Category Archives: Actuarial

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 specified a Cobb Douglas utility curve for each person in MIES. I also demonstrated a situation in which a person might choose to not fully insure. However, I’ve gone a few chapters ahead in my readings and found out that under certain assumptions, a risk-averse person who is offered a fair premium will choose to fully insure. In MIES, since each company charges the pure premium without loading for profit or expenses, each person is getting a fair premium – so there’s something missing from my current model that makes it inconsistent with economic theory.

Risk aversion is not yet implemented in MIES, and will have to wait a few weeks before I get to it, since there’s quite a bit of work to do. But I’m mentioning the issue here, just in case someone reading this knows more about the subject than I do.

This week, I’m going to demonstrate a set of tools to examine consumer choice – the offer, Engel, and personal demand curves. I don’t recall using the first two curves very much in my economics courses, but the latter will be very important and will serve as a bridge between personal demand and market demand. Surprisingly, these curves were very quick to implement, since they all rely on the same method I wrote last week for the Cobb Douglas class.

The topic of this post roughly corresponds to chapter 6 of Varian.

Offer Curve

The offer curve for a consumer depicts their optimal consumption bundle at each level of income. Since the offer curve is unique to a particular consumer, I decided to define the methods that generate and plot the offer curve within the Person class. Luckily, the CobbDouglas class that I defined last week has a method called optimal_bundle, which returns the optimal consumption bundle given a set of prices and income. Since this is exactly what we need given the definition of the offer curve, we can simply use this method to generate each person’s offer curve:

Note that while I only have one utility curve defined in MIES at the moment (Cobb Douglas), the definition of the offer curve doesn’t need to have anything specific to the Cobb Douglas utility function. This means in the future, I should be able to abstract this method to accept other utility functions without too much modification.

I’ve also added a method to plot a person’s offer curve:

This method takes a preset range of income values, and uses the get_offer method to plot the optimal consumption bundle for each income value in the range. For example if we’ve already run a few iterations of a market simulation, we can examine what combinations of insurance and non-insurance a person can afford at different income levels. Let’s do this for the person with id=1:

Imagine what would happen if you were to shift the blue budget line inward and outward. The optimal consumption bundle would the the point of tangency with the corresponding utility function. We can see that the orange offer curve is the set of all these points.

Engel Curve

The Engel curve is similar to the offer curve, but plots the optimal choice of a good at various levels of income. Its definition within the Person class is also similar, except we only need to return the first good of the optimal bundle:

Let’s see what the Engel curve looks like for person 1:

Demand Curve

The demand function depicts how much of a good a person would buy if it were at a certain price. This one’s important since we’ll need it to derive industry demand, which will then be used to answer many fundamental questions about the insurance market. Like the other curves, defining this one was simple, we just get the optimal bundle at each price and return the quantity demanded of the first good:

Let’s see what the demand curve looks like for person 1:

Note that the demand curve slopes downward as it should, since we’d expect a person to buy more insurance the cheaper it is. However, note that there is no price such that the demand equals zero. The demand curve asymptotically approaches zero as the premium increases, but this particular person will never go uninsured. This is due the property of the Cobb Douglas utility function that the exponent of the good equals the percent of income spent on that good, which is hard coded as 10% at the moment. However, in the real world people do go uninsured, and this is a subject of great interest to me, so we’ll need to revisit this later.

Further Improvements

I’ve added quite a few features to the person class, but I haven’t integrated them to the point where I can perform more than two market simulations. I’m also several chapters ahead in my readings than what I’ve posted about, and I’ve encountered an interesting demonstration on risk aversion and intertemporal choice concerning assets, which will take quite an effort to both implement and reconcile with what I’ve written so far.

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 demonstrated how MIES can be used to calculate the budget constraint for each insured in the marketplace. This answered the question – how much insurance can each person afford during each underwriting period?

Although we now have the set of all possible consumption bundles that each person can afford, we still have no method for determining what bundle they will ultimately select – that is, how much insurance will each person actually buy? To answer this question, we turn to the concept of utility. Utility is a measurement of the satisfaction a person receives from a course of action, and we will assume that rational people seek to maximize their utility under scarcity. In this case, the course of action is purchasing insurance, and people seek to maximize their utility subject to the constraint of what they can afford.

Thus, for today’s post I will demonstrate how MIES can assign a utility function to each person in the market and determine each how much insurance each person will purchase.

By introducing utility, I would eventually like to answer certain questions I had about the insurance industry when I thought about building MIES. None of these will be answered today, but this should take us one step closer:

1. How much insurance will be purchased in total at market equilibrium?
2. Who will go uninsured?
3. Should there be a mandatory minimum amount of insurance required by law? If so, should there be a state-run high-risk pool? And if so, how should it be funded?

More detailed treatment of utility can be found in chapters 4 and 5 of Varian.

Utility

To model each person’s preferences for insurance consumption, I’ve decided to use the Cobb-Douglas utility function:

Where c and d represent the percentage of income spent on goods and when , and each x represents the quantity of each good. While other utility functions may eventually prove to be more realistic for our simulation, Cobb-Douglas utility functions are a good candidate to start with since they have many convenient features. For example, in order to find the optimal consumption bundle for a person, we need to find the bundle of goods such that the marginal rate of substitution (MRS) equals the slope of the budget constraint, while satisfying the budget constraint itself. For the Cobb-Douglas utility function, Varian provides a derivation for the MRS:

This can then be used with the budget constraint, , to solve for the quantities of and given the income and price of each good:

and, when , and . This means that using this result, we can find the optimal consumption bundle algebraically. This is very useful since 1) I have simply forgotten a lot of calculus since leaving school and 2) I won’t have to program calculus into MIES for the time being.

The convexity of the Cobb-Douglas utility function also satisfies certain assumptions underlying consumer behavior. Chapter 3 of Varian provides an in-depth explanation of these assumptions.

Utility Module

I hinted last week that I thought it might be a good idea to break up the econtools.py module into more specific modules. I’ve decided to do this to improve readability and to keep things organized. I’ve created a new folder called econtools to house these modules. The Budget class is now in the budget.py module and the utility function classes are now in a file called utility.py.

The utility.py module contains a single class, called CobbDouglas:

The CobbDouglas class takes two arguments, c and d, which correspond to the and parameters in the function definition. The class provides three methods: optimal_bundle() calculates the optimal consumption bundle using the results derived by Varian, trace() defines the curve as it will appear when plotted, and show_plot() plots the utility function.

Here’s an example on how to use the class to plot the function:

Here, we’ve provided a simple example where c = d = .5, which means that the consumer will allocate 50% of their income to each good. If the price of each good is the same, the line connecting all optimal consumption bundles will go through the origin (more on that for a later post).

MIES Integration

The Person Class

Now that we’ve got a class defined for our Cobb Douglas function, we now need to find a way to get it working with MIES. I’d like to be able to access the utility function for each consumer in the simulation. Since I’ve started to examine more person-specific characteristics, I’ve created a new class in the entities.py module, Person:

Since there is already a Person class in the schema.py module referencing a SQLite table, I’ve renamed that class to PersonTable. The Person class takes a database connection, along with the person table in the database, queries the details of a person, and generates a utility function for that person. The Person class provides additional methods for attaching policy information, calculating that person’s budget, and determining how much insurance they will buy.

To keep things simple, I’ve assumed that each person’s parameter equals, .1, which means that the population has homogeneous preferences for insurance, and will spend 10% of their income on it. The parameters.py module and environment class have been updated accordingly to reflect this. We can loosen these assumptions later on.

Optimal Consumption

Let’s find one person’s optimal bundle for purchasing insurance. To do this, we run two iterations of the simulation and examine the results for person_id = 1, like we did last week:

This person has an income of about 32k. You can also see additional columns for their Cobb Douglas parameters. If all goes well, we would expect them to want to spend about 3200 on insurance. Let’s query their renewal quote to see how much premium they need to pay:

Since their premium is about 8k, we’d expect them to consume roughly 3.2/8 = .4 units of insurance upon renewal. Let’s solve for their optimal bundle to confirm:

Indeed, the point of tangency between the budget constraint and red utility curve is around .4 (ish) units of insurance. Interestingly, since the insured will only purchase .4 units of insurance upon renewal, they are no longer fully insured. From an actuarial standpoint, it is desirable for customers to fully insure things like homes, and a penalty is typically built into the premium if a customer decides not to do so. More on that topic (much) later.

Further Improvements

I’ve introduced quite a few concepts into the consumption aspect of MIES, and have yet to rerun the simulation for more than two iterations. Before I can do this, I’ll need to revise certain aspects of the insureds, such as personal wealth, and partial insurance. I have not even incorporated wealth for each person, so the idea of having only part of it covered under the case of partial insurance currently has no meaning.

Appendix: SymPy

Since the optimization problem discussed here involves calculus, I thought maybe I should eventually have some calculus programmed in MIES. There’s a library called SymPy that facilitates symbolic computation in Python. Let’s try it out by computing the partial derivatives of the Cobb-Douglas utility function, required to solve for the MRS:

The console prints -c*x2/(d*x1), which is MRS = , the same as that derived in Varian. That looks pretty useful. However, Sympy’s documentation is massive, at over 2000 pages. It might take some time to learn it, even if it just involves me grabbing what I need. Since I can make do without calculus for the time being, I’ll save this for another day.

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 demonstrated the first simulation of MIES. Although it can run indefinitely without intervention on my part, I’ve made a lot of assumptions that aren’t particularly realistic – such as the ability of insureds to buy as much insurance as they wanted without any kind of constraint on affordability. I’ll address this today by implementing a budget constraint, a concept typically introduced during the first week of a second year economics course.

Most of the economic theory that I’ll be introducing to MIES for the time being comes from two books: Hal Varian’s Intermediate Microeconomics, a favorite of mine from school, and Zweifel and Eisen’s Insurance Economics, which at least according to the table of contents and what little I’ve read so far, seems to have much of the information I’d want to learn about for MIES.

I’m going to avoid repeating much of what can already be read in these books, so just an fyi, these are the sources I’m drawing from. My goals are to refresh my knowledge of economics as I read and then implement what I see in Python to add features to MIES.

The Economics Module

I’ve added a new module to the project, econtools.py. For now, this will contain the classes for exploring economics concepts with MIES, but seeing how much it has grown in just one week, it’s likely I’ll break it up in the near future. I’ll go over the first classes I’ve written for the module, those written for the budget constraint: Budget, Good, Tax, and Subsidy.

Budget Constraint

Those of you who have taken an economics course ought to find the following graph, a budget constraint, familiar:

As in the real world, there are only two goods that a person can possibly buy:

1. Insurance
2. Non-insurance

The budget constraint represents the set of possible allocations between insurance and non-insurance, aka all other goods. This can be represented by an equation as well:

Where each x represents the quantity of each good and each p represents the prices per unit for each good, and m represents income. People can only buy as much insurance as their incomes can support, hence the need for introducing this constraint into MIES for each person. The budget constraint simply shows that in order to buy one dollar more of insurance, you have to spend one dollar less on anything else that you were going to buy with your income.

The price of insurance is often thought of as a rate per unit of exposure, exposure being some kind of denominator to measure risk, such as miles driven, house-years, actuarial exams taken, or really anything you can think of that correlates with a risk that you’d like to charge money for.

Interestingly, there is nothing in the budget constraint as shown above that would prevent someone from insuring something twice or purchasing some odd products like a ‘lose-1-dollar-get-5-dollars back’ multiplier scheme. I’m not sure if these are legal or simply just discouraged by insurers as I’ve never tried to buy such a product myself or seen it advertised. I could imagine why insurers would not to sell these things – maybe due to the potential for fraud or the fact that an insured thing is 100% correlated with itself. On the other hand, a company might be able to compensate for these risks by simply charging more money to accept them. Regardless, if these products are really undesirable, I’d rather let the simulations demonstrate the impact of unrestricted product design than to have it constrained from the start. I’ll save that for another day.

The budget constraint is modeled with the Budget class in econtools.py. It accepts the goods to be allocated along with other relevant information, such as their prices and the income of the person for whom we are modeling the budget. You’ll see that the class contains references to the other component classes (Good, Tax, Subsidy) which are explained later:

Goods

The Good class represents things consumers can buy. We can set the price, as well as apply any taxes and subsidies:

Taxes

Taxes can be added to goods via the Tax class:

The tax class has three attributes, the amount, which specifies the amount of tax to be applied, style, which (for lack of a better word) specifies whether the tax is a quantity, value (or ad valorem) tax, or lump-sum tax. A quantity tax is a fixed amount of tax applied to each unit of a good purchased, a value tax is a tax that is proportional to the price of a good, and a lump-sum tax is a one-time tax paid for participating in the market for that good. Finally, the base attribute specifies over what quantities a tax applies (for example, a tax applied to the first 5 units purchased).

To demonstrate, we’ll add three different types of taxes to our insurance, first by specifying the tax details, and then adding them to the goods, we then specify the relevant budget constraints:

We can now plot the resulting budget constraints on a single graph:

As expected, the no-tax scenario (blue line) allows the insured to purchase the most insurance, indicated by the x-intercept at 10. Contrast this with the most punitive tax, the value tax shown by the green line with an x-intercept at 2. In this scenario, we increase the price of each unit of insurance form 1 to 5, so that the insured can at most afford 2 units of insurance.

Between these two extremes are the two ad valorem taxes that double the price of insurance. However, the purple line is less punitive as it only taxes the first two units of insurance purchased.

Subsidies

Subsidies behave very similarly to taxes, so much so that I have considered either defining them as a single class or both inheriting from a superclass. Instead of penalizing the consumer, subsidies reward the consumer either by lowering the price of a good or by giving away free units of a good:

Just like with taxes, we can specify the subsidy amount, style, and quantities to which it applies. For instance, let’s suppose that as part of a risk-management initiative, the government grants 2 free units of insurance to a consumer in the form of a lump sum subsidy:

The subsidy has allowed the consumer to have at least 2 units of insurance without impacting the maximum amount of all other goods they can buy. We also see that the maximum amount of insurance the consumer can purchase has increased by the same amount, from 10 to 12.

MIES Integration

To integrate these classes and make use of the econtools.py module, I’ve made some changes to the existing MIES modules. In last week’s example, income was irrelevant for each person, and therefore they were unconstrained with respect to the amount of insurance they could purchase. Since a budget constraint is only relevant in the form of some kind of limited spending power, I’ve decided to introduce consumer income into MIES.

Income is now determined by a Pareto distribution, though certainly other distributions are possible and more realistic. In the parameters.py module, I’ve added a value of 30000 as the scale parameter for each person:

The person SQLite table has also been updated to accept income as a field:

And finally, I’ve added a method to the environment class to assign incomes to each person by drawing from the Pareto distribution using the scale parameter:

This results in a mean income of five figures with some high-earning individuals making a few million dollars per year. To examine the budget constraint for a single person in MIES, we can do so by running two iterations of the simulation and then querying the database:

When we query the person table, we can see that the person we are interested in (id=1) has an income of about 56k per year:

We can also see that their premium upon renewal was about 21k, almost half their income and much higher than the original premium of 4k:

Let’s look at the events table to see why their premium is so high. It looks like they had a loss of about 174k, so the insurance up to this point has at least been worth their while:

We can now query the relevant information about this person, and use econtools.py to graph their budget constraint prior to and during renewal: