Why a Cash-Flow Map Matters
A universal life policy can have several amounts recorded in the same period: a premium, a charge deducted from the account, an interest credit, and a benefit or surrender payment. Those amounts do not all represent separate cash received or paid by the insurer. A cash-flow map makes the distinction explicit: it records what moves between the policyholder and insurer, when it moves, and under which event.
This tutorial uses the insurer’s perspective. An inflow to the insurer is positive and an outflow is negative. The map is a valuation tool for the specified transactions; it is not itself a policy-account projection or an insurer reserve calculation.
Classifying the Lines Before Valuing Them
Start by naming the valuation perspective and deciding which transactions are included. A gross premium is recorded once as an insurer inflow. If part of it is allocated to a premium load, that allocation is not a second receipt: the gross premium already records the cash received. A charge taken from the policy account may be shown as a separate fee receipt in a transaction-based map, but it must not also be added to the gross premium as though the policyholder paid it a second time.
An account credit is different. It changes the amount credited to the policy account, but it is not an additional cash payment from the policyholder to the insurer at the time it is credited. Likewise, a policy-account debit is not automatically a separate insurer expense. Whether a charge is included as insurer income depends on the stated cash-flow convention; the map must say which convention it uses.
| Item | Typical map treatment | Check before including it |
|---|---|---|
| Gross premium | Positive when received | Do not add a premium load as a second receipt. |
| Policy charge | Positive if modeled as a fee transferred to the insurer | Do not count the same amount again as a separate premium. |
| Account credit | Usually not an external transaction in the map | It changes account value, not cash received from the policyholder. |
| Expense | Negative when paid or incurred, according to the stated timing | Do not assume an assessment date and payment date are identical unless specified. |
| Death benefit | Negative when paid | Weight it by the probability of death at the specified time. |
| Surrender payment | Negative when paid | Use the stated payment amount; do not add the account value separately unless it is also paid. |
A map also needs a timing convention. A premium due at the beginning of a year is not discounted like a benefit paid at year-end. If different outcomes occur, record each outcome at its own payment time and weight it by its probability.
Why Expected Cash Flows Can Be Discounted Period by Period
Suppose the cash flow at time \(t\) depends on the outcome of the policy, and denote that random cash flow by \(C_t\). At a fixed deterministic interest rate, its present value is \(v^t C_t\). By linearity of expectation,
The discount factor is deterministic, so it can be taken outside the expectation. This justifies mapping expected cash flows one period at a time and then discounting them. It does not mean that probabilities can be ignored: each \(E[C_t]\) must still reflect the probability that the policy is in the state or outcome that produces that transaction.
Worked Examples
Worked Example: Mapping a One-Year Policy with Competing Outcomes
A hypothetical one-year universal life policy receives a premium of $900 at time 0. An acquisition expense of $80 is paid at time 0. At year-end, exactly one of three outcomes occurs: death with probability 0.02, surrender with probability 0.08, or persistence with probability 0.90. The death benefit is $20,000, the surrender payment is $3,000, and a $120 policy charge is collected only if the policy persists. Assume an effective annual discount rate of \(i=5\%\). There are no other cash flows. Calculate the expected present value to the insurer. The specified outcome probabilities are the complete one-year distribution; the death probability is 0.02 and surrender and persistence are the other mutually exclusive outcomes.
Method. Map the certain time-0 transactions first, then calculate the expected year-end transaction. The policy charge is counted only for persistence, as specified.
At time 0, the insurer receives $900 and pays $80:
At year-end, the expected death-benefit outflow is \(0.02(20{,}000)=\$400\). The expected surrender outflow is \(0.08(3{,}000)=\$240\). The expected charge receipt is \(0.90(120)=\$108\). Therefore,
Since \(v=1/1.05\), the expected present value is
The result is positive from the insurer’s perspective. The year-end expected outflow exceeds the expected charge receipt, but the time-0 net inflow is larger. The result is rounded to the nearest cent. Check: the probabilities sum to \(0.02+0.08+0.90=1\), and every year-end line is included only for its specified outcome. A common mistake is to apply the $120 charge to all policies; it is collected only on the 0.90 persistence outcome.
Worked Example: Building a Two-Year Expected Cash-Flow Map
A hypothetical policy has a $600 premium at time 0 and a further $600 premium at time 1 only if the policy persisted through year 1. At the end of each policy year, conditional on being in force at the start of that year, death has probability 0.02, surrender has probability 0.08, and persistence has probability 0.90. The death benefit is $20,000, the surrender payment is $4,000, and a charge of $80 is collected at year-end only if the policy persists. An expense of $40 is paid at the end of each year for every policy in force at the start of that year. Assume an effective annual discount rate of \(i=5\%\), and no other cash flows. Find the expected present value to the insurer. The stated conditional outcome probabilities are the mortality and termination assumptions for each year.
Method. Calculate each year’s unconditional probability by multiplying the probability of reaching the year by its conditional outcome probability. Then combine all transactions occurring at the same time.
At time 0, the premium produces \(C_0=\$600\). For year 1, the policy is in force with probability 1. The time-1 premium is received only if the policy persisted through year 1, so its expected amount is \(0.90(600)=\$540\). The expected year-1 charge is \(0.90(80)=\$72\); the expected death benefit is \(0.02(20{,}000)=\$400\); the expected surrender payment is \(0.08(4{,}000)=\$320\); and the expected expense is $40. Thus,
The probability that the policy reaches the start of year 2 is 0.90. The unconditional probability of persistence through year 2 is \(0.90(0.90)=0.81\). Therefore, the expected year-2 charge is \(0.81(80)=\$64.80\). The unconditional probability of death in year 2 is \(0.90(0.02)=0.018\), giving an expected death benefit of \(0.018(20{,}000)=\$360\). The unconditional probability of surrender in year 2 is \(0.90(0.08)=0.072\), giving an expected surrender payment of \(0.072(4{,}000)=\$288\). The year-2 expected expense is \(0.90(40)=\$36\). There is no premium at time 2. Hence,
The expected present value is
The result is rounded to the nearest cent. Check: the time-1 premium is weighted by 0.90, while the year-2 outcomes and expense are weighted by the 0.90 probability of reaching year 2. A common mistake is to apply the year-2 probabilities as though every policy were still in force; doing so overstates the year-2 expected payments and receipts.
Worked Example: Keeping an Account Credit Out of the Transaction Total
A hypothetical policy receives a $700 premium at time 0, and the insurer pays a $45 expense at time 0. At time 1, the policy account has a value of $2,500, which already includes a stated $28 account credit. The policy then surrenders, and the contractual surrender payment is 92% of that account value. Assume an effective annual discount rate of \(i=4\%\). Surrender is certain at time 1, so there is no mortality or other termination risk; the outcome distribution assigns probability 1 to surrender. Calculate the expected present value to the insurer.
Method. Map only the insurer’s transaction flows. The $28 is already included in the account value used to determine the surrender payment; it is not a separate payment from the policyholder.
The time-0 net flow is
The time-1 surrender payment is \(0.92(2{,}500)=\$2{,}300\), so \(C_1=-\$2{,}300\). The expected present value is
The result is rounded to the nearest cent. Check: \(0.92(2{,}500)=2{,}300\), and \(2{,}300/1.04=2{,}211.5385\ldots\). A common mistake is to add the $28 account credit as an insurer inflow. It is an account movement, and the surrender payment already uses the stated account value that includes it.
A Reliable Mapping Process
The same sequence works whether the valuation covers one period or many. Keeping the process explicit helps prevent errors in timing, policy status, and sign.
Choose the insurer or policyholder perspective and apply one sign convention consistently.
Separate premiums, charges, expenses, benefits, and surrender payments. Identify account credits and debits that are internal account movements rather than new external transactions.
Record when each amount is received or paid and whether it depends on persistence, death, or surrender.
Multiply each contingent amount by the probability of the relevant outcome, including the probability of reaching that period.
Discount each expected net flow using its own time, then check for omitted flows, duplicated charges, and account values counted as separate payments.
Common Exam Traps
- Counting an account credit as cash received. An account credit changes the policy account; it is not automatically a separate insurer inflow.
- Counting the same premium twice. If the map records the gross premium, a premium load is an allocation of that receipt, not another receipt.
- Using the wrong exposure probability. A year-2 cash flow must be weighted by the probability that the policy reaches year 2 as well as the conditional probability of the year-2 outcome.
- Mixing timing conventions. A premium at the beginning of a year and a claim at year-end occur at different times and must not receive the same discount factor.
- Adding an account value to a surrender payment. If the surrender payment is defined as a percentage of the account value, use the resulting payment once; do not add the account value again.
A cash-flow map is useful because it makes the valuation auditable: each amount has a perspective, sign, time, and condition. It also clarifies the boundary between policy-account mechanics and insurer transaction flows. Later valuation steps can use the map, but they cannot repair a map that has duplicated or omitted a cash flow.
This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, use the insurer’s perspective, identify the timing and conditions of each flow, and calculate the requested expected present value.
- A hypothetical one-year policy receives a $500 premium and incurs a $30 expense at time 0. At year-end, death occurs with probability 0.01 and produces a $10,000 benefit; surrender occurs with probability 0.09 and produces a $2,000 payment; otherwise the policy persists with probability 0.90 and a $50 charge is collected. At \(i=4\%\), what is the expected present value? (Answer: $244.04)
- A policy receives an $800 premium and incurs a $25 expense at time 0. It surrenders for $1,800 at time 1. The account value used to determine the surrender payment includes a $60 credit, which is not a separate external payment. At \(i=3\%\), what is the expected present value? (Answer: -$972.57)
- A one-year policy receives a $1,000 premium and incurs a $50 expense at time 0. At year-end, death with probability 0.03 produces a $12,000 benefit, surrender with probability 0.07 produces a $2,500 payment, and persistence with probability 0.90 produces a $70 charge. At \(i=5\%\), what is the expected present value? (Answer: $500.48)
- A hypothetical two-year policy receives a $400 premium at time 0 and another $400 at time 1 only if it persisted through year 1. At each year-end, conditional on being in force at the start of that year, death has probability 0.01, surrender has probability 0.04, and persistence has probability 0.95. Death pays $10,000, surrender pays $1,500, and a $40 charge is collected only on persistence. An expense of $20 is paid at each year-end for each policy in force at the start of that year. At \(i=4\%\), find the expected present value. (Answer: $504.12)