Lesson 36 – Counts: The language of Poisson distribution

I want you to meet my friends, Mr. Able and Mr. Mumble.

Unlike me who talks about risk and risk management for a living, they take and manage risk and make a killing.

True to his name, Mr. Mumble is soft-spoken and humble. He is the go-to person for numbers, data, and models. Like many of his contemporaries, he checked off the bucket list given to him and stepped on the ladder to success.

Some of you may be familiar with this bucket list.

Mumble was a “High School STEM intern,” was part of a “High School to College Bridge Program for STEM,” has a “STEM degree,” and completed his “STEM CAREER Development Program.” Heck, he even was called a “STEM Coder” during his brief stint with a computer programming learning center. A few more “STEMs,” and he’d be ready for stem cell research.

These “STEM initiatives” have landed him a risk officer job for an assets insurance company in the financial district.

On the contrary, Mr. Able is your quintessential American pride who has a standard high school education, was able to pay for his college through odd jobs, worked for a real estate company for some time, learned the trade and branched off to start his own business that insures properties against catastrophic risk.

He is very astute and understands the technical aspects involved in his business. Let’s say you cannot throw in some lingo into your presentation and get away without his questions. He is not your “I don’t get the equations stuff” person. He is the BOSS.

Last week, while you were learning the negative binomial distribution, Able and Mumble were planning a new hurricane insurance product. Their company would sell insurance against hurricane damages. Property owners will pay an annual premium to collect payouts in case of damages.

As you know, the planning phase involves discussion about available data and hurricane and damage probabilities. The meeting is in their 61st-floor conference room that oversees the Brooklyn Bridge.

Mumble, is there an update on the hurricane data? Do you have any thoughts on how we can compute the probabilities of a certain number of hurricanes per year?

 

Mr. Able, the National Oceanic and Atmospheric Administration’s (NOAA) National Hurricane Center archives the data on hurricanes and tropical storms. I could find historical information on each storm, their track history, meteorological statistics like wind speeds, pressures, etc. They also have information on the casualties and damages.

That is excellent. A good starting point. Have you crunched the numbers yet? There must be a lot of these hurricanes this year. I keep hearing they are unprecedented.

 

Counting Ophelia, we had ten hurricanes this year. Take a look at this table from their website. I am counting hurricanes of all categories. I recall from our last meeting that Hurin will cover all categories. By the way, I never liked the name Hurin for hurricane insurance. It sounds like aspirin.

Don’t worry about the name. Our marketing team has it covered. Funny name branding has its influence. You will learn when you rotate through the sales and marketing team. Tell me about the counts for 2016, 2015, etc. Did you count the number of hurricanes for all the previous years?

 

Yes. Here are the table and a plot showing the counts for each year from 1996 to 2017.

 

Based on this 22-year data, we see that the lowest number per year is two hurricanes and the highest number is 15 hurricanes. When we are designing the payout structure, we should have this in mind. Our claim applications will be a function of the number of hurricanes. Can we compute the probability of having more than 15 hurricanes in a year using some distribution?

Absolutely. If we assume hurricane events are independent (the occurrence of one event does not affect the probability that a second event will occur), then the counts per year can be assumed a random variable that follows a probability distribution. Counts, i.e., the number of times an event occurs in an interval follows a Poisson distributionIn our case, we are counting events that occur in time, and the interval is one year.

Let’s say the random variable is X and it can be any value zero hurricanes, one hurricane, two hurricanes, ….. What is the probability that X can take any particular value P(X = k)? What are the control parameters?

 

Poisson distribution has one control parameter 

It is the rate of occurrence; the average number of hurricanes per year. Based on our data, lambda is 7.18 hurricanes per year. The probability P(X = k) for a unit time interval t is

The expected value and the variance of this distribution are both

We can compute the probability of having more than 15 hurricanes in a year by adding P(X = 16) + P(X = 17) + P(X = 18) and so on. Since 15 happened to be in the extreme, the probability will be small, but our risk planning should include it. Extreme events will create a catastrophic damage. I see you have more slides on your deck. Do you also have the probability distribution plotted?

 

Yes, I have them. I computed the P(X = k) for k = 0, 1, 2, …, 20 and plotted the distribution. It looks like this for = 7.18.

Let me show you one more probability distribution. This one is for storms originating in the western Pacific. They are reported to the Joint Typhoon Warning Center. Since we also insure assets in Asia, this data and the probability estimates will be useful to design premiums and payouts there. The rate of events is higher in Asia; an average of 14.95 typhoons per year. The maximum number of typhoons is 21.

 

Very impressive Mumble. You have the foresight to consider different scenarios.

 

As the meeting comes to closure, Mr. Able is busy checking his emails on the phone. A visibly jubilant Mumble sits in his chair and collects the papers from the table. He is happy for having completed a meeting with Mr. Able without many questions. He is already thinking of his evening drink.

The next meeting is in one week. Just as Mr. Able gets up to leave the conference room, he pauses and looks at Mumble.

“Why is it called Poisson distribution? How is this probability distribution different from the Binomial distribution? Didn’t you say in a previous meeting that exactly one landfall in the next four hurricanes is binomial?”

Mumble gets cold feet. His mind already switched over to the drinks after the last slide; he couldn’t come up with a quick answer. As he begins to mumble, Able gets sidetracked with a phone call. “See you next week Mumble.” He leaves the room.

Mumble gets up and watches over the window — bright sunny afternoon. He refills his coffee mug, takes a sip and reflects on the meeting and the question.

To be continued…

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