Assignment ST3074

AY2024-2025 Semester 2

Instructions:

• 100 marks are available.

• Please answer both questions, each worth 50 marks.

• Suggested time is 2 hours.

• You must upload a completed report (word file/pdf), copying and

pasting supporting calculations/output/code from Excel and R into the spaces provided.

• You should also upload any accompanying excel files and R scripts with your report to Canvas.

• Each uploaded file should be named with your name and student number, e.g.

“Forename_Surname_123456789_Report”

“Forename_Surname_123456789_Q1 Excel”

“Forename_Surname_123456789_Q2 R”

• This is an open book assignment. You are permitted the use of any notes, Canvas materials and the R help function.

Q1.

You are a trainee reserving actuary at XYZ Ltd, an established Irish insurance company. You are currently managing two Lines of Business (LoBs):

• Employers' Liability Insurance (LoB X): A well-established LoB with multiple years of claims data.

• Medical Malpractice Liability Insurance (LoB Y): A newer LoB with limited historical data.

Claims for both lines are subject to inflation, and you have been provided with  historical inflation values as per the CPI Index. Your task is to estimate the required reserves as of year- end  2024, considering both lines of business. The necessary data are provided in  the accompanying excel file.

(a)  Describe at least two examples (for each line of business) of events that could lead to claims for Employer’s Liability and Medical Malpractice.

(b) You are provided with the incremental and cumulative incurred claims triangles for LoB X in the accompanying excel file. Calculate the Basic Chain Ladder Estimate of the reserves required for LoB X. Note any assumptions required.

(c)  You are provided with historical inflation data in the accompanying excel file. Research and  apply the Inflation-Adjusted Chain Ladder Method to estimate the required reserve.  Note any assumptions required.  Comment on  the impact of using the Inflation-Adjusted Chain Ladder Method versus the Chain Ladder Method to estimate the reserve.

(d)  LoB Y is a new line of business with limited historical data (€2,000 of incurred claims in the current year). In order to establish a reserve, it should be assumed that the development pattern for LoB Y can be derived from   LoB X with appropriate adjustments. In an effort to modernise the company, XYZ has laid off its underwriting staff and replaced them with  an AI chatbot, arguing that the AI model has been appropriately trained on company data and can emulate a discussion with an underwriting expert accurately. Engage in a conversation with an AI chatbot (Chat GPT or similar) to determine reasonable assumptions for reserving LoB Y, including:

•   The claims development assumptions.

•    Inflation assumptions.

•   Other relevant actuarial assumptions.

Copy and paste below your conversation history with the AI chatbot. Without further discussion with the chatbot, provide a critical assessment of the chatbot’s responses.

(e)  Based on your discussion in (d), document your assumptions and construct a reserve model in Excel for LoB Y. This model should apply the Inflation-Adjusted Chain Ladder Method to project the claims development for LoB Y to Ultimate and estimate the required reserves.

(f)  The Chief Actuary has become concerned that the excel reserving model takes a long time to update. She has asked you to develop a model in R to implement the Inflation- Adjusted Chain Ladder Method for LoB X. Create an implementation that replicates your answer for part (c) above, pasting your code and output below.

(g)  The Chief Actuary has further commented that the current model is limited in that it only provides a point estimate for the reserves and gives no indication as to their potential variability.  Extend your model from (e) to bootstrap the future inflation assumption from the historical inflation data to generate a  distribution of 50,000 reserve estimates. Report below:

•   The best estimate (mean of the bootstrap distribution).

•   The 99.5th percentile reserve estimate.

•   An appropriate visualisation of the reserve distribution to illustrate the reserve variability by varying the inflation assumption.

•   Comment on your results, using your actuarial judgement to suggest (with justification) the level at which the reserves should be set for LoB X.

• Note:  to ensure reproducibility of the bootstrap reserves, the regulator requires that the R model uses the set.seed(100) command.

Q2.

You have joined a startup company which plans to sell Property Damage and Business Interruption Insurance Policies. The regulator has provided you with a dataset of 1,000 annual aggregate claims  (Gross of any reinsurance arrangement) expressed  in  2024 terms in the accompanying excel file Property_BI_Claims.csv.

(a)  Read the data into R, generate appropriate visualisations and comment on the features of the data.

(b) Your task is to parameterise a suitable model to assess the solvency of the business in

2025. The company’s reinsurance programme is as follows:

•   The Property Damage LoB is reinsured via a Quota Share (QS) arrangement where 40% of each claim is transferred to the reinsurer.

•   The Business Interruption LoB is reinsured via an individual Excess of Loss (XoL) arrangement with retention of 10,000.

The insurer’s profit may be modelled as:

profit = Initial capital + premiums — Net claims — Expenses

•   Assume the initial Capital is €150,000 and the annual aggregate  premium intake is €70,000.

•   Assume fixed expenses of €10,000 and that claims handling expenses are 5% of claims gross of reinsurance.

•   Select (giving your rationale) a suitable model for the claims portfolio and use the regulator’s data to estimate its   parameters.  Hence create 50,000 simulations of the insurer’s profit.

•   Assess the probability that the insurer will be solvent (have a profit greater than zero).

(c)  The  regulator  has  asked for you to  provide sensitivity analysis to demonstrate the robustness of the company’s solvency.

•    Produce a plot to illustrate the change in probability of solvency for a range of values of initial capital.

•    How much capital would be required to ensure a 99.5% probability  of solvency?

(d) Assess (with supporting statistical tests, calculations and output) the goodness of fit of your chosen claims model, commenting on your answer.