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Higher Applications of Mathematics

Insurance

Premiums, excesses, risk and policy decisions.

Formula referenceFinance mixed quiz

Formula reference

Check the key Finance formulae and when to use each one.

Mixed revision quiz

Review the whole Finance section with adaptive feedback.

Before you start

  • Calculate percentages and probabilities as decimals.
  • Understand expected value as a long-run average.
  • Know the meanings of premium, excess and payout.
  • Compare financial and non-financial factors.

Method chooser

Which method do I use?

Are you finding interest on the original amount only?

Use simple interest.

Is the amount increasing by a percentage repeatedly?

Use compound interest.

Are you finding what future money is worth today?

Use present value.

Are you finding what today's money will be worth later?

Use future value.

Are there regular payments, deposits or withdrawals?

Use a payment timeline.

Are you tracking a loan balance after repayments?

Use a repayment schedule.

Are you comparing insurance policies or risk?

Use expected value and excess comparisons.

Are you working with gross pay, deductions and take-home pay?

Use net pay.

Are you comparing prices over time with inflation?

Use inflation / real value.

Finance lesson

Key idea

  • Insurance transfers some financial risk from a person to an insurer. The customer pays a premium. If an insured event happens, the insurer may pay out, often after the customer pays an excess.
  • Higher Applications questions may ask you to compare policies using expected value, risk, excess, probability and non-financial factors.
  1. Identify premium, excess, payout and claim probability.
  2. Calculate expected claim or expected excess where relevant.
  3. Add fixed costs such as premium.
  4. Compare policies using expected cost.
  5. State a limitation, such as risk tolerance or cover exclusions.

Key formulae

  • Expected value = sum of each outcome value x its probability
  • Expected claim cost to insurer = probability of claim x payout
  • Customer cost after a claim = premium + excess + any uninsured loss

Worked examples

Worked example 1

Comparing expected claim value

A phone worth £900 has a 6% chance of being damaged in a year. If damaged, repair costs £300.

  1. Expected repair cost = probability x cost
  2. Expected repair cost = 0.06 × 300 = 18

So: The expected repair cost is £18. This does not mean the actual repair will cost £18; it is a long-run average.

Worked example 2

Comparing two car insurance policies

Policy A has premium £640 and excess £250. Policy B has premium £720 and excess £100. A driver estimates a 20% chance of making one claim.

  1. Expected cost A = premium + probability x excess = 640 + 0.20 × 250 = 690
  2. Expected cost B = 720 + 0.20 × 100 = 740
  3. Compare expected costs and consider risk tolerance.

Final step: Policy A has the lower expected cost by £50, but Policy B has a lower cost if a claim actually happens.

Worked example 3

Expected value for travel insurance

A traveller estimates a 5% chance of needing a £600 medical claim. The policy premium is £42 and excess is £80.

  1. Expected excess cost = 0.05 × 80 = 4
  2. Expected customer cost = premium + expected excess
  3. Expected customer cost = 42 + 4 = 46

So: The expected customer cost is £46, ignoring uninsured losses.

Worked example 4

Comparing contents policies

Policy A costs £180 with £250 excess. Policy B costs £235 with £75 excess. Claim probability is estimated at 18%.

  1. A expected cost = 180 + 0.18 × 250 = 225.
  2. B expected cost = 235 + 0.18 × 75 = 248.50
  3. Compare and consider risk.

Final step: Policy A has the lower expected cost, but Policy B has a smaller excess if a claim happens.

Watch out

  • Treating expected value as a guaranteed result for one person.
  • Forgetting to include the excess when a claim is made.
  • Choosing only the cheapest premium without checking cover limits.
  • Ignoring non-financial factors such as legal cover, courtesy car, or exclusions.

Spreadsheet connection

Spreadsheet connection

Set up expected value and policy comparison tables for premiums, excesses and claim probabilities.

Open spreadsheet skill

Next step

Move into practice

Use the method notes to choose the correct financial model, then try varied rates, time periods, tables and decision contexts.

Finance mixed quiz