Integrated Disability Income Profit and Reserve Case

Introduction

A disability income projection must track more than whether a policyholder is alive. Incidence determines when disability begins, recovery may return a claimant to active status, and death terminates the contract. A waiting period adds another distinction: a person may be disabled without yet being eligible for periodic benefits. If these states or their timing are handled incorrectly, reserves and profits can be wrong even when the transition probabilities are individually correct.

This tutorial develops a hypothetical annual model with active, waiting-period, claim-paying, and death states. It uses the model to calculate prospective reserves, expected annual profits under experience assumptions, and the discounted NPV of those profits. The setup is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Model and timing: At each year-start, the policy is in one of four states: active $A$, disabled in the waiting period $W$, disabled and eligible for benefits $C$, or dead $D$. Premiums and expenses are paid at the start of the year. Transitions occur at year-end. A benefit is paid at year-end if the policy was in state $C$ at the start of that year, before the year-end transition. A person in $W$ receives no benefit for that year; if still disabled at year-end, the person enters $C$ for the next year. Death is absorbing, and no cash flows occur in state $D$.

State Transitions and the Waiting Period

Use a two-year policy term, with valuation dates $t=0,1,2$. The hypothetical valuation transition probabilities for each year are as follows. A row gives the state at the start of the year; a column gives the state at year-end.

Start state	End state and probability	Interpretation
$A$	$A: 0.78,\ W: 0.20,\ D: 0.02$	Active at year-end, new disability, or death
$W$	$A: 0.25,\ C: 0.70,\ D: 0.05$	Recovery, continued disability through the waiting year, or death
$C$	$A: 0.20,\ C: 0.70,\ D: 0.10$	Recovery, continued claim, or death
$D$	$D: 1.00$	Death is absorbing

Each row sums to 1. The waiting-period state $W$ is essential: combining $W$ with $C$ would incorrectly pay benefits before the waiting period has been completed. In this annual approximation, the waiting period occupies one full annual transition interval. Recovery or death during that interval prevents entry into the claim-paying state.

Worked Example: Timing the First Benefit

Suppose a policyholder is active at time 0. Under the valuation probabilities above, find the probability that the policyholder is in the claim-paying state at time 2, and identify when the first benefit can be paid. Use the annual transition timing specified in the model.

Method. To be in $C$ at time 2, the policyholder must first enter $W$ at time 1 and then remain disabled through the waiting year. From $A$, the probability of entering $W$ is 0.20. From $W$, the probability of entering $C$ is 0.70.

P(X_2=C\mid X_0=A)=0.20(0.70)=0.14.

Thus the probability is 0.14. The person who enters $C$ at time 2 was in $W$ at the start of year 2, so no benefit is paid at the end of that waiting year. The first benefit for that path is payable at the end of year 3, if the person is in $C$ at the start of year 3. The two-year policy term in the reserve calculations below ends before that payment date.

Check. The result uses the path $A\to W\to C$, not $A\to C$. Treating a new disability as immediately claim-paying would bypass the waiting period.

Prospective Reserves by State

Let $V_t(s)$ be the prospective reserve at time $t$, immediately before that year's start-of-year cash flows, conditional on state $s$. A positive reserve is a net liability; it is the present value of future benefits and expenses less future premiums. The hypothetical contract charges a premium of $120 at each year-start in $A$, and has a $10 expense at each year-start in any living insured state $A,W,C$. A $100 benefit is paid at each year-end for a policy in $C$ at the start of the year. No benefit is paid in $A$ or $W$. The annual effective interest rate is $i=0.05$, so $v=1/1.05$.

Reserve recursion: For a state $s$ at time $t$, let $p_s$ be the premium at the start of the year, $e_s$ the start-of-year expense, and $b_s$ the year-end benefit. If $q_{sj}$ is the probability of transitioning from $s$ to $j$, then $$ V_t(s)=e_s-p_s+v\left(b_s+\sum_j q_{sj}V_{t+1}(j)\right). $$ At the end of the term, $V_2(s)=0$ for every state: there are no cash flows or obligations after time 2.

The recursion follows by valuing cash flows from the reserve date. At the start of the year, the insurer's net outgo is $e_s-p_s$. At year-end it pays $b_s$ and holds the reserve for the state then reached. Discounting those year-end amounts by $v$ gives the displayed equation. This also shows why transition probabilities must be applied to the next-state reserves, not to the current-state reserve.

Worked Example: Calculating the State Reserves

Calculate the reserves at times 1 and 0 for the hypothetical contract. Use the valuation transition probabilities, $i=0.05$, and the two-year term. Keep unrounded values through the calculations and round final monetary answers to the nearest cent.

Step 1: Reserves at time 1. There is one year remaining, and every time-2 reserve is zero. In state $A$, the start-of-year net amount $e_A-p_A$ is $10-120=-110$, and there is no benefit:

V_1(A)=-110+1.05^{-1}(0)=-110.00.

In state $W$, there is an expense of $10, no premium, and no benefit:

V_1(W)=10+1.05^{-1}(0)=10.00.

In state $C$, the expense is $10 and the $100 benefit is paid at year-end:

V_1(C)=10+\frac{100}{1.05}=105.238095\ldots=\$105.24.

Step 2: Reserve at time 0 in state $A$. The start-of-year net amount is again $-110$. The expected time-1 reserve is $0.78(-110)+0.20(10)+0.02(0)=-83.80$. Therefore,

V_0(A)=-110+\frac{-83.80}{1.05} =-189.809523\ldots=-\$189.81.

Step 3: Reserve at time 0 in state $W$. The expected time-1 reserve is $0.25(-110)+0.70(105.238095\ldots)+0.05(0)=46.166666\ldots$. Thus,

V_0(W)=10+\frac{46.166666\ldots}{1.05} =53.968253\ldots=\$53.97.

Step 4: Reserve at time 0 in state $C$. The current year's net start-of-year amount is $10, and the benefit is $100 at year-end. The expected time-1 reserve is $0.20(-110)+0.70(105.238095\ldots)+0.10(0)=51.666666\ldots$. Hence,

V_0(C)=10+\frac{100}{1.05}+\frac{51.666666\ldots}{1.05} =154.444444\ldots=\$154.44.

Check and interpretation. The active-state reserve is negative because the projected premiums exceed projected net outgo over this short term. This is a net prospective reserve; no minimum-reserve floor has been imposed. The positive $C$ reserve includes both the current claim benefit and the expected time-1 liability.

Annual Profit and Discounted NPV

For profit testing, distinguish the valuation assumptions used to calculate reserves from the experience assumptions used to project actual expected cash flows and transitions. In the next calculation, the experience probabilities differ only for an active policy: the incidence probability is 0.25 rather than 0.20, the death probability remains 0.02, and the probability of remaining active is therefore 0.73. The $W$ and $C$ transition probabilities are unchanged. Premiums, expenses, benefits, and interest remain as specified above.

End-of-year profit: If the policy starts the year in state $s$ and ends in state $j$, with reserve $V_t(s)$ at the start and $V_{t+1}(j)$ at year-end, profit at year-end is $$ \Pi_{t+1}(s,j)=\bigl(V_t(s)+p_s-e_s\bigr)(1+i)-b_s-V_{t+1}(j). $$ The initial reserve and start-of-year premium, less expense, accumulate at interest. The benefit is then paid and the ending reserve is held. When the valuation transition probabilities and cash-flow assumptions match the projection assumptions, the expected profit conditional on the starting state is zero by the reserve recursion. A difference in assumptions can produce nonzero expected profit.

Worked Example: Expected Profits and NPV Under Higher Incidence

For a policy starting in $A$ at time 0, calculate expected profits at the ends of years 1 and 2 under the experience assumptions above, then calculate their NPV at time 0. Use the reserves from the previous example, a 5% effective annual interest rate, and unrounded intermediate values.

Step 1: Determine the year-1 accumulated amount. The year-0 reserve is $-189.809523\ldots$. The premium less expense at the start of the year is $120-10=110$. Accumulating the sum to year-end gives

\bigl(-189.809523\ldots+110\bigr)(1.05)=-83.80.

There is no benefit for a policy starting in $A$. The possible ending reserves are $V_1(A)=-110$, $V_1(W)=10$, and $V_1(D)=0$. The year-1 profits by transition are therefore

\begin{aligned} \Pi_1(A,A)&=-83.80-(-110)=26.20,\\ \Pi_1(A,W)&=-83.80-10=-93.80,\\ \Pi_1(A,D)&=-83.80-0=-83.80. \end{aligned}

The experience probabilities are 0.73, 0.25, and 0.02, respectively. Thus expected year-1 profit is

E[\Pi_1\mid X_0=A] =0.73(26.20)+0.25(-93.80)+0.02(-83.80) =19.126-23.450-1.676 =-\$6.00.

Step 2: Calculate year-2 expected profit. For any state at time 1, the time-2 reserve is zero. In $A$, the accumulated reserve plus premium less expense is $(-110+110)(1.05)=0$. In $W$, it is $(10-10)(1.05)=0$. In $C$, it is $(105.238095\ldots-10)(1.05)-100=0$. State $D$ has no cash flows. Consequently, every year-2 transition produces zero profit, and expected year-2 profit is $0.00. This conclusion uses the stated terminal boundary condition; it would not follow if time 2 had a continuing obligation.

Step 3: Discount the profit signature. Profits are measured at year-end, so the year-1 profit is discounted once:

\operatorname{NPV} =\frac{-6.00}{1.05}+\frac{0}{1.05^2} =-5.714285\ldots=-\$5.71.

Check. Higher incidence shifts probability from the active state, whose time-1 reserve is $-110$, to the waiting state, whose reserve is $10$. The expected ending reserve is therefore higher than on the valuation basis, reducing profit. The negative expected profit is consistent with that direction.

Common Traps in an Integrated Projection

Paying during the waiting period. A policy in $W$ is disabled but not yet claim-paying. Apply the benefit only to policies in $C$ at the start of the year under this model.
Mixing transition timing and benefit timing. The benefit is paid for the year if the policy starts in $C$; the year-end transition then determines the next state. Do not remove the current year's benefit merely because recovery or death is the year-end destination.
Using experience probabilities to calculate valuation reserves without instruction. Reserves use the stated valuation basis. Profit projection uses experience assumptions and compares the resulting ending reserves with the valuation reserves.
Reversing the reserve sign. Here reserve is outgo less income. The start-of-year amount in the recursion is expense minus premium, while the accumulated amount in the profit formula is reserve plus premium minus expense.
Forgetting the terminal boundary. The final reserve is zero only because the hypothetical term ends at time 2 with no further obligations. A continuing product requires a nonzero terminal reserve when future benefits remain.

Key takeaway: A reliable disability income profit test starts with explicit state definitions and payment timing. Calculate reserves on the valuation basis, project transitions and cash flows on the experience basis, reconcile profit to the change in reserves, and discount each end-of-year profit using its actual time point.

Check Your Understanding

Use the timing and sign conventions stated in each question. Monetary answers are rounded to the nearest cent unless a probability is requested.

A policy starts active. The incidence probability in year 1 is 0.30. A person entering the waiting state remains disabled through the next year with probability 0.60. What is the probability of being claim-paying at time 2 along the path of incidence in year 1 followed by continued disability? (Answer: 0.1800)
A one-year claim-paying policy has an annual effective interest rate of 4%, a start-of-year expense of $8, and a $90 benefit at year-end. There are no premiums and the terminal reserve is zero. Calculate its reserve at the start of the year. (Answer: $94.54)
For a policy starting in $W$, the start reserve is $15, the start expense is $5, and there is no premium or current benefit. At year-end, experience probabilities are 0.30 to $A$, 0.65 to $C$, and 0.05 to $D$. The corresponding ending reserves are $-\$40$, $80, and $0. At 4% interest, calculate expected end-of-year profit. (Answer: $-\$29.60$)
A hypothetical policy has expected end-of-year profits of $-\$10$, $4, and $12 at the ends of years 1, 2, and 3. At an annual effective interest rate of 5%, calculate the NPV at time 0. (Answer: $4.47)

Start state	End state and probability	Interpretation
\(A\)	\(A: 0.78,\ W: 0.20,\ D: 0.02\)	Active at year-end, new disability, or death
\(W\)	\(A: 0.25,\ C: 0.70,\ D: 0.05\)	Recovery, continued disability through the waiting year, or death
\(C\)	\(A: 0.20,\ C: 0.70,\ D: 0.10\)	Recovery, continued claim, or death
\(D\)	\(D: 1.00\)	Death is absorbing

What You'll Learn