Chapter 19: Methods of calculating risk premium

Question 1

Techniques used under the cost-plus approach to calculate the premium:

Answer 1

• The burning cost approach
• The frequency-severity approach
• Multivariate models, including GLM
• Original loss curves

Question 2

Burning cost data requirements:

Answer 2

• Dates on cover
• All rating factors and exposure measure details
• Details of premium charged unless can be calculated from the rating factors & exposure measures

Question 3

Advantages of the burning cost approach to calculating the risk premium:

Answer 3

• Simplicity
• Needs relatively little data
• Quicker than other methods to perform
• Allows for the experience of individual risks or portfolios

Question 4

Disadvantages of the burning cost approach to calculating the risk premium:

Answer 4

• Harder to spot trends so it provides less understanding of changes impacting the individual risks
• Adjusting past data is hard
• Adjusting for changes in cover, deductibles and so on is hard as we often lack individual claims data
• It can be a very crude approach depending on the adjustments for trends made to data

Question 5

Causes of frequency trends include changes in:

Answer 5

• Accident frequency
• The propensity to make claims and other changes in the social and economic environment
• Legislation
• The structure of the risk

Question 6

Drivers of severity trends include:

Answer 6

• Economic inflation
• Changes in court awards and inflation
• Economic conditions
• Changes to the structure of the risk

Question 7

Methods of developing individual losses for IBNER:

Answer 7

• Apply an incurred development factor to each individual loss, reflecting it maturity, to estimate the ultimate settlement value
• Develop open claims using case estimate development factors. These case estimate factors are usually higher than the incurred factors to offset the effect of not developing closed claims
• Use stochastic development methods to allow for the variation that may occur in individual ultimate loss amounts around each of their expected values

Question 8

Common components found in insurance arrangements:

Answer 8

• Aggregate deductible
• Non-ranking deductible
• Ranking deductible
• Trailing deductible
• Per occurrence limit
• Annual aggregate limit

Question 9

Loss distributions often used with frequency-severity method:

Answer 9

• Frequency: Poisson, Negative binomial
• Severity: LogNormal, Weibull, Pareto, Gamma

Question 10

Common underlying fitting algorithms:

Answer 10

• Maximum likelihood estimates
• Method of least squares
• Method of moments

Question 11

Statistical goodness of fit tests usually used:

Answer 11

• Chi-squared statistic
• Kolmogorov-Smirnov statistic
• Anderson-Darling statistic

Question 12

Advantages of frequency-severity approach to calculating the risk premium:

Answer 12

• The approach mirrors the underlying approach, so it is more readily understood by underwriters
• We can use the approach for complex insurance structures, e.g. deductibles/limits
• By separately assessing information on loss frequency and severity, we gain additional insight into aggregate amounts
• It helps us identify trends in frequency and severity of claims

Question 13

Disadvantages of frequency-severity approach to calculating the risk premium:

Answer 13

• Assessing the compound frequency-severity distribution has more onerous data requirements than assessing aggregate amounts
• This approach can be time-consuming for a single risk and requires a high level of expertise

Question 14

Main types of multivariate models:

Answer 14

• Minimum bias methods
• Generalised linear models
• Generalised non-linear models
• Generalised additive models

Question 15

Approaches to classification:

Answer 15

• Spatial smoothing methods (distance-based or adjacency-based)
• Vehicle classification techniques
• Decision trees
• CHAID

Question 16

Considerations when modelling using multivariate models:

Answer 16

• Choosing which factors to include in the model
• Analysing the significance of factors
• Measuring uncertainty in model estimators
• Comparisons with time
• Consistency checks with other factors
• Restrictions on the use of factors in the model
• Parameter smoothing

Question 17

Original loss curves are more commonly called:

Answer 17

• First loss scales, exposure curves or loss elimination functions in the property business
• Excess of loss scales when showing allocation to the excess layer in property business
• Increased limit factors in liability business

Question 18

When using exposure curves, we may need to adjust for:

Answer 18

• The heterogeneity of the underlying business, e.g. by using separate curves for each peril or risk size band
• The effect of claims inflation

Question 19

Assumptions within each risk group when using increased limit factors:

Answer 19

• The ground-up loss frequency is independent of the limit purchased
• The ground-up severity is independent of the number of losses and of the limit purchased

Question 20

Advantages of using original loss curves to calculate the risk premium:

Answer 20

• Relatively simple to implement
• Relatively easy to explain to a non-technical audience
• The loss costs obtained should be internally consistent
• It can be used where there is little to no credible loss data available

Question 21

Disadvantages of using original loss curves to calculate the risk premium:

Answer 21

• Application is difficult in practice, often due to uncertainty in estimating or selecting the appropriate curves

Question 22

Factors to consider when choosing the most appropriate techniques for calculating the risk premium:

Answer 22

• Validity of current method
• The amount of data available
• Presence of trends in the data
• The need to review rating factors
• The frequency of model use
• The expertise available
• Access to modelling software (including cost)
• Flexibility of the model

Question 23

When evaluating proprietary catastrophe models for property catastrophe reinsurance business, we should consider:

Answer 23

• Model robustness: Identify which models perform better for specific perils and regions. Stay current through industry publications and discussions with model providers. Be cautious of marketing bias.
• Model assumptions: How they differ behind the models and how often they’re updated
• Input requirements: Variations in required input data as these can impact results
• Output differences: Compare outputs. Outputs usually include simulated event data, event probabilities, expected losses, and uncertainty measures. Some models account for both primary and secondary uncertainty, while others do not. This affects perceived volatility and the required risk loading.
• Source of models runs: If receiving only model output (not raw data), clarify how it was produced, by speaking to the cedant’s or broker’s catastrophe modelling team and involving your own experts where possible.

Question 24

Converting catastrophe model output to risk premium for property catastrophe reinsurance business

Answer 24

1. Model Output: Catastrophe models provide a distribution of events, usually in two formats:
   * OEP (Occurrence Exceedance Probability)
   * AEP (Aggregate Exceedance Probability)
2. Simulation: These files are used within a stochastic frequency-severity model to simulate potential annual loss scenarios for the cedant.
3. Applying Contract Terms: The reinsurance contract terms (e.g., limits, retentions) are applied to the simulated losses to calculate the reinsurance recoveries for each scenario.
4. Premium Calculation:
   * From the recoveries, derive the distribution of annual recoveries.
   * Calculate the expected annual recovery (i.e., the technical risk premium).
   * Add a risk load based on a volatility measure to account for uncertainty.

Question 25

Burning cost method for calculating the risk premium for property and liability per-risk non-proportional covers

Answer 25

* Trend the claims data * Aggregate by year to give triangles of paid/incurred losses * Develop to ultimate using benchmarks if necessary * Apply the reinsurance contract terms to give the loss to the layer * Adjust exposure (often premium) for past rate and exposure changes * Divide losses by exposure to get burning cost

Question 26

Frequency-severity method for calculating the risk premium for property and liability per-risk non-proportional covers:

Answer 26

* Fit statistical models to historical loss frequency and severity. * Use trends and combine distributions to estimate reinsurance recoveries. * This is similar to modeling direct insurance risk net of XL layers.

Question 27

Using exposure curves for calculating the risk premium for property per-risk non-proportional covers:

Answer 27

* Estimate gross risk premium from cedant’s premium and assumed loss ratio. * For each risk: o Use risk size to determine deductible and excess %. o Apply exposure curve to estimate proportion of losses in reinsurance layer. o Multiply this by the expected loss (loss ratio × premium) to get expected reinsurance losses. * Aggregate across risks to get total risk premium to the layer.

Question 28

Using ILFs for calculating the risk premium for liability per-risk non-proportional covers:

Answer 28

* Use ILFs to determine how much of each risk's losses fall within the reinsured layer. o Reinsurance risk premium proportion =(ILF(LR+ER)-ILF(ER))/(ILF(L+E)-ILF(E)) o LR = reinsurance limit, ER= reinsurance excess, L= policy limit, E= policy excess. * Estimate expected losses using premiums and a selected loss ratio. * Apply the proportion above to these losses to get expected losses to the reinsurance layer.

Question 29

Deriving a risk premium (ceding commission rate) for quota share reinsurance:

Answer 29

* Adjust claims for inflation and premiums for rate/exposure changes * Use triangulations to get ultimate historic loss ratios * Decide on an estimated loss ratio for the period in question * Calculate a suitable commission, bearing in mind other outgo, e.g. expenses * Use a stochastic model if there is a profit or sliding scale commission

Question 30

Deriving a risk premium (ceding commission rate) for surplus reinsurance:

Answer 30

* Simulation-based approach using: o Gross loss severity distribution. o Distribution of limits/cession rates. o Randomly allocate each simulated loss to a risk and apply the relevant cession rate. * Simplified approach using historical ceded loss ratios and projecting forward (with assumption that future cessions resemble the past).

Question 31

Key considerations in determining risk premium for stop loss reinsurance:

Answer 31

* Regulatory requirements for risk transfer must be met. * Specific contract terms (e.g. loss ratio thresholds, limits). * Any inuring reinsurance (i.e. reinsurance that applies before or alongside the stop-loss).