Two-State Active-and-Disabled Reserve Model

Why Reserves Depend on the Insured’s State

For a contract that covers disability, the future cash flows depend on whether the insured is active or disabled at the valuation date. An active insured may pay premiums and may become disabled during the next year. A disabled insured may receive income benefits, recover, or die. A single reserve cannot represent these different conditional future cash flows without first specifying the current state.

This tutorial uses a discrete annual model. The valuation is immediately before any premium due at that time. Premiums are paid at the beginning of the year, if payable in the insured’s current state. Transition benefits, disability income, and death benefits are paid at the end of the year under the stated event convention. This timing convention is part of the model: a contract with benefits paid at other times requires different discounting.

Definition: The active reserve $V_A(t)$ is the expected present value, at time $t$ and conditional on the insured being active, of future benefits and expenses less future premiums. The disabled reserve $V_D(t)$ is defined in the same way, conditional on the insured being disabled. Both reserves are measured immediately before any premium due at time $t$.

The prospective value is conditional on the state at the valuation date, not on the insured’s state at issue. A disabled reserve can therefore include the value of a future recovery into the active state, while an active reserve can include the value of benefits triggered by a future transition into disability.

Annual Transition Model and Reserve Equations

During a year, an active insured can remain active, become disabled, or die. A disabled insured can recover to active status, remain disabled, or die. Write the corresponding one-year probabilities as $p_{AA}, p_{AD}, p_{A\dagger}$ for an active insured and $p_{DA}, p_{DD}, p_{D\dagger}$ for a disabled insured. The dagger denotes death. For each starting state, the probabilities sum to one.

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Let $P_A(t)$ and $P_D(t)$ be premiums due at time $t$ in the active and disabled states. Let $K(t)$ be a lump-sum transition benefit on becoming disabled, $b(t)$ be disability income paid at year-end if the insured is disabled then, and $C_A(t)$ and $C_D(t)$ be the death benefits following death from the active and disabled states. The equations below use these stated payment conditions; in particular, there is no disability income on recovery or death.

V_A(t)=-P_A(t)+v\left[p_{AA}V_A(t+1)+p_{AD}\{K(t)+b(t)+V_D(t+1)\}+p_{A\dagger}C_A(t)\right]

V_D(t)=-P_D(t)+v\left[p_{DA}V_A(t+1)+p_{DD}\{b(t)+V_D(t+1)\}+p_{D\dagger}C_D(t)\right]

A premium is subtracted at the start of the year only in a state where it is payable. The active-to-disabled branch includes the transition benefit, the year-end disability income, and the reserve conditional on being disabled next year. The disabled-to-disabled branch includes the income benefit and next year’s disabled reserve. If the contract ends after the year’s payments, both reserves at that endpoint are zero.

Why the Equations Hold

The equations follow by conditioning on the event that occurs during the next year. At time $t$, the immediate premium is a certain inflow, so it reduces the reserve. After that, each possible transition contributes its probability multiplied by the discounted amount associated with that transition. For example, starting active, the probability of becoming disabled is $p_{AD}$; on that branch, the end-of-year cash benefit is $K(t)+b(t)$, and the future liability is $V_D(t+1)$.

More formally, if $Z$ denotes the end-of-year benefit plus the next reserve for the realized transition, the present value of that contingent amount is $vE[Z\mid\text{state at }t]$. Partitioning the expectation by the mutually exclusive transition events gives $v\sum_j p_{Aj}E[Z\mid j]$ for an active insured, or $v\sum_j p_{Dj}E[Z\mid j]$ for a disabled insured. Subtracting the premium due at time $t$ gives the displayed recursions. This argument also explains why the reserve attached to each branch must be the reserve for the state reached on that branch.

Fix the valuation time and payment timing.
Record whether the reserve is before or after a premium, and when each benefit is paid.

List the possible next states.
For each current state, include every transition probability and confirm that the probabilities sum to one.

Attach cash flows to the correct branches.
Include a benefit only when its specified trigger occurs; add the reserve for the state reached.

Discount and combine.
Multiply each branch amount by its transition probability, discount one year, and subtract any premium due immediately.

Worked Reserve Calculations

Worked Example: Reserves One Year Before the Next Valuation

A hypothetical annual contract is valued immediately before the start-of-year premium. The effective annual interest rate is $4\%$. The premium is $\$350$ if active and zero if disabled. At year-end, the disability income benefit is $\$900$ if disabled, the transition benefit on becoming disabled is $\$500$, and the death benefit is $\$12{,}000$ from either insured state. One year later, the active and disabled reserves are $\$900$ and $\$2{,}400$, respectively. The transition probabilities are $p_{AA}=0.82$, $p_{AD}=0.13$, $p_{A\dagger}=0.05$, and $p_{DA}=0.25$, $p_{DD}=0.65$, $p_{D\dagger}=0.10$. Calculate both current reserves.

Method. Apply the state-contingent prospective reserve equations. The discount factor is $v=1/1.04=0.961538\ldots$.

For an active insured, the active-to-active branch contributes $0.82(900)=738$. The active-to-disabled branch includes the transition benefit, disability income, and next disabled reserve: $500+900+2{,}400=3{,}800$, so its probability-weighted amount is $0.13(3{,}800)=494$. The death branch contributes $0.05(12{,}000)=600$. Thus the expected end-of-year amount is $738+494+600=1{,}832$, and

V_A=-350+\frac{1{,}832}{1.04}=-350+1{,}761.53846\ldots=\$1{,}411.54.

For a disabled insured, recovery leads to the active reserve and no disability income, so that branch contributes $0.25(900)=225$. Remaining disabled leads to the income payment and disabled reserve: $900+2{,}400=3{,}300$, contributing $0.65(3{,}300)=2{,}145$. Death contributes $0.10(12{,}000)=1{,}200$. Therefore

V_D=\frac{225+2{,}145+1{,}200}{1.04} =\frac{3{,}570}{1.04} =\$3{,}432.69.

Both values are rounded to the nearest cent only at the final step. The calculation avoids the common mistake of adding disability income to the recovery or death branches. As a check, the branch probabilities sum to $1$ in each state, and every future reserve used is conditional on the state reached.

Worked Example: Backward Calculation Over Two Years

A hypothetical two-year policy has an effective annual interest rate of $5\%$. A premium of $\$300$ is payable at the start of each year in which the insured is active, and none is payable while disabled. The year-end disability income is $\$900$ if disabled, a $\$1{,}000$ transition benefit is paid on becoming disabled, and the death benefit is $\$10{,}000$ from either state. In each year, the active transition probabilities are $0.85$ to active, $0.10$ to disabled, and $0.05$ to death. From disability, the probabilities are $0.25$ to active, $0.65$ to disabled, and $0.10$ to death. The policy ends after the second year. Find the active and disabled reserves at times $1$ and $0$.

Method. Start at time $1$, when only one year remains and the reserves after the final year’s payments are zero. Then substitute those time-1 reserves into the time-0 equations. Here $v=1/1.05$.

At time $1$, the active reserve is

V_A(1)=-300+\frac{0.10(1{,}000+900)+0.05(10{,}000)}{1.05} =-300+\frac{690}{1.05} =\$357.14.

The active-to-active branch has zero terminal reserve and no benefit. The disabled reserve at time $1$ is

V_D(1)=\frac{0.65(900)+0.10(10{,}000)}{1.05} =\frac{1{,}585}{1.05} =\$1{,}509.52.

Recovery has zero terminal reserve, and no disability income is paid on that recovery branch. Next, at time $0$, the active-to-active branch contributes $0.85V_A(1)$. The active-to-disabled branch contributes $0.10[1{,}000+900+V_D(1)]$, while death contributes $0.05(10{,}000)=500$. Using unrounded intermediate reserves,

\begin{aligned} V_A(0) &=-300+\frac{0.85(357.142857\ldots)+0.10(1{,}900+1{,}509.523810\ldots)+500}{1.05}\\ &=-300+\frac{1{,}144.523810\ldots}{1.05}\\ &=\$790.02. \end{aligned}

For an insured disabled at time $0$, recovery leads to $V_A(1)$, continued disability leads to the income payment plus $V_D(1)$, and death leads to the death benefit:

\begin{aligned} V_D(0) &=\frac{0.25(357.142857\ldots)+0.65(900+1{,}509.523810\ldots)+0.10(10{,}000)}{1.05}\\ &=\$2{,}529.02. \end{aligned}

The time-0 reserves are rounded to the nearest cent; unrounded values were carried through the recursion. The check is chronological: the final-year reserves were calculated first, and only then used as future values at time $0$.

Worked Example: Solving for a Level Premium

A hypothetical two-year contract pays a $\$500$ transition benefit and $\$1{,}000$ of disability income at year-end when an active insured becomes disabled. Disability income is also paid at year-end if a disabled insured remains disabled. The death benefit is $\$8{,}000$ from either state. The effective annual interest rate is $0\%$. At the start of each year, an active insured pays a level premium $P$; a disabled insured pays no premium. In each year, the active probabilities are $0.80$ to active, $0.15$ to disabled, and $0.05$ to death. The disabled probabilities are $0.20$ to active, $0.70$ to disabled, and $0.10$ to death. The policy ends after year two. Find $P$ such that the time-0 active reserve is zero.

Method. Calculate the one-year remaining reserves first, retaining $P$ in the active reserve because an active insured pays the second premium at time $1$. At time $1$, the terminal reserves after year-end payments are zero. Therefore

V_A(1)=-P+0.15(500+1{,}000)+0.05(8{,}000)=625-P, \qquad V_D(1)=0.70(1{,}000)+0.10(8{,}000)=1{,}500.

At time $0$, an active insured pays $P$. The active-to-active branch has reserve $625-P$; the active-to-disabled branch has the transition benefit, disability income, and disabled reserve; and death pays $\$8{,}000$. Thus

\begin{aligned} V_A(0) &=-P+0.80(625-P)+0.15(500+1{,}000+1{,}500)+0.05(8{,}000)\\ &=-P+500-0.80P+450+400\\ &=1{,}350-1.80P. \end{aligned}

Set the reserve equal to zero: $1{,}350-1.80P=0$, so $P=1{,}350/1.80=\$750.00$. As a substitution check, $V_A(1)=625-750=-125$, and

V_A(0)=-750+0.80(-125)+0.15(3{,}000)+400 =-750-100+450+400=0.

The negative time-1 active reserve is possible: given this premium, the remaining premium income exceeds the expected remaining benefit outgo for an active insured. The calculation avoids omitting the second-year premium from $V_A(1)$, which would change the time-0 premium equation.

Common Timing and State Errors

The transition probabilities alone do not determine the reserve. The payment convention determines which cash flows belong on each branch. In particular, “disability income” must be tied to a precise condition, such as being disabled at year-end. If the contract instead pays income for disability that occurred at any time during the year, the model must specify that different trigger and the appropriate payment timing.

A frequent error is to apply the disability income to every branch out of the disabled state. For example, suppose a disabled insured has probabilities $0.20$ of recovery, $0.70$ of remaining disabled, and $0.10$ of death; the next active reserve is $\$400$, the income payment is $\$700$ only if still disabled, and the death benefit is $\$8{,}000$. At $i=0\%$, the incorrect calculation that pays income on recovery as well is

$$ 0.20(400+700)+0.70(700)+0.10(8{,}000)=1{,}510. $$

The correct calculation assigns no income to recovery or death:

V_D=0.20(400)+0.70(700)+0.10(8{,}000) =80+490+800 =\$1{,}370.00.

A second check is to confirm whether the premium is due in each state. A premium waived during disability must not be subtracted in the disabled equation. Conversely, if premiums continue while disabled, they must be included there. Finally, if a benefit is paid at the beginning of the year rather than at year-end, it should not be discounted by a full year in the same way as the year-end amounts in these examples.

Key takeaway: Calculate a reserve conditional on the current state. For each possible next state, combine only the cash flows triggered on that branch with the reserve for the state reached, then probability-weight and discount. State-specific premiums and the valuation-time convention must be applied consistently.

Check Your Understanding

For each problem, use the stated annual timing convention: premiums are due at the beginning of the year, and specified benefits are paid at year-end.

A disabled insured has one year remaining. The effective annual interest rate is $4\%$. Recovery, continued disability, and death probabilities are $0.25$, $0.65$, and $0.10$. The next active reserve is $\$600$, the income benefit is $\$800$ if still disabled, and the death benefit is $\$12{,}000$. There is no payment on recovery. Find the disabled reserve. (Answer: $\$1{,}798.08$)
At $i=0\%$, an active insured pays a $\$250$ premium now. The probabilities of remaining active, becoming disabled, and dying are $0.75$, $0.20$, and $0.05$. The next active and disabled reserves are $\$300$ and $\$1{,}500$. A $\$400$ transition benefit and $\$600$ income benefit are paid on becoming disabled, and the death benefit is $\$10{,}000$. Find the active reserve. (Answer: $\$975.00$)
A two-year contract has $i=0\%$, no premium while disabled, and a level premium $P$ at the start of each year while active. The active transition probabilities are $0.70$ to active, $0.20$ to disabled, and $0.10$ to death. The disabled probabilities are $0.20$ to recovery, $0.70$ to continued disability, and $0.10$ to death. A $\$300$ transition benefit and $\$700$ income benefit are paid when active becomes disabled; $\$700$ is paid if disabled remains disabled; and the death benefit is $\$9{,}000$. Find the premium that makes the time-0 active reserve zero. (Answer: $P=\$1{,}263.53$)
A disabled insured has one year remaining and no premium. At $i=5\%$, recovery, continued disability, and death probabilities are $0.30$, $0.60$, and $0.10$. The next active and disabled reserves are $\$500$ and $\$2{,}000$. The income benefit is $\$1{,}000$ only if still disabled, and the death benefit is $\$8{,}000$. Find the disabled reserve. (Answer: $\$2{,}619.05$)
An active insured pays a $\$400$ premium now. The effective annual interest rate is $4\%$; the probabilities of remaining active, becoming disabled, and dying are $0.82$, $0.13$, and $0.05$. The next active and disabled reserves are $\$900$ and $\$2{,}500$. A $\$300$ transition benefit and $\$700$ income benefit are paid on becoming disabled, and the death benefit is $\$12{,}000$. Find the active reserve. (Answer: $\$1{,}324.04$)

What You'll Learn