### Insuring your Bed

November 12, 2013

“What is your most valuable possession?” asked Raissa Schickel, our guest instructor.

“Probably my bed,” responded a student. Raissa went on to use “bed insurance” as an example as she explained the job of an actuary. That job is to price insurance in a way that is

- not discriminatory (fair),
- not excessive (not so expensive that customers wouldn’t buy it), and
- not inadequate (not so cheap that the insurance company would go out of business).

Before jumping into bed insurance, however, Raissa set up a decision matrix with our game* from last week. The group discussed expected value, the rule of large numbers, and the resemblance between this matrix and the “red pill/blue pill” reference from the film The Matrix.**

Students then hypothetically insured a bed against calamities such as cat damage, fire, and destruction via airplane crash. We discussed the mathematics of insurance:

- statistical likelihood of each event,
- how the probability of each risk impacted the premium (price) of the policy,
- the pooling of risk, and
- the influence of the amount of historical data on confidence and therefore on premium (profit load).

Then, with the students, Raissa generated an equation useful in pricing insurance. Not everyone understood the equation, so we discussed different ways of doing the calculations. We also discussed my conjecture that in the fields of business and science, people tend to use percents, versus fractions, to express numerical relationships.

At this point, class was nearing an end, and students’ interests in this multifaceted topic diverged. One student was intrigued when Raissa mentioned regression, and the discussion moved into an explanation of that.*** Another student had a deep interest in the philosophical/policy aspects of the insurance industry (“Are insurance companies for profit” “Would universal health care put insurance companies out of business?”) Other students were interested in the various factors that affect risk. I think that everyone wanted more time to discuss which factors that mathematically affect risk are actually illegal to consider in price setting. I think we needed more time to explore everyone’s area of interest. But sadly we did not. Next time, we’ll discuss Pascal’s wager, which - using the language of today’s discussion - we can frame as “whether you can insure your soul.”

My thanks to Raissa for the very engaging conversation about insurance. It was a rare treat to explore applied mathematics.

Rodi

** Suppose you have an opportunity to play a game that costs $1 to play. You have a 50% chance of winning. If you do win, you get $3, but if you lose, you get nothing. Should you play?*

** “You take the blue pill, the story ends, you wake up in your bed and believe whatever you want to believe. You take the red pill, you stay in Wonderland, and I show you how deep the rabbit hole goes.” (The Matix) This is not the first time that the content of a Math Circle has overlapped with this film. When I discussed Plato’s Cave with a younger group a while back, one of the parents informed me that The Matrix is a retelling of that ancient allegory. Some day, I have to see this movie!

***After class, Raissa did email me more information on regression for those who are interested: *“This link below will offer some explanation. I would recommend the top section "Linear Regression" as an explanation for students who are still interested. *

*http://www.stat.yale.edu/Courses/1997-98/101/linreg.htm*

*For pricing car insurance, I might use regression to determine if there is a relationship between age of driver (x-axis) and car accident cost history (y-axis). This could be used to determine if there is mathematical support for our "hypothesis" that a youthful driver is going to cost the insurance company more money than a 40 year old driver, for example. *

*What comes out of this is possibly a factor that an actuary uses in their formula for calculating premium. If a linear curve is "fit" to the actual data points and the r^2 (correlation coefficent) shows that it is has high correlation (that it's a "good" fit so the actual data points are pretty close to the fitted curve), then we can use the fitted curve to "predict" loss experience for different age groups. For example, we could use the fitted curve to determine relativities between different age groups. The fitted curve may show that 18 year old drivers (x-axis) "cost" (y-axis) 60% more than 40 year old drivers and so they would get a factor of 1.60, while 40 yr old would get a factor of 1.00. These factors are one of many that get multiplied together to determine premium. So the difference in premium between these two age groups as a result of age characteristic alone of 60%. (There are many other factors as well as a base rate that go into the formula for determining premium. Also, multiple regression would be used if there are dependencies between some of the factors, so as not to "double count" their impact. Of course, that's a very difficult concept to grasp and usually not covered until college I think!)*

*Another topic which could be fascinating for the kids is the idea of credit scoring, which is highly debated. The math actually finds it to be HIGHLY correlated (using regression) to accident history. But it's very unfairly discriminatory as well, and many states do not allow it.*

*Lastly, I never got a chance to sum up the expected value discussion. I wanted to use a bicycle example. Probability of total loss ($30) is 25%, probability of losing only a tire ($5) is 50%, and probability of losing only a seat($10) is 25%. Then expected value = $30 * 25% + $5 * 50% + $10 * 25% = $12.50.”*