Real-World Mathematics: Insurance, Risk, Premiums, Expected Loss and Pooling

Application of Mathematics in Real-World Usage · Guide 28 · BTT Mathematics Hub

A loss with only a 1% chance can still matter if its severity is enormous. A portfolio of many independent risks can have a more stable average loss even while its total expected loss grows. A deductible changes expected insurer payment without changing the underlying event probability. Insurance Mathematics is built from probability distributions, payment functions and aggregation.

This page is educational only. It is not insurance, actuarial, financial, legal or purchasing advice. Every premium, claim, probability and policy term below is fictional. Real pricing includes regulation, expenses, capital, reinsurance, taxes, underwriting and dependence structures that these teaching models omit.

Expected loss is probability-weighted loss

A fictional risk has 90% chance of $0 loss and 10% chance of $1,000 loss. Expected loss is 0.9×0+0.1×1,000=$100.

The expected value is not the most likely realised loss. Here the most likely loss is zero, while the long-run mean across many comparable exposures is $100.

Several outcomes require summing probability-weighted values

Suppose losses are $0 with probability 0.8, $500 with 0.15 and $2,000 with 0.05. Expected loss is 0+75+100=$175.

The probabilities sum to one, so the distribution is complete under the teaching model.

Frequency and severity can be separated

If expected claim frequency is 0.2 claims per exposure-year and mean claim severity is $800, a simple frequency-severity model gives expected annual loss 0.2×800=$160.

This factorisation depends on the model. If claim count and severity are strongly related, conditioning may be needed.

Expected claim cost is not the same as the final premium

Suppose expected covered loss is $160. Charging exactly $160 would cover claims only in expectation, with no allowance for expenses, capital, uncertainty or profit.

Real premium terminology varies. This page uses the decomposition only to teach the Mathematics of separating expected loss from other components.

A deductible changes the payment function

With deductible d, insurer payment for loss X in a simple per-loss model is max(X−d,0).

If d=$200 and loss is $150, payment is $0. If loss is $700, payment is $500. If loss is $2,000, payment is $1,800.

Expected payment after a deductible transforms every outcome

Using losses $0, $500 and $2,000 with probabilities 0.8, 0.15 and 0.05, and deductible $200, the corresponding insurer payments are $0, $300 and $1,800.

Expected payment is 0.15×300+0.05×1,800=45+90=$135. The deductible reduces expected insurer payment from $175 to $135 under the model.

A policy limit caps payment

With a $1,000 limit and no deductible, payment is min(X,1,000). A $2,000 loss produces a $1,000 insurer payment under the simplified contract.

With deductible $200 and payment limit $1,000 applied after the deductible, the payment function can be written min(max(X−200,0),1,000).

Linearity of expectation survives aggregation

If each of 100 policies has expected covered loss $160, expected aggregate loss is 100×160=$16,000.

Independence is not required for adding expected values. It becomes crucial when we study variance and tail behaviour.

Variance measures spread around expected loss

For the binary loss $0 with 90% probability and $1,000 with 10%, mean is $100. E[X²]=0.1×1,000,000=100,000.

Variance is 100,000−100²=90,000 dollars², so standard deviation is $300.

Independent pooling stabilises average loss

For n independent identical risks with individual SD $300, aggregate SD is 300√n. Average loss per policy has SD 300/√n.

At n=100, average-loss SD is $30. At n=10,000 it is $3. The average becomes more stable even though total expected claims increase in proportion to portfolio size.

Correlation weakens diversification

For two equal-variance risks with variance σ² and correlation ρ, variance of their sum is 2σ²+2ρσ²=2σ²(1+ρ).

If ρ=0, variance is 2σ². If ρ=1, variance is 4σ². Perfect positive dependence removes the diversification benefit between the pair.

Common shocks create dependence

Many properties can be affected by the same storm, flood or earthquake. Treating all policies as independent can understate aggregate variability.

Insurance Mathematics therefore cares not only about each risk’s individual distribution but also about shared drivers and dependence.

Loss ratio compares claims with premium on a defined basis

Suppose fictional earned premium is $100,000 and incurred claims are $65,000. Simple loss ratio is 65%.

This does not imply a 35% profit margin. Expenses, commissions, capital costs and other items are outside this ratio.

Expense ratio and combined ratio use additional components

If underwriting expenses are $25,000 on $100,000 earned premium, expense ratio is 25%. A simplified combined ratio is 65%+25%=90%.

Under this limited definition, a combined ratio below 100% means included underwriting inflows exceed included claims and expenses. It is not a complete measure of company profitability.

A loading is an additional modelled component

Suppose expected loss is $160 and a fictional rule adds 20% of expected loss as a combined loading. The indicated amount is 160×1.2=$192.

This is not an actuarial pricing standard. It simply demonstrates multiplication by a defined loading factor.

Percentiles answer tail questions the mean cannot

A portfolio may have expected annual loss $1 million and a 95th-percentile loss of $1.8 million under a model. The percentile is a distribution threshold, not an extra expected loss.

Risk management often needs both centre and tail summaries because two distributions can have the same mean and very different extreme behaviour.

Value at Risk is a quantile, not the worst possible loss

In a teaching distribution, 95% VaR of $1.8 million means the 95th percentile is $1.8 million under the chosen convention. Losses can still exceed that value.

VaR does not state the average loss beyond the threshold.

Expected shortfall summarises the tail beyond a threshold

Suppose the worst 5% of modelled outcomes have average loss $2.4 million. A 95% expected-shortfall-style quantity is $2.4 million under that simplified definition.

Tail metrics require precise conventions, especially for discrete distributions and ties at the quantile boundary.

Exposure volume affects statistical credibility

A group with 10 exposure-years and one claim has observed frequency 0.10. Another with 10,000 exposure-years and 1,000 claims also has frequency 0.10.

The point estimates match, but the second dataset usually contains much more information about the underlying frequency if the exposures are comparable.

Selection effects can bias observed claim rates

If higher-risk customers are more likely to buy a particular contract, the insured population may not represent the general population.

Comparing claim frequency across products without accounting for who selected each product can confuse portfolio composition with contract effect.

Severity can change while frequency stays stable

If average claim severity rises from $1,000 to $1,080 while frequency remains 0.2, expected loss rises from $200 to $216, an 8% increase.

Separating frequency from severity helps identify which component changed.

A complete insurance calculation states the payment function and dependence assumptions

State the loss distribution, deductible, limit, exposure period and dependence assumptions. Separate expected loss from premium, and total portfolio risk from average risk per policy.

The return path is: event distribution → contractual payment transformation → aggregate distribution → decision metric. Skipping a stage can make a familiar percentage refer to the wrong object.

Practice: twenty insurance Mathematics questions

1. Loss $0 with 90% and $1,000 with 10%. Find expected loss.
2. For losses $0, $500 and $2,000 with probabilities .8, .15 and .05, find expected loss.
3. Frequency .2 and severity $800. Find expected loss.
4. Deductible $200 and loss $150. Find payment.
5. Deductible $200 and loss $700. Find payment.
6. Using the distribution in question 2 and deductible $200, find expected payment.
7. With limit $1,000 and loss $2,000, find payment before any deductible.
8. 100 policies each have expected loss $160. Find aggregate expectation.
9. For the binary loss in question 1, find variance.
10. Find standard deviation.
11. For 100 independent identical risks with individual SD $300, find aggregate SD.
12. Find SD of average loss per policy.
13. Earned premium $100k and claims $65k. Find loss ratio.
14. Expenses $25k on premium $100k. Find expense ratio.
15. Find simplified combined ratio.
16. Expected loss $160 with 20% loading. Find indicated amount.
17. Frequency .2 and severity rises from $1,000 to $1,080. Find old expected loss.
18. Find new expected loss.
19. For two equal-variance risks with correlation 1, compare sum variance with correlation 0.
20. Why can pooling reduce variability of average loss without reducing expected loss per policy?

Worked answers

1. $100.
2. $175.
3. $160.
4. $0.
5. $500.
6. $135.
7. $1,000.
8. $16,000.
9. 90,000 dollars².
10. $300.
11. $3,000.
12. $30.
13. 65%.
14. 25%.
15. 90%.
16. $192.
17. $200.
18. $216.
19. 4σ² with perfect positive correlation versus 2σ² when independent.
20. Because expected value averages linearly while, under independence, variance of the average falls with 1/n.

Sources and connected applications

For foundational actuarial probability and loss-model context, see the Society of Actuaries Probability exam syllabus. All policies, losses and pricing quantities on this page are fictional teaching examples.

Continue with Healthcare Screening, Diagnostic Tests, Risk and Rates; Queues, Waiting Times, Capacity and Service Systems; and Elections, Voting, Apportionment and Ranking. Return to the BTT Mathematics Hub.