Why Profit-Vector Entries Can Vary by Duration
A policy’s cash flows do not necessarily repeat from one year to the next. An insurer might pay a higher commission when a policy is new, charge different expenses at renewal, increase the premium after a specified duration, or provide a benefit that changes over time. The annual profit vector must represent those changes year by year. Repeating one annual amount would erase the duration pattern and can materially misstate the results.
To make the calculations precise, this tutorial uses a particular annual timing convention. A policy is in force at the start of policy year \(t\). The premium, commission, and stated expenses occur at the start of that year. The net amount is invested for one year at effective annual rate \(i\). Death benefits and surrender benefits, if any, are paid at the end of the year. Unless stated otherwise, there are no reserve movements, taxes, or other cash flows in the examples.
For a year with premium \(P_t\), commission rate \(c_t\), start-of-year expense \(e_t\), conditional death probability \(q_t\), and end-of-year death benefit \(B_t\), the basic calculation is
Here \(q_t\) is the probability of death during policy year \(t\), given that the policy is in force at its start. It is a duration-indexed shorthand for this tutorial, not a replacement for age-based mortality notation. The equation assumes that the only benefit is payable on death and that there is no other termination payment. If the benefit or any other cash flow occurs at a different time, it must be valued using that timing rather than inserted unchanged into this equation.
Why the Annual Formula Holds
The formula follows by separating the year’s cash flows by timing and taking an expectation conditional on being in force at the start. The premium less commission and expense is \(P_t-c_tP_t-e_t=P_t(1-c_t)-e_t\). Because that net amount is received or paid at the start of the year, its end-of-year accumulated value is multiplied by \(1+i\). The death benefit is paid only if death occurs during the year, so its expected end-of-year cost is \(q_tB_t\). Subtracting that expected cost gives the stated profit.
This derivation uses linearity of expectation: the expected profit is the accumulated start-of-year cash flow less the expected benefit payment. It does not require the premium or expense to be the same in different policy years. Rather, each duration’s values are substituted into the corresponding entry.
Worked Example: New-Business Commission and Changing Benefits
A hypothetical three-year contract has a 5% effective annual investment return. Premiums, commissions, expenses, death probabilities, and death benefits are given below. A policy is in force at the start of each year considered; calculate the conditional annual profit entries using the stated one-year timing convention.
| Policy year \(t\) | \(P_t\) | \(c_t\) | \(e_t\) | \(q_t\) | \(B_t\) |
|---|---|---|---|---|---|
| 1 | $700 | 30% | $80 | 0.010 | $12,000 |
| 2 | $700 | 8% | $50 | 0.012 | $12,000 |
| 3 | $760 | 0% | $50 | 0.014 | $16,000 |
Method. For each duration, calculate the premium remaining after commission, subtract that year’s expense, accumulate the result at 5%, and subtract the expected death benefit for that year.
Year 1. Commission is \(0.30(700)=210\). The start-of-year net cash flow is \(700-210-80=410\), with end-of-year value \(410(1.05)=430.50\). The expected death benefit is \(0.010(12{,}000)=120\). Thus \(\Pi_1=430.50-120=\$310.50\).
Year 2. Commission is \(0.08(700)=56\). The start-of-year net cash flow is \(700-56-50=594\), with end-of-year value \(594(1.05)=623.70\). The expected death benefit is \(0.012(12{,}000)=144\). Thus \(\Pi_2=623.70-144=\$479.70\).
Year 3. Commission is zero. The start-of-year net cash flow is \(760-0-50=710\), with end-of-year value \(710(1.05)=745.50\). The expected death benefit is \(0.014(16{,}000)=224\). Thus \(\Pi_3=745.50-224=\$521.50\).
The resulting vector is \((\$310.50,\ \$479.70,\ \$521.50)\). The increasing entries reflect several changes, including the lower renewal commission, the higher third-year premium, and the larger third-year death benefit. The calculation checks because each expected claim cost uses the probability and benefit from the same row as its profit entry.
Building the Vector from a Duration Schedule
A reliable construction method is to make a separate row for every duration. Record each amount and probability against the year in which it applies, then calculate that year’s profit without carrying forward an earlier year’s values by habit. This is especially important when a product has a commission schedule, an expense allowance that changes after issue, or a benefit that steps up at a specified duration.
Identify the policy years in the projection and the profit entry associated with each one.
Enter the applicable premium, commission rate or amount, expenses, and benefits for each duration.
Use probabilities conditional on the policy being in force at the start of that year, and distinguish end-of-year claim payments from start-of-year cash flows.
Apply the timing convention to the row’s values and place the result in the corresponding vector position.
Confirm that no commission, expense, probability, or benefit from one row was inadvertently used in another.
Worked Example: Premium-Linked Commission and a Changing Expense
A hypothetical policy earns 4% effective annual investment return. Its premium is $800 in each of three years. Commission is 25% in year 1 and 10% in years 2 and 3. Expenses are $90, $60, and $60, respectively. The conditional death probabilities are 0.010, 0.012, and 0.015, and the death benefits are $15,000, $15,000, and $18,000. Find all three profit-vector entries.
Use \(\Pi_t=(P_t(1-c_t)-e_t)(1.04)-q_tB_t\), applying the duration-specific inputs.
Year 1. Commission is \(0.25(800)=200\), so the net start-of-year cash flow is \(800-200-90=510\). Its end-of-year value is \(510(1.04)=530.40\). Expected death cost is \(0.010(15{,}000)=150\). Therefore \(\Pi_1=530.40-150=\$380.40\).
Year 2. Commission is \(0.10(800)=80\), and net cash flow is \(800-80-60=660\). Its end-of-year value is \(660(1.04)=686.40\). Expected death cost is \(0.012(15{,}000)=180\). Therefore \(\Pi_2=686.40-180=\$506.40\).
Year 3. Commission is \(0.10(800)=80\), and net cash flow is again \(800-80-60=660\). Its end-of-year value is \(660(1.04)=686.40\). Expected death cost is \(0.015(18{,}000)=270\). Therefore \(\Pi_3=686.40-270=\$416.40\).
The vector is \((\$380.40,\ \$506.40,\ \$416.40)\). Years 2 and 3 have the same premium, commission, and expense, but their profit entries differ because the mortality probability and death benefit differ. A repeated profit amount would miss that change in the expected claim cost.
Including More Than One Termination Outcome
Some policies pay different amounts depending on how coverage ends. For example, a death benefit may be paid on death, while a surrender amount is paid on lapse. If \(l_t\) is the lapse probability conditional on surviving to the lapse decision during year \(t\), then the unconditional probability of lapse during that year is \((1-q_t)l_t\), provided death and lapse occur in that order as modeled. The expected end-of-year termination payments are consequently \(q_tB_t+(1-q_t)l_tS_t\), where \(S_t\) is the surrender payment.
This expression does not apply unchanged if the timing or ordering of events differs. In particular, a lapse probability already stated as unconditional must not be multiplied by \(1-q_t\) again. The probability basis must be identified before calculating expected payments.
Worked Example: Duration-Dependent Death and Surrender Payments
A hypothetical two-year policy has a 3% effective annual investment return. At the start of year 1, its premium is $1,200, commission is 30% of premium, and expense is $150. In year 2, the corresponding amounts are $1,250, 5%, and $100. Death probabilities are 0.010 in year 1 and 0.015 in year 2. Conditional on surviving the year, lapse probabilities are 0.040 and 0.030. Death benefits are $30,000 and $32,000, and surrender payments are $500 and $1,000. Calculate the two conditional profit entries.
Year 1. The start-of-year net cash flow is \(1{,}200-0.30(1{,}200)-150=1{,}200-360-150=690\). Its end-of-year value is \(690(1.03)=710.70\). The death probability is 0.010. The unconditional lapse probability is \((1-0.010)(0.040)=0.99(0.040)=0.0396\). Expected termination payments total \(0.010(30{,}000)+0.0396(500)=300+19.80=319.80\). Thus \(\Pi_1=710.70-319.80=\$390.90\).
Year 2. The start-of-year net cash flow is \(1{,}250-0.05(1{,}250)-100=1{,}250-62.50-100=1{,}087.50\). Its end-of-year value is \(1{,}087.50(1.03)=1{,}120.125\). The unconditional lapse probability is \((1-0.015)(0.030)=0.985(0.030)=0.02955\). Expected termination payments total \(0.015(32{,}000)+0.02955(1{,}000)=480+29.55=509.55\). Thus \(\Pi_2=1{,}120.125-509.55=\$610.575\), or $610.58 to the nearest cent.
The year 2 event probabilities check: death probability is 0.015, lapse probability is 0.02955, and persistence probability is \(0.985(1-0.030)=0.95545\). Their sum is \(0.015+0.02955+0.95545=1.00000\). This check confirms that the conditional lapse probability has been converted correctly.
Common Mistake: Repeating the First-Year Amounts
A common error is to calculate year 1 correctly and then reuse its commission or expense in later entries, even when the schedule changes. In the first worked example, applying the year 1 commission rate of 30% to year 2 would give the incorrect calculation
The correct year 2 commission rate is 8%, not 30%. Using it gives \[ \bigl(700-0.08(700)-50\bigr)(1.05)-0.012(12{,}000) =594(1.05)-144 =\$479.70. \] The incorrect entry understates year 2 profit by $161.70. The correction is not to choose one “average” commission rate for the whole contract; it is to use the rate applicable to each duration.
Duration-specific entries are also important when the vector is later used in a larger profit test. Keep each entry’s definition and conditioning basis clear: an expected profit conditional on being in force at the start of a policy year is not itself a cohort total or a present value. Those quantities require additional steps and should not be confused with the annual entries constructed here.
Check Your Understanding
For each problem, use the stated start-of-year premium, commission, and expense, accumulate the net amount for one year, and subtract expected end-of-year payments. Amounts are hypothetical.
- A policy has a $500 premium, 10% commission, $30 expense, 2% effective annual return, death probability 0.020, and death benefit $5,000. Find the conditional annual profit. (Answer: $328.40)
- In a particular renewal year, premium is $1,000, commission is 5%, expense is $80, investment return is 3%, death probability is 0.020, and death benefit is $25,000. Find that year’s profit entry. (Answer: $396.10)
- A year has premium $800, commission 10%, expense $40, and investment return 4%. Death probability is 0.020; conditional on surviving, lapse probability is 0.050. The death benefit is $10,000 and the surrender payment is $600. Find the annual profit. (Answer: $477.80)
- A two-year policy has a 2% investment return. In year 1, premium is $600, commission is 20%, expense is $50, death probability is 0.010, and benefit is $10,000. In year 2, premium is $650, commission is 5%, expense is $60, death probability is 0.015, and benefit is $12,000. Find both entries. (Answer: \(\Pi_1=\$338.60,\ \Pi_2=\$388.65\))
- A two-year policy has a 5% investment return. Year 1 has premium $1,000, commission 20%, expense $100, death probability 0.010, conditional-on-survival lapse probability 0.040, death benefit $20,000, and surrender payment $500. Year 2 has premium $1,100, commission 5%, expense $80, death probability 0.020, conditional-on-survival lapse probability 0.030, death benefit $22,000, and surrender payment $900. Find both annual entries, converting lapse probabilities to unconditional probabilities first. (Answer: \(\Pi_1=\$515.20,\ \Pi_2=\$546.79\))