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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling

Gross Premiums and Reserves for CM1

A gross premium covers expected benefits plus expenses, and a reserve is the expected present value of future outgo minus future income on a stated valuation basis. You solve both with the equation of value: write future cash flows, discount them with survival probabilities, and solve for the unknown premium or reserve.

What this chapter covers

This chapter takes the net premium ideas from earlier CM1 work and makes them realistic. A net premium ignores expenses and uses a simple basis. A gross premium adds initial and renewal expenses, commission and sometimes profit. A reserve (policy value) is what the insurer must hold at a given time to meet future liabilities, given what it expects to receive.

The tools are the same throughout. You build an equation of value for a policy: present value of future premiums on one side, present value of future benefits and expenses on the other. You then change the time point (at issue for premiums, at time t for reserves) and the basis (pricing basis or valuation basis). Net premium reserves and gross premium reserves differ only in which premium and which expenses you include.

The chapter then moves from formulas at discrete times to recursion and to continuous time with Thiele's differential equation. It ends with bonuses, surrender values and profit analysis. These link back to the interest rate and decrement topics and forward to pricing and profit testing questions that tend to be long written questions. Pricing and reserving is the largest part of the 2026 CM1 syllabus, so this chapter sits in the heaviest area.

Pricing and reserving carries the largest syllabus weighting in CM1 (35%), and this chapter is the core of it. Questions are often multi-part written questions where early parts feed later ones, so a clean method earns method marks even if one number slips. The same skills also appear in short MCQs and in Paper B computer work, where you implement recursions or solve Thiele's equation numerically. Once you can set up an equation of value under pressure, many other CM1 questions become easier.

Gross premiums and reserves: topics in the order to study them

  1. 1Gross Premium Principles and Expense LoadingsStart here because it defines the premium you are valuing and the expense and profit terms every later topic reuses.
  2. 2Gross Premium Policy Value CalculationsNext, apply the same equation of value at time t, using the gross premium and expenses on a stated basis.
  3. 3Net Premium Reserves and Valuation BasesThen compare with net premium reserves, where the valuation basis differs from the pricing basis and expenses are ignored.
  4. 4Recursive Reserves and Thiele's Differential EquationOnce reserve definitions are clear, learn how reserves move from one year to the next, then in continuous time.
  5. 5Reserves with Bonuses, Surrender Values and Profit AnalysisFinish with with-profits features and profit analysis, which use everything above and are the hardest to assemble.

How to prepare Gross premiums and reserves

Work from the equation of value outward. Every topic in this chapter is a variation on it, so method practice matters more than memorising formulas.

  1. Write the equation of value for a simple policy (term, endowment, whole life) in your own words and notation, for premiums at issue, until you can do it without notes.
  2. Practise gross premium questions with explicit lists of the initial, renewal and claim expenses. Tick each item off as you put it in the equation.
  3. Recalculate the same policy's reserve at several durations on both a gross premium basis and a net premium basis. Note exactly which terms drop out.
  4. Derive the recursion for a reserve from one year to the next and use it to check a prospective calculation. Then derive Thiele's equation from a small time step, and solve a simple case by Euler's method.
  5. Do past-style written questions on bonuses, surrender values and profit analysis. Define the basis and notation at the top before any numbers.
  6. Finish by repeating a few full written questions under time. Check the units, the time point and the basis in each part before you move on.

Common mistakes in Gross premiums and reserves

  • Using the pricing basis to value the reserve when the question gives a separate valuation basis.

    Fix: Underline the basis given for each part and rewrite it at the top of your working before you calculate.

  • Leaving expenses out of a gross premium reserve, or including them in a net premium reserve.

    Fix: Keep a one-line rule: net ignores expenses and uses the net premium; gross includes expenses and uses the gross premium.

  • Getting the timing wrong in the recursion, for example mixing start-of-year and end-of-year cash flows.

    Fix: Draw a one-year timeline for each question and mark premium, expense, interest and claim dates before writing the recursion.

  • Using the total sum assured instead of the sum at risk in Thiele's equation.

    Fix: Write the death term as μ × (S − tV) and say in words that the reserve released on death reduces the extra cost.

  • Skipping the boundary condition, so the reserve is not anchored at the policy end.

    Fix: State the terminal value first (maturity benefit or zero), then work backwards.

  • Giving a final number with no stated assumptions in written answers on bonuses and profit analysis.

    Fix: List your assumptions in one or two lines, then show each step so method marks are available even if the final value is off.

Last-day revision: Gross premiums and reserves

  • Gross premium: solve PV of premiums = PV of benefits + PV of expenses (+ profit loading) at issue.
  • A reserve is PV of future outgo minus PV of future income, at time t, on a stated basis.
  • Prospective and retrospective reserves agree only when the same basis is used for both and the premium is consistent with that basis.
  • Net premium reserve ignores expenses; gross premium reserve includes them and uses the gross premium.
  • State the valuation basis: mortality, interest, expenses. Do not mix it with the pricing basis.
  • Recursion: (tV + P − e)(1 + i) = q × (benefit) + p × (t+1)V, for one-year steps with payments at the start of the year and claims at the end.
  • Thiele: d/dt (tV) = δ × tV + P − e − μ(x+t) × (S − tV), with S the sum assured and δ the force of interest.
  • The term μ × (S − tV) is the cost of the death strain, called the sum at risk times the force of mortality.
  • Check boundary conditions: reserve at maturity equals the maturity benefit, and for a term policy it is zero.
  • Profit analysis compares expected and actual cash flows each year, from the reserve change, interest and decrements.
  • Surrender value should not exceed the reserve held if the insurer is to avoid a loss on surrender.
  • Always write the time point, basis and premium timing before you start the arithmetic.

Gross premiums and reserves practice questions

Gross premiums and reserves in other exams

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

Gross premiums and reserves: frequently asked questions

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

A net premium covers only the expected cost of benefits on a given basis. A gross premium also covers expenses, commission and any profit loading. In CM1, always check which one the question wants and which basis to use.

Is a policy value the same as a reserve?

In CM1 the two terms are used closely together. A policy value is the value of a policy at a given time on a stated basis, and a reserve is the amount held. They can differ if the insurer holds more than the policy value, so read the wording of the question.

Do I need to memorise Thiele's differential equation?

You should know it and be able to derive it from a short time step, so you can rebuild it under pressure. Practise applying it to term, endowment and whole life policies, and to solving it numerically.

How is this chapter tested in Paper A and Paper B?

Paper A tests set-up, derivation and written explanation of premiums and reserves, usually in multi-part questions. Paper B is computer-based and tests whether you can implement recursions or numerical solutions accurately. Practise both styles.

Which topic should I revise first if time is short?

Revise the equation of value for gross premiums and policy values first, because every other topic depends on it. Then practise the recursion and Thiele's equation, since they are easy to lose marks on through small timing errors.