Loss Distributions, With and Without Risk Sharing

What this chapter covers

Loss distributions, with and without risk sharing: topics in the order to study them

1. 1Loss Distributions and Their PropertiesEverything else uses these distributions, their moments and tail behaviour, so you learn them first.
2. 2Fitting Loss Distributions: Method of Moments and MLEOnce you know the distributions, you learn how to estimate their parameters from claim data.
3. 3Deductibles, Policy Limits and Excess of LossThis is your first risk-sharing case. It needs only one fitted distribution and a truncated or capped variable.
4. 4Proportional and Excess of Loss ReinsuranceIt builds on the deductible and limit ideas and adds the insurer and reinsurer split, so it follows them.
5. 5Effect of Inflation on Loss DistributionsIt comes last because it changes the effect of fixed deductibles and retentions you have already studied.

How to prepare Loss distributions, with and without risk sharing

1. Make a one-page sheet for each distribution: density, mean, variance, and whether the tail is light or heavy. Check it against the Tables.
2. Practise method of moments and MLE for the exponential, gamma, Pareto and lognormal. Write the likelihood, take logs, differentiate and solve. Check that you have a maximum.
3. For every risk-sharing arrangement, write the payment variable first. For example, the insurer pays Y = max(X − d, 0) with a deductible d. Then find E[Y] using the survival function or a direct integral.
4. Do the same exercise for the reinsurer and the insurer. Check that the two shares add back to X.
5. Redo a deductible question after inflation. Scale X by (1 + i), find the new parameters, and compare the answer with the old one.
6. Practise in R: fit a distribution with a built-in optimiser, read the parameter estimates, and compare fitted and empirical values. Write what each line of code does.
7. Finish with past written questions under timed conditions. Show your formula and your assumptions in each answer.

Common mistakes in Loss distributions, with and without risk sharing

• Mixing up payment per loss and payment per payment.

  Fix: Write down which one the question asks for. Per payment is the per-loss mean divided by P(X > d).

• Adding the deductible back, or forgetting to subtract it, when finding the insurer's payment.

  Fix: Define Y = max(X − d, 0) first. Then integrate the right range with the right limits.

• Assuming inflation raises a claim with a deductible by the same percentage as the ground-up claim.

  Fix: Inflate X first, then apply the deductible to the inflated claim. Compare the new payment with the old one.

• Treating a proportional share and an excess of loss share as if they behave the same way.

  Fix: Proportional cover scales the distribution. Excess of loss truncates it. Write the payment variable for each separately.

• Not checking that an MLE is a maximum, or ignoring censored and truncated data in the likelihood.

  Fix: Check the second derivative. For censored or truncated data, use the survival function or the conditional density in the likelihood.

• Giving R output without interpretation in Paper B.

  Fix: State the method, the parameter estimates, and what they say about the fit. Link the result to the question asked.

Last-day revision: Loss distributions, with and without risk sharing

• ✓Deductible d: the insurer pays Y = max(X − d, 0). Per loss, E[Y] = ∫ from d to ∞ of S(x) dx.
• ✓Policy limit M: the insurer pays min(X, M). E[min(X, M)] = ∫ from 0 to M of S(x) dx.
• ✓Excess of loss with retention M: the insurer pays min(X, M) and the reinsurer pays max(X − M, 0). The two add to X.
• ✓Proportional reinsurance with retention α: the insurer pays αX and the reinsurer pays (1 − α)X.
• ✓Proportional reinsurance scales the mean by α and the variance by α². It does not change the shape of the distribution.
• ✓Exponential: mean 1/λ, and the memoryless property means the excess over d is again exponential with the same λ.
• ✓Pareto and lognormal are heavy tailed. The exponential has a light tail compared with them.
• ✓Method of moments: equate sample moments to the model moments and solve for the parameters.
• ✓MLE: write the log-likelihood, set the derivative to zero, solve, then check the second derivative.
• ✓Inflation by a factor (1 + i) multiplies X by that factor. For a scale family, the scale parameter is multiplied and the shape parameter is unchanged.
• ✓With a fixed deductible, inflation raises the insurer's expected payment by more than the inflation rate on the ground-up claim.
• ✓Always state whether your answer is per loss or per payment.

Loss distributions, with and without risk sharing practice questions

• Claims follow a lognormal distribution with mu = 8 and sigma = 1.2. Claims inflate uniformly by 25%. What are the parameters of the inflated…
• A claim X is lognormal with parameters mu = 8 and sigma^2 = 0.5. Which expression gives E[X^2]?
• Claims are lognormal with mu = 8.0 and sigma = 0.5. Over the next year claims inflate uniformly by 20%. Which describes the inflated claim d…
• An insurer has a loss-per-payment random variable after introducing an ordinary deductible d on a loss X with a heavy-tailed Pareto distribu…
• Claims X are lognormal with parameters μ and σ². Five observed claims (₹) are e^1, e^2, e^3, e^4 and e^5. What is the MLE of σ² (using the M…
• Ground-up losses are exponential with mean Rs 50,000. An insurer pays losses in excess of a deductible of Rs 20,000 (payment per payment). N…
• Claim sizes are gamma with parameters α and λ. A sample has mean ₹800 and variance 320,000 (₹²). Using method of moments, what is the estima…
• A general insurer's motor policy has an ordinary (non-franchise) deductible of Rs 20,000 per loss. A loss of Rs 55,000 occurs. Which stateme…

Loss distributions, with and without risk sharing in other exams

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

• CMA FinalStructure and Type of Re-insurance
• FRM Part IOperational Risk
• ACCA Applied SkillsLimitations of interpretation techniques
• CS ExecutiveDistribution of Profits