Future Benefits, Expenses, and Premiums

Introduction

A prospective reserve is based on what is expected to happen after the valuation date. In a multi-state contract, that future is not a single fixed sequence: a policy may remain active, enter a disabled state, recover, or terminate. The reserve therefore depends both on the cash flows specified by the contract and on which future policy states make those cash flows payable.

A reliable setup starts by listing future benefits, expenses, and premiums separately. For each item, ask what amount is due, when is it due, and which state or transition triggers it? Only after these questions are answered should the cash flows be combined and discounted. The discussion and examples below use hypothetical contracts.

Definition: In a discrete-time model, let $S_k$ denote the policy state at the end of year $k$, and let $C_k$ be the insurer’s net outgo at that time: benefits plus expenses minus premiums. At a valuation date when the policy is in state $j$, the prospective reserve for $n$ years is $$ V_j(0)=E\left[\left.\sum_{k=1}^{n}v^k C_k\right|S_0=j\right], \qquad v=\frac{1}{1+i}. $$ The reserve is conditional on the current state. Positive values represent expected net outgo; negative values represent expected net income under the specified assumptions.

What Belongs in the Future Cash-Flow List?

The reserve is not a list of every amount ever associated with a policy. It includes future cash flows arising after the valuation date under the chosen valuation basis. A premium already paid is not a future receipt. An acquisition expense already incurred is not a future expense. Conversely, a future maintenance expense can belong in the reserve even if it is not a benefit promised to the policyholder.

For a multi-state policy, a useful inventory separates the items as follows:

Benefits: amounts payable on a specified future state, such as income while disabled, or on a transition, such as a lump sum when disability begins.
Expenses: future amounts incurred to administer or maintain the contract. They may apply in all in-force states or only in particular states.
Premiums: future amounts expected under the contract and valuation assumptions. Their inclusion depends on whether the policy remains eligible to pay them.

A cash flow may depend on the state at the payment date, on the transition into that state, or on both. For example, an income payment made whenever the policy is disabled depends on the end-of-year state. A one-time disability benefit paid only when the policy first becomes disabled depends on a transition into disability. These triggers must not be confused: remaining disabled for a second year may generate another income payment, but it does not necessarily generate another “new disability” benefit.

State probabilities must also be conditional on the current state. In a Markov model, the transition probabilities describe the next state given the present state; probabilities for later years are obtained by following the possible intervening states. A policy that has terminated cannot continue paying premiums or receiving in-force expenses unless the contract explicitly specifies a later payment.

From Cash-Flow Items to Expected Present Value

Suppose the policy can occupy states $1,\ldots,m$. Write $p_{ab}$ for the probability of moving from state $a$ at the start of a year to state $b$ at its end. A payment that depends only on the end state can be calculated using the probability of that state. A payment that depends on a particular transition $a\to b$ uses the probability of that transition, including the probability of being in state $a$ at the start of the year.

The principle behind the setup is the linearity of expectation. If net outgo is $C_k=B_k+E_k-P_k$, where $B_k$, $E_k$, and $P_k$ are the benefit, expense, and premium amounts, then

E[v^k C_k\mid S_0=j] = v^k\left(E[B_k\mid S_0=j]+E[E_k\mid S_0=j]-E[P_k\mid S_0=j]\right).

Discounting is deterministic here, so it factors out of the expectation. Summing over years gives the reserve. This also explains why it is safe to calculate the expected present values of benefits, expenses, and premiums separately and then combine them with the appropriate signs.

Fix the valuation state.
State which policy state is known at the valuation date. The reserve is conditional on that information.

List each future cash flow.
Record its amount and payment time, and classify it as a benefit, expense, or premium.

Specify its trigger.
Identify whether it is payable in a future state, on a transition, or under another stated condition.

Apply the appropriate probabilities and discounting.
Use probabilities conditional on the valuation state, then combine discounted expected outgo and income.

Worked Example: One-Year Reserve With State-Dependent Premiums

A hypothetical policy is active at the valuation date. At the end of one year, it is active with probability 0.80, disabled with probability 0.15, or dead with probability 0.05. These are the complete one-year transition probabilities from the active state. The effective annual interest rate is $i=5\%$. All listed amounts are paid at year-end. A premium of $120 is received only if the policy is active at year-end. Expenses are $20 if active and $50 if disabled. A $300 lump-sum benefit is paid on the transition from active to disabled, and a $1,000 benefit is paid on death. Find the reserve.

Method. For each possible end-of-year outcome, identify the net outgo, counting premiums as income. There is no disability benefit in the active or death outcome, and no premium is received in the disabled or dead outcome.

If the policy is active, net outgo is $20-120=-100$. If it is disabled, net outgo is $50+300=350$. If it is dead, net outgo is $1{,}000$. Thus the expected net outgo before discounting is

$$ 0.80(-100)+0.15(350)+0.05(1{,}000) =-80+52.50+50 =22.50. $$

The one-year discount factor is $v=1/1.05=0.95238095$. Therefore,

V_{\mathrm{active}}(0)=22.50v =\frac{22.50}{1.05} =\$21.43.

As a check, expected benefits are $0.15(300)+0.05(1{,}000)=\$95.00$, expected expenses are $0.80(20)+0.15(50)=\$23.50$, and expected premiums are $0.80(120)=\$96.00$. The net is $95+23.50-96=\$22.50$ before discounting, as above. The common mistake is to include the $120 premium in every outcome; it is received only when the policy is active.

Worked Example: Following State Probabilities Over Two Years

A hypothetical policy starts active. At each year-end, an active policy remains active with probability 0.75, becomes disabled with probability 0.20, or dies with probability 0.05. From disability, the next state is active with probability 0.10, disabled with probability 0.80, or dead with probability 0.10. Death is absorbing and produces no later cash flows. The effective annual interest rate is $5\%$, and all amounts are paid at year-end. A $100 premium is received if active; expenses are $15 if active and $40 if disabled. A $200 benefit is paid when a transition into disability occurs, and an $800 benefit is paid on death. Find the two-year reserve.

Year 1. Starting active, the expected benefit is $0.20(200)+0.05(800)=\$80.00$. Expected expenses are $0.75(15)+0.20(40)=\$19.25$, and expected premiums are $0.75(100)=\$75.00$. Year 1 expected net outgo is $80+19.25-75=\$24.25$.

Year 2 state probabilities. The probability of being active at the start of year 2 is 0.75, and the probability of being disabled is 0.20. Therefore the end-of-year probabilities are:

\begin{aligned} P(S_2=\text{active})&=0.75(0.75)+0.20(0.10)=0.5825,\\ P(S_2=\text{disabled})&=0.75(0.20)+0.20(0.80)=0.3100,\\ P(\text{death during year 2})&=0.75(0.05)+0.20(0.10)=0.0575. \end{aligned}

The remaining probability, 0.05, represents policies that died in year 1; they produce no year 2 cash flow. A new disability benefit is due only on the active-to-disabled transition in year 2, whose probability is $0.75(0.20)=0.15$. Expected year 2 benefits are $0.15(200)+0.0575(800)=\$76.00$. Expected expenses are $0.5825(15)+0.3100(40)=\$21.1375$, and expected premiums are $0.5825(100)=\$58.25$. Thus expected year 2 net outgo is $76+21.1375-58.25=\$38.8875$.

The reserve is

V_{\mathrm{active}}(0) =\frac{24.25}{1.05}+\frac{38.8875}{1.05^2} =23.095238+35.272109 =\$58.37.

The calculation is checked by discounting each year’s expected net outgo separately. A frequent error is to use 0.31 as the probability of a new disability in year 2. That is the probability of ending year 2 disabled, including policies already disabled at the start of that year. The new-disability benefit instead uses the transition probability $0.75(0.20)=0.15$.

Worked Example: Starting in Disability With State-Contingent Income

A hypothetical policy is disabled at the valuation date. At each year-end, a disabled policy remains disabled with probability 0.70, recovers to active with probability 0.20, or terminates with probability 0.10. An active policy remains active with probability 0.80, becomes disabled with probability 0.10, or terminates with probability 0.10. Termination is absorbing. The effective annual interest rate is $5\%$. At each year-end, the contract pays $250 if disabled and a premium of $80 if active; expenses are $30 if disabled and $10 if active. A $500 termination benefit is payable upon transition to termination. Find the reserve for the next two years.

Year 1. The end-state probabilities are 0.20 active, 0.70 disabled, and 0.10 terminated. Expected benefits are $0.70(250)+0.10(500)=\$225$. Expected expenses are $0.20(10)+0.70(30)=\$23$. Expected premiums are $0.20(80)=\$16$. Year 1 net outgo is $225+23-16=\$232$.

Year 2 state probabilities. From the year 1 active probability of 0.20, the year 2 outcomes contribute 0.16 active, 0.02 disabled, and 0.02 terminated. From the year 1 disabled probability of 0.70, they contribute 0.14 active, 0.49 disabled, and 0.07 terminated. The 0.10 already terminated has no further cash flows. Hence year 2 probabilities are 0.30 active, 0.51 disabled, and 0.09 newly terminated.

Expected year 2 benefits are $0.51(250)+0.09(500)=\$172.50$. Expected expenses are $0.30(10)+0.51(30)=\$18.30$, and expected premiums are $0.30(80)=\$24.00$. Year 2 net outgo is $172.50+18.30-24=\$166.80$. Therefore,

V_{\mathrm{disabled}}(0) =\frac{232}{1.05}+\frac{166.80}{1.05^2} =220.952381+151.292517 =\$372.24.

The result is checked by confirming that the year 2 state probabilities sum to $0.30+0.51+0.09=0.90$, with the remaining 0.10 having terminated in year 1. A common mistake is to include year 2 income or premiums for policies already terminated.

Assumptions and Common Traps

The examples use a fixed effective annual interest rate and discrete year-end payments. Their transition probabilities are specified for each relevant starting state, and future outcomes are treated according to those stated probabilities. A different timing pattern or transition model requires a correspondingly different setup; the numerical results above do not automatically apply to continuous-time payments or to another valuation basis.

Using an unconditional probability for a conditional reserve. If the policy is known to be disabled at the valuation date, start from disability, not from a population mixture of active and disabled policies.
Confusing a state with a transition. “Pay while disabled” uses the probability of being disabled at payment time. “Pay on becoming disabled” uses the probability of a transition into disability.
Assuming every premium remains payable. Include a premium only in outcomes where it is due under the stated contract and assumptions.
Continuing cash flows after termination. An absorbing terminated state has no later in-force premiums or expenses unless a separate contract provision says otherwise.
Combining signs too early. Keep expected benefits, expenses, and premiums distinct until each has been attached to the correct trigger and probability.

Getting the cash-flow inventory right is essential: a correct discount factor cannot repair a benefit assigned to the wrong transition or a premium counted in an ineligible state. This state-by-state setup is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated year-end timing and calculate the reserve as expected discounted benefits plus expenses minus premiums.

A policy starts active. At year-end it is active with probability 0.90, disabled with probability 0.08, and dead with probability 0.02. At $i=4\%$, the premium is $90 if active; expenses are $10 if active and $30 if disabled; benefits are $200 on transition to disability and $600 on death. Find the one-year reserve. (Answer: $-\$40.00$)
A policy starts disabled. At year-end it is active with probability 0.25, disabled with probability 0.65, or terminated with probability 0.10. At $i=5\%$, it pays $180 of income if disabled and a $400 benefit on termination; expenses are $12 if active and $24 if disabled; a $60 premium is received if active. Find the one-year reserve. (Answer: $152.95)
A term policy starts active, with a 0.10 probability of death in each year conditional on being alive at the start of that year. Death is absorbing. At $i=5\%$, a $500 death benefit is paid at year-end on death; a $100 premium and $10 expense are paid at year-end only if alive. Find the two-year reserve. (Answer: $-\$54.83$)
A policy starts active. In each year, an active policy remains active with probability 0.70, becomes disabled with probability 0.20, or dies with probability 0.10. From disability, it becomes active with probability 0.10, remains disabled with probability 0.80, or dies with probability 0.10. Death is absorbing. At $i=4\%$, a $100 premium is received if active, expenses are $10 if active and $30 if disabled, a $250 benefit is paid only on a transition from active to disabled, and a $500 benefit is paid on death. Find the two-year reserve. (Answer: $81.19)

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