Benefits at Contract Termination

Why Termination Conditions Matter

A contract may pay different amounts when it ends because of maturity, surrender, death, or another specified event. The phrase “benefit at termination” is not enough to value the payment: the actuary must identify which event ends the contract, when that event occurs, and what other payments may be triggered at that same time. A benefit payable at maturity to a disabled policyholder, for example, should not be omitted merely because it is not payable to a policyholder who died before maturity.

A termination payment is also different from a payment triggered by a transition between states. A transition benefit may be paid when a policyholder enters disability even if the contract remains in force. If entry into disability coincides with maturity, the contract terms determine whether both payments are made. The valuation must reflect those terms rather than treating all payments at a date as automatically exclusive or automatically additive.

Definition: Let $T$ be the time at which the contract terminates, and let $J$ identify the cause of termination, such as maturity, surrender, or death. A termination benefit is a payment made because the event $(T,J)$ occurs. A state-contingent maturity benefit may depend on the policyholder’s state at the maturity date, while a transition benefit is triggered by a specified change from one state to another. The contract terms determine whether different benefits triggered at the same time are alternatives or are payable together.

Representing Termination Events

For a discrete-time contract with payments at the ends of years, write $T=t$ when termination occurs at the end of year $t$. Let $J=j$ indicate the cause. If $b_{t,j}$ is the amount payable for cause $j$ at time $t$, and $i$ is the effective annual interest rate, the expected present value of termination benefits is

\operatorname{EPV}_{\mathrm{term}} =\sum_t v^t\sum_j P(T=t,J=j)b_{t,j}, \qquad v=\frac{1}{1+i}.

This expression assumes that the events $(T=t,J=j)$ have been defined consistently with the contract. For example, if surrender and death are alternative causes of termination, they cannot both be counted as the cause of termination for the same policy at the same time. One way to enforce that is to define the contract’s termination cause as the first event that ends coverage. Maturity applies only if the contract has not already terminated.

A benefit triggered by a transition is handled separately unless the terms make it part of an alternative termination benefit. If the transition benefit is payable in addition to the termination benefit, both cash flows belong in the valuation, even when they occur at the same time. In a model with multiple possible benefits at a payment date, define each trigger explicitly and include its payment only when that trigger occurs.

Why the Expected Present Value Formula Holds

For each time $t$ and cause $j$, define an indicator $I_{t,j}$ that equals 1 if termination occurs at time $t$ for cause $j$, and 0 otherwise. The termination payment at that time is $b_{t,j}I_{t,j}$. Its present value is $v^t b_{t,j}I_{t,j}$. Taking expectations and using $E[I_{t,j}]=P(T=t,J=j)$ gives

E\!\left[v^t b_{t,j}I_{t,j}\right] =v^t b_{t,j}P(T=t,J=j).

Adding these expected present values over all times and causes gives the stated formula. The sum is valid because expectation is additive. If two benefits can both be paid on the same path, each benefit has its own payment indicator and both are included. If the contract instead specifies one payment in place of another, the events must be partitioned into mutually exclusive alternatives before applying the sum.

Worked Example: Maturity Benefits for More Than One State

A hypothetical one-year contract is in force until the end of year 1. At that date, the probabilities of being active, disabled, dead, and surrendered are 0.70, 0.20, 0.08, and 0.02, respectively. These outcomes are mutually exclusive and exhaustive. The contract pays $6,000 at maturity if the policyholder is active or disabled, $3,000 on death, and $1,000 on surrender. Payments are made at the end of year 1. The effective annual interest rate is 4%. Find the expected present value.

Method. Partition the possible year-end outcomes by the payment each one triggers. Maturity is payable in two states, so its probability is the sum of the active and disabled probabilities:

P(\text{maturity payment})=0.70+0.20=0.90.

The expected payment at the end of year 1 is therefore

0.90(6{,}000)+0.08(3{,}000)+0.02(1{,}000) =5{,}400+240+20 =\$5{,}660.

With $i=0.04$, $v=1/1.04$. Thus

\operatorname{EPV}=\frac{5{,}660}{1.04} =5{,}442.3077\ldots =\$5{,}442.31.

Check. The outcome probabilities sum to $0.70+0.20+0.08+0.02=1.00$, and the expected payment includes all four outcomes. The amount is rounded to the nearest cent. A common error is to use only the active probability for the maturity payment, giving $[0.70(6{,}000)+0.08(3{,}000)+0.02(1{,}000)]/1.04=\$4{,}288.46$. That calculation omits the disabled policyholders, even though the stated maturity condition includes them.

Include Each Triggered Payment Once

A reliable valuation starts by listing payment triggers, not by listing amounts alone. For each payment, record the event that makes it payable, the time of payment, and whether another benefit can be paid at the same time. This makes the distinction between “either this benefit or that benefit” and “this benefit as well as that benefit” explicit.

Define when coverage ends.
Specify the termination events and, where necessary, which event takes precedence if more than one could occur at the same time.

State the payment trigger.
For each benefit, say whether it depends on the termination cause, the state at termination, a transition, or a combination of these.

Check for simultaneous payments.
Apply the contract’s payment rules to determine whether benefits are alternatives or are paid together.

Calculate probabilities and discount.
Use probabilities for the stated events, include only payments triggered on those events, and discount each payment from its actual payment time.

Worked Example: Transition and Termination Payments in a Two-Year Contract

A hypothetical contract starts with a policyholder active. At the end of year 1, the probabilities of being active, disabled, or dead are 0.60, 0.30, and 0.10. At the end of year 2, conditional on being active at the end of year 1, the probabilities of being active, disabled, or newly dead are 0.70, 0.20, and 0.10. Conditional on being disabled at the end of year 1, the probabilities of still being disabled or newly dead at the end of year 2 are 0.80 and 0.20. Death is absorbing.

The contract pays $1,000 on each entry into disability. At the end of year 1, a death benefit of $3,000 is paid if death has occurred. At the end of year 2, the contract pays a maturity benefit of $6,000 if the policyholder is active or disabled, or a death benefit of $3,000 if death occurs during year 2. A year-2 transition benefit is paid in addition to any year-2 maturity benefit. The effective annual interest rate is 5%. Find the expected present value of these payments.

Year 1. The probability of entering disability is 0.30, so the expected transition benefit is $0.30(1{,}000)=\$300$. The probability of death is 0.10, so the expected death benefit is $0.10(3{,}000)=\$300$. The total expected year-1 payment is $\$600$.

Year 2 probabilities. The unconditional probabilities of being active and disabled at the end of year 2 are

P(A_2)=0.60(0.70)=0.42, \qquad P(D_2)=0.60(0.20)+0.30(0.80)=0.12+0.24=0.36.

The probability of dying during year 2 is

P(\text{death in year 2}) =0.60(0.10)+0.30(0.20) =0.06+0.06 =0.12.

The remaining probability, 0.10, is death during year 1; those contracts already terminated and receive no year-2 payment. The year-2 maturity probability is $0.42+0.36=0.78$, so the expected year-2 termination payment is

0.78(6{,}000)+0.12(3{,}000) =4{,}680+360 =\$5{,}040.

A new entry into disability during year 2 occurs only on the path active at year 1 and disabled at year 2. Its unconditional probability is $0.60(0.20)=0.12$, giving an additional expected transition payment of $0.12(1{,}000)=\$120$. Thus, the expected total year-2 payment is $\$5{,}040+\$120=\$5{,}160$.

Present value. Discount the expected payments from their respective year-end payment dates:

\operatorname{EPV} =\frac{600}{1.05}+\frac{5{,}160}{(1.05)^2} =571.4286+4{,}680.2721 =\$5{,}251.70.

The calculation is checked by noting that the year-2 transition benefit is included, and the year-1 death probability is not counted again at year 2. The maturity benefit and transition benefit are both paid for the active-to-disabled transition at the end of year 2, as specified by the contract. The common error of omitting that transition benefit would understate the expected year-2 payment by $0.12(1{,}000)=\$120$.

Termination Benefits Can Depend on the Final State

A maturity payment may vary by state, and a surrender or death benefit may be payable only for a specified cause. The probability of a payment must then be matched to its exact condition. If a maturity benefit is $4,000 in the active state and $2,500 in the disabled state, the expected maturity payment is the sum of each amount multiplied by the probability of being in its corresponding state at maturity. It is not generally correct to multiply a single maturity amount by the probability of being in force unless that amount is the same in every eligible state.

The same care applies to repeated transitions. If a transition benefit is payable on every entry into disability, its expected amount depends on the expected number of qualifying entries, not merely on whether the policyholder is disabled at the end of the term. A policyholder can enter disability, recover, and enter disability again. Each entry is included only if the contract specifies a payment for each entry and the transition occurs while the contract remains in force.

Worked Example: Surrender, Maturity, and a Same-Date Transition Benefit

A hypothetical two-year contract starts in the active state. At the end of year 1, the probabilities of surrender, disability, and remaining active are 0.10, 0.20, and 0.70. At the end of year 2, conditional on being active at the end of year 1, the probabilities of surrender, entry into disability, and remaining active are 0.10, 0.20, and 0.70. Conditional on being disabled at the end of year 1, the probabilities of remaining disabled or dying during year 2 are 0.80 and 0.20.

Surrender pays $2,000; death pays $5,000; and maturity at the end of year 2 pays $4,000 in the active state or $2,500 in the disabled state. A $500 transition benefit is paid on each entry into disability. The transition benefit is payable in addition to maturity when both are triggered at the end of year 2. Use an effective annual interest rate of 4%.

Year 1. The surrender payment has expected amount $0.10(2{,}000)=\$200$. The disability transition payment has expected amount $0.20(500)=\$100$. Therefore, the expected year-1 payment is $\$300$.

Year 2 event probabilities. The unconditional probabilities of surrender, new disability, and remaining active are

P(S_2)=0.70(0.10)=0.07, \qquad P(\text{new }D_2)=0.70(0.20)=0.14, \qquad P(A_2)=0.70(0.70)=0.49.

The probability of remaining disabled is $0.20(0.80)=0.16$, and the probability of death in year 2 is $0.20(0.20)=0.04$. These probabilities total $0.07+0.14+0.49+0.16+0.04=0.90$, the probability that the contract was still in force after year 1.

The year-2 termination payments have expected total

0.07(2{,}000)+0.49(4{,}000)+ (0.14+0.16)(2{,}500)+0.04(5{,}000) =140+1{,}960+750+200 =\$3{,}050.

The transition payment is triggered on the 0.14 probability of new entry into disability, so its expected amount is $0.14(500)=\$70$. The total expected year-2 payment is $\$3{,}050+\$70=\$3{,}120$. Thus,

\operatorname{EPV} =\frac{300}{1.04}+\frac{3{,}120}{(1.04)^2} =288.4615+2{,}884.6154 =\$3{,}173.08.

The state and cause probabilities have been checked against the in-force probability, and each payment trigger has been applied once. In particular, the 0.14 probability of entering disability at the end of year 2 produces both a disabled-state maturity payment and a transition payment because the terms explicitly allow both.

Common Traps

Omitting an eligible maturity state. If maturity is payable in active or disabled states, add both probabilities before applying the common benefit amount, or value each state-specific amount separately.
Counting terminated contracts again. A contract surrendered or terminated by death in an earlier year cannot receive a later maturity payment unless the terms explicitly provide otherwise.
Confusing state occupancy with a transition. Being disabled at a date does not by itself identify when, or how often, entry into disability occurred.
Assuming all same-date benefits are alternatives. A transition benefit and a maturity benefit may both be payable; read and apply the stated payment rules.
Adding mutually exclusive termination benefits on one outcome. A policy that terminates by death cannot also terminate by surrender at the same date when the causes are defined as mutually exclusive.

Key takeaway: Write each payment as an amount paired with its trigger condition and payment time. Partition alternative termination causes, include every state eligible for the stated benefit, and add simultaneous payments only when the contract terms make them payable together. This event-based approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the applicable payment triggers before calculating. Amounts are paid at the stated year-end unless the question says otherwise.

A hypothetical one-year contract has year-end probabilities 0.50 active, 0.30 disabled, 0.15 dead, and 0.05 surrendered. It pays $8,000 at maturity if active or disabled, $2,000 on death, and $500 on surrender. At an effective annual interest rate of 5%, find the expected present value. (Answer: $6,404.76)
At the end of year 1, a hypothetical contract terminates by death with probability 0.08, by surrender with probability 0.12, and remains in force with probability 0.80. Death pays $10,000 and surrender pays $1,500. If both payments are made at the end of year 1 and $i=4\%$, find the expected present value. (Answer: $942.31)
A two-year contract starts active. At the end of year 1, the probabilities of entering disability and remaining active are 0.25 and 0.75. At the end of year 2, conditional on active at year 1, the probabilities of entering disability and remaining active are 0.10 and 0.90. Conditional on disabled at year 1, the probabilities of remaining disabled and recovering to active are 0.85 and 0.15. A $400 transition benefit is paid on each entry into disability; maturity pays $5,000 if active and $3,000 if disabled. With $i=5\%$, find the expected present value. (Answer: $4,136.05)
A two-year contract starts active. At year 1, the probabilities of surrender, death, disability, and remaining active are 0.10, 0.05, 0.20, and 0.65. At year 2, conditional on active at year 1, the probabilities of surrender, entry into disability, and remaining active are 0.10, 0.20, and 0.70; conditional on disabled at year 1, the probabilities of remaining disabled or dying are 0.90 and 0.10. Surrender pays $1,000, death pays $5,000, and maturity pays $4,000 active or $2,500 disabled. A $300 benefit is paid on entry into disability, including when entry coincides with maturity. At $i=4\%$, find the expected present value. (Answer: $2,982.06)

What You'll Learn