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No. 144: MIES – Intertemporal Choice

26 July, 2020 10:11 PM / Leave a Comment /

This entry is part of a series dedicated to MIES – a miniature insurance economic simulator. The source code for the project is available on GitHub.

Current Status

This week, I’ve been continuing my work on incorporating risk into the consumer behavior component of MIES. The next step in this process involves the concept of intertemporal choice, an interpretation of the budget constraint problem whereby a consumer can shift consumption from one time period to another by means of savings and loans. The content of this post follows chapter 10 of Varian.

For example, a person can consume more in a future period by saving money. A person can also increase their consumption today by taking out a loan, which comes at a cost of future consumption because they have to pay interest. When making decisions between current and future consumption, we also have to think about time value of money. When I was reading through Varian, I was happy to see that many of the concepts I learned from the financial mathematics actuarial exam were also discussed by Varian – such as bonds, annuities, and perpetuities – albeit in much less detail.

This inspired me to create a new repo to handle time value of money computations, which is not yet ready for its own series of posts, but for which you can see the initial work here. I had intended to make this repo further out in the future, but I got excited and started early.

Also relevant, is the concept of actuarial communication. Now that I’m blogging more about actuarial work, I will need to be able to write the notation here. There are some \LaTeX packages that can render actuarial notation, such as actuarialsymbol. Actuaries are still in the stone age when it comes to sharing technical work over the Internet, not out of ignorance, since many actuaries are familiar with \LaTeX, but out of corporate inertia in getting the right tools at work (which I can suppose be due to failure to persuade the right people) and lack of momentum and willingness as many people simply just try to make do with using ASCII characters to express mathematical notation. I think this is a major impediment to adding rigor to practical actuarial work, which many young analysts complain about when they first start working, as they notice that spreadsheet models tend to be a lot more dull than what they see on the exams.

I was a bit anxious in trying to get the actuarialsymbol package working since, although I knew how to get it working on my desktop, I wasn’t sure if it would work with WordPress or Anki, a study tool that I use. Fortunately, it does work! For example, the famous annuity symbol can be rendered with the command \ax{x:\angln}:

Rendered by QuickLaTeX.com

That was easy. There’s no reason why intraoffice email can’t support this, so I hope that it encourages you to pick it up as well.

The Statics Module

Up until now, testing new features has been cumbersome since many of the previous demos I have written about required existing simulation data. That is, in order to test things like intertemporal choice, I would first need to set up a simulation, run it, and then use the results as inputs into the new functions, classes, or methods.

That really shouldn’t be necessary, especially since many of the concepts I have been making modules for apply to economics in general, and not just to insurance. To solve this problem, I created the statics module, which is named after the process of comparative statics, which examines how behavior changes when an exogenous variable in the model changes (aka all the charts I’ve been making about MIES).

The statics module currently has one class, Consumption, which can return attributes such as the optimal consumption of a person given a budget and utility function:

A lot of the code here is the same as that which can be found in the Person class. However, instead of needing to instantiate a person to do comparative statics, I can just use the Consumption class directly from the statics module. This should make creating and testing examples much easier.

Since much of the code in statics is the same as in the Person class, that gives me a hint that I can make things more maintainable by refactoring the code. I would think the right thing to do is to have the Person class use the Consumption class in the statics module, rather than the other way around.

The Intertemporal Class

The intertemporal budget constraint is:

    \[c_1 + c_2/(1+r) = m_1 + m_2/(1+r)\]

Note that this has the same form as the endowment budget constraint:

    \[p_1 x_1 + p_2 x_2 = p_1 m_1 + p_2 m_2 \]

With the difference being that the two endowment goods are now replaced by consumption in times 1 and 2, represented by the cs and the prices, the ps are now replaced by discounted unit prices. The subscript 1 represents the current time and the subscript 2 represents the future time, with the price of future consumption being discounted to present value via the interest rate, r.

The consumer can shift consumption between periods 1 and 2 via saving and lending, subject to the constraint that the amount saved during the first period cannot exceed their first period income, and the amount borrowed during the first period cannot exceed the present value of the income of the second period.

Since the intertemporal budget constraint is a form of the endowment constraint, we can modify the Endowment class in MIES to accommodate this type of consumption. I have created a subclass called Intertemporal that inherits from the Endowment class:

The main difference here is that the Intertemporal class can accept an interest rate and an inflation rate to adjust the present value of future consumption.

Example

As an example, suppose we have a person who makes 5 dollars in each of time periods 1 and 2. The market interest rate is 10% and their utility function takes the Cobb Douglas form of:

    \[u(x_1, x_2) = x_1^{.5} x_2^{.5}\]

which means they will spend half of the present value of the endowment as consumption in period 1:

The main thing that sticks out here is that the slope of the budget constraint has changed to reflect the adjustment of income to present value. The x-axis intercept is slightly less than 10 because the present value of income is slightly less than 10, and the y-axis intercept is slightly more than 10 because if a person saved all of their time 1 income, they would receive interest of 5 * .1 = .5, making maximum consumption in period 2 10.5.

Since the person allocates half of the present value of the endowment to time 1 consumption, this means they will spend (5 + 5/1.1) * .5 = 4.77 in period one, saving 5 – 4.77 = .23, which then grows to .23 * (1 + .1) = .25 in period 2, which allows for a time 2 consumption of 5 + .25 = 5.25. This is verified by calling the optimal_bundle() method of the Consumption class:

Further Improvements

The Varian chapter on intertemporal choice briefly explores present value calculations for various payment streams, such as bonds and perpetuities. I first made a small attempt at creating a tvm module, but quickly realized that the subject of time value of money is much more complex than what is introduced in Varian, since I know that other texts go further in depth, and hence it may be necessary to split a new repo off from MIES so that it can be distributed separately. This repo is called TmVal, the early stages of which I have uploaded here.

Neither of these are ready for demonstration, but you can click on the links if you are interested in seeing what I have done. The next chapter of Varian covers asset markets, which at first glance seems to just be some examples of economic models, so I’m not sure if it has any features I would like to add to MIES. There is still more work to be done on refactoring the code, so I may do that, or move further into risk aversion, or do some more work on TmVal.

Posted in: Actuarial, MIES / Tagged: , , ,

No. 143: MIES – Endowments

19 July, 2020 11:57 PM / Leave a Comment /

This entry is part of a series dedicated to MIES – a miniature insurance economic simulator. The source code for the project is available on GitHub.

Current Status

Last week, I took a break from MIES to focus on PCDM, a relational database specification for the P&C insurance industry. This week, I’m back to making progress on the consumer behavior portion of MIES, by shifting the focus from personal income to the personal endowment as the main financial constraint underlying purchasing decisions.

In short, an endowment is the consumer’s assets. When making consumption choices, people can use their income to purchase goods and services, but they can also draw from assets that they have accumulated over time, such as from savings and checking accounts, and by selling goods that they own. Furthermore, by taking the endowment into consideration, we will now be able to model situation when a person might not have a regular income, but can still make purchases using their assets (such as unemployed or retired persons who are not working).

In the context of insurance, the endowment is important because people purchase insurance to indemnify themselves against events that might damage or reduce the value of their assets. In the absence of the endowment, we would ignore an important determinant of insurance purchasing behavior. Incorporating wealth into MIES will take some time, and the textbook material I need to work on spans five chapters of Varian. Therefore, I estimate that this process will take me at least a month to do:

  1. Endowment
  2. Intertemporal Choice
  3. Asset Markets
  4. Uncertainty
  5. Risky Assets

These concepts will involve making some substantial changes to the utility functions as well. For now I’ll start with the endowment, which required me to modify the Budget, Slutsky, and Hicks classes of MIES.

The Endowment Class

An endowment is a bundle of goods or services that has a value based on the sum product of their prices and quantities:

    \[p_1 \omega_1 + p_2 \omega_2 = m \]

Where m represents income, each omega represents the quantity of each good, and each p represents the price. Rather than treat income as a flow quantity from an external source, in this interpretation of consumer choice theory we, we treat income as a stock quantity that includes the assets of of the consumer – that is, what the consumer has to spend at a certain point of time depends on the valuation of their assets.

This definition of income loosens the assumption of fixed income that I had made until now. This is because changes in asset values can now impact a person’s income. For example, if a person has a car and a house, their depreciation or appreciation changes the amount the person can sell them for on the market.

The good news is that MIES already has much of the machinery already coded up to allow us to work with endowments, so the new class definition is quite simple:

An endowment takes two goods, and their quantities. Upon initialization, Python will automatically calculate the Endowment’s value by multiplying the prices of the goods by the quantities supplied. I wrote this function as a property decorator, which was introduced to fix a bug I discovered when working with the Budget class. Earlier, changing the price of a good failed to change the budget constraint of a consumer, but the property decorator will now dynamically calculate certain attributes that depend on the price, such as income in the case of an endowment.

To illustrate, we can define two goods, each with a price of 1. We then initialize an endowment with a quantity of 5 for each of these goods:

Now we can check that the income was properly calculated by calling endowment.income. Since each good has a price of 1, and there are 5 of each good, the income should be 5 x 1 + 5 x 1 = 10:

Now that we have the Endowment class defined, we need to modify the other classes that used goods, such as the Budget class. Previously, the Budget class accepted two goods, an income amount, and a name to refer to the budget. Now I would like the Budget class to an accept an endowment as an alternative to specifying each good individually. The tricky part here is that in the former case, the class needs to be able to keep income fixed when the prices of goods change, but in the latter case, the income needs to change dynamically based on the prices of the goods.

To handle this, I created an alternative constructor called from_endowment() that lets you pass an endowment to the Budget class to initialize a budget object. I also created another constructor called from_bundle() that lets you define a Budget the old way more explicitly, to make it more obvious to anyone reading the code whether the budget was initialized with an endowment or individual goods:

And lastly, I added some consistency checks to make sure that the endowment value equals the sum product of the prices and quantities of the goods provided. The reason why these checks are here is because a person can still use the default constructor to specify each good individually along with an endowment, just based on how the arguments are defined. While this is possible, I would discourage doing this since 1) it’s less explicit than using the alternative constructors, 2) supplying the individual goods along with the endowment is redundant, and 3) it can lead to errors being thrown.

To initialize a budget by passing an endowment, simply use the alternative constructor:

Slutsky Decomposition

The loosening of assumptions brought about by the endowment introduces some changes to the Slutsky equation. In the examples I provided a few weeks ago, we assumed that income remained fixed when prices changed. Since changes in price now change the value of the endowment, we must now account for this change in the Slutsky equation. The derivation of this modified form can be found in Varian, so I’ll skip to the result:

    \[\frac{\Delta x_1}{\Delta p_1} = \frac{\Delta x_1^s}{\Delta p_1} + (\omega_1 - x_1)\frac{\Delta x_1^m}{\Delta m}\]

The Slutsky equation can now be explained by three effects: the substitution and ordinary income effects, which are the same as before, and an endowment effect, which models how consumer choice changes when the value of the endowment changes.

Like the Budget class, the Slutsky class has been modified to take budgets that were constructed from individual goods or an endowment. Plotting the Slutsky class is now quite bit messier, since a new budget line, bundle, and utility curve are now added to an already crowded plot.

I have not yet gotten endowments to work within the context of insurance, so the image below comes from a modified version of an example provided in Varian where a milk producer faces a $1 increase in the price of milk – his endowment increases in value, and hence income. However with the graph as cluttered as it is, it can be hard to visually isolate the effects:

It does look better with a larger plotting area if you try it with MIES, but not so much when I have to shrink the image to fit it within the margins here.

Posted in: Actuarial, MIES

No. 142: The Property Casualty Data Model Specification

12 July, 2020 10:31 PM / 4 Comments /

Introduction

A few weeks ago, I stumbled across something neat – the Property Casualty Data Model (PCDM). PCDM is a relational database specification that covers all major parts of an insurance company’s operations. At first glance, the web page on which it is located seemed mundane to me, so I almost overlooked it, but when I opened the accompanying documentation, I realized I had stumbled upon a goldmine of useful information. This document contains enough information to implement an entire data warehouse and then tweak it to an organization’s specific needs.

Although we actuaries have a reputation for being able to handle data, most of us have not received any kind of formal training in handling relational database systems, and have had little interaction with standards organizations outside of our own governing organizations like the CAS, SOA, and AAA. When I tried searching for PCDM in the CAS library, I was astonished to find just a handful of references to the specification, and flabbergasted that there had been a document floating around on another profession’s website for the last seven years that almost no actuaries had ever heard about, but contained the exact type of information that many actuaries wanted but thought had never existed in the public domain (and in my case, it contains more than what I need for parts of the MIES backend).

One benefit that I’ve had from working several jobs, and also as a consultant, is that I’ve been able to witness widely varying levels of maturity of data warehouse systems at commercial insurance companies of different sizes and functions (specialty, commercial, and reinsurance) as well as those of a major stock exchange to get some perspective of how insurance database implementations compare to those of other industries. I have seen many actuaries at small and midsize carriers struggle with how to create OLAP data warehouses, not knowing what tables, relationships, and fields to define, how to do it in a way that conforms to commonly accepted data management practices (if they even knew what they were), or even where to look or whom to talk to to find out what data are stored at the company, and in what form.

Earlier in my career, at a small insurer, a few years before this document existed, I would sometimes encounter databases with hundreds of undocumented tables, not knowing what any of the tables stored, if they were the right tables I needed, or if they were designed appropriately. I was one of the few actuaries in my department who knew SQL (and even then, I wasn’t good at it). So, with my introductory textbook on database management systems at my side and a landline I used to contact people in the IT deparment, I embarked upon the long journey of understanding insurance information systems.

In those days, the game kind of went like this. I would ask my boss if they knew anyone in IT, then I’d pick up the phone and call them to ask if they maintained the database I was looking at or knew of anyone who did or anyone who might know anyone who did. That chain of phone calls typically went 5 people deep until I finally reached someone who actually worked on the relevant database, and if I was lucky, they’d have some documentation, and I’d slowly figure out what kind of data I was working with.

It might have looked like this. What the hell? Boss, this is gonna take me a while to understand. (Actually this is PCDM, but the good news about PCDM is that it’s documented – but imagine what it would be like if it weren’t and the last person who touched it left 3 years ago).

When I moved to a larger organization, I came to realize how large of a knowledge/talent gap there was between companies when I actually got to use a well-documented data warehouse that had all the entities, relationships, and fields defined as well a data dictionary detailing what all the values were – a rare practice that even many established insurers don’t follow. Unfortunately, when I left that organization, I also lost access to that documentation, and with it, a wealth of knowledge on what a proper data warehouse should look like.

I had thought to myself if only the CAS had a standard that actuaries and students could access, we’d have something that would actually resemble what a database at a real insurer actually looked like, instead of just having just theoretical papers from which to learn that might have some code copy-pasted in. If we had a standard, knowledge of proper database design and implementation would then not be employer dependent, and new research papers could reference the standard rather than just imagining what a data warehouse would look like or omitting proprietary parts of a company’s database.

When I came across PCDM, I realized the significance of a document I randomly found while eating breakfast on holiday. Therefore, I immediately got to work in writing an implementation of it in Python so that other actuaries could finally have what I wish I had when I was younger. I completed the first release at the end of the July 4th weekend, and made it available on GitHub. My hope is that somebody out there finds this post and clones my repository – and understands how important it is to spread word of this specification.

Subject Area Models

PCDM details 13 subject area models (SAMs), each of which comes with an entity-relationship (ER) diagram. Each subject area model represents a major portion of an insurance company’s operations:

  1. Party
  2. Account and Agreement
  3. Policy
  4. Claim
  5. Assessment
  6. Agreement Role
  7. Claim Role
  8. Staffing Role
  9. Party Subtype
  10. Insurable
  11. Money
  12. Event
  13. Product Coverage

For example, the Party SAM contains information on all parties that are involved in insurance transactions, such as households, employees, vendors, etc. Other SAMs contain information on underwriting, claims, and accounting operations. The rest of this post will show you what those SAMs look like – the images are crowded and contain quite a bit of information, so it’s best to click on them to see them in full resolution. If you want more details, the PCDM document is available on the object management group website.

Party

As stated before, the Party SAM contains information about households, firms, and staff:

Account and Agreement

The Account and Agreement SAM contains information on legal agreements, such as policies, reinsurance contracts, etc.

Policy

The Policy SAM contains information on…policies. Policy number, effective date, expiration date, limits, deductibles, coverage, etc.

Claim

The Claim SAM contains information on claims, relevant dates, adjusters, lawyers, damage amounts, etc.

Assessment

The Assessment SAM contains information on how the insurer goes about gathering information, such as credit scores, appraisals, and investigations carried out during the claim adjustment and underwriting processes.

Agreement

The Agreement Role SAM contains information on the different types of parties that might be involved in insurance – mostly an expansion of the Party Role superclass found in the Party SAM.

Claim Role

The Claim Role SAM contains information on the parties involved in the adjustment process, such as adjusters, claimants, lawyers, etc.

Staffing Role

The Staffing Role SAM contains information on insurance company staff and contractors.

Party Subtype

The Party Subtype SAM contains information on company subdivisions.

Insurable Object

The Insurable Object SAM contains information on things that can be insured, like cars and buildings.

Money

The Money SAM contains information on transactions, like premium, loss payments, and case reserves.

Event

The Event SAM contains information on policy and claim events, like policy inception, cancelation, and claim occurrence.

Product Coverage

The Product Coverage SAM contains inforamtion on product types and coverages.

Python/SQLAlchemy Implementation

I figured if I want to claim credit for anything besides just copying and pasting the diagrams above from the specification, I should actually make some kind of contribution. The good news is I haven’t found any kind of implementation on GitHub – what I mean by that is the PDF is nice to have and all, but you can’t deploy a datawarehouse with the PDF, it needs to be translated into code which people can then use to deploy the warehouse. Therefore, I’ve written my own implemntation of PCDM in Python using the SQLAlchemy ORM.

Below is an example of how I translated the Party SAM into code. I did this for the 12 other SAMs as well, and then wrote another module to combine them all together into a single deployable data warehouse:

I have taken a few liberties in my own implementation, such as creating surrogate keys whenever I came across a composite identifier, which should make joining easier. I’ve also chosen my own data types for ambiguous fields (such as integers for identifiers and strings for descriptions) based on my own interpretation, and chose to represent superclass/subclass relationships by making one table for each superclass and subclass, which can be joined with primary/foreign key relationships.

Installation and Deployment

To use PCDM, you need to install SQLAlchemy if you don’t already have it:

You can install and deploy PCDM by cloning the repo from my GitHub. The deployment script is displayed below:

You can either run this script, or use the terminal to clone and then deploy:

If the deployment succeeds, you should see a SQLite database appear with 256 tables in it:

Although SQLite is the default, you can use this repo to make your own deployments for other RDBMSs (Postgres, SQL Sever, etc.). I have not made these ports yet, but I’m assumming if you have read this far, you’re a technically savvy person who can help me out by writing your own.

Documentation

I am working on making my own documentation, which is not yet available. But for now you can refer to the official OMG PCDM document.

Bugs

Warning – this repo is open source, and hence contains no warranty and may contain bugs. You can help contribute to the effort by making an issue or pull request should you encounter a bug.

Posted in: Actuarial / Tagged: , , , ,

No. 141: MIES – Premium and Claim Transactions

5 July, 2020 10:28 PM / Leave a Comment /

This entry is part of a series dedicated to MIES – a miniature insurance economic simulator. The source code for the project is available on GitHub.

Current Status

Up until now, I’ve been able to demonstrate basic consumer behavior and certain market phenomena seen in the insurance industry. However, there’s been a major problem with MIES in that no money has actually changed hands in any of the simulations we’ve seen so far. Consumers have simply switched carriers depending on price, which allowed me to demonstrate adverse selection, but not much else. Consumption decisions involving insurance depend on wealth, but since there was no way to calculate wealth in MIES, its ability to model these decisions was limited.

The next view chapters in Varian place a heavy emphasis on wealth and risk tolerance, so this week, I made the decision to work on incorporating transactions before diving deeper into consumer behavior.

At first glance, transactions might seem like a simple thing to implement, after all, why not just keep a running cash balance for each entity, and then add and subtract payments as needed? The problem with this method is the same problem that leads companies to use double-entry accounting. Transactions are more complicated than simply sending money from one place to another. Loans are generated and capital is invested, which creates liabilities that must be considered when trying to calculate the wealth of an entity. Even something as simple as a premium payment is effectively a loan to an insurance company that needs to have a liability recorded (the unearned premium reserve), in addition to an increase in cash to the insurer.

Therefore, I’ve had to draw on my basic knowledge of accounting, which made me uneasy since I’m not an accountant myself. However, in order to get MIES to model the phenomena I want to model, and to answer the questions I have about insurance, I need to implement double-entry accounting, and eventually, statutory accounting rules. I ran across a post on Hacker News titled, ‘It’s OK for your open source library to be a bit shitty,’ which encouraged me to keep moving forward with the project despite the amount of discomfort I have.

I have certainly found many errors in MIES from past versions and there will likely be many more, including in this post. However, there’s not a lot of open source actuarial stuff out there, or in particular, open source actuarial simulations incorporating both economics concepts and double-entry accounting. Or, if there are packages out there, they aren’t easy to find. Thus, I’ve taken the step to put something out there, awaiting any feedback for things that need to fixed, and then making improvements. If this ever proves to be something useful, younger generations will create even better tools in the future.

The Bank Class

Banks facilitate transactions between insurers, brokers, customers, and other banks. While it may eventually be possible to define more than one bank per simulation, all the examples in the near future will have a single bank at which entities can deposit money and send payments to each other.

Schema

As with all the other entities, each bank comes with its own database:

Here, a bank can accept three types of customers. The customer table represents the customer superclass and references the person, insurer, and bank subclasses. Each customer can have an optional number of accounts. Any of these customers can send transactions to any other customer, including themself. Notice that the transaction table has two relationships to the account table, since the debit and credit account fields for each transaction both point to the account table.

The SQLAlchemy mapping is defined below:

Bank Methods

Upon initialization, a bank will register itself as a customer, this allows it to have its own accounts, as well as to accept deposits from other customers. The reason why it needs to have its own accounts is because each transaction will need to have corresponding debit and credit accounts, and transactions between a bank and its customers will sometimes have one of the bank’s own accounts involved.

The method bank.get_customers() takes a list of IDs, which can be either those of people, insurers, or banks. For each of these entity IDs, the bank will create its own identifier for the customer, which can differ between that customer’s underlying ID. For example, a person with a person ID of 1, and an insurer with an insurer ID of 1, will have separate customer ids.

The method bank.assign_accounts() takes a list of customer IDs, a type of account (such as cash), and then creates that type of account for each customer.

The method bank.make_transactions() will take a list of transactions, each of which have a debit account, credit account, transaction date, and transaction volume defined, and then store them in the transactions table.

Now that we have our bank defined, I’ll walk through the insurance underwriting/claim cycle and show how the transactions that are currently available in MIES work. First, we’ll import the necessary modules, create an environment, make a population of 1000 people, and then create a bank called ‘blargo’ that has 4B in starting capital:

Starting Wealth

Before we can issue policies to customers, we need to have a starting amount of wealth for each person so that they can actually pay for their policies. Furthermore, we also need each person to have a bank account from which they can issue payments to their insurers. To take care of these two steps, we’ll first send the person IDs to the bank, which will then create corresponding customer IDs and then establish one cash account for each customer:

This action populates two tables. The person table contains a record of all 1000 people, and the customer table has 1001 records, since each person becomes a customer, but the bank itself is already its own customer:

In the next step, we’ll use a new method called grant_wealth() which the environment uses to give each person a starting amount of wealth. Since wealth is not evenly distributed in society, I’ve drawn these values from the Pareto distribution:

The method is defined as follows:

This action deposits wealth in each person’s cash account. This is marked as the debit side of the transaction, the credit side is the liability that the bank takes on by accepting the deposits, since deposits are loans to the bank:

Policy Inception

Before we issue policies, we need to create two more entities in the simulation. One, a broker called ‘rayon’ and an insurer called ‘company_1.’ Initializing a company now requires two new arguments, a bank with which it associates, and its inception date. These new arguments are used to create bank accounts for the insurer:

We’ll now use the broker rayon to place the customers with the insurer. This method also has a bank as a new argument, which will be used to facilitate the transactions:

This action creates a policy record for each insurer, which I won’t show here since I’ve already done so in a previous post. What has changed however, is that each person needs to pay the premium to the insurer up front. The broker is able to tell the bank to create a transaction for each premium payment. This time, the debit account is the insurer’s cash account, and the credit account is the person’s cash account. In the picture below, we see that there are 1000 additional transactions starting with transaction_id 1003 (1002 was used to seed insurer capital). The debit account on all these transactions is account ID 1004, which is the insurer’s cash account:

In reality, a corresponding liability should also be created in the insurer’s accounting system. This is called the unearned premium reserve which is what the insured is entitled to recieve if their policy gets canceled. This feature is not yet implemented in MIES, but it’s an important liability to consider in insurance.

Loss Occurrence

Now that we have our policies issued, we’re ready to simulate some losses:

This action produced 57 loss events, which can be found in the events table in the universe database:

Claim Reporting

These losses are not considered claims until they are reported to the insurer. Otherwise, the insurer has no knowledge that they occurred:

This method has changed from last week. The loss amounts are now reported as case reserves, which are estimates made by an insurer on how much they will need to pay for the claim. This is now distinguished from paid losses, which are the actual payments the insurer makes to the insured:

This action creates two transactions for each claim. One transaction, called ‘open claim’ signals that a claim has been created. Another transaction, called ‘set case reserve,’ sets a case reserve for each claim. Since there are 57 losses, there are 57 claims, and 114 transactions:

Notice that we have some fairly restrictive assumptions for the simulation. The case reserves are equal to the ground up losses, and all 57 losses are reported to and known by the insurer immediately. This is not the case in the real world, where an insurer does not know how much claims will cost until they are settled, and may not know about claims until many years after they have occurred. We’ll need to revisit this problem in the future, since estimating claim amounts, including those on unreported claims, is a core function of actuarial science.

Claim Settlement

Let’s close these claims by issuing payments:

Insurer.pay_claims() is a new method used to send checks to the insureds for indemnification:

There’s a lot going on here, so I’ll break it down. The payments to the insureds are handled first. We can see this by going to the transactions table in the bank database:

Notice that there are 114 additional transactions, two for each of the 57 claims. One transaction sends a payment from the insurer to the customer. You can see this since the debit account is the cash account of the customer, and the credit accont (1004) is the cash account of the insurer. The other transaction is a payment from the insured to whomever they owe money to due to the loss. The debit side of the transaction is now a reduction in the bank’s liability account, and the credit amount is a reduction in cash equal to the claim amount for the insured:

Next, three transactions are entered for each claim into the insurer’s database:

  1. Reduce case reserve
  2. Claim payment
  3. Close claim

‘Reduce case reserve’ is a reduction in the claims reserve to zero, signifying that the insurer no longer owes money to the insured. The ‘claim payment’ is a corresponding transaction representing the actual payment, and ‘close claim’ is a transaction that indicates that the claim is now closed. The insurer now has 5 claims transactions for each claim, 57 x 5 = 285 transactions in total. These five transactions are: 1) open claim, 2) set case reserve, 3) reduce case reserve 4) claim payment, and 5) close claim.

One source of confusion I had is that this way of recording transactions does not have the double-entry accounting that you’d see in a general ledger. Indeed, this form is more common for actuarial pricing modelers who do not usually need to get into the finer details of debits and credits. However, I’m leaning towards changing the claims transaction tables to also be double-entry, since doing so also makes things easier to program and avoids questions with negative values. For example, I had to think about whether to record case reserve reductions as a positive or negative value. This confusion is not present if I record them as credit and debit amounts.

Consumer Income

Each person now gets a paycheck during each period. This can be issued by the environment class:

There’s nothing too special about this method, it just debits each person’s cash account by their income amount:

An additional 1000 transactions have been recorded, one for each customer. As with wealth, the debit side of the transaction is the person’s cash account, with a corresponding credit to the bank’s liability account:

Policy Pricing

Now that the claims have been reported and settled, the insurer can use this information to recalibrate the premium for each customer. However, unlike last week, claim amounts are not stored as a single column called ‘incurred_loss’, but now must be calculated from the transaction amounts. To handle this, I created a set of queries that can be used to return the case reserves, paid losses, and incurred amounts for each claim:

We mostly need to be concerned about the last one of these, query_pricing_model_data() which uses the first three to combine policy and claim information together, which can be priced with a GLM:

Since most people don’t have a claim, most incurred loss amounts are zero. This data set can then be used by the insurer to reprice with a GLM, and the new pricing algorithm is used by the broker to quote and place business:

Further Improvements

Now that I’ve got transactions modeled, I can calculate the wealth for each entity in the simulation. This is key piece required to make further changes to the way consumer preferences work in MIES, which will soon incorporate risk tolerance and wealth.

Posted in: Actuarial, Mathematics, MIES