Chapter 15: Quantifying uncertainty in reserves Flashcards
We can consider the run-off of claims reserves to be a random process, with many random factors influencing the final outcome. These uncertain factors include:
- the occurence and severity of claims
- the notification delays on individual claims
- legal changes that affect the size of awards
- legal changes that affect the “heads of damage” changes in the litigiousness of society
- levels of claims inflation
- court rulings on liability or quantum of individual claims not foreseen by claims handlers and/or not in the historical data
- changes in the mix of claim types
- changes in claims handling
The projection of the run off of claims often introduces futher uncertainties, including:
- historic data only provides a limited sample
- quality of data may have varied over time
- there are many ways of deriving an estimate of the claims reserve and many judgemets required in each method
Terms used to identify sources of uncertainty or error
- process uncertainty
- parameter uncertainty
- model error
- systemic error
Process uncertainty
The uncertainty in what the future outcome will be, arising from the inherent variability of the data being analysed. This is the randomness of the underlying process.
Parameter uncertainty
The uncertainty in selecting the parameters within the reserving process and hence the results. Arises, for example, from the uncertainty around what the past data is telling us about the future.
Model error
The error/uncertainty arising from the fact that we might select an inappropriate model to derive our reserve estimates, or the model used may not be the only possible model
Systemic error
The uncertainty arising from unforeseen trends or shifts away from the current claims environment,
Types of ranges that may be used
- range of best estimates
- range of possible outcomes
- range of reasonable/probable/plausible outcomes
Range of possible outcomes
This represents the actuary’s estimate of the complete range of outcomes for future claims. It would be considerably wider than the range of best estimates
Range of reasonable/probable/plausible outcomes
Typically wider than the range of best estimates, but narrower than the range of possible outcomes, because it’ll allow for outcomes that cannot be reasonably regarded as an estimate of the mean outcome, but can still be regarded as a plausible outcome.
Advantages of using alternative sets of assumptions to quantify uncertainty in reserves
- it is simple to perform on deterministic or stochastic models
- we can use judgement when we select possible parameters. We can therefore allow for atypical volatlity in the historical data.
Disadvantages of using alternative sets of assumptions to quantify uncertainty in reserves
- we assign no explicit probability to each set of parameters. It’s not possible to estimate the distribution of future outcomes unless we assign a probability to each set of assumptions
- we ignore model uncertainty using this method
- we don’t allow for process uncertainty if use alternative sets of assumptions for a deterministic model. There is no reason why we cannot use alternative sets of assumptions for a stochastic model
Scenario analysis
We examine the likely impact of catastrophic events on a firm’s outstanding liabilities.
Ways of deriving a scenario for scenario analysis
- basing it on a historical event
- thinking up a hypothetical event using our judgement
- from the results of a stochastic model
Scenario analysis:
Typical scenarios affecting outstanding liabilities
- claims outstanding from a single catastrophe
- claims outstanding on major individual contracts
- latent claims
- reinsurance bad debt
- interest rate changes (if discounting reserves)
- inflation levels affecting the ultimate size of claims paid
- expense levels
- exchange rate movements if claims are paid in a foreign currency
Advantages of scenario testing
- advantage over stochastic modelling by allowing a more detailed analysis of the tail end of the reserve
- in performing a scenario test, we pay particular attention to the likely coincidence of these adverse factors
- a scenario analysis is more focused, whereas a stochastic approach provides a full analysis - time consuming and expensive
- because it is aimed at a specific question, we can construct a scenario test and produce reliable results more quickly than for a stochastic model
- easier to communicate the results of a scenario test than the results of a stochastic model, as they are more transparent
- model uncertainty is much less of a problem when we construct scenario tests because we consider the driving factors explicitly. Stochastic models may fail to capture some features of the real life process, especially under extreme circumstances
Disadvantages of scenario testing
- there is no specific probability associated with the outcomes and so it’s not possible to construct a distribution of outcomes (might be better than spurious accuracy of a stochastic model)
- scenarios only give information on the extremes of the distribution of the eventual outcomes, while the actuary may want to disclose information on the overall distributon to stakeholders too
- method is more subjective than stochastic models and alternative sets of assumptions, since the actuary makes the decision on which extreme scenarios are to be investigated
Relative merits of stochastic and deterministic approaches when quantifying the uncertainty in reserves
- deterministic approaches only consider limited number of factors and one result from each, while a stochastic model generates a number of potential scenarios that may not be thought of under a deterministic approach
- failure is often due to the interaction of many differing factors which could not be modelled deterministically. The stochastic model can allow for the interdependency of these key factors. Deterministic scenarios can be chosen in such a way as to estimate the interdependencies
- analysis of the impact of atypical scenarios aids understanding of variation around expected outcomes, and assigns a distinct value to them. This can be done to scenarios generated by a stochastic model or scenarios generated for a deterministic model
Stochastic claims reserving can be used to:
- assess reserve adequacy
- compare different estimates and datasets
- monitor performance
- allocate capital
- provide information to investors
- facilitate discussions with regulators
Reserve risk
The risk in respect of financial losses that could arise is the actual claim payments from expired business turn out to be higher than reserved for.
Main benefits of using a stochastic approach for reserving
- we can estimate the reliability of the fitted model and likely magnitude of random variation
- we may apply statistical tests to the modelling process to verify any assumptions and gain understanding of the variability of the claims process
- we can develop models in which the influence of each data point in determining the fitted model depends on the amount of random variation within that datapoint
Drawbacks of using a stochastic approach for reserving
- it takes more time
- it requires a high level of skill and training
- the methods are more complicated, so the risk of making mistakes is greater and they are harder to explain to a non-technical audience
- considerable element of judgeent is needed in the choice of model and in selecting the prior distribution with Bayesian methods
- using more sophisticated methods may lead to spurious accuracy and false confidence in the results
Components of model error
- specification error
- selection (or “systemic”) error
- estimation/parameter error
- process error
Specification error
This is the error arising from the specification of the model
Selection (or “systemic”) error
This arises from incorrect selection of the underlying data used. Trends may cause systematic changes to experience making the chosen data irrelevant.
Estimation/parameter error
This arises from the fact that the estimated parameters are random variables
Process error
This reflects the inherent random noise in the process
Ways to test the appropriateness of any model used
- use plots or triangles of residuals - standardised residuals should be randomly distributed
- F tests - appropriateness of the number of variables
- fit the model to old data
Distributions which might be spcified for the claims process
Analytical methods
- over-dispersed Poisson (ODP)
- negative binomial
- normal approximation to negative binomial
Mack model
Analytical methods
The Mack model reproduces chain ladder estimates and makes limited assumptions about the distribution of the underlying data, specifying the first two moments only.
Key assumptions of the Mack model
Analytical methods
- the run-off patten is the same for each origin period
- the future development of a cohort is independent of historical factors
- the variance of the cumulative claims to development time t is proportional to the cumulative claims amount to time t-1
Bootstrapping
Involves sampling (with replacement) multiple times from an observed data set to create a number of pseudo data sets. We then refit the model to each new dataset and obtain distributions of the parameters.
Key assumptions of Bootstrapping/ODP model
- the run-off pattern is the same for each origin period
- incremental claim amounts are statistically independent
- the variance of the incremental claim amount is proportional to the mean
- incremental claims are positive for all development periods
Why might the claims run-off between lines of business be correlated?
- reporting delays may change across all classes
- the same claims team may handle claims from several lines of business and so changes to claims handling may impact more than one line
- problems with data that may affect more than one line
Over-dispersed Poisson (ODP) model
- bootstrap model
- assumed the variance exceeds the best estimate by a constant factor greater than one that’s estimated from the data
- model cannot handle negative values that might be present of incremental claims data is used
Why would assuming the run-offs between lines are independent underestimate the variability of the aggregate distribution?
It’s likely the claims from different classes will be positively correlated. This leads to an increase in combined variance.
Stochastic claims reserving models can be broadly split into:
- analytical methods
- simulation methods
- Bayesian methods
Issues with stochastic claims reserving models
- model forms - mismatch between type of model and data to be used (e.g. presence of negative increments)
- latent claims - not suitable since unable to reflect variability in claims data available
- sparse data and data peculiarities - missing/erroneous data
- the extremes - parameterise distribution based onfinite amount of historic data, which may not be representative of the tail
Bayesian stochastic reserving method
Under the Bayesian theory framework, the prior distribution of the predicted variable is first chosen based on judgement or experiece. Then the posterior distribution of the predicted variable is calculated using Bayes’ formula.
Advantages of the Bayesian method
- provides complete predictive distribution of the ultimate reserve - depends on prior distribution but gives more information
- it explicitly shows the impact of judgements, which is reflected in the prior distribution
- could give closed-form results when appropriate prior distribution is chosen
Disadvantages of the Bayesian method
- the choice of prior distribution is subjective and the posterior distribution may be over-reliant on the choice of prior distribution
- may not give closed-form results and numerical integration may be needed to get results - use of Markov Chain Monte Carlo (MCMC) method to calculate integration
Methods that can be used before claims experience is available (absense of past claims experience)
- use of cedant or market figures/reinsurers’ expertise
- policy limits used as an upper limit for the reserve
- intuition/professional judgement
Objectives in communicating uncertainty
- ensuring stakeholders understand the level of uncertainty
- being consistent with the vocabulary used by other actuaries and other professionals and explaining any terms that may not be understood by the audience to which the report is directed
- emphasising the bigger issues
- explaining what has been allowed for in the best estimate and what has not
- emphasising the unusual issues
- commenting in the context of the scope and purpose
- avoiding misunderstandings
