Actuarial Mathematics for Modelling · Projecting expected future cashflows and profit testing

Profit Testing Basics and Cashflow Projection

Updated 11 October 2026 · Fact-checked

Profit testing projects a policy's expected cashflows year by year under stated assumptions. Each year you take premiums less expenses and commission, add interest, then subtract expected claims and the cost of setting up reserves. That gives the profit vector. Weight it by the probability the policy is in force and discount to get the NPV.

Understand Profit Testing Basics and Cashflow Projection

Profit testing is a cashflow method. You pick a policy, assume how it behaves (mortality, interest, expenses, commission, lapses), and project the money the insurer expects to receive and pay in each year of the term. The result shows when the product makes or loses money and whether it earns enough for the shareholders.

You project per policy in force at the start of each year. Premiums usually come in at the start of the year. Expenses and commission are paid with the premium or at the start of the year. Claims are paid at the end of the year in the standard model. Interest is earned over the year on money held after the start-of-year cashflows.

The insurer must also hold reserves against future liabilities. Money put into the reserve is not available as profit, so a rise in reserves reduces profit that year. Releasing a reserve adds to profit. With zero reserves, the profit is simply income less outgo plus interest.

The year-by-year profit for a policy in force at the start of the year is the profit vector. Because not every policy survives, you multiply each entry by the probability that the policy is still in force at the start of that year. That gives the profit signature. You then discount the signature at the risk discount rate to get the net present value (NPV) of the profits.

Always write down your assumptions first: the timing of each cashflow, the basis for each item and the rate used for each purpose. Examiners give marks for clear structure as much as for the final figure.

Key rules to remember

Profit vector (with reserves): Pr(t) = [(t−1)V + P(t) − e(t) − c(t)] × (1 + i) − q(x+t−1) × [S(t) + E(t)] − p(x+t−1) × tV; Per policy in force at the start of year t. (t−1)V is the reserve at the start of the year, tV the reserve at the end for survivors. P is premium, e expenses, c commission, S death benefit, E any extra death cost. Here E is zero unless the contract pays reserve on top; for a sum assured paid on death use S(t) + E(t) = S(t) when the death benefit is the whole payment.
Profit vector (zero reserves): Pr(t) = [P(t) − e(t) − c(t)] × (1 + i) − q(x+t−1) × S(t); Use when the question says reserves are zero or ignored. Death benefit assumed paid at end of year.
In-force probability: (t−1)p(x) = p(x) × p(x+1) × … × p(x+t−2), with 0p(x) = 1; Probability the policy is in force at the start of year t, allowing for deaths (and lapses if given). Year 1 uses 1.
Profit signature: Π(t) = (t−1)p(x) × Pr(t); Expected profit at the end of year t per policy sold. Any cashflow at time 0 is not multiplied by a survival probability.
Net present value: NPV = Σ Π(t) ÷ (1 + r)^t; r is the risk discount rate. Sum over all years of the term.
Profit margin: Profit margin = NPV ÷ PV of premiums; The PV of premiums uses the same in-force probabilities and the same risk discount rate.

How to solve Profit Testing Basics and Cashflow Projection questions

Use this order for any question that asks you to project cashflows or build a profit test.

1List the assumptions: term, premium, benefit, expense and commission timing, mortality, interest rate, reserves and risk discount rate.
2Set up a table with a column for each item for each year: premium, expenses, commission, interest, expected claims, reserve changes.
3For each year, take premium less expenses and commission at the start of the year. Add the opening reserve if reserves are held.
4Add interest on that amount for the year, using the stated interest rate.
5Subtract expected claims at the year end: q × benefit. Then subtract the reserve needed at year end for survivors: p × reserve.
6The result is Pr(t), the profit per policy in force at the start of year t.
7Multiply by the in-force probability to get Π(t). Include any time 0 amount without a survival factor.
8Discount at the risk discount rate to find the NPV. If asked, divide by the PV of premiums for the profit margin.

Quickest way: Row-by-row table method

When to use it: Use in exams when you must give a short numerical profit test over a few years and time is tight.

Write the four lines for each year: net start-of-year cash, interest, expected claim, reserve effect.
Compute net start-of-year cash once and reuse it. Renewal years usually repeat the same number.
Compute the in-force probabilities in one line before the table, so each signature is a single multiplication.
Calculate discount factors once at the risk discount rate and reuse them for NPV and PV of premiums.
Check sign and size: a negative first-year profit is normal with high initial costs and reserves. A very large profit suggests a missed claim or reserve term.

Common mistakes in Profit Testing Basics and Cashflow Projection

Forgetting to multiply the profit vector by the in-force probability.
The profit vector looks like a finished answer, so students discount it straight away.
Fix: Always do the extra step: Π(t) = (t−1)p × Pr(t). Say it in your working so the examiner sees it.
Applying interest to the wrong cashflows.
Claims are at year end, but students put interest on them too.
Fix: Earn interest only on money held from the start of the year: premium less expenses and commission, plus opening reserve. Claims and closing reserves are year-end items.
Paying the reserve out on top of the sum assured on death.
Students treat the reserve as an extra benefit.
Fix: Use the death benefit as the claim cost. The reserve on death is simply no longer needed, so only survivors need a closing reserve: p × tV.
Using the risk discount rate for investment interest, or the other way round.
Both rates are given and both are percentages.
Fix: Use the interest rate on assets to project cashflows and the risk discount rate only to discount signature entries. Label them in your assumptions.
Ignoring the sign convention for reserves.
Setting up a reserve feels like income, but it is cash held back.
Fix: A rising reserve reduces profit. A falling reserve increases profit. Check the sign of the reserve term each year.
Applying a survival probability to time 0 expenses.
Students multiply every entry in the signature by a probability by habit.
Fix: At time 0 the policy is certainly issued, so the probability is 1. Only later entries carry (t−1)p.

Worked examples

Example 1

A 3-year term assurance pays ₹10,00,000 at the end of the year of death. The premium is ₹5,000 at the start of each year. Initial expenses are ₹1,500 at the start of year 1. Renewal expenses are ₹100 at the start of years 2 and 3. Commission is 30% of the first premium and 5% of each renewal premium. Interest is 6% a year. Mortality rates in years 1, 2 and 3 are 0.002, 0.003 and 0.004. Assume zero reserves. Find the profit vector, the profit signature and the NPV at a risk discount rate of 10%.

Show the solution

Year 1: premium ₹5,000. Expenses ₹1,500 and commission 30% × 5,000 = ₹1,500. Net start cash = 5,000 − 3,000 = ₹2,000.
Year 1 interest: 2,000 × 0.06 = ₹120. Total = ₹2,120. Expected claim = 0.002 × 10,00,000 = ₹2,000. Pr(1) = 2,120 − 2,000 = ₹120.
Year 2: premium ₹5,000, expenses ₹100, commission 5% × 5,000 = ₹250. Net = ₹4,650. With interest: 4,650 × 1.06 = ₹4,929.
Year 2 claim = 0.003 × 10,00,000 = ₹3,000. Pr(2) = 4,929 − 3,000 = ₹1,929.
Year 3: net is again ₹4,650, so 4,650 × 1.06 = ₹4,929. Claim = 0.004 × 10,00,000 = ₹4,000. Pr(3) = ₹929.
In-force probabilities at the start of years 1, 2, 3 are 1, 0.998 and 0.998 × 0.997 = 0.995006.
Profit signature: Π(1) = ₹120. Π(2) = 0.998 × 1,929 = ₹1,925.14. Π(3) = 0.995006 × 929 = ₹924.36.
NPV = 120 ÷ 1.1 + 1,925.14 ÷ 1.21 + 924.36 ÷ 1.331 = 109.09 + 1,591.03 + 694.49 = ₹2,394.60.

Answer: Profit vector: ₹120, ₹1,929, ₹929. Profit signature: ₹120, ₹1,925.14, ₹924.36. NPV at 10% is about ₹2,394.60 per policy sold.

Example 2

For a policy in force at the start of year 2, the opening reserve is ₹4,000. The premium is ₹8,000 at the start of the year, expenses are ₹300 and commission is ₹400, all at the start of the year. Interest is 5%. The death benefit is ₹2,00,000 at the end of the year. The mortality rate is 0.004. The year-end reserve for each survivor is ₹12,500. The probability that the policy is in force at the start of year 2 is 0.99. Find the profit vector entry and the profit signature entry for year 2.

Show the solution

Start-of-year cash: 4,000 + 8,000 − 300 − 400 = ₹11,300.
Add interest: 11,300 × 1.05 = ₹11,865.
Expected death cost: 0.004 × 2,00,000 = ₹800.
Reserve needed for survivors at year end: 0.996 × 12,500 = ₹12,450.
Pr(2) = 11,865 − 800 − 12,450 = −₹1,385.
Π(2) = 0.99 × (−1,385) = −₹1,371.15.

Answer: The profit vector entry is −₹1,385 and the profit signature entry is −₹1,371.15. The loss comes from building the reserve faster than the margin in the premium releases cash.

Exam tips

Write the assumptions list and timing of every cashflow before you start. Marks are given for stating them even when the numbers go wrong.
Keep unrounded figures through the table and round only at the end. The NPV is sensitive to rounding in the in-force probabilities.
Read whether the question asks per policy sold or per policy in force. The first needs the signature and the second needs the profit vector.
Comment briefly on the result. A negative year 1 followed by positive years is a typical new business strain pattern, and examiners reward a one-line interpretation.
In the computer-based paper, set up the table with formulas that reference the assumption cells so you can change the interest rate or mortality and see the NPV update.

Practice questions from Projecting expected future cashflows and profit testing

Profit Testing Basics and Cashflow Projection: frequently asked questions

What is profit testing in actuarial science?

It is a way of projecting the expected cashflows of an insurance policy and then finding the profit the insurer expects from it. You use assumptions for mortality, interest, expenses and reserves. The output is a profit vector, profit signature and measures such as NPV and profit margin.

What is the difference between the profit vector and the profit signature?

The profit vector is the profit in each year per policy in force at the start of that year. The profit signature multiplies each entry by the probability that the policy is still in force at that time. The signature is what you discount to get the NPV.

Why do reserves affect profit?

Reserves are money the insurer must hold back to meet future claims. The increase in reserves in a year is cash that cannot be paid to shareholders, so it reduces profit. A fall in reserves releases cash and increases profit.

Which interest rate do I use to discount profits?

Use the risk discount rate given in the question. It reflects the return the insurer requires on the capital it commits. The interest rate on investments is used only to project the interest earned on the cashflows.