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Actuarial Mathematics for Modelling · Gross premiums and reserves

Gross Premium Calculation with Expenses and Commission

Updated 11 October 2026 · Fact-checked

A gross premium is the premium the insurer actually charges. It is found by the equivalence principle: the expected present value of premiums equals the expected present value of benefits plus initial, renewal and claim expenses, commission and any profit margin. Write the equation, put G on both sides, then solve for G.

Understand Gross Premium Principles and Expense Loadings

A net premium pays for the benefits only. It ignores the cost of running the business. A gross premium is the amount the policyholder really pays. It must cover the benefits, the expenses, the commission paid to the agent, and a margin for profit or contingencies.

The usual method is the equivalence principle. You set the expected present value (EPV) of all premiums equal to the EPV of all outgo. Outgo means benefits plus expenses plus commission plus any profit loading. You value everything at the policy start date, using the pricing basis given in the question (mortality, interest, expenses).

Expenses come in three main kinds. Initial expenses occur once at the start, such as underwriting and issue costs. Renewal expenses occur each year the policy stays in force, usually at the start of each year from year 2. Claim expenses occur when a claim is paid, so they are valued like a benefit. Each can be a fixed rupee amount or a percentage of premium or sum assured.

Commission is normally a percentage of premium, often higher in year 1 than later. Because it depends on G, G appears on both sides of the equation. Collect all the G terms on one side and solve. The gap between G and the net premium is the expense loading.

The key skill is timing. Premiums and most expenses are paid at the start of the year, so you value them with an annuity-due. Benefits and claim expenses are paid on death, so you value them with an assurance. Match each item to the right function.

Key rules to remember

Equivalence principle (gross premium)
EPV(premiums) = EPV(benefits) + EPV(expenses) + EPV(commission) + EPV(profit loading)
Valued at the policy start on the pricing basis given. Only use a profit loading if the question states one.
Net premium
P = S × A ÷ ä
For sum assured S, assurance value A and premium annuity-due ä over the premium term. Benefits only, no expenses.
Gross premium with expenses as a percentage of premium
G × ä = S × A + I + F × ä(renewal) + c₁G + c₂G × (ä − 1) + eG × ä
Illustrative layout for a level-premium policy with premiums for the whole term. I is the initial expense, F the annual renewal expense, c₁ and c₂ the first-year and renewal commission rates, e any percentage-of-premium expense or profit rate. Adjust to the question.
Renewal items from year 2
EPV = amount × (ä − 1)
Use when the item is paid at the start of years 2, 3, … while premiums are payable. Here ä is the premium-paying annuity-due.
Claim expense
EPV = claim expense × A
Value it as part of the death benefit, using the same assurance function as the benefit.
Solving for G
G = (EPV of benefits + fixed expenses) ÷ (ä − first-year and renewal % items)
Collect every term containing G on one side, then divide.

How to solve Gross Premium Principles and Expense Loadings questions

Use this routine for any gross premium question. It keeps the timing and the G terms organised.

  1. 1List the benefits, the premium term and payment pattern, and the pricing basis (mortality, interest).
  2. 2List every expense and commission item. For each, note the amount, whether it is a fixed sum or a percentage, and when it is paid (start, each year from year 2, or on claim).
  3. 3Write the EPV of premiums: G × ä over the premium-paying term.
  4. 4Write the EPV of outgo item by item. Benefits use assurance functions. Items at the start of years 2 onwards use (ä − 1). Claim expenses use the assurance function.
  5. 5Put percentage-of-premium items as a rate times G times the correct annuity factor. Add any profit loading in the same way.
  6. 6Set income equal to outgo. Move all G terms to the left side and fixed amounts to the right.
  7. 7Solve for G. Check that G is above the net premium and round only at the end.
  8. 8If asked, state the expense loading as G minus the net premium.

Quickest way: Split into G terms and fixed terms

When to use it: Use this for any level-premium question with several expense and commission items under time pressure.

  1. Make two columns: terms that contain G, and terms that do not.
  2. Fixed column: benefit EPV, initial expense, renewal expense × (ä − 1), claim expense × A.
  3. G column: ä on the premium side, then subtract c₁ for first-year commission, c₂(ä − 1) for renewal commission, and e × ä for any other percentage item.
  4. Compute the coefficient of G first: ä − c₁ − c₂(ä − 1) − eä.
  5. Divide the fixed total by that coefficient. Check G exceeds the net premium S × A ÷ ä.

Common mistakes in Gross Premium Principles and Expense Loadings

  • Valuing renewal expenses with ä instead of (ä − 1) when they start in year 2.

    Students value every annual item with the same annuity factor.

    Fix: Read the timing. If the item starts at the beginning of year 2, use ä − 1. Only an item paid from time 0 uses ä.

  • Treating a percentage-of-premium expense as a fixed rupee amount, or leaving G out of it.

    The expense is written beside the other costs and the G is forgotten.

    Fix: Any rate of premium must be written as rate × G × the annuity factor. Put it in the G column.

  • Adding claim expenses to the annuity side or valuing them as annual expenses.

    Students see 'expense' and use the renewal pattern.

    Fix: Claim expenses are paid on death. Value them as (claim expense) × A, or add them to the sum assured in the assurance term.

  • Forgetting the first-year commission rate applies only to the first premium.

    The commission rate is applied to the whole annuity.

    Fix: First-year commission is c₁ × G. Later commission is c₂ × G × (ä − 1).

  • Subtracting profit or expenses from the net premium instead of solving the equation.

    Students try a shortcut loading on P, which is wrong when expenses are percentages of G.

    Fix: Always write the full equation. Then solve for G.

  • Rounding annuity and assurance values early.

    Values are copied to two decimal places mid-calculation.

    Fix: Keep full values given in the question and round G only at the end.

Worked examples

Example 1

A 20-year endowment assurance with sum assured ₹10,00,000 is issued with level annual premiums payable at the start of each year for 20 years. On the pricing basis, A(x:20) = 0.5192 and ä(x:20) = 12.50. Expenses: initial ₹2,500; renewal ₹300 at the start of each of years 2 to 20. Commission: 30% of the first premium and 4% of each later premium. Find the gross premium using the equivalence principle.

Show the solution
  1. Income: G × ä = 12.5G.
  2. Benefit EPV: 10,00,000 × 0.5192 = ₹5,19,200.
  3. Initial expense: ₹2,500.
  4. Renewal expense: 300 × (12.5 − 1) = 300 × 11.5 = ₹3,450.
  5. Commission: 0.30G + 0.04G × 11.5 = 0.30G + 0.46G = 0.76G.
  6. Equation: 12.5G = 5,19,200 + 2,500 + 3,450 + 0.76G = 5,25,150 + 0.76G.
  7. So (12.5 − 0.76)G = 11.74G = 5,25,150.
  8. G = 5,25,150 ÷ 11.74 = ₹44,731.7 (approx.).
  9. Check: net premium = 5,19,200 ÷ 12.5 = ₹41,536, so G is higher. The loading is about ₹3,196.

Answer: Gross premium ≈ ₹44,732 per year.

Example 2

A 10-year term assurance has sum assured ₹5,00,000 payable at the end of the year of death, with level annual premiums at the start of each year for 10 years. On the pricing basis, A¹(x:10) = 0.0300 and ä(x:10) = 8.20. Expenses: initial ₹1,500; 25% of the first premium; 3% of each later premium; a claim expense of ₹2,000 on each death claim. A profit margin of 5% of every premium is also included. Find the gross premium.

Show the solution
  1. Income: 8.2G.
  2. Benefit plus claim expense: (5,00,000 + 2,000) × 0.0300 = 5,02,000 × 0.03 = ₹15,060.
  3. Initial fixed expense: ₹1,500. Fixed total = 15,060 + 1,500 = ₹16,560.
  4. First-year percentage expense: 0.25G.
  5. Later percentage expense: 0.03G × (8.2 − 1) = 0.03G × 7.2 = 0.216G.
  6. Profit margin: 0.05G × 8.2 = 0.41G.
  7. G terms total: 0.25 + 0.216 + 0.41 = 0.876G.
  8. Equation: 8.2G = 16,560 + 0.876G, so 7.324G = 16,560.
  9. G = 16,560 ÷ 7.324 = ₹2,261.15 (approx.).

Answer: Gross premium ≈ ₹2,261 per year.

Exam tips

  • Write the full equation of value before substituting numbers. Examiners give marks for the correct structure even if the arithmetic slips.
  • Underline the timing words in the question: 'at the start of each year', 'on death', 'from the second year'. They decide whether you use ä, ä − 1 or A.
  • Check that your gross premium is higher than the net premium from the same basis. If not, you have an error.
  • In written answers, state assumptions, such as premiums payable annually in advance and claims paid at the end of the year of death, if the question is not clear.
  • In the computer-based paper, set up cells for each EPV item and build G from the fixed and G-dependent parts, so you can change a rate quickly.

Practice questions from Gross premiums and reserves

Gross Premium Principles and Expense Loadings in other exams

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

Gross Premium Principles and Expense Loadings: frequently asked questions

What is the difference between a net premium and a gross premium?

A net premium covers only the expected cost of benefits. A gross premium also covers expenses, commission and any profit or contingency margin. The gross premium is what the policyholder pays.

Why does G appear on both sides of the gross premium equation?

Commission and many expenses are a percentage of the premium, so they depend on G. You collect the G terms on one side and solve for G.

Do I use ä or ä − 1 for renewal expenses?

It depends on when they start. If renewal expenses are paid at the start of each year from year 2, use ä − 1 over the premium term. If they start at time 0, use ä.

How do I treat claim expenses?

Claim expenses are incurred when a claim is paid. Value them like a death benefit, using the assurance function, not an annuity.

Does the equivalence principle include a profit margin?

Only if the question says so. A profit margin can be added as an extra outgo item, often as a percentage of premium, or built into the basis by using cautious assumptions.