Prospective Reserves by State in a Profit Test

Why Reserves Must Be Conditional on State

A single policy may have very different future cash flows depending on whether the insured is active, disabled, receiving long-term care, or in another in-force state. A prospective reserve therefore cannot generally be calculated from the policy alone: it must be conditional on the state occupied at the valuation time. The reserve for an active insured may reflect future premiums, while the reserve for a disabled insured may reflect substantial future benefits and no further premiums.

This tutorial uses a discrete annual model. At time $t$, the policy is in an in-force state $i$. A premium and an expense are paid at the beginning of the year, and a benefit is paid at the end of the year if the transition warrants one. The reserve at time $t$ is measured immediately before the beginning-of-year premium and expense. This convention makes the reserve recursion and the profit calculation line up directly.

Definition: The prospective reserve $V_t(i)$ is the expected present value, conditional on being in state $i$ at time $t$, of future benefits and expenses less future premiums. Under the timing convention used here, the premium and expense at time $t$ are included. A positive reserve is a net liability; a negative reserve means the present value of future premiums exceeds the present value of future benefits and expenses.

Let $P_t(i)$ be the premium received at time $t$ in state $i$, and let $E_t(i)$ be the expense paid then. Write $p_{ij}(t)$ for the probability of moving from state $i$ at time $t$ to state $j$ at time $t+1$. Let $B_{t+1}(i,j)$ be the benefit paid at the end of the year for that transition. A death state can be included as a destination state. If the benefit depends only on the destination, write it as $B_{t+1}(j)$.

V_t(i)=-P_t(i)+E_t(i)+v\sum_j p_{ij}(t)\left[B_{t+1}(i,j)+V_{t+1}(j)\right], \qquad v=\frac{1}{1+i}.

The formula assumes that the valuation and projection use the same transition probabilities, interest rate, and cash-flow amounts. It also assumes the stated timing: premium and expense at the start of the year, benefit at the end. If an expense or premium is paid at the end of the year, it must be moved into the discounted end-of-year term. At the end of the coverage period, $V_{t+1}(j)=0$ if no further payments are due after that year’s benefit. A state with a later settlement or continuing benefit may instead have a nonzero reserve.

Why the Recursion Holds

Condition on the state reached at the end of the year. If the transition is from $i$ to $j$, the end-of-year amount is the benefit $B_{t+1}(i,j)$ plus the reserve $V_{t+1}(j)$ for the future cash flows after arrival in state $j$. Its expected value, given state $i$, is the probability-weighted sum over destinations. Discount that expectation one year. At time $t$, add the expense and subtract the premium paid immediately. This gives the recursion.

Fix the in-force state.
Use the transition probabilities for the state occupied at the valuation time; do not substitute probabilities from another state.

Specify the payment timing.
Identify which premiums and expenses are immediate and which benefits are paid at year-end.

Value each destination.
For each possible next state, add the transition benefit to the reserve conditional on arriving in that state.

Take the conditional expectation.
Weight destination amounts by the transition probabilities, discount as required, and include the immediate premium and expense.

Worked Example: One-Year Reserves in Active and Disabled States

A hypothetical contract has one year remaining. The annual effective interest rate is $i=5\%$, so $v=1/1.05=0.95238095$. At the end of the year, a benefit of $400 is paid if the insured is disabled and $1,000 if the insured dies; no benefit is paid if the insured remains active. After this year’s payments, no further cash flows are due, so all next-year reserves are zero. The premium at the start of the year is $100 in the active state and $0 in the disabled state. The beginning-of-year expense is $20 in either in-force state.

From the active state, the transition probabilities are 0.80 to active, 0.15 to disabled, and 0.05 to death. From the disabled state, they are 0.40 to active, 0.50 to disabled, and 0.10 to death.

Method. Apply the prospective recursion separately for each starting state. For an active insured, the expected end-of-year benefit is $0.80(0)+0.15(400)+0.05(1{,}000)=60+50=110$. Therefore,

V_1(A)=-100+20+\frac{110}{1.05} =-80+104.7619 =\$24.7619.

For a disabled insured, the expected benefit is $0.40(0)+0.50(400)+0.10(1{,}000)=200+100=300$. There is no premium in this state, so

V_1(D)=0+20+\frac{300}{1.05} =20+285.7143 =\$305.7143.

Rounded to the nearest cent, the reserves are $24.76 in the active state and $305.71 in the disabled state. The difference is plausible: the disabled state has a higher probability of a benefit and no premium income. The calculation also avoids the common mistake of applying the active-state transition probabilities to a disabled insured.

Reserves Across More Than One Year

For a multi-year projection, calculate reserves backward from the terminal time. At each step, the reserve in a state depends on both the current year’s cash flows and the reserves in all possible next states. These next-state reserves must remain conditional: a reserve in the disabled state is not interchangeable with a reserve in the active state.

A negative reserve is not automatically an error. It can occur when future premiums, particularly premiums due only while active, have greater expected present value than the future benefits and expenses. The sign must be interpreted together with the cash-flow convention and the state in which the policy is valued.

Worked Example: Two-Year Reserves by State

A hypothetical two-year contract has an annual effective interest rate of $5\%$, so $v=1/1.05$. At the end of each year, the benefit is $200 if the insured is disabled and $500 if the insured dies; there is no benefit on transition to active. Each year, the beginning-of-year expense is $10 in either living state. A $100 premium is due at the beginning of each year in the active state and no premium is due in the disabled state. The annual transition probabilities are: from active, 0.75 to active, 0.20 to disabled, and 0.05 to death; from disabled, 0.30 to active, 0.60 to disabled, and 0.10 to death. After the second year’s benefits, no further payments are due, so $V_2(A)=V_2(D)=0$.

Method. First calculate the reserves at time 1, when one year remains. In the active state, expected benefit in the final year is $0.20(200)+0.05(500)=40+25=65$. Thus,

V_1(A)=-100+10+\frac{65}{1.05} =-90+61.904762 =-\$28.095238.

In the disabled state, expected benefit is $0.60(200)+0.10(500)=120+50=170$, giving

V_1(D)=0+10+\frac{170}{1.05} =10+161.904762 =\$171.904762.

Now calculate the time-0 reserves. Starting active, the expected sum of year-end benefit and time-1 reserve is

$$ 0.75(-28.095238)+0.20(200+171.904762)+0.05(500) =-21.071429+74.380952+25 =78.309524. $$

Consequently,

V_0(A)=-100+10+\frac{78.309524}{1.05} =-90+74.580499 =-\$15.419501.

Starting disabled, the corresponding expected end-of-year amount is

$$ 0.30(-28.095238)+0.60(200+171.904762)+0.10(500) =-8.428571+223.142857+50 =264.714286, $$

V_0(D)=0+10+\frac{264.714286}{1.05} =10+252.108844 =\$262.108844.

Rounded to the nearest cent, $V_0(A)=-\$15.42$, $V_0(D)=\$262.11$, $V_1(A)=-\$28.10$, and $V_1(D)=\$171.90$. The negative active-state reserves reflect the future active-state premiums. A common mistake is to average the active and disabled reserves before applying a transition probability; instead, each destination probability must be multiplied by the reserve conditional on that destination.

Using State Reserves in a Profit Projection

A state-conditional reserve supplies the closing liability for a profit calculation. Let $g_t(i,j)$ be the profit for a policy that begins the year in state $i$ and transitions to state $j$. Under the same basis and timing as the reserve recursion, the amount available after accumulating the opening reserve, premium, and expense adjustment is compared with the end-of-year benefit and the reserve required in the destination state:

g_t(i,j)=(1+i)\bigl[V_t(i)+P_t(i)-E_t(i)\bigr] -\bigl[B_{t+1}(i,j)+V_{t+1}(j)\bigr].

The expected profit conditional on starting in state $i$ is $\sum_j p_{ij}(t)g_t(i,j)$. Substituting the reserve recursion shows why the expected value is zero when the projection and valuation assumptions match:

\begin{aligned} \sum_j p_{ij}(t)g_t(i,j) &=(1+i)\bigl[V_t(i)+P_t(i)-E_t(i)\bigr]\\ &\quad-\sum_j p_{ij}(t)\bigl[B_{t+1}(i,j)+V_{t+1}(j)\bigr]\\ &=0. \end{aligned}

This is an expected-profit check, not a claim that profit is zero for every transition. A transition to death may produce a loss while remaining active may produce a gain; the probability-weighted average is zero on the matching basis. If the projection uses different assumptions from the reserve valuation, or includes margins or other cash flows not included in the reserve, expected profit need not be zero.

Worked Example: A Transition-Specific Profit Vector

Use the one-year contract in the first example, starting in the active state. The annual effective interest rate is $5\%$, the opening reserve is $V_1(A)=\$24.761905$, the beginning premium is $100, and the beginning expense is $20. The transition probabilities are 0.80 to active, 0.15 to disabled, and 0.05 to death. The end-of-year benefits are $0, $400, and $1,000, respectively, and all destination reserves after settlement are zero.

Method. Accumulate the opening reserve net of the beginning cash flows. The amount is

1.05(24.761905+100-20) =1.05(104.761905) =\$110.

Subtract the benefit and destination reserve for each transition. Thus the profit vector, ordered as active, disabled, and death, is

\bigl(g(A,A),g(A,D),g(A,\text{death})\bigr) =(110-0,\;110-400,\;110-1{,}000) =(\$110,-\$290,-\$890).

Check the conditional expected profit:

0.80(110)+0.15(-290)+0.05(-890) =88-43.5-44.5 =\$0.

The expected profit is zero, as required by the matching prospective valuation basis. The individual transition profits are not zero. The common mistake this example avoids is reporting only the expected profit and treating it as the profit on every path; the profit vector preserves the distinct outcomes by destination state.

Common Checks and Exam Traps

Condition on the correct state. Use the transition row and cash flows for the actual in-force state at the valuation time.
Keep state reserves distinct. The destination reserve in a transition calculation is $V_{t+1}(j)$, not an average across possible states.
Respect payment timing. An immediate premium is not discounted; an end-of-year benefit is discounted in the reserve recursion.
Include premiums only where payable. A premium waived during disability must not be subtracted from the disabled-state reserve.
Use the same basis for the check. Zero expected profit follows only when the reserve assumptions and projected cash flows agree.

Key takeaway: Calculate prospective reserves conditional on each in-force state, carry those reserves forward by destination state, and use them in the profit vector. The probability-weighted profit conditional on the starting state is zero when valuation and projection use the same assumptions and cash flows.

Check Your Understanding

For each problem, use the timing convention stated in the question and calculate reserves separately by state when required.

A one-year hypothetical contract has $i=4\%$. From active, the probabilities of ending active, disabled, and dead are 0.70, 0.20, and 0.10. The end-year benefits are $0, $300, and $800. At the start of the year, the active premium is $120 and the expense is $15. There are no next-year reserves. Find the active-state reserve. (Answer: $29.62)
A one-year hypothetical contract has $i=5\%$. From disabled, the probabilities of ending active, disabled, and dead are 0.25, 0.65, and 0.10. The end-year benefits are $0, $240, and $600. The beginning expense is $12 and there is no premium while disabled. There are no next-year reserves. Find the disabled-state reserve. (Answer: $217.71)
A one-year contract has $i=5\%$, an active premium of $80, and a beginning expense of $10. From active, the probabilities of ending active, disabled, and dead are 0.60, 0.30, and 0.10. End-year benefits are $0, $100, and $700, and all next-year reserves are zero. Calculate the active-state reserve and the three transition profits. (Answer: Reserve = $25.24; profit vector = ($100.00, $0.00, -$600.00))
A hypothetical two-year contract has $i=0\%$. Each year’s beginning expense is $5 in either living state; the beginning premium is $50 in active and zero in disabled. End-year benefits are $100 on transition to disabled, $200 on transition to death, and zero on transition to active. From active, probabilities to active, disabled, and death are 0.80, 0.15, and 0.05; from disabled they are 0.30, 0.60, and 0.10. No payments follow the second year’s benefits. Find $V_1(A)$, $V_1(D)$, $V_0(A)$, and $V_0(D)$. (Answer: $V_1(A)=-\$20.00$, $V_1(D)=\$85.00$, $V_0(A)=-\$23.25$, $V_0(D)=\$130.00$)

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