Chapter 16: Stochastic Reserving Methods

Question 1

9 Random factors influencing the run-off of claims reserves

(sources of process error)

Answer 1

We can consider the run-off of claims reserves to be a random process, with many random factors influencing the outcome. These uncertain factors include:
1. the occurrence and severity of claims
2. the notification delays on individual claims
3. legal changes that affect the size of awards
4. legal changes that affect the ‘heads of damage’ awarded. This can change the types of loss recognised in compensation awards for serious injuries, for example loss of income, medical and nursing costs
5. changes in the litigiousness of society
6. levels of claims inflation which is often related to levels of price inflation and wage inflation in the economy
7. court rulings on liability or quantum of individual claims not foreseen by claims handlers and/or not in the historical data
8. changes in the mix of claim types, either caused by an underlying change in claim type experience or by changes in the mix of business written
9. changes in claims handling, either because of policy changes or because of external events, for example a catastrophe leading to claims handlers being over-stretched
10. the emergence of new types of claim
changes in the way claims are settled, for example if more claims are settled in the form of a series of payments rather than as lump sums (in the UK this is referred to as a PPO).
These factors contribute to the uncertainty underlying the process of the run-off of claims.

Question 2

“Heads of damage”

Answer 2

Types of loss recognised in compensation awards for serious injuries, such as loss of income, medical and nursing costs, etc.

Question 3

Further uncertainties in using historic data to project the run-off of claims (3)

Answer 3

• The historic data only provides a limited sample
• The quality of data may have varied over time.
• “model uncertainty” because there are many ways of deriving the reserve estimates and many judgements are required.

Question 4

4 Terms used to identify the sources of uncertainty

Answer 4

• Parameter uncertainty - estimation error
• Process uncertainty - inherent random noise in the process
• Model error - choice/specification of model
• Systemic error - data selection error

PREDICTION ERROR OR STANDARD ERROR = PARAM/ESTIMATION ERROR + PROCESS ERROR

Question 5

Process uncertainty

Answer 5

The uncertainty in what the future outcome will be.
This is the randomness of the underlying process.

Question 6

Parameter uncertainty

Answer 6

The uncertainty in selecting parameters within the reserving process, and hence the results.

Question 7

Model error

Answer 7

The error/uncertainty arising from the fact that we might select an inappropriate model to derive our reserve estimates.

Question 8

Systemic error

Answer 8

The uncertainty arising from unforeseen trends or shifts away from the current claims environment.

Question 9

Stochastic claims reserving can be used to: (6)

Answer 9

• assess reserve adequacy
• Compare the reasonableness of different sets of reserve estimates.
• Compare datasets at different as at dates.
• monitor performance of claims
• inform management so that decisions to contract or expand business is taken.
• allocate capital
• provide information to investors
• facilitate discussions with regulators
• Price insurance and reinsurance policies

Question 10

3 Main benefits of using a stochastic approach for reserving

Answer 10

• We can estimate the RELIABILITY OF FITTED MODEL, and likely the MAGNITUDE OF RANDOM VARIATION
• We may apply STATISTICAL TESTS TO VERIFY ASSMPTIONS 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 data point.

Question 11

5 Drawbacks to stochastic reserving

Answer 11

• It takes more time
• It requires a higher level of skill and training
• The methods are more complicated, so the risk of mistakes is greater and they are harder to explain to a non-technical audience
• A considerable element of judgement is required in the choice of model and in selecting a prior (Bayesian methods)
• Using more sophisticated methods may lead to spurious accuracy and false confidence in the results.

Question 12

3 ways in which the appropriateness of any model might be tested

Answer 12

• Examine plots or triangles of residuals
• Use F tests to check the appropriateness of the number of parameters.
• Fit the model to past data

Question 13

3 Types of stochastic claims reserving models

Answer 13

• analytical methods - Mack, ODP, negative binomial, normal appox to negative binomial, log normal
• simulation methods - ODP, bootstrapped form
• Bayesian methods - BF method, Bayesian form

Question 14

3 relative merits of stochastic and deterministic approaches

Answer 14

• Deterministic approaches only consider a 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.
• Analysis of the impact of atypical scenarios aids understanding of variation around expected outcomes, and assigns a distinct value to them.

Question 15

Define “reserve risk”

Answer 15

The risk in respect of financial losses that could arise if the actual claim payments from expired business turn out to be higher than reserved for.

Question 16

Analytical methods:
4 Distributions which might be specified for the claims process:

Answer 16

• over-dispersed Poisson
• negative binomial
• normal approximation to negative binomial
• lognormal

Question 17

Mack model

Answer 17

The Mack model reproduces chain ladder estimates, and make limited assumptions about the distribution of the underlying data, specifying the first two moments only.

Question 18

3 Key assumptions of the Mack model

Answer 18

• the run-off pattern is the same for each 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.

Question 19

Bootstrapping

Answer 19

Involves sampling (with replacement) multiple times from an observed data set in order to create a number of pseudo data sets. 
We can then refit the model to each new data set, and obtain a distribution of the parameters. 

In the context of claims reserving, “bootstrapping” often refers to bootstrapping the ODP (over-dispersed Poisson) model.

Question 20

4 Keys assumptions of “Bootstrapping/ODP model”

Answer 20

• the run-off pattern is the same for each origin period
• incremental claim amounts are statistically independent
• the variance of the incremental claim amounts is proportional to the mean
• incremental claims are positive for all development periods.

Question 21

Why might claims run-off between lines of business be correlated? (6)

Answer 21

• they are impacted by similar events, eg a windstorm could impact both household and commercial property accounts
• legal changes often affect several lines of business, eg a change to the Ogden discount rate would affect both employers’ liability and motor classes
• inflationary trends will affect many adjacent origin periods
• 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 may affect more than one line of business.

Question 22

5 Issues with stochastic claims reserving models

Answer 22

• Claims need to be aggregated across lines of business. This process needs to allow for correlations.
• Certain models are limited by the type of model or data that can be fitted. A key problem is with instances of negative increments in incurred data.
• Stochastic models can be unreliable when applied to latent claims.
• Models fitted using sparse data can be very sensitive to small changes.
• Care is required in the tail of the claims distribution because data may be inadequate and assumptions may not be valid at the extremes.
• stochastic methods tend to underestimate reserve variability

Question 23

Bayesian stochastic reserving method

Answer 23

Bayesian methods use
… a prior distribution for the variable
… in combination with the data
… to produce a posterior distribution for the predicted variable

Question 24

3 Advantages of the Bayesian method

Answer 24

• they provide a complete predictive distribution
• they explicitly state the subjective judgement used
• closed-form distributions can often be obtaines

Question 25

2 Disadvantages of the Bayesian method

Answer 25

• the choice of prior distribution is subjective
• numerical methods are required when there is no closed-form distribution

Question 26

3 Methods to use in the absence of any past claims data

Answer 26

• Using market data or data from reinsurers
• Applying a percentage to the policy limit
• using professional judgement and experience

Question 27

Advantages and disadvantages of stochastic and deterministic reserving methods:

Answer 27

“Deterministic:
less costly/cheaper
less effort
Less expertise required compared to stochastic
less data and
fewer assumptiosn required eg. new companies or new line of business
Easier to deal with correlations
Easy to communicate both results and methods
Lesser scope for errors such as what is introduced by model error, paramter error in stochastic resrving.
Helps in validating a stochastic model
Allows for actuarial judgement to be applied for specific scenarios
Helps in stress testing, scenario testing and sensitivity tests
appropriate to use in certain cases such as as a resonableness check on other analysis or on projects that are not significant to the organization.
Stochastic methods can underestimate extreme events int he tail of the distribution
As a check for stochastic model
Stochastic:
A deterministic reserving model, such as the chain ladder method, only provides a single best estimate of the reserve required.
A deterministic reserve gives no information about the uncertainty present in the reserve estimate.
In reality the actual amount ultimately required to pay the claims may differ from the best estimate, eg because of:
variations in the occurrence and severity of claims
notification delays
legal changes, levels of litigiousness, court rulings
inflation
changes in the mix of claims
changes in claims handling procedures.
regarded as best practice ni the industry/market
Regulators require to use stochastic eg. captial requirements
Uncertainty surrounding the reserve estimates are reflected in stochastic methods
Automatically results in communicating that uncertainty to the others as we have a range of estimates
Can help in finding the best estimate
Useful in fitting distributions to the data (eg. freq and sev distributions)
dealts with correlations better eg. copula espeically useful for GI, Cat events
Stochastic is useful for low freq, high sev events like cat
Volatility is better modeled by stochastic methods than deterministic
Helps in feeding directly into a capital model
Helps in decision making such as RI management
Helps in capital allocation
Better computational power and methods today make it easier to implement
Issues surrounding stochastic reserving:
Stochastic methods are not very suitable for latent claims as past data may not have much info in it.
Stochastic methods that do not allow for negative increments are also a problem. However, certain analytical methods such as Mack overcome this problem.
Sparse data is an issue as evwen a small data point can skew up the whole stochastic results.
Variability may be understated
Also tails may not reflect the true nature of the distribution as the stochastic method is based on limited historical data which may not have sufficient data points in the tail.”

Question 28

Explain the components of the variance of stochastic reserve predictions

Answer 28

“One of the reasons for using a stochastic reserving method is to assess the variance or the level of uncertainty from best estimate that occurs in real life. This is quantified using the prediction variance.
The components of the prediction variance are as follows:
Prediction variance = Estimation variance + Process variance
PRocess variance refers to the inherent uncertainty in the process. the actual vale of the reserves will almost always vary from the predicted best estimate because of process variance.
Estimation variance occurs because of the inaccurate choice of the parameters that we are choosing to model the claim amounts. “

Question 29

Mack Method

Answer 29

”*Best known analytic model
*Uses past data to estimate the mean and variance
*Distribution free method
*The run-off pattern is the same for each origin period
*the future development of a cohort is not dependent on past factors
*variance of the aggregated claims at time ““t”” is proportional to the aggregated claims in total to time ““t-1””.”

Question 30

Data required for Mack Model and the output it produces

Answer 30

“1. Requires incremental or cumulative claim triangle by origin and development year

2. IT could be incurred or paid claims information
3. This data is used to produce a best estimate or mean of the reserves and variance estimates through the Mack method. standard errors are obtained for both individual origin years as well as the collective development period.
4. Mack method is distribution free. But using mean and variance distribution can be fitted
5. A two parameter distribution can be fitted to these parameters eg. Log normal or normal can be fitted using the mean and the variance. This can be used to stochastically model the claims reserves”

Question 31

how stochastic claims reserves for several lines of business should be amalgamated

i.e. outline two methods for deriving an aggregate distribution covering all lines of business

State why line of business can be correlated.

Answer 31

1. Stochastic best estimates can be added together for serveral lines of business to obtain the overall best estimate.
2. However, that cannot be done for the variance. The lines of business are correlated to each other. Most of the times, it is positive correlation between classes.
3. We need to use stochastic methods separately for obtaining the claims distribution of all the classes together to find the variability of results at an overall level.
4. For this simulation results from individual lines of business can be collected to obtain the aggregate claims distribution.
5. It is not possible to get the aggregate claims distribution using only analytical methods.

Question 32

Bootstrapping the ODP - assumptions

Answer 32

”*Run off pattern is the same for each origin period
*incremental claim amounts are statistically independent
*variance of incremental claims are proportional when compared to the mean
*incremental claim amount are +Ve for all development periods”

Question 33

Bootstrapping a ODP model

Answer 33

“1. Fit a Basic Chain Ladder to cumulative claims, obtaining fitted values for the observed data.

2. Calculate the residuals of your fitted model against the actual entries in the traingle (to do this use the leading diagonal i.e. the most recent entries diagonal and apply the developments factors to know what the historical cumulative claims should have been given the BCL link ratios) i.e. actual less expected/ back-fitting
3. Take a sample from the residuals (with replacement), and invert these to generate a pseudo-dataset.
4. Refit the BCL using this pseudo-dataset, to get the ultiimate reserves.
5. Repeat steps 3 and 4 many times to derive a reserve amount for each pseudo-dataset. This gives a distribution of reserves
6. moments, percentiles and other statitical properties can now be arrived at.”

Question 34

Meaning of bootstrapping in reserving context

Answer 34

“when we have a dataset from a population, this sample can be used to create psuedo datasets.
The purpose of psuedo datasets is to exploit the randomness in the sample.
This is done by sampling with replacement from the initial dataset.
Sampled data is assumed to be iid
Several such datasets are created and statistical properties of the results are analyzed.
when this is used in stochastic claims reserving, a distributuion of reserve estimates can be obtained.
“

Question 35

Bayesian methods in the context of reserving

Answer 35

“Bayesian method involves a prior distribution
We assume that parameters in a model are from a prior distribution which we know.
We combine this assumed prior distribution for the parameters with a model for the claims development to arrive at the posterior distirbution. I.e. our assumption about the parameter is combined with the additona perspective provided by the current data.
Posterior distribution is then used to calculate statistical measures like mean or percentiles or variance.
Eg. Bayesian measure of BF method
Obtain past data of claims by origin and development years
Then, use the prior distribution’s parameters by assigning suitable values to the parameters
determine the posterior distribution for the projection of the claim reserves
use this posterior distribution to determine mean/variance/percentiles for claims resesrves”

Question 36

Over dispersed Poisson

Answer 36

“In a regular poisson distribution, the variance equals mean.
However, in a ODP, the variance is larger than the mean and hence we call it over dispersed.
Most claims distributions are over dispersed and hence a ODP model is used in stochastic reserving
in this model the variance is estimated as Phi multiplied by the deterministic estimate or the mean, and Phi >1
and Phi is a constant multiplier estimated from past data. Generally the Phi is a contant for each developmental period. For developmetn from year 0 to1, we could use Phi 1, then for 1 to 2 phi 2, etc.
We can get the Phi by making an assumption about the distribution involved. this will result in a analytical model.
Alternatively, we can use residuals to bootstrap the prior data.”

Question 37

Comparison between Mack and Bayesian approaches (2015A, Q6)

Answer 37

“1. Mack is distribution free, whereas in Bayesian we have an assumption about the prior distribution and it provides a complete posterior distribution

2. The prior distribution assumptions is quite subjective and requires judgment.
3. This subjective selection of the prior distribution has a heavy influence on the resulting posterior distribution
4. Bayesian method can be used in stochastic reserving, whereas Mack method is analytical. Though, a distribution can be fit using the first two derived moments.
5. For Bayesian method, deriving the results may need more complex methods.
6. Negative increments are handled by both methods.
7. Mack methods is quite sensitive to the data quality but Bayesian is not as much influenced.
8. Mack method is more prevalent in use than Bayesian.”

Question 38

Outline 4 approaches to derive a resrve distribution net of reinsurance under non-proportional covers, when using stochastic reserving methods

Answer 38

1. Derive a gross distribution and scale down the distribution such that themean equals the net best estimate. However, this wil overestimate the uncertainty because reinsurance reduces the volatility of individual large claims
2. Estimate a distribution of reinsurance to gross reserve ratios, and apply this to the gross reserve distribution
3. Use a net claims triangle to derive a distribution. This will underestimate the volatility if reinsurance retentions are increaseing and vice versa.
4. Simulate individual claims and net down explicitly before aggregation.