Disability Income Reserve Worked Problems

Disability Income Reserves by State

A disability income policy can have different cash flows depending on whether the insured is active, disabled but still in a waiting period, receiving disability income, or dead. A reserve calculated without distinguishing these states can misstate both the premium income and the benefits expected from the contract. The key is to specify the state at the valuation time, the possible transitions over the next period, and which cash flows occur on each transition.

This tutorial uses an annual, discrete-time model. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The examples are hypothetical and use probabilities and contract terms supplied in each problem.

Definition: Let $A$ denote active, $W$ disabled but within a waiting period, $B$ disabled and eligible for income benefits, and $D$ dead. A one-year transition probability $p_{rs}(t)$ is the probability that a policy in state $r$ at time $t$ is in state $s$ at time $t+1$. The reserve $V_r(t)$ is the prospective reserve at the start of the year, conditional on being in state $r$, just before any premium due at that time.

Cash-Flow Convention and Reserve Recursion

Use the following timing convention throughout: a premium $P_r(t)$, if due, is received at the beginning of the year; expenses $e_r(t)$, benefits $b_{rs}(t)$, and the next reserve are valued at the end of the year. The expense $e_r(t)$ is paid at year-end for a policy starting in state $r$, regardless of the state at year-end. The benefit $b_{rs}(t)$ is paid at year-end if the transition from $r$ to $s$ occurs. A death benefit is therefore included on a transition to $D$, while the reserve after death is normally zero when no further contractual liability remains.

V_r(t)=-P_r(t)+v\left[e_r(t)+\sum_s p_{rs}(t)\bigl(b_{rs}(t)+V_s(t+1)\bigr)\right], \qquad v=\frac{1}{1+i}.

If the expense depends on the destination state, use a transition-specific expense $e_{rs}(t)$ inside the sum instead of a single $e_r(t)$. Premiums collected at a different time also require a timing adjustment. In particular, this recursion is not a continuous-time formula; the one-year probabilities and end-of-year payment convention must match the problem.

For disability income, state definitions encode benefit eligibility. A waiting state $W$ allows the model to represent a period during which disability has begun but income benefits are not yet payable. A transition $W\to B$ can mark the end of that waiting period. Whether a benefit is payable in the year of transition must still be read from the contract terms: the state labels alone do not decide payment timing.

Identify the valuation state and time.
Record whether the policy is active, waiting, or benefit-paying at the start of the year, and identify the reserve time being requested.

List premiums and expenses.
Apply a premium only if it is due in the starting state. Include expenses once, using the stated payment timing.

Enumerate transitions and payments.
For each possible next state, multiply its probability by the transition benefit plus the reserve in that next state.

Discount and check.
Discount end-of-year expected amounts by $v$, then check that no waiting-period income, premium, or death payment has been omitted or counted twice.

Why the Recursion Holds

The prospective reserve is the expected present value of future outgo less the expected present value of future income, conditional on the current state. Over the next year, the premium due at the beginning is an immediate inflow, so it enters with a negative sign. At year-end, the insurer pays the specified expense and, depending on the transition, the corresponding benefit and the reserve for the policy’s next state.

Condition on the next state $s$. The probability of that state is $p_{rs}(t)$, and the amount due at year-end on that branch is $b_{rs}(t)+V_s(t+1)$. Summing the branch amounts weighted by their probabilities gives the expected end-of-year liability; the starting-state expense is added once. Discounting this expectation by $v$, then subtracting the premium received immediately, gives the stated recursion. If death occurs, the death claim is included in $b_{rD}(t)$; setting $V_D(t+1)=0$ does not remove that claim.

Worked Example: Active Policy with a Waiting-State Transition

A hypothetical policy is valued at the start of a year while the insured is active. The annual effective interest rate is $4\%$. A premium of $\$900$ is collected immediately, and a year-end expense of $\$100$ applies. The next-state probabilities are $p_{AA}=0.85$, $p_{AW}=0.10$, and $p_{AD}=0.05$. No disability income is payable during the year if the insured enters $W$. A death benefit of $\$20{,}000$ is paid on a transition to $D$. The next-year reserves are $V_A= \$1{,}200$, $V_W=\$3{,}500$, and $V_D=0$. Calculate the current active-state reserve.

Method. Apply the one-year prospective recursion. Since the premium is received at the beginning of the year, subtract it without discounting. The death benefit is included on the $A\to D$ branch.

The discount factor is $v=1/1.04$. The expected next reserve is $0.85(1{,}200)+0.10(3{,}500)+0.05(0)=1{,}020+350+0=1{,}370$. The expected transition benefit is $0.05(20{,}000)=1{,}000$. Including the year-end expense, total expected end-of-year outgo is $1{,}370+1{,}000+100=\$2{,}470$.

V_A=-900+\frac{2{,}470}{1.04} =-900+2{,}375 =\$1{,}475.00.

Check. The expected reserve and the expected death benefit are both included, and the expense is counted once. A common mistake is to discount the beginning-of-year premium with the end-of-year amounts; under the stated timing convention, that premium is an immediate $\$900$ inflow and is subtracted directly.

Waiting Periods, Recovery, and Death

A waiting period affects the reserve through both the cash flows during the waiting state and the reserve in the state reached when the waiting period ends. If no income benefit is payable during the year, do not insert one merely because a transition to the benefit-paying state occurs at year-end. Conversely, when benefits are payable during a year that starts in the benefit-paying state, include them according to the payment condition given.

Recovery returns the insured to an active or recovered state, depending on the model. The next reserve on that branch can be negative: for example, it may represent the present value of future premiums in excess of future benefits. Do not replace a negative reserve by zero unless the contract or valuation method specifically requires that. Death is an absorbing state in the usual model, but the immediate death claim is a separate transition cash flow.

Worked Example: Reserve During a Waiting Year

A hypothetical insured is in state $W$ at the start of the year. The annual effective interest rate is $5\%$. The transition probabilities are $p_{WA}=0.20$, $p_{WB}=0.75$, and $p_{WD}=0.05$. A year-end expense of $\$100$ is incurred. No income benefit is payable during this waiting year, including on a transition to $B$. A death benefit of $\$10{,}000$ is payable on transition to $D$. The next-year reserves are $V_A=-\$200$, $V_B=\$4{,}000$, and $V_D=0$. There is no premium while in $W$. Find $V_W$.

Method. Separate the probability-weighted next reserves from the transition benefit. The expected next reserve is $0.20(-200)+0.75(4{,}000)+0.05(0)=-40+3{,}000+0=\$2{,}960$. The expected death benefit is $0.05(10{,}000)=\$500$. Add the waiting-state expense of $\$100$, giving expected end-of-year outgo of $2{,}960+500+100=\$3{,}560$.

V_W=\frac{3{,}560}{1.05} =3{,}390.476\ldots =\$3{,}390.48.

Check and trap. The expected next reserve is $\$2{,}960$, not $\$3{,}000$, because the $\$200$ reserve in state $A$ is negative. Omitting the death claim would give $3{,}060/1.05=\$2{,}914.29$, which is too low. The death claim is paid on the transition to $D$, even though the reserve after death is zero. No waiting-year income benefit is added because the problem explicitly says none is payable.

Worked Example: Benefit-Paying State with Recovery

A hypothetical policy is valued at the start of a year in state $B$, where disability income is payable. The annual effective interest rate is $4\%$. The next-state probabilities are $p_{BA}=0.30$, $p_{BB}=0.65$, and $p_{BD}=0.05$. A year-end expense of $\$240$ applies. An income benefit of $\$5{,}000$ is paid only if the insured is still in $B$ at year-end. A death benefit of $\$15{,}000$ is paid on transition to $D$. There is no premium in $B$, and the next-year reserves are $V_A=-\$400$, $V_B=\$7{,}500$, and $V_D=0$. Calculate $V_B$.

Method. Weight each next reserve by its transition probability, then separately weight benefits by the conditions that trigger them. The expected next reserve is $0.30(-400)+0.65(7{,}500)+0.05(0)=-120+4{,}875+0=\$4{,}755$. The income benefit expectation is $0.65(5{,}000)=\$3{,}250$, and the death benefit expectation is $0.05(15{,}000)=\$750$. Including the expense, expected end-of-year outgo is $4{,}755+3{,}250+750+240=\$8{,}995$.

V_B=\frac{8{,}995}{1.04} =8{,}649.038\ldots =\$8{,}649.04.

Check. The income benefit is paid only on the $B\to B$ branch, so its expected amount is $\$3{,}250$, not $\$5{,}000$. The recovery branch uses $V_A=-\$400$, which reduces expected outgo. A common error is to pay the income benefit on the death branch as well, despite the stated trigger.

Practical Checks for Disability Income Reserves

Before accepting a calculated reserve, make a branch-by-branch cash-flow check. In particular, verify that the state at the start of the year determines whether a premium is collected, while the specified transition or end-of-year state determines whether each benefit is payable. For a contract with premium waiver during disability, a premium may be due in $A$ but not in $W$ or $B$. For a waiting period, a transition into $B$ may affect the next reserve without triggering a benefit in the current year.

Item	Question to check
Premium	Is it collected at the start of the year, and is it due in the current state?
Waiting period	Does the stated benefit condition pay during this year, or only after eligibility begins?
Recovery	Is the reserve in the recovered state used, even if it is negative?
Death	Is the death benefit included on the transition to $D$, separately from $V_D$?
Expenses	Are expenses applied once with the stated timing and state dependence?

The annual recursion is useful because it makes the contract’s state-contingent cash flows visible. It is not appropriate without adjustment if the problem instead specifies continuous benefit payments, premiums at another time, or cash flows that vary within the year. In those cases, the payment timing and model must be reflected explicitly rather than forced into an annual end-of-year calculation.

Key takeaway: A disability income reserve is conditional on the insured’s current state. Apply the premium, expenses, benefits, transition probabilities, and next-state reserves using their stated triggers and timing; do not infer a payment solely from a state label.

Check Your Understanding

For each question, use the annual beginning-premium and end-of-year outgo convention stated above. Round money to the nearest cent.

An active policy has $i=3\%$, a beginning-of-year premium of $\$500$, and a year-end expense of $\$40$. Its transitions are $A\to A$ with probability $0.80$, $A\to W$ with probability $0.15$, and $A\to D$ with probability $0.05$. Next reserves are $\$600$ in $A$, $\$1{,}800$ in $W$, and zero in $D$. A death benefit of $\$8{,}000$ is paid on $A\to D$, and no benefit is paid on $A\to W$. Find $V_A$. (Answer: $\$655.34$)
In state $W$, $i=4\%$, the year-end expense is $\$60$, and there is no premium or waiting-year income benefit. Transition probabilities to $A,B,D$ are $0.30,0.60,0.10$. Next reserves are $-\$100,\$2{,}500,0$, respectively, and a $\$5{,}000$ death benefit is payable on transition to $D$. Find $V_W$. (Answer: $\$1{,}951.92$)
In state $B$, $i=5\%$, the year-end expense is $\$150$, and no premium is due. Transition probabilities to $A,B,D$ are $0.20,0.70,0.10$. Next reserves are $-\$250,\$3{,}200,0$. A $\$2{,}000$ income benefit is paid only if the insured remains in $B$, and a $\$10{,}000$ death benefit is paid on transition to $D$. Find $V_B$. (Answer: $\$4{,}514.29$)
A waiting-state policy has $i=2\%$, no premium, no income benefit during the year, and a year-end expense of $\$80$. Transitions to $A,B,D$ have probabilities $0.50,0.40,0.10$. Next reserves are $\$400,\$3{,}000,0$, and a $\$12{,}000$ death benefit is paid on transition to $D$. Find the waiting-state reserve. (Answer: $\$2{,}627.45$)
A policy in $B$ has $i=4\%$, no premium, and a year-end expense of $\$100$. Transition probabilities to $A,B,D$ are $0.25,0.70,0.05$, with next reserves $-\$200,\$5{,}000,0$. A $\$3{,}000$ income benefit is payable only if the insured remains in $B$, and a $\$9{,}000$ death benefit is payable on transition to $D$. Find the reserve. (Answer: $\$5{,}865.38$)

What You'll Learn