Why Decompose Cash Flows?
A participating policy may involve several amounts that arise for different reasons and have different levels of certainty. Premiums are received under the contract, expenses are incurred in administering it, and benefits may include a guaranteed amount, bonuses already declared, and a possible future discretionary addition. Combining these into one benefit figure too early makes it difficult to see which amounts are contractual, which depend on an event, and which are only projections.
Cash-flow decomposition means listing each amount separately at the time it is expected to occur, then aggregating amounts only when their timing and conditions match. It is a preparation step for valuation: a correctly discounted total cannot repair an incorrect classification or an omitted cash flow. The approach below is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Build the Schedule Before Valuing It
Use one perspective consistently. The examples use the insurer’s perspective, so a positive amount increases net receipts and a negative amount represents an outflow. A policyholder’s schedule would reverse the signs. Record each component at its payment time, not merely when it is calculated or declared. When a payment depends on survival, death, maturity, or another event, state that condition and the probability assumption used to aggregate it.
The following classification helps prevent unlike amounts from being merged prematurely:
| Component | How to record it | Key distinction |
|---|---|---|
| Premium | Positive receipt on its scheduled payment date, subject to any stated condition | Do not assume a future premium is received if the policy will not be in force then. |
| Expense | Negative outflow on the date it is incurred | State whether the expense is certain or depends on the policy remaining in force. |
| Guaranteed benefit | Negative outflow on the applicable claim or maturity date | Include amounts contractually payable under the stated conditions. |
| Declared bonus | Separate line, or an identified part of the benefit, once its guaranteed status is established | Do not count the declared addition both within a stated benefit total and again as a separate addition. |
| Possible future discretionary benefit | Separate, explicitly projected outflow, if included in the projection | Do not describe an illustration as a guarantee. |
The earlier material on bonus declarations distinguishes a declared addition that has become guaranteed from an illustration of a possible future bonus. For a cash-flow schedule, that distinction determines the label and certainty attached to the amount; it does not justify silently treating a projection as contractual. Also distinguish an amount already included in a quoted total from an amount that must still be added.
A Basic Decomposition and Discounting Check
For a specified scenario, let \(C_t\) be the insurer’s net cash flow at time \(t\), with receipts positive and outflows negative. At effective annual interest rate \(i\), the present value of those specified cash flows is the sum of each dated amount discounted to time 0. If the schedule includes mutually exclusive outcomes, expected cash flows are formed by weighting each outcome by its probability before summing; equivalently, the expected present value is the probability-weighted average of the outcome present values, when the timing and discount assumptions are the same.
This identity follows from discounting each cash flow separately: one dollar paid at time \(t\) has time-0 value \(v^t\), so an amount \(C_t\) has value \(C_tv^t\). Present value is additive, and therefore the value of a schedule split into components is the sum of the component values, provided every component uses the same dates, sign convention, and assumptions. For a random cash flow, finite additivity of expectation gives \(E[\sum_t C_t v^t]=\sum_t E[C_t]v^t\). This is a decomposition identity, not a claim that a discretionary projection is guaranteed or that every future valuation basis is appropriate.
Worked Example: Separating a Maturity Benefit and a Projection
A hypothetical endowment policy is known to remain in force to maturity at time 2. The insurer receives a premium of $1,000 at time 0 and $1,000 at time 1. Expenses are $120 at time 0 and $60 at time 1. At maturity, the guaranteed basic benefit is $5,000 and a previously declared bonus of $300 is guaranteed. A further $400 terminal addition is a projection only. Using an effective annual interest rate of 4%, construct the dated schedule and calculate its present value, first excluding and then including the projected addition. The specified in-force-to-maturity scenario is deterministic; no mortality distribution is needed for this conditional calculation.
Method. First separate receipts, expenses, guaranteed benefits, and the projection by date. Then discount the net cash flow at each date. This avoids treating a projected addition as guaranteed.
At time 0, the net receipt is \(1{,}000-120=880\). At time 1, it is \(1{,}000-60=940\). The guaranteed maturity outflow is \(5{,}000+300=5{,}300\) at time 2. The projected addition is a separate possible outflow of $400 at time 2.
With \(v=1/1.04\), the guaranteed-only present value is
The separate projected addition has present value \(-400/1.04^2=-369.822485\). Including it gives \(-3{,}116.301775-369.822485=-3{,}486.124260\), or -$3,486.12 to the nearest cent. The check is that the combined time-2 outflow is $5,700, and discounting it directly gives the same total: \(880+940/1.04-5{,}700/1.04^2=-$3{,}486.12\).
Common mistake avoided. Calling the full $5,700 maturity amount guaranteed would misstate the contract: only $5,300 is guaranteed under the stated facts. The remaining $400 is included only as a projection.
Keep Declaration Status and Payment Type Visible
A bonus can affect a benefit without being a new payment at the date it is declared. The schedule should follow the actual payment condition and date. If a declared reversionary addition increases a claim benefit, record it as a distinct component of that claim benefit or show it within a clearly labelled total. If a cash dividend is paid earlier, record that payment at its own date. This distinction helps avoid counting a bonus once at declaration and again at claim, or confusing an earlier cash payment with an addition payable only at a claim.
Worked Example: A Prior Cash Dividend and a Claim Benefit
A hypothetical participating policy is assumed to produce a death claim at time 2. The insurer receives $600 at time 0 and $600 at time 1, and incurs expenses of $40 at each of those times. A $75 cash dividend is paid at time 1. The death benefit at time 2 is a guaranteed basic amount of $4,000 plus a declared, guaranteed addition of $200. A further $300 terminal bonus is projected, but not guaranteed. Use an effective annual interest rate of 5%. Calculate the present value of the specified net cash flows, separating guaranteed amounts from the projection. The calculation is conditional on the stated claim at time 2; no mortality distribution is being applied.
Method. Place the dividend at its payment date, and place both claim components at time 2. The projected terminal bonus is shown separately from the guaranteed claim.
The time-0 net cash flow is \(600-40=560\). The time-1 net cash flow is \(600-40-75=485\). The guaranteed time-2 claim is \(4{,}000+200=4{,}200\), and the projected time-2 outflow is $300.
The projected bonus adds a separate present-value outflow of \(300/1.05^2=272.108844\). The total including the projection is \(-2{,}787.619048-272.108844=-3{,}059.727892\), or -$3,059.73 to the nearest cent. A direct check combines the time-2 amounts: \(560+485/1.05-4{,}500/1.05^2\) gives the same answer.
Common mistake avoided. The $75 dividend is already paid at time 1, so it is not added to the time-2 claim. The $200 declared addition is included once in the guaranteed claim; adding it a second time would double count it.
When Outcomes Are Contingent
A schedule for one specified outcome is not an expected schedule for a population or a policy with uncertain outcomes. To aggregate contingent amounts, state the probability model and the event condition. For example, a claim paid only on death in a year contributes its benefit amount multiplied by the probability of death in that year to the expected outflow for that date. A maturity payment contributes only for policies that survive to maturity. Premiums and expenses also require their stated in-force conditions; a premium due only while the policy is active is not received on a path where the policy has already terminated.
The next example uses a simple discrete-year mortality model. It is included to show how probability weighting fits into decomposition, not to introduce a new benefit valuation method.
Worked Example: Expected Cash Flows with Death and Maturity Outcomes
A hypothetical two-year policy has a premium of $1,000 at time 0 and a $100 expense at time 0. At time 1, each policy still in force pays a $1,000 premium and incurs a $30 expense. A death during year 1 produces a guaranteed claim of $5,200 at time 1. At time 2, a policy surviving to the start of year 2 either dies during that year or survives to maturity; in either case, the guaranteed benefit is $5,200 and a projected terminal addition of $300 is also assumed for this illustration. The annual death probabilities are \(q_1=0.02\) in year 1 and \(q_2=0.03\) in year 2 conditional on survival to that year. Thus the probabilities of survival through years 1 and 2 are 0.98 and \(0.98(0.97)=0.9506\). Use an effective annual interest rate of 4%. Premiums at time 1 are paid only by policies surviving year 1; claims for death in year 1 occur at time 1. Calculate the expected present value including the projected addition.
Method. Find the probability attached to each dated cash-flow component, form the expected net cash flow at each time, and then discount. This prevents a claim paid on death from being counted for every policy.
At time 0, every policy contributes \(1{,}000-100=900\). At time 1, the probability of being in force is 0.98, so expected premiums are \(0.98(1{,}000)=980\), and expected expenses are \(0.98(30)=29.40\). The probability of death during year 1 is 0.02, so the expected claim is \(0.02(5{,}200)=104\). Therefore the expected net cash flow at time 1 is \(980-29.40-104=846.60\).
At time 2, the probability of death during year 2 is \(0.98(0.03)=0.0294\), and the probability of survival to maturity is \(0.98(0.97)=0.9506\). The two probabilities sum to 0.98, the probability in force at the start of year 2. The guaranteed benefit is paid under either stated outcome, so its expected amount is \(0.98(5{,}200)=5{,}096\). The projected addition’s expected amount is \(0.98(300)=294\). Thus the expected time-2 outflow including the projection is \(5{,}096+294=5{,}390\).
The expected present value is -$3,269.32 to the nearest cent. As a check, the time-2 guaranteed component alone has expected amount $5,096, while the projected component has expected amount $294; together they equal the $5,390 used above. The survival and death probabilities also reconcile: \(0.02+0.0294+0.9506=1\).
Common mistake avoided. Multiplying the time-2 benefit by 1 would assume every policy reaches the second-year claim or maturity date. Only 0.98 of the policies remain in force at the start of year 2 under the specified mortality model.
Schedule Checks and Exam Traps
Choose insurer or policyholder signs, and state when each premium, expense, dividend, and claim is paid.
Separate guaranteed contractual amounts, additions already declared and guaranteed, and possible future discretionary amounts.
Identify which policies or scenarios produce each amount, then use the corresponding probabilities if calculating expected cash flows.
Sum amounts only after their date and conditions are clear. Reconcile component totals to any stated benefit total and confirm the sign of each net amount.
A frequent error is to discount an undisclosed lump sum without first checking what it contains. For example, if a benefit total already includes a declared addition, the wrong calculation is to add that addition again as a separate outflow. The correction is to use either the total benefit as one line or its underlying components as separate lines, but never both. Another trap is to treat an illustrated terminal addition as guaranteed merely because it appears in a projection. Preserve its separate label even if a later calculation assigns it a projected amount.
Check Your Understanding
For each problem, use the insurer’s perspective unless the question says otherwise. Round monetary totals to the nearest cent.
- A hypothetical policy has a $750 premium and $50 expense at time 0, followed by a $750 premium and $25 expense at time 1. A guaranteed benefit of $1,600 is paid at time 2. At an effective annual interest rate of 4%, find the present value of the net cash flows. (Answer: \(700+725/1.04-1{,}600/1.04^2=-$82.17\).)
- A hypothetical death benefit is stated as a guaranteed basic amount of $4,000 plus a declared addition of $240. A further $180 is only a possible future discretionary addition. State the guaranteed claim amount and the total claim amount if the projection is realized. (Answer: $4,240 guaranteed; $4,420 including the projected amount.)
- Under a discrete annual mortality model, the probability of death in year 1 is 0.01 and the conditional probability of death in year 2 is 0.04. A claim of $3,000 is paid at time 1 on death in year 1, and $3,300 is paid at time 2 on death in year 2. Find the expected claim payment at each date. (Answer: time 1, \(0.01(3{,}000)=$30\); time 2, \(0.99(0.04)(3{,}300)=$130.68\).)
- A hypothetical policy has a $900 premium and $60 expense at time 0; at time 1 it has a $900 premium, $40 expense, and a $50 cash dividend paid. At time 2, the guaranteed benefit is $4,500 plus a declared guaranteed addition of $250; a further $200 is projected but not guaranteed. At an effective annual interest rate of 5%, find the present value including the projection. (Answer: \(840+810/1.05-4{,}950/1.05^2=-$2{,}878.37\).)
- A policy has annual death probabilities \(q_1=0.02\) and \(q_2=0.03\), conditional on being in force at the start of each year. At time 2, a $5,000 benefit is paid on either death in year 2 or survival to maturity. What is the expected time-2 benefit payment per policy issued at time 0? (Answer: \([0.98(0.03)+0.98(0.97)](5{,}000)=0.98(5{,}000)=$4,900.)