Course Overview. This workshop will show you how to use techniques from probability, statistics, and optimization to build algorithms that construct risk insurable portfolios that are optimal under budget constraints. The analytical approach, case studies, and code will be drawn primarily from a recently published text, Constructing Insurable Risk Portfolios, written by the workshop presenter. An online (html) version of the book is freely available.
Firms face many risks, such as damage to their buildings due to fire, liability resulting from management misbehavior, and threats external to the organization like cyber attacks. Because the financial impact of these risks can be mitigated by using risk transfer tools like insurance, we think of them as insurable risks. Firms are financially responsible for these risks, essentially “owning” them. This naturally leads to the idea of managing them as a portfolio. This book shows how risk managers can manage a risk portfolio cost-effectively by adapting financial concepts of diversification to insurable risk management.
Who is this workshop for? It is for financial analysts who advise risk managers, such as commercial insurers, risk brokers, and reinsurers, who are responsible for risk retention and risk portfolio management. Risk managers are accustomed to using their knowledge of industry practices and experience in the field to make retention recommendations. This workshop introduces techniques that provide an objective framework to supplement, and corroborate, an advisor’s thoughts. Although no prior industry knowledge is required, familiarity with the applications will enhance the experience.
What is this workshop about? It introduces a novel approach for constructing data-driven insurable risk portfolios. Drawing inspiration from Markowitz portfolio theory, it leverages techniques from probability, statistics, and optimization to build algorithms that create optimal risk insurance portfolios under budget constraints. As such, outputs of the algorithm include a frontier that portrays the uncertainty of a portfolio versus a cost of transferring risks. A visual display of this frontier, mirroring familiar Markowitz investment tools, allows for informed decision-making and seamless adoption by risk advisors.
Course Format. The format of the course will consist of alternating blocks between presentations of the underlying principles and practical applications. In a typical block, the instructor will spend around 45 minutes reviewing the insurance motivation and key mathematical underpinnings. This will be followed by about 45 minutes in which participants will actively explore a selected case study.
Thus, it is anticipated that participants will bring a laptop.
For this workshop, you only need a (free) Google account.
For this workshop, you will need a free Google account (if you have a Gmail address, then you have a Google account). If you do not already have one, please create an account at https://accounts.google.com/signup. You will use your Google account to access Google Colab, Gemini Notebook, and if you would like to save intermediate work, Google Drive.
To facilitate active learning, you will explore case studies that utilize the statistical package ‘R’. This package is written by statisticians and its approach is tied directly to statistical applications. Participants with some familiarity with ‘R’ will benefit most from the course. We will use Google Colab, so you do not need to have ‘R’ installed on your laptop nor will you need to download datasets in advance. Just a laptop and a Google account!
Target Audience: Practicing actuaries, students, and educators interested in exposure to the foundations of insurance analytics.
Schedule
Time
Activity
9:00–9:20 am
Welcome, Introduction to the workshop,
How it will work, What you can get out of it.
9:20–9:50 am
Topic 1. Introduction to Constructing Insurable Risk Portfolios
9:50–10:30 am
Participants learn how to construct an optimal investment portfolio
Participants will work with code to generate dependent variables as well as to quantify risk measures and risk retention functions. You also learn to optimize basic portfolios.
12:30–2:00 pm
Lunch
2:00–2:45 pm
Topic 3. Two Case Studies: Optimizing Risk Retention with Australian and Wisconsin Data
2:45–3:30 pm
Participants will explore optimizing risk portfolios using Australian large company data and Wisconsin commercial insurance data
3:30–3:50 pm
Break
3:50–4:20 pm
Topic 4. Incorporating Spatial Dependence
4:20–4:50 pm
Participants will explore Florida flood insurance data
Topic 2. Basic Tools: Dependence, Risk Retention Functions and Risk Measures. Optimal Portfolio Construction. Topic 2 Pdf Slides
11:45–12:30 pm
Participants will work with code to generate dependent variables as well as to quantify risk measures and risk retention functions. You also learn to optimize basic portfolios. Topic 2 Jupyter Notebook -Basic Tools
12:30–2:00 pm
Lunch
2:00–2:45 pm
Topic 3. Two Case Studies: Optimizing Risk Retention with Australian and Wisconsin Data Topic 3 Pdf Slides
Edward (Jed) Frees is a Fellow of the American Statistical Association and a former Fellow of the Society of Actuaries. He served as founding chairperson of the Society of Actuaries’ Education and Research Section as well as a member of the Board of Directors of the Society of Actuaries; he has also served as a Trustee of the Actuarial Foundation. Professor Frees served as an actuarial representative to the Social Security Advisory Board’s Technical Panel on Methods and Assumptions. Professor Frees was the Editor and is currently on Honorary Advisory Boards for the North American Actuarial Journal and the Annals of Actuarial Science. He is also an Editor for Insurance: Mathematics and Economics.
Professor Frees is an emeritus professor at the University of Wisconsin-Madison and was a professor of actuarial studies with the Australian National University (2019-2024). He is
the editor of the Spanish version of the open actuarial textbook, Loss Data Analytics,
Here is a link to a read-only Gemini Notebook that is being shared. This tool is like a smart study assistant that only knows the content uploaded — so it is focused, grounded, and safe to use!
Gemini Notebook is your personal learning assistant. Use it to go beyond just recalling information and build a deeper understanding of course materials. Use the prompts below to guide your learning and prepare for assignments.
Core Learning: The Basics (Bloom’s Taxonomy) 🧠
Use these prompts to build and test your knowledge.
Remembering
Prompt Idea: What are the five top risks from the Aon study?
Prompt Idea: What is the definition of an “insurable risk”?
Prompt Idea: Name the joint distribution function of two uniform random variables.
Understanding
Prompt Idea: Explain the differences between assets and liability risk portfolio strategies.
Prompt Idea: Based on the foundations laid out in Chapter 2, describe how contractual forms are used to decompose a firm’s risk. Explain how a risk advisor uses these tools to summarize both the uncertainty of the risk being retained and the financial cost of the risk being transferred.
Prompt Idea: In Chapter 4, the textbook transitions from analyzing a single risk to managing multivariate risks. Explain why copulas are introduced at this stage and how they help a financial analyst understand and model the dependence among multiple insurable risks under budget constraints
Prompt Idea: Describe how the book adapts Markowitz investment asset allocation theory to the management of liabilities and insurable risk portfolios. Explain how this quantitative framework helps financial analysts provide risk managers with an objective, data-driven framework to supplement and corroborate traditional industry experience.
Applying
Prompt Idea: Using the actuarial R packages listed in the book’s preliminary setup (specifically library("actuar")), write a functional R script to calculate the expected retained claims (\(RC_1(u)\)) for a corporate risk modeled by a Gamma distribution at a specific retention limit \(u\). Test your code using the sample functions provided in the book’s introductory supplements.
Prompt Idea: Using the basic summary functions outlined in the textbook’s R code supplement, construct a customized R function to calculate the combined Additive Risk Transfer Cost (RTC) for two independent risk exposures. Ensure your program properly handles a portfolio where the first risk follows a Gamma distribution and the second follows a Pareto distribution.
Prompt Idea: Imagine you are hired as a financial analyst to advise a multi-line general insurer on their risk retention strategy. Identify which of the book’s primary datasets—the Australian National University (ANU) Case Study or the Wisconsin Property Fund Case Study—is designed for this specific industry application. Outline the steps you would take to load this dataset and apply its variables to model the insurer’s liability portfolio.
Analyzing
Prompt Idea: Using the basic retention functions defined in the textbook’s R preliminaries, analyze the structural differences between modeling an exposure with a Gamma distribution (\(RC_1(u)\)) versus a Pareto distribution (\(RC_2(u)\)) at a given retention limit \(u\). Break down how the tail behavior of each distribution impacts the expected retained claims, and explain how these differences subsequently alter the behavior of the Additive Risk Transfer Cost (\(RTC(u_1, u_2)\)).
Prompt Idea:Analyze the financial trade-off between the uncertainty of the risk a firm retains versus the cost of the risk it transfers. How does adjusting a contract’s retention limit mathematically alter both of these components simultaneously?
Prompt Idea:Analyze the structural differences between the Australian National University (ANU) Case Study and the Wisconsin Property Fund Case Study. How do their differing organizational roles (advising medium/large corporations vs. insuring multi-line general exposures) influence how you apply and interpret the Chapter 7 simulation strategy to their portfolios?
Evaluating
Prompt Idea: A risk manager decides to use only stress testing to evaluate the reliability of an optimal insurable risk portfolio, arguing that sensitivity analysis and robustness planning are redundant. Critique this decision. Under what specific portfolio conditions would stress testing fail to capture the structural vulnerabilities that the other two perspectives are designed to reveal?
Prompt Idea: For a corporate client with a highly constrained risk-retention budget, an analyst defends the choice of a Gamma distribution over a heavy-tailed Pareto distribution to model severe losses, claiming it produces a safer optimal frontier. Evaluate the validity of this claim. To what extent does choosing a lighter-tailed distribution compromise the safety of the optimal retention limits when the budget is tightly constrained?
Prompt Idea: Contrast the mathematically rigorous, analytical optimization method used in two-risk portfolios with the simulation-based strategy implemented for multivariate portfolios. Evaluate which approach is more defensible for a multi-line general insurer, such as the Wisconsin Property Fund, and defend your choice based on the trade-off between mathematical precision and real-world business applicability.
Creating
Prompt Idea:Formulate a customized risk-sharing contract for a corporate client facing a major risk modeled by a specific loss distribution (such as a Gamma or Pareto distribution). Your design must specify both a unique retention limit and a risk transfer mechanism that minimizes their total risk transfer cost under a strict, binding budget constraint.
Prompt Idea:Construct a step-by-step mathematical framework for resolving a multi-risk portfolio allocation problem that incorporates a copula to model dependency among the underlying risks. Outline how you would structure the objectives and constraints to map out a functional optimal risk-retention frontier under a fixed corporate budget.
Prompt Idea:Develop a comprehensive, data-driven risk advisory plan for either the Wisconsin Property Fund or the Australian National University (ANU) Case Study. Your plan must synthesize the empirical data, integrate the Chapter 7 simulation algorithm, and build a tailored portfolio strategy that addresses the unique organizational role of your chosen entity.
Welcome! As your teaching assistant for this course on constructing insurable risk portfolios, I would be delighted to help you adapt these learning prompts for your students. Focusing on these practical tasks is an excellent way to interact with the material, organize thoughts, and master the quantitative techniques.
1. Outline & Draft
Use these prompts to structure technical writing, design reports, or prepare for corporate presentations based on actuarial frameworks.
Prompt Idea: Create an outline for an essay on how modern peer-to-peer (P2P) risk exchanges and robo-advising platforms can be quantitatively modeled using the liability portfolio tools discussed in Chapter 6.
Prompt Idea: Draft a 5-point executive summary for an actuarial advisory report analyzing the optimal risk retention strategy for the Wisconsin Property Fund Case Study.
Prompt Idea: Generate a presentation script for a pitch to a corporate board of directors explaining why they should adopt the constrained optimization framework of Chapter 3 to balance the uncertainty of their retained risk against their risk transfer costs.
2. Summarize & Synthesize
These prompts will help you quickly digest dense mathematical concepts, compare methodologies, and visualize the relationships between different chapters of the textbook.
Prompt Idea: Summarize the key differences between managing exactly two risks analytically (Chapter 5) and solving the multivariate risk retention problem using the simulation-based strategy advocated in Chapter 7.
Prompt Idea: Synthesize the main findings of how the introduction of a copula-based dependence structure alters the optimal risk frontier compared to assuming complete independence (Chapter 4), in a single paragraph.
Prompt Idea: Create a table that compares the Australian National University (ANU) Case Study and the Wisconsin Property Fund Case Study in terms of their target audience, data structure, and real-world risk management applications.
3. Build a Glossary
Actuarial science and portfolio optimization have a highly specialized vocabulary. These prompts will help build customized reference tools to keep track of key terms, software packages, and mathematical formulas.
Prompt Idea: Find and list all the key portfolio optimization and risk retention terms (such as ‘optimal frontier’, ‘retained claims’, and ‘risk exchange’) and their definitions from the foundational material in Chapters 1 through 4.
Prompt Idea: Create a list of the specialized statistical R packages loaded in our preliminary setups (such as actuar, copula, and CVXR) and define what each package is used for in our portfolio calculations.
Prompt Idea: Find and define the core mathematical functions used for individual and combined risk calculations in our introductory supplements, including the retained claims functions RC1(u), RC2(u), and the Additive Risk Transfer Cost function RTC(u1, u2).
Use these prompts to test your knowledge and prepare for exams.
1. Generate a Quiz
These prompts allow students to test their baseline recall of the mathematical setups, distributions, and case studies covered in the foundations of the course.
Prompt Idea: Create a 5-question multiple-choice quiz on the organizational differences, target audiences, and data formats of the Australian National University (ANU) Case Study and the Wisconsin Property Fund Case Study.
Prompt Idea: Generate five true or false questions about the functions of the specialized R packages loaded in our preliminaries—such as actuar and copula—and their specific roles in solving multivariate risk portfolios.
Prompt Idea: Write a short-answer quiz to test my knowledge of how the expected retained claims functions (\(RC_1(u)\) and \(RC_2(u)\)) are mathematically constructed for Gamma and Pareto loss distributions.
2. Create Flashcards
Perfect for helping students memorize core actuarial terminology, mathematical notations, and dataset parameters before they sit for their exams.
Prompt Idea: Generate 10 flashcards on key concepts from Chapters 1 through 4, featuring terms on one side (such as ‘optimal frontier’, ‘risk decomposition’, and ‘retained claims’) and their definitions on the other.
Prompt Idea: Create flashcards summarizing the key characteristics of the three main datasets in the text, detailing what type of risk advisor or insurer each dataset represents.
Prompt Idea: Generate a set of flashcards with mathematical formulas on one side (such as the Additive Risk Transfer Cost\(RTC(u_1, u_2)\)) and their corresponding distributional assumptions and definitions on the back.
3. Simulate a Discussion
These role-play scenarios challenge students to synthesize their knowledge and practice communicating complex quantitative portfolio strategies to corporate boards or clients.
Prompt Idea: Act as a financial analyst advising a corporate risk manager. Explain to me why we must incorporate a copula-based dependence structure in our portfolio rather than assuming complete independence, using the budget constraints of Chapter 4.”
Prompt Idea: Debate the pros and cons of analytical two-risk optimization (Chapter 5) versus the multivariate simulation-based strategy (Chapter 7) from the perspective of an academic actuary and an on-the-ground risk broker.”
Prompt Idea: Role-play as a representative from a multi-line general insurer (like the Wisconsin Property Fund). Ask me to justify how using the book’s core Chapter 7 simulation strategy will help your organization set optimal retention limits under a strictly binding budget.
Remember: The more specific are your prompts, the better Gemini Notebook can assist you. Use this tool not just for answers, but to think critically and deeply about the material.
Use Gemini Notebook to help interpret mathematical expressions
Prompt Idea: “Interpret equation (1.1).”
If you ask a question in English, it is designed to respond in English. In the same way, if you ask in Spanish, it should respond in Spanish.
Google Colaboratory and Jupyter Notebooks
To deliver this course, we will utilize some resources that may be unfamiliar to some participants.
Google colaboratory (colab for short) is a cloud-based system of servers designed to process machine learning code.
We will use the free base system - you only need a (free) Google account.
Colab handles both R and python code. We will not use python for this workshop but it is a very handle tool for industry implementations.
Machine learning applications often depend upon large datasets and utilize computationally intensive algorithms - Colab is designed to accommodate these demands.
Jupyter notebooks provide a handy way to combine executable code, code outputs, and text into one connected file.
For this workshop, you will find links to the data embedded in our Jupyter notebooks (that you will retrieve on the fly). So, you will not need to download data in advance. However, should you be interested in experimenting further, here are some helpful sources of data:
In Google Colab, you will see a Gemini chatbot sitting on the right next to the code. I think you will find this very handy for helping you adjust and understand the purpose of the code.
If you want, you can also ask Gemini Notebook about the code.
Gemini Notebook only knows what is in the uploaded sources, it does not browse the internet.
Because there is ‘R’ code in the uploaded sources, it can give some insights in to ‘R’ questions.
However, the Gemini chatbot in Colab has more resources available and is especially aware of the Colab environment, so I recommend going to there first for ‘R’ questions.
So, why bother with Gemini Notebook?
An important feature is that it only knows what is in the uploaded sources, it does not browse the internet.
This means it can give responses with direct attribution in the uploaded sources, very handy.
In addition, because of its limited sources, it is much less likely to be prone to “hallucination” (making things up) when compared to more general chatbots.
Edward (Jed) Frees is a Fellow of the American Statistical Association and a former Fellow of the Society of Actuaries. He served as founding chairperson of the Society of Actuaries’ Education and Research Section as well as a member of the Board of Directors of the Society of Actuaries; he has also served as a Trustee of the Actuarial Foundation. Professor Frees served as an actuarial representative to the Social Security Advisory Board’s Technical Panel on Methods and Assumptions. Professor Frees was the Editor and is currently on Honorary Advisory Boards for the North American Actuarial Journal and the Annals of Actuarial Science. He is also an Editor for Insurance: Mathematics and Economics.