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

Pricing Using Profit Tests and Sensitivity Analysis

Updated 11 October 2026 · Fact-checked

Pricing with a profit test means finding the premium that makes the discounted profit meet a target, such as NPV equal to a set percentage of the present value of premiums. Project profit per policy, write it as a linear function of the premium, solve for it, then change assumptions one at a time to see the effect.

Understand Pricing Using Profit Tests and Sensitivity Analysis

A profit test projects the cashflows of a policy year by year on a set of best-estimate assumptions. You get a profit vector, then a profit signature, and then a discounted profit at the company's risk discount rate. Until now you may have been given the premium and asked for the profit. In pricing you reverse this.

You are given a profit criterion. Examples: NPV = 0, NPV = 5% of the present value of premiums, or an IRR of at least 12%. You then find the premium that just meets it. The key fact is that every profit item is a straight-line function of the premium. If you double the premium, the premium-related terms double and the claim and fixed-expense terms stay the same. So NPV = a × P − b, where a and b are numbers you calculate. Setting that equal to the target gives P directly, with no trial and error.

If the criterion is an IRR, the premium is no longer linear in the answer. Calculate the NPV at the target IRR used as the discount rate, and set it to zero. That is again a linear equation in P.

Sensitivity analysis asks how much the result changes if an assumption is wrong. You change one assumption at a time (mortality, lapses, interest, expenses, risk discount rate), keep the premium fixed, and recalculate the NPV or profit margin. A large change in profit tells you the product is exposed to that assumption. Lapses are a good example. A policy with heavy first-year expenses often makes a loss in year 1, so a lapse removes the later profits that were meant to repay that loss. Higher lapses therefore usually reduce NPV. But this depends on the surrender value and reserves. If lapses release reserves or a surrender value is below the reserve, the effect can be the opposite, so always calculate it.

Key rules to remember

Profit in year t (per policy in force at start of year t): Pr_t = (V_(t-1) + P_t − e_t) × (1 + i) − q_(x+t-1) × (S_t + E_t) − p_(x+t-1) × V_t − (surrender cost, if lapses are modelled); Premium P_t and expense e_t are paid at the start of the year, claims at the end. S_t is the sum assured, E_t any claim expense, V the reserve per policy in force. Add the cost of lapses if modelled.
Profit signature: Π_t = (t−1)p_x × Pr_t; This converts profit per policy in force into profit per policy issued. Include lapses in the survival probability if you model them: use the probability of being in force at the start of year t.
Net present value: NPV = Σ Π_t × v_r^t, where v_r = 1 ÷ (1 + r); r is the risk discount rate. Note that the interest rate used in the projection is a different rate.
Profit margin: Profit margin = NPV ÷ PV of premiums; The PV of premiums is per policy issued, discounted at the risk discount rate, and includes survival (and lapse) probabilities.
Premium for a target profit margin m: Set a × P − b = m × (PV of premiums per unit of P) × P, so P = b ÷ (a − m × A); Here NPV = a × P − b and PV of premiums = A × P. For m = 0 the premium is b ÷ a.
Sensitivity: Change in NPV = NPV(new assumption) − NPV(base); Change one assumption at a time and keep the premium fixed at the priced value.

How to solve Pricing Using Profit Tests and Sensitivity Analysis questions

Use this method for any question that asks for a premium from a profit criterion or for the effect of changing an assumption.

1Write down the basis: interest, mortality, expenses, lapses, reserves, the risk discount rate and the profit criterion. Check the timing of each item (start or end of year).
2Write Pr_t for each year with P as an unknown. Keep the P terms and the constant terms separate.
3Find the in-force probabilities (survival and lapse) and multiply to get Π_t, still with P unknown.
4Discount at the risk discount rate. Collect terms so that NPV = a × P − b.
5Find the PV of premiums as A × P, using the same in-force probabilities. Express the criterion in the same form, for example NPV = m × A × P.
6Solve the linear equation for P. Check it by substituting back to see that the criterion is met.
7For sensitivity, fix this P, change one assumption, recalculate the profit vector and the NPV, and compare it with the base case.
8State the direction and the size of the change, and give a reason for it: for example, which years make a loss and which make a profit.

Quickest way: Linear-in-P shortcut

When to use it: Use it when the criterion is NPV = 0 or a percentage of PV of premiums and you must find the premium in a timed question.

Calculate the profit signature in two columns: the coefficient of P and the constant term.
Discount each column at the risk discount rate and add up to get a and b.
Break-even premium is b ÷ a. For a margin m, P = b ÷ (a − m × A).
For sensitivity, only recalculate the columns that change. If only lapses change, the per-policy profit Pr_t stays the same and only the probabilities change.
Do a quick check: at the answer, NPV should be m × PV premiums.

Common mistakes in Pricing Using Profit Tests and Sensitivity Analysis

Using the risk discount rate to project the investment of cashflows, or the interest rate to discount profits.
Both are called interest rates and both are in the question.
Fix: Use the interest earned on assets in Pr_t and the risk discount rate only when discounting Π_t. Label them i and r in your working.
Forgetting to convert Pr_t into Π_t using the probability of being in force.
Pr_t looks like the answer once it has been calculated.
Fix: Always multiply by the probability of being in force at the start of year t. Include lapses if they are in the basis.
Taking the profit margin as NPV ÷ annual premium, or the PV of premiums without survival probabilities.
The premium is a single known figure in many earlier questions.
Fix: Discount the premium at the risk discount rate with in-force probabilities. Note that the premium at time 0 is always paid with probability 1.
Changing several assumptions together in a sensitivity test, or letting the premium change too.
Students try to give a realistic scenario and save time.
Fix: Change one assumption at a time and keep the premium at the priced value, unless the question says otherwise. Then the effect can be attributed to one cause.
Stating that higher lapses always reduce profit.
Students remember the typical new-business-strain case.
Fix: Work it out. Look at the sign of the future profits and at the surrender value against the reserve. Then state the result for that contract.
Leaving out reserve changes or the change in reserve basis when solving for the premium.
Reserves may depend on the premium (for example, gross premium reserves) and this is missed.
Fix: If the reserve depends on P, include that dependence in the P coefficient. If reserves are on a fixed basis, treat them as constants.

Worked examples

Example 1

A 3-year term assurance pays ₹1,00,000 at the end of the year of death. Level annual premium P is paid at the start of each year. Expenses are 20% of the first premium and 5% of each renewal premium, paid at the start of the year. Mortality is q = 0.01 each year, interest on assets is 5% a year, and there are no lapses and no reserves. The risk discount rate is 10% a year. Find the premium that gives a profit margin of 5% of the present value of premiums.

Show the solution

Profit per policy in force in year t: Pr_t = (P − e_t) × 1.05 − 0.01 × 1,00,000 = 1.05 × (1 − f_t) × P − 1,000, where f_t is the expense rate.
Year 1: Pr_1 = 0.84P − 1,000. Years 2 and 3: Pr_2 = Pr_3 = 0.9975P − 1,000.
In-force probabilities at the start of each year: 1, 0.99, 0.9801.
Π_1 = 0.84P − 1,000. Π_2 = 0.99 × Pr_2 = 0.987525P − 990. Π_3 = 0.9801 × Pr_3 = 0.97764975P − 980.1.
Discount at 10%. Coefficient of P: 0.84 ÷ 1.1 + 0.987525 ÷ 1.21 + 0.97764975 ÷ 1.331 = 0.763636 + 0.816136 + 0.734523 = 2.314295.
Constant term: 1,000 ÷ 1.1 + 990 ÷ 1.21 + 980.1 ÷ 1.331 = 909.09 + 818.18 + 736.36 = 2,463.64. So NPV = 2.314295P − 2,463.64.
PV of premiums = P × (1 + 0.99 ÷ 1.1 + 0.9801 ÷ 1.21) = P × (1 + 0.9 + 0.81) = 2.71P.
Set NPV = 0.05 × 2.71P = 0.1355P. Then (2.314295 − 0.1355)P = 2,463.64, so 2.178795P = 2,463.64.
P = 2,463.64 ÷ 2.178795 = 1,130.7.

Answer: The premium is about ₹1,131 a year. For reference, the break-even premium (NPV = 0) is about ₹1,065.

Example 2

A 2-year term assurance has sum assured ₹1,00,000 paid at the end of the year of death, and annual premium ₹1,200 paid at the start of each year. Expenses are ₹400 at the start of year 1 and ₹50 at the start of year 2. Mortality is 0.01 in each year, interest is 5%, reserves are zero, and the risk discount rate is 10%. Find the NPV with no lapses. Then find the NPV if 10% of surviving policyholders lapse at the end of year 1 and receive no surrender value. Comment on the effect of lapses.

Show the solution

Year 1 profit per policy in force: Pr_1 = (1,200 − 400) × 1.05 − 0.01 × 1,00,000 = 840 − 1,000 = −160.
Year 2 profit: Pr_2 = (1,200 − 50) × 1.05 − 1,000 = 1,207.50 − 1,000 = 207.50.
No lapses: Π_1 = −160. Π_2 = 0.99 × 207.50 = 205.425.
NPV = −160 ÷ 1.1 + 205.425 ÷ 1.21 = −145.45 + 169.77 = 24.32.
With lapses: Pr_1 and Pr_2 are unchanged, because the lapse happens at the end of year 1 with no payment. The probability of being in force at the start of year 2 is 0.99 × 0.9 = 0.891.
Π_2 = 0.891 × 207.50 = 184.8825. Discounted: 184.8825 ÷ 1.21 = 152.80.
NPV with lapses = −145.45 + 152.80 = 7.34.
Change in NPV = 7.34 − 24.32 = −16.98.

Answer: NPV with no lapses is about ₹24.32 and with lapses about ₹7.34. Lapses reduce NPV by about ₹16.98. Year 1 makes a loss because of the initial expense, and the profit that repays it comes in year 2. Lapses remove some of that year 2 profit, and no surrender value is paid to offset it.

Exam tips

Write the profit vector with P as a symbol first. A question that asks for a premium cannot be answered from numbers alone, and the marks are for a clear linear form.
Always label the two rates: interest on assets (i) and the risk discount rate (r). Examiners often check that they are used in the right place.
For sensitivity questions, show the base case and the changed case side by side, and give the numerical change and a one-line reason. A number without a comment loses marks.
In MCQs, check the criterion first (NPV = 0, or a percentage of PV premiums, or an IRR) and the timing of expenses. These are where wrong options are built.
In Paper B (computer-based), build the projection with premium as an input cell, so you can use goal seek or solve linearly. Keep the assumptions in separate cells so each sensitivity can be done by changing one cell.

Practice questions from Projecting expected future cashflows and profit testing

Pricing Using Profit Tests and Sensitivity Analysis in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

ACCA Applied SkillsSensitivity Analysis

Pricing Using Profit Tests and Sensitivity Analysis: frequently asked questions

How do I calculate a premium using a profit test?

Write the profit in each year as a function of the premium P and multiply by the probability of being in force. Discount at the risk discount rate to get NPV = a × P − b. Then set it equal to the target, such as zero or a percentage of the PV of premiums, and solve for P.

What is the effect of lapses on profit test results?

It depends on the contract. If early years make a loss (because of initial expenses) and later years make a profit, lapses usually reduce NPV since they remove the later profits. A surrender value below the reserve, or reserve releases, can change this, so you must recalculate.

How do I do a sensitivity analysis in a profit test?

Keep the premium fixed. Change one assumption at a time, such as mortality, lapses, interest, expenses or the risk discount rate. Recalculate the profit vector, the signature and the NPV or profit margin, and compare each with the base case.

Is the premium always linear in the profit criterion?

For NPV equal to zero or to a fixed percentage of the PV of premiums, yes, provided reserves do not depend on the premium in a non-linear way. For an IRR criterion, discount at the target IRR and set NPV to zero, which is also linear in the premium.