^k=(1+i)^{-k}$. Under a constant force $\delta$, use $e^{-\delta t}$. These are related by $\delta=\ln(1+i)$ when they represent the same annual accumulation, but they are not interchangeable without conversion.

Key takeaway: For a state-contingent benefit, value the cash flow that the contract actually specifies: identify its payment time or payment rate, find the probability of occupying the eligible state then, and apply the matching discount factor. State probabilities and timing must be correct before the arithmetic can be correct.

Check Your Understanding

For each problem, state the relevant occupancy probability before calculating the EPV. Round dollar answers to the nearest cent.

A policy starts active. At each year-end, active policies move to active with probability 0.75, disabled with probability 0.25; disabled policies move to active with probability 0.40 and remain disabled with probability 0.60. A $400 benefit is paid if disabled at year 1, and an $800 benefit if disabled at year 2. There is no termination, and $i=5\%$. Find the EPV. (Answer: $340.14)
A policy starts disabled. At each annual transition, disabled policies move to active with probability 0.4, remain disabled with probability 0.5, and terminate with probability 0.1. Active policies move to active with probability 0.7, disabled with probability 0.2, and terminate with probability 0.1; termination is absorbing. A $500 benefit is paid if active at either of the first two year-ends. If $i=4\%$, find the EPV. (Answer: $414.20)
A two-state policy starts active and transitions from active to disabled at rate 0.1 per year and from disabled to active at rate 0.1 per year. It pays continuously at $1,000 per year while disabled for three years. The force of interest is $\delta=0.2$. Find the EPV. (Answer: $254.46)
A policy starts active, and termination is absorbing. Annual transitions from active are 0.5 to active, 0.4 to disabled, and 0.1 to termination. From disabled, they are 0.2 to active, 0.6 to disabled, and 0.2 to termination. At the end of year 2, it pays $300 if disabled or $1,000 if terminated. At $i=6\%$, find the EPV. (Answer: $322.18)
A two-state policy starts active. It moves from active to disabled at rate 0.2 per year and from disabled to active at rate 0.1 per year. A benefit is paid continuously at $600 per year while disabled for two years. The force of interest is $\delta=0.1$. Find the EPV. (Answer: $174.41)

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