Deferred State-Contingent Benefits

Why a Deferred State Benefit Needs Two Calculations

A benefit described as “payable after three years if the policyholder is disabled” involves two separate questions. First, what is the probability that the policyholder is in the required state at the end of the deferral period? Second, what is the present value of the payment made at that date? The state probability and the discount factor must refer to the same payment date.

This tutorial considers a one-time benefit payable at a specified future time if a stated condition on the policyholder’s state is met then. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The calculations use a discrete-time state model: state changes occur at the ends of annual periods, and eligibility is assessed at the benefit payment date.

Definition: A deferred state-contingent benefit is a benefit scheduled for a future payment date $n$ that is paid only if the policyholder satisfies a specified state condition at time $n$. If the benefit amount is $B$ and the eligible-state set is $E$, the payment at time $n$ is $B$ when the state at time $n$ belongs to $E$, and zero otherwise.

The condition “is disabled at time $n$” is not the same as “has ever been disabled by time $n$.” A policyholder who becomes disabled and then recovers before time $n$ satisfies the first condition only if disabled again at the payment date; the earlier disability alone does not make the payment due. If eligibility depends on the full path of states, rather than the state at payment, state probabilities alone may not be enough unless the model also records the relevant history.

State Probabilities and Present Value

Suppose the model has states $1,\ldots,r$. Let $p_j(k)$ be the probability of being in state $j$ at the end of year $k$. Let $q_{ij}(k)$ be the probability of moving from state $i$ at the end of year $k$ to state $j$ at the end of year $k+1$. The state process is Markov if the next state probabilities, given the current state, do not depend on earlier states.

Under this model, the probabilities are updated one year at a time. The initial probabilities $p_i(0)$ must describe the policyholder’s state at issue; if the initial state is known, its probability is 1 and the others are 0.

p_j(k+1)=\sum_{i=1}^{r}p_i(k)q_{ij}(k).

For a benefit $B$ paid at the end of year $n$ only if the state at that date belongs to $E$, let $i$ be a deterministic effective annual interest rate and $v=(1+i)^{-1}$. The expected present value (EPV) is:

Theorem: If the benefit is paid once, at the end of year $n$, and the eligible states are $E$, then

\operatorname{EPV}=v^n B\sum_{j\in E}p_j(n).

If the amount varies by eligible state, use $B_j$ for the benefit payable in state $j$, giving $\operatorname{EPV}=v^n\sum_{j\in E}p_j(n)B_j$. These formulas assume payment at the end of year $n$, a known deterministic interest rate, and a state condition assessed at that date. They do not value a benefit paid at the first transition into a state or a continuous payment stream.

Why the Valuation Formula Holds

Let $I_j(n)$ equal 1 if the policyholder is in state $j$ at the end of year $n$, and 0 otherwise. With one benefit amount $B$, the payment at time $n$ is $B\sum_{j\in E}I_j(n)$. Discounting gives a present value of $v^nB\sum_{j\in E}I_j(n)$. Taking expectations and using $E[I_j(n)]=p_j(n)$ gives:

E\left[v^nB\sum_{j\in E}I_j(n)\right] =v^nB\sum_{j\in E}E[I_j(n)] =v^nB\sum_{j\in E}p_j(n).

The formula therefore combines the expected payment at the payment date with the discount factor for that same date. It does not require the policyholder to remain in an eligible state during the deferral period unless the contract separately says so.

A Reliable Calculation Sequence

For a multi-year deferral, calculate the state distribution at each year needed to reach the payment date. Then use only the probability of the eligible state or states at that date. Keep the state-probability calculation separate from the discounting until the final step.

Fix the payment date and condition.
Write down the year of payment and the state or states that trigger it.

Find the state distribution at that date.
Start with the initial-state probabilities and apply the transition probabilities for each year.

Calculate the expected payment.
Multiply each eligible-state probability by its benefit amount and add the results.

Discount from the payment date.
Multiply the expected payment by $v^n$ for an end-of-year-$n$ payment.

Worked Valuations

Worked Example: A Two-Year Deferral in a Two-State Model

A hypothetical policyholder starts in the active state. In each year, an active policyholder has probability 0.20 of becoming disabled; a disabled policyholder has probability 0.30 of recovering to active. The other transition from each state is remaining in that state. A benefit of $\$9{,}000$ is payable at the end of year 2 if the policyholder is disabled then. The effective annual interest rate is 5%. Find the EPV.

Method. Calculate the disabled-state probability at the end of year 2 by applying the annual transition probabilities twice. Then discount the resulting expected payment for two years.

At the end of year 1, the probability of being disabled is $0.20$, and the probability of being active is $0.80$. At the end of year 2, the policyholder is disabled either by remaining disabled after year 1 or by moving from active to disabled:

$$ p_D(2)=0.20(0.70)+0.80(0.20)=0.14+0.16=0.30. $$

The expected payment at the end of year 2 is $0.30(\$9{,}000)=\$2{,}700$. The two-year discount factor is $v^2=1/(1.05)^2=1/1.1025$. Therefore:

\operatorname{EPV}=\frac{\$2{,}700}{1.1025} =\$2{,}448.9796\ldots =\boxed{\$2{,}448.98}.

Check. The discounted amount is less than the expected payment of $\$2{,}700$, as expected at a positive interest rate. The result is rounded to the nearest cent.

Common mistake avoided. Using the one-year disability probability of $0.20$ as if it were the year-2 probability would give $0.20(\$9{,}000)/1.1025=\$1{,}632.65$. That omits policyholders who become disabled during the second year, including those who were active at the end of year 1.

Worked Example: A Three-Year Deferral With Death as a Competing State

A hypothetical long-term care contract has three states: active $A$, receiving care $C$, and dead $X$. The annual transition probabilities are constant. From active, the probabilities of ending the year active, receiving care, or dead are 0.75, 0.20, and 0.05. From care, the probabilities of ending the year active, receiving care, or dead are 0.25, 0.65, and 0.10. Death is absorbing. A benefit of $\$20{,}000$ is payable at the end of year 3 only if the policyholder is then receiving care. The effective annual interest rate is 4%. Find the EPV.

Method. Track the active and care probabilities from year to year. Death must be included as a competing state because a policyholder who has died cannot be receiving care at the payment date.

The policyholder starts active, so at the end of year 1 the probabilities are $p_A(1)=0.75$, $p_C(1)=0.20$, and $p_X(1)=0.05$. At the end of year 2:

\begin{aligned} p_A(2)&=0.75(0.75)+0.20(0.25)=0.5625+0.0500=0.6125,\\ p_C(2)&=0.75(0.20)+0.20(0.65)=0.1500+0.1300=0.2800,\\ p_X(2)&=0.05+0.75(0.05)+0.20(0.10)=0.1075. \end{aligned}

The three probabilities sum to $0.6125+0.2800+0.1075=1.0000$. At the end of year 3, the probability of receiving care is:

$$ p_C(3)=p_A(2)(0.20)+p_C(2)(0.65) =0.6125(0.20)+0.2800(0.65) =0.1225+0.1820=0.3045. $$

The expected payment at the end of year 3 is $0.3045(\$20{,}000)=\$6{,}090$. The three-year discount factor is $1/(1.04)^3=1/1.124864$, so:

\operatorname{EPV} =\frac{\$6{,}090}{1.124864} =\$5{,}413.9878\ldots =\boxed{\$5{,}413.99}.

Check. The care probability of 0.3045 is no greater than 1, and the EPV is below the expected payment of $\$6{,}090$ because the payment is deferred and interest is positive. The result is rounded to the nearest cent.

Common mistake avoided. Discounting the expected payment by only two years because the year-3 probability was calculated from the year-2 distribution would mismatch the payment date. The benefit is paid at the end of year 3, so it is discounted by $v^3$.

Worked Example: An Uncertain Initial State

A hypothetical contract begins with a policyholder who is active with probability 0.60 and disabled with probability 0.40. The annual transition probabilities are: active to active 0.80, active to disabled 0.15, and active to dead 0.05; disabled to active 0.20, disabled to disabled 0.70, and disabled to dead 0.10. Death is absorbing. A benefit of $\$18{,}000$ is payable at the end of year 2 if the policyholder is disabled then. The effective annual interest rate is 5%. Find the EPV.

Method. Carry the initial probabilities through the transition model, rather than treating the policyholder as certainly active or certainly disabled.

At the end of year 1, the active probability is $0.60(0.80)+0.40(0.20)=0.48+0.08=0.56$. The disabled probability is $0.60(0.15)+0.40(0.70)=0.09+0.28=0.37$. The death probability is $0.60(0.05)+0.40(0.10)=0.03+0.04=0.07$. These probabilities sum to 1.

At the end of year 2, only transitions from active or disabled can lead to disabled:

$$ p_D(2)=0.56(0.15)+0.37(0.70)+0.07(0) =0.084+0.259=0.343. $$

The expected payment at the end of year 2 is $0.343(\$18{,}000)=\$6{,}174$. Since $(1.05)^2=1.1025$:

\operatorname{EPV} =\frac{\$6{,}174}{1.1025} =\boxed{\$5{,}600.00}.

Check. The transition probabilities from each living state sum to 1, the year-1 state probabilities sum to 1, and the discounted amount is below the expected payment. No rounding is needed.

Common mistake avoided. Applying the transition probabilities to a single assumed initial state would discard the stated 60%/40% initial mixture. The initial distribution must be incorporated before calculating the payment-date probability.

Timing and Eligibility Traps

Deferred benefits are particularly sensitive to the meaning of the contract condition. In a state-at-payment condition, the benefit amount is multiplied by the probability of being eligible at the payment date. A past transition into the state is not enough unless the contract makes prior entry relevant. Likewise, the benefit is discounted from its actual payment date, not from the beginning of the deferral period.

Before calculating, check these points:

Payment date: An end-of-year-$n$ payment uses $v^n$, not $v^{n-1}$.
State at payment: Use the probability of the eligible state at time $n$, not its probability at an earlier date.
Initial distribution: If the initial state is uncertain, include each possible starting state with its stated probability.
Competing states: Include transitions into states such as death when they affect the probability of being eligible later.
Eligibility rule: Do not replace “is in state $j$ at time $n$” with “has ever been in state $j$” or “has remained in state $j$” unless the contract explicitly requires that different condition.

The calculation matters because the deferred benefit depends jointly on state evolution and financial discounting. A correct payment amount with the wrong state probability is still an incorrect valuation, as is a correct state probability discounted for the wrong number of years. A useful final check is to confirm that the eligible-state probability lies between 0 and 1, that all state probabilities at each date sum to 1, and that a positive-interest EPV does not exceed the expected payment at the future date.

Key takeaway: For a one-time deferred benefit conditional on the state at payment, first calculate the probability of eligibility at the payment date, then multiply by the benefit amount and discount from that date. Keep the state condition, payment date, and discount period aligned.

Check Your Understanding

For each problem, use the state condition as written and discount from the stated payment date.

A policyholder starts active. Each year, active changes to disabled with probability 0.10, while disabled recovers to active with probability 0.40. A benefit of $\$8{,}000$ is paid at the end of year 2 if disabled then. The effective annual interest rate is 5%. Find the EPV. (Answer: $\$1{,}088.44$)
A policyholder starts active. In each year, active changes to disabled with probability 0.20 and to dead with probability 0.10; otherwise active continues. Disabled changes to active with probability 0.30 and to dead with probability 0.10; otherwise disabled continues. A benefit of $\$10{,}000$ is paid at the end of year 2 if disabled then. The interest rate is 0%. Find the EPV. (Answer: $\$2{,}600.00$)
At issue, a policyholder is active with probability 0.25 and receiving care with probability 0.75. Over the next year, active changes to care with probability 0.10; care changes to active with probability 0.20 and otherwise remains care. A benefit of $\$16{,}000$ is paid at the end of year 1 if the policyholder is receiving care. The effective annual interest rate is 4%. Find the EPV. (Answer: $\$9{,}615.38$)
A policyholder starts active in a model with active, disabled, and dead states. Each year, an active policyholder moves to active, disabled, or dead with probabilities 0.60, 0.30, and 0.10. A disabled policyholder moves to active, disabled, or dead with probabilities 0.20, 0.70, and 0.10; dead is absorbing. A benefit of $\$25{,}000$ is paid at the end of year 3 if disabled then. The effective annual interest rate is 5%. Find the EPV. (Answer: $\$8{,}616.78$)
At the end of year 2, the state probabilities are active 0.45, disabled 0.35, and dead 0.20. A contract pays $\$2{,}000$ if active or $\$12{,}000$ if disabled at that date, and nothing if dead. The effective annual interest rate is 0%. Find the EPV. (Answer: $\$5{,}100.00$)

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