Projecting Bonus Payments Before Valuation
A projected bonus scale is not yet an expected present value. Each projected amount must be matched to the date it is paid and to the event that makes it payable. A bonus payable only if a policy is in force at an anniversary has a different expected value from a claim bonus payable on death during the preceding year, even if the two amounts are equal.
This tutorial treats the projected bonus amounts and eligibility conditions as given. The task is to determine the expected value at the valuation date of the resulting future payments. The policy’s eligibility probabilities may come from survival, a claim event, lapse, or a specified combination of outcomes.
The condition \(C_j\) must be stated precisely. For example, “survives to the end of year 2” and “dies during year 2” are different events. The first requires survival through both years; the second requires survival through year 1 followed by death in year 2. A conditional probability for year 2 alone is not necessarily the probability of the whole event from the valuation date.
Expected Present Value of a Projected Sequence
Suppose each projected payment has a fixed amount \(B_j\), a time \(t_j\), and an eligibility event \(C_j\). The expected present value of the sequence is the sum of the expected present values of its individual payments:
If several different payments may occur at the same time, list their conditions separately. For instance, a death bonus and a maturity bonus can both be dated at year 3, but one may be payable only on death during that year and the other only on survival to maturity.
The formula applies when the discount rate is deterministic and the payment amounts and eligibility conditions have been specified. If a payment amount varies by outcome, sum over the possible outcomes, multiplying each outcome-specific amount by the probability of that outcome. If interest rates are stochastic or depend on the same events as the bonus, a single deterministic discount factor cannot generally be taken outside the expectation.
Why the Probability-Weighted Formula Holds
For a fixed bonus amount \(B_j\), the actual payment is \(B_j\) if \(C_j\) occurs and zero otherwise. Its present value is therefore \(v^{t_j}B_j\mathbf{1}_{C_j}\). Taking expectations gives the value of that payment. Since expectation is linear, the expected present value of a finite sequence is the sum of those individual expected values; the payment events do not need to be independent.
This argument also clarifies an important point: probabilities for different payments may overlap. For example, a policyholder may qualify for a year-1 bonus and later qualify for a maturity bonus. Both payments belong in the sum. By contrast, if two outcomes at the same date are mutually exclusive, their separate probability-weighted payments are alternative contributions, not two payments made on the same policy.
Worked Example: Annual Bonuses and a Year-3 Claim or Maturity Bonus
A hypothetical policy has a fixed annual effective interest rate of 10%. Mortality is the only decrement, with a conditional death probability of 0.10 in each policy year for a policy in force at the start of that year. The projected bonus is $110 at time 1 if the policyholder survives to that anniversary, and $121 at time 2 if the policyholder survives to that anniversary. At time 3, a $1,331 death bonus is payable if death occurs during year 3; otherwise, a $1,331 maturity bonus is payable if the policyholder survives to time 3. Calculate the expected present value at issue.
Method. First calculate the probability of each payment condition from issue, then multiply each eligible amount by its probability and discount it to issue. The conditional survival probability for each year is \(p=1-0.10=0.90\).
Time 1 bonus. The probability of surviving to time 1 is \(0.90\). The expected payment is \(110(0.90)=99\), so its present value is \(99/1.10=\$90\).
Time 2 bonus. The probability of surviving to time 2 is \(0.90^2=0.81\). The expected payment is \(121(0.81)=98.01\), and the discount factor is \(1/(1.10)^2=1/1.21\). Thus, the present value is \(98.01/1.21=\$81\).
Death bonus at time 3. The policyholder must survive the first two years and die during the third. The probability is \(0.90^2(0.10)=0.081\). The expected payment is \(1{,}331(0.081)=107.811\). Since \(1.10^3=1.331\), the present value is \(107.811/1.331=\$81\).
Maturity bonus at time 3. The probability of surviving all three years is \(0.90^3=0.729\). The expected payment is \(1{,}331(0.729)=970.299\). Its present value is \(970.299/1.331=\$729\).
Adding all four contributions gives \(\operatorname{EPV}=90+81+81+729=\boxed{\$981.00}\). The result includes both earlier bonuses and exactly one of the two time-3 outcomes. The two time-3 bonus conditions are mutually exclusive.
Using Conditional Probabilities and Competing Outcomes
A payment condition is often described using probabilities conditional on being in force at the beginning of a period. To use the expected present value formula from an earlier valuation date, convert those probabilities into probabilities from that date. For example, if the probability of being in force at time 1 is \(0.90\), and the conditional probability of death during year 2 is \(0.20\), then the probability of death during year 2 from issue is \(0.90(0.20)=0.18\).
If death, lapse, and continued insurance are the possible outcomes during a year, their conditional probabilities must be interpreted as a coherent set of alternatives. A death bonus uses the probability of reaching the relevant period and then dying in it. A lapse bonus uses the probability of reaching that period and then lapsing in it. A bonus payable to a policy still in force at the end of the period uses the probability of reaching that point without a terminating event.
Worked Example: A Survival Bonus and a Death Bonus
A hypothetical policy has an annual effective interest rate of 5%. Mortality is the only decrement. The conditional death probability is 0.10 in year 1 and 0.20 in year 2 for a policy in force at the start of each year. A $40 bonus is payable at time 1 if the policyholder survives year 1. At time 2, a $60 bonus is payable if the policyholder survives year 2, while a separate $250 death bonus is payable if death occurs during year 2. Find the expected present value.
Method. Treat each payment condition separately, using an unconditional probability from issue. The probability of surviving year 1 is \(0.90\); the probability of surviving both years is \(0.90(0.80)=0.72\); and the probability of dying during year 2 is \(0.90(0.20)=0.18\).
The time-1 bonus has expected payment \(40(0.90)=36\). Its present value is \(36/1.05=\$34.2857\), rounded to four decimal places.
The time-2 survival bonus has expected payment \(60(0.72)=43.20\). Its present value is \(43.20/(1.05)^2=43.20/1.1025=\$39.1837\), rounded to four decimal places.
The time-2 death bonus has expected payment \(250(0.18)=45\). Its present value is \(45/1.1025=\$40.8163\), rounded to four decimal places.
Therefore, the total is \(34.2857+39.1837+40.8163=\$114.2857\), or \(\boxed{\$114.29}\) to the nearest cent. This includes the survival bonus only for policies surviving to time 2 and the death bonus only for deaths during year 2.
A Projection With Several Payment Conditions
A useful working layout records, for each projected payment, its time, amount, exact condition, probability from the valuation date, discount factor, and expected present value. Separating these entries makes it easier to identify a misplaced conditional probability or a bonus assigned to the wrong date.
Record the amount and the date it would be paid if the eligibility condition is met.
State whether the payment requires survival, death, lapse, maturity, or another specified outcome.
Multiply conditional probabilities along the required path, or use a supplied unconditional probability directly.
Multiply the amount by its eligibility probability, then by the discount factor for its payment time.
Add the contributions and verify that each payment appears once, at its stated time, under its stated condition.
Worked Example: Survival, Death, Lapse, and Maturity Bonuses
A hypothetical policy is valued at issue using an annual effective interest rate of 5%. In year 1, the probability of survival is 0.96. Conditional on being in force at time 1, the year-2 outcomes are death with probability 0.03, lapse with probability 0.07, and survival through year 2 with probability 0.90. Conditional on survival through year 2, the probability of survival through year 3 is 0.95. A $25 bonus is payable at time 1 if the policyholder survives year 1. At time 2, a $50 bonus is payable if the policyholder survives year 2, a $400 bonus if death occurs during year 2, and a $120 bonus if lapse occurs during year 2. A $600 maturity bonus is payable at time 3 if the policyholder survives through year 3. Calculate the expected present value.
Method. Construct the unconditional probability of each condition. The year-2 death and lapse outcomes are distinct from survival through year 2, so their probabilities are calculated as the probability of reaching time 1 multiplied by the relevant conditional probability.
The time-1 survival bonus has probability \(0.96\). Its present value is \(25(0.96)/1.05=24/1.05=\$22.8571\).
The probability of surviving through year 2 is \(0.96(0.90)=0.864\). The time-2 survival bonus has expected payment \(50(0.864)=43.20\), and present value \(43.20/1.1025=\$39.1837\).
The probability of death during year 2 is \(0.96(0.03)=0.0288\). The death bonus has expected payment \(400(0.0288)=11.52\), and present value \(11.52/1.1025=\$10.4490\).
The probability of lapse during year 2 is \(0.96(0.07)=0.0672\). The lapse bonus has expected payment \(120(0.0672)=8.064\), and present value \(8.064/1.1025=\$7.3143\).
The probability of surviving through year 3 is \(0.96(0.90)(0.95)=0.8208\). The maturity bonus has expected payment \(600(0.8208)=492.48\), and present value \(492.48/(1.05)^3=492.48/1.157625=\$425.4227\).
The total, retaining unrounded contributions until the final step, is \(22.8571429+39.1836735+10.4489796+7.3142857+425.4227405=\$505.2268222\). To the nearest cent, the expected present value is \(\boxed{\$505.23}\). The death and lapse bonuses are not multiplied by the probability of surviving year 2; they apply to the alternative events during that year.
Common Mistakes and Checks
A frequent error is to use a conditional event probability as though it were unconditional. In the first example, the conditional probability of death during year 3 is 0.10, but the policyholder must first survive years 1 and 2. The correct probability from issue is \(0.90^2(0.10)=0.081\). Using 0.10 directly would give a death-bonus present value of \(1{,}331(0.10)/1.331=\$100\), rather than the correct \(1{,}331(0.081)/1.331=\$81\).
A second error is discounting from the wrong date. A payment at time 3 is discounted by \(v^3\), even when eligibility is determined by an event in year 3. The event date and the payment date should both be read from the projection; they need not be assumed to be the same unless the contract specifies that they are.
Before finalizing a calculation, check that probabilities refer to the whole required path, that mutually exclusive outcomes have not been combined as if they all occur, and that no payment has been counted twice. A simple table is often sufficient:
| Payment | Condition | Probability from valuation date | Discount factor |
|---|---|---|---|
| Survival bonus | In force at anniversary | Probability of reaching that anniversary in force | \(v^t\) at payment time \(t\) |
| Death bonus | Death in a specified period | Probability of reaching the period multiplied by its conditional death probability | \(v^t\) at payment time \(t\) |
| Maturity bonus | Survival to maturity | Probability of surviving to the maturity date | \(v^t\) at maturity time \(t\) |
Check Your Understanding
For each problem, calculate the expected present value at the stated valuation date. Retain full precision until the final answer.
- A hypothetical policy earns an annual effective interest rate of 4%. A $104 bonus is paid at time 1 only if the policyholder survives the first year; the probability of survival is 0.90. (Answer: $90.00)
- A hypothetical policy uses an annual effective interest rate of 5%. Survival probabilities are 0.95 through year 1 and 0.90 in year 2 conditional on survival to time 1. Bonuses of $40 and $80 are paid at times 1 and 2, respectively, if the policyholder survives to each payment date. (Answer: $98.23)
- A hypothetical policy has an annual effective interest rate of 4%. The conditional death probabilities are 0.02 in year 1 and 0.05 in year 2. A $500 bonus is payable at time 2 only if death occurs during year 2. (Answer: $22.65)
- A hypothetical policy has an annual effective interest rate of 5%. The probability of being in force at time 1 is 0.90. During year 2, conditional on being in force at time 1, lapse has probability 0.10, death has probability 0.04, and survival through year 2 has probability 0.86. At time 2, a $250 bonus is payable on lapse and a $600 bonus on death; there is no bonus on survival. (Answer: $40.00)
- A hypothetical policy has an annual effective interest rate of 5%, and a bonus is payable at time 2. Under scenario A, with probability 0.40, eligibility for a $1,000 bonus has conditional probability 0.80. Under scenario B, with probability 0.60, eligibility for a $500 bonus has conditional probability 0.50. The scenarios are mutually exclusive. (Answer: $426.30)