Why Premium Cessation Changes a Reserve
A premium waiver changes the future cash flows of a contract when the insured enters a specified state, commonly disability. The insurer may continue to pay benefits while the insured is disabled, but stop collecting premiums. A reserve therefore depends not only on the probability of each future state, but also on which state applies at each premium due time.
The timing rule matters. If a premium is due while the policy is still in the premium-paying state, a transition to disability later that day or later in the policy year does not normally cancel the premium already due. If disability has begun before a later premium date, that premium is waived when the contract says premiums cease upon disability. The policy’s stated timing convention determines the answer.
This tutorial is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. It focuses on calculating prospective reserves under state-dependent premium rules, using explicit discrete-time timing conventions.
Prospective Reserve with State-Dependent Premiums
At a valuation time, the prospective reserve in a given state is the expected present value of future benefits and expenses less the expected present value of future premiums, conditional on the policy being in that state at that time. For the examples below, expenses are zero unless stated, and all cash flows are discounted at an effective annual interest rate \(i\), with \(v=1/(1+i)\).
For a payment at time \(k\), let \(B_k\) denote the benefit amount and \(P_k\) the premium amount. A premium payable only in the active state contributes to expected present value in proportion to the probability that the policy is active immediately before its due time. In indicator notation, its value at time 0 is \(v^k P_k I(A_k)\), where \(I(A_k)=1\) if the policy is active at the premium date and \(0\) otherwise.
The state \(J_t\) is the policy state at time \(t\). The formula applies when the future cash flows and transition assumptions are specified and the reserve is defined prospectively. A premium already due at the valuation instant must be included or excluded according to whether the reserve is measured just before or just after that premium is paid.
Why the State Indicator Belongs in the Premium Value
A premium that is payable only while active is not a fixed future cash flow. Its payment depends on the state at its due date. Consider a single premium of amount \(P\) due at time \(k\). Conditional on the valuation state, its expected present value is the expectation of its discounted payment indicator:
The last equality follows because \(v^k\) and \(P\) are fixed, while the indicator equals 1 exactly when the policy is active. Thus its expected value is \(P(A_k)\). If the insured has entered disability before time \(k\), then \(I(A_k)=0\), and the premium contributes nothing. If disability begins only after the premium is paid, the indicator at the due time is 1 and the premium remains payable.
This argument depends on the contract’s precise premium rule. If premiums are paid in all states, the indicator is always 1. If waiver begins only after a waiting period or after approval, the relevant state or eligibility condition must reflect that contract wording; simply treating every disability transition as an immediate waiver would be incorrect.
Worked Calculations
Worked Example: Waiver After Disability at a Policy-Year Boundary
A hypothetical two-year contract starts in the active state \(A\). At each year-end, an active policy transitions to disabled state \(D\) with probability 0.25 and otherwise remains active. Once disabled, the policy stays disabled through the end of the contract. A premium of $90 is due at times 0 and 1 if the policy is active immediately before payment. A benefit of $700 is paid at time 2 if the policy is disabled then. The effective annual interest rate is \(i=5\%\). Find the reserve at time 0 just before the first premium.
Method. Calculate the probability of disability at time 2 for the benefit value, then value each premium using the probability of being active at its due time. The discount factor is \(v=1/1.05=0.95238095\).
At time 0 the policy is active, so the premium of $90 is payable immediately. By time 1, the policy is active with probability \(1-0.25=0.75\), so the expected premium at time 1 is \(0.75(90)=\$67.50\). Disability by time 2 can occur in either of two mutually exclusive ways: transition to \(D\) at time 1, with probability 0.25, or remain active through time 1 and transition at time 2, with probability \(0.75(0.25)=0.1875\). Therefore,
The expected benefit at time 2 is \(0.4375(700)=\$306.25\), with present value \(306.25v^2=306.25/1.1025=\$277.7778\). The premiums have present value \(90+67.50v=90+64.2857=\$154.2857\). Hence,
Rounded to the nearest cent, the reserve is $123.49. The calculation includes the time-1 premium only for policies still active at that date; it does not charge a premium in state \(D\). The method avoids the common mistake of treating the time-1 premium as certain.
Worked Example: Disability Begins Between Premium Dates
A hypothetical one-year contract begins active at time 0. A premium of $150 is due at time 0 and again at time 1, but the contract ends at time 1, so only the time-0 premium is a contract payment unless a renewed one-year coverage period is specified. Instead, consider the stated two-payment arrangement: the coverage and benefit period runs to time 1, and a continuation premium is due at time 1 only for policies still active immediately before that payment; the associated benefit is also paid at time 1. By time 0.5, the insured has entered disability with probability 0.30 and otherwise remains active. A benefit of $1,200 is paid at time 1 if disabled. The effective annual interest rate is \(i=4\%\). Find the time-0 reserve just before the first premium.
Method. The contract explicitly includes a premium due at time 1 if the policy is active immediately before that payment. The time-0 premium is certain. Since disability begins by time 0.5, it is already known whether the time-1 premium is payable. The probability of disability at time 1 is 0.30, and the probability of remaining active is 0.70.
The expected benefit at time 1 is \(0.30(1,200)=\$360\), so its present value is \(360/1.04=\$346.1538\). The time-0 premium has present value $150. The expected time-1 premium is \(0.70(150)=\$105\), with present value \(105/1.04=\$100.9615\). Thus,
Rounded to the nearest cent, the reserve is $95.19. A common incorrect calculation subtracts the full time-1 premium, as if it were payable in both states. That gives \(346.1538-(150+150/1.04)=\$51.92\). It understates the reserve by the present value of the waived premium, \(0.30(150)/1.04=\$43.2692\). The correct calculation uses the state immediately before the time-1 payment.
Worked Example: Comparing Active and Disabled Reserves
A hypothetical contract has two years remaining, from time 1 to time 3. State \(D\) is absorbing: a disabled policy remains disabled. At each year-end, an active policy enters \(D\) with probability 0.20 and otherwise stays active. A premium of $120 is due at times 1 and 2 if active immediately before payment. A benefit of $500 is paid at times 2 and 3 whenever the policy is disabled at that year-end. The effective annual interest rate is \(i=5\%\). Find the reserves at time 1 just before the premium, conditional on being active and conditional on being disabled.
Method. Calculate each reserve conditional on its time-1 state. For the active reserve, include the time-1 premium with certainty, but make the time-2 premium contingent on still being active at time 2. For the disabled reserve, include both future benefits and no premiums.
If the policy is active at time 1, the time-1 premium is $120. The probability of disability at time 2 is 0.20, so the expected time-2 benefit is \(0.20(500)=\$100\). The probability of disability by time 3 is \(0.20+0.80(0.20)=0.36\), so the expected time-3 benefit is \(0.36(500)=\$180\). The probability of remaining active at time 2 is 0.80, making the expected time-2 premium \(0.80(120)=\$96\). Therefore,
The discounted benefits are \(100/1.05=\$95.2381\) and \(180/1.1025=\$163.2653\). The discounted premiums are $120 at time 1 and \(96/1.05=\$91.4286\) at time 2. Thus, \({}_1V_A=95.2381+163.2653-120-91.4286=\$47.0748\), or $47.07 to the nearest cent.
If the policy is disabled at time 1, it remains disabled and no further premiums are due. Both benefits are certain, so
Rounded to the nearest cent, \({}_1V_D=\$929.71\). The substantially higher disabled-state reserve reflects both certain future benefits and the complete cessation of premiums. The common mistake to avoid is applying the active-state premium stream to the disabled reserve.
Practical Checks and Common Traps
A state-dependent premium calculation is easier to audit when each scheduled payment is listed with its due time and the state test that controls it. A compact table can make the timing visible before any discounting is done.
| Cash flow | State condition | Valuation treatment |
|---|---|---|
| Premium due at time \(t\) | Active immediately before payment | Include only for paths active at \(t\) |
| Premium due at time \(t\) | Disabled before payment | Exclude if waiver is already effective |
| Premium due before disability begins | Active at the due instant | Include, even if disability begins later |
| Benefit due at time \(t\) | Benefit trigger specified by contract | Use the state or event at the stated benefit time |
The most frequent errors are timing errors rather than discounting errors. In particular:
- Waiving a premium too early. A transition after a premium due instant does not generally cancel a premium already due.
- Charging a waived premium. A premium due after disability has begun is excluded when waiver is effective immediately upon entering disability.
- Using the wrong state probability. The probability relevant to a premium is the probability of being in a premium-paying state at its due time, not necessarily the probability at the start of the projection.
- Mixing up reserve instants. A reserve just before a premium includes that premium if payable; a reserve just after it does not include the same payment as future cash flow.
- Ignoring the contract definition. A waiting period or a delayed waiver changes the state condition used to decide whether a premium is payable.
These checks support multi-state prospective reserve work without changing the underlying principle: value each future cash flow under the stated transition and payment rules. State-dependent premiums must be projected with the same care as state-dependent benefits.
Check Your Understanding
For each problem, use the stated interest rate and timing convention. Premiums are payable only if the policy is active immediately before their due time.
- A one-year contract starts active. Disability occurs before time 1 with probability 0.30. A premium of $100 is due at time 0 and at time 1 if active; a benefit of $500 is paid at time 1 if disabled. At \(i=5\%\), find the reserve at time 0 just before the first premium. (Answer: −$23.81)
- A two-year contract starts active. At each year-end, an active policy enters absorbing disability with probability 0.20. A premium of $100 is due at times 0 and 1 if active. A benefit of $400 is paid at each year-end if disabled at that year-end. At \(i=5\%\), find the time-0 reserve before the premium. (Answer: $30.61)
- A policy is disabled at the valuation time and remains disabled for three years. A benefit of $200 is paid at each year-end, and no premium is payable while disabled. At \(i=4\%\), find the reserve. (Answer: $555.02)
- A two-year contract starts active. At each year-end, an active policy enters absorbing disability with probability 0.25. A premium of $90 is due at times 0 and 1 if active. A benefit of $600 is paid at each year-end if disabled then. At \(i=5\%\), find the time-0 reserve before the first premium. (Answer: $226.67)
- A premium is due at time 2. The insured is active immediately before the time-2 payment, pays it, and enters disability immediately afterward. If waiver is effective upon entering disability, is the time-2 premium payable? (Answer: Yes; disability begins after the due payment, so the premium is not waived.)