Polya: Actuaries Good or Bad

As an actuary, were you born 'Good' or 'Bad'? The answer to this question can be given with help of mathematics!

Let's start with a simple model. When you, as a prospective actuarial talent, were born, you had only a limited number of experiences. Let's assume you came to earth quite neutral, with one 'Good' (G) and one "Bad" (B) experience.
At this point in time, your (still unconscious) attitude and therefore expectation of a 'Good' (B) outcome of your next experience, will be 50%.

In line with the expression "You'll always reap what you sow" (Gal 6:7), or associative translated "You'll become what you X" (with X ='Think', 'Eat', 'Are', etc.)", your next experience will indeed turn out to be equally G or B.

Let's assume that providence decided, the outcome is G. Now you've become a more optimistic baby actuary. Your experience-bucket is now filled with two G's and one B (experience), so your subjective 'colored' outlook on G's is 66,66% (2/3=[2 G's/(2 G's + 1 B)]) . You also look back on a relatively Positive Life Score of PLS=66,66% G's.
Would you have experienced a 'B score' instead, it would be the other way around and as a potential pessimist your outlook and PLS would have been lowered to 33,33% .

But happily you're a 66,66% (!) G-Score-optimist and life goes on. According to the same principles, the probability of scoring a new G-experience is now 66,66% instead of 50%.

As you may already notice, your PLS will more and more develop according your personal historical G- an B-experience track record.

A few questions that may rise, are:

• Does your Positive Life Score (PLS) has a limit? And if so, what's that limit?
• Once you're in a pessimistic phase, what are the changes of getting out?

Here is were the help of a great mathematician, George Pólya,

comes in, by modeling the above situation in what is called:

Polya's Urn model
An urn contains G₀ Green (Good) and B₀ Black (Bad) balls. One ball is drawn randomly from the urn and is then placed back in the urn together with an (extra) ball of the same color.

Our Good&Bad exercise turns out to be a simplified two color Polya Urn Model (G₀=1,B₀=1) that is part of a large family of General Urn Models.

Properties
It turns out that this model has the following (translated)properties:

• On any given moment in your life if you do not know what kind of balls have been drawing before, the expectation of drawing a Good or Bad ball (experience) is always G₀ =G0/(G₀ +B₀) =50%.
  
  
• On any given moment in your life, gaining a Good or a Bad experience depends on the track record of G&B experiences in your life. So if you've experienced G Good experiences and B Bad experiences, your changes of experiencing a next Good experience are equal to the track record of your Positive Life Score : PLS_(G+B)=G/(G+B)
  
  
• The relative influence of a G or B experience on the PLS decreases rapidly as the number of total experiences increases. Your PLS has a definitive limit in (life)time with equal changes of outcome on the interval [0,1].
  
  
• As is clear from some simulations, the first 10 to 20 experiences in our life determine whether we'll become an optimist (PLS_(∞)> 0.75) or an pessimist (PLS_(∞)<0,25).
  
  
  
  
  
  
  
  
  
  
• Moreover, the first 5 to 10 experiences in your life already determine the direction of our PLS in life. This means that our parents and teachers have an important role in guiding us in our baby and youngster phase to a positive balanced number of experiences (a more than average PLS).
  
  For example if on a given moment in life you have had 4 Bad experiences and 1 Good, the probability of having a next Good experience is 20%. What's more frustrating is that the probability in this case to get in three steps to a 50% level is only about 3% (=1/5*2/6*3/7) . This illustrates the heavy responsibility of our parents and teachers.
  
  That's why it's for example so difficult to change your religion. Once the first 50 religion experiences have been brought in by your parents, it's hard to change from Budha to Allah or Christ, or the other way around.
  
  The same is true with regard to our actuarial education and experience. Once we've experienced more than 10 years in a row that longevity increases slowly, it will hard to be convinced that longevity will explode one day. As a consequence, the way we are formed - per definition - causes that we will always underestimate the risk of a change, as we unconsciously relate risk more to our paste experience more than (we can) to the future. .
  
  
• Once a more than average PLS in our life is achieved, we're more likely to absorb a Bad experience without getting unbalanced. Parents and teachers can 'let go'.

Keep in mind, Polya's Urn is only a think-model to help you to become aware of the important mechanisms that play a role in becoming 'who you are' or 'what you'll be'.

Change?
Once you become experienced in life and your PLS direction has been set, you can only change this by either a Professional De or Re-programming (PDR) or a, what is called, Life Changing Experience (LCE). In PDR Bad experiences are taken away (i.e. out of the urn) and replaced by Good experiences, to regain trust and a higher confidence (PLS) level. In LCE's, your environmental or physical circumstances suddenly chance in such a way that you are forced to experience only just B (or just G) experiences. Another LCE is created by the change of context. What before were B experiences now turn out to be G experiences (or the other way around).

What if?
There are many other aspects that could be studied in relation to the Polya model. For example:

• What would be the effect if an experience is not just only Good or Bad, but a mix.
• What if a 'Good experience' doesn't trigger extra positive confidence (an extra G) but a negative experience (an extra B).

The answer in both cases is that almost always the PLS-limit=50% !, in other words: You'll become average.

But how does a little bit of extra Bad (or Good) influence the PLS limit? If you want to experiment (online) and learn more about Good and Bad, go and visit

Polya's Urn Experiment

and look up the Math behind Polya's Urn (attachments).

Perhaps Polya's Urn is also a good start to model the stock market.
I'll leave that up to you.
Math helps us to discover who we are or what we become...

God must be an Actuary

Let's dive back in history and take a look at a unusual 'biblical' article in The Actuary of 1986 (Vol. 20, nr. 6 ; page 8).

In a amusing article Mark W. Campbell develops a simple lifespan equation with regard to our 'Greatⁿ Grandfathers'.

This is the original (somewhat restyled) article:

You Should Live So Long

Sir:

In the January issue, Murray Projector quotes Genesis 6:3 as follows:

“And the Lord said: ‘My Spirit shall not abide in man forever, for that he also is flesh; therefore shall his days be a hundred and twenty years.’ “

Mr. Projector suggests the interpretation that 120 years is the maximum age or “omega” for man. This is an interesting idea when one considers the recorded life spans of Noah (of whose generation Genesis 6:3 speaks) and his descendants. The enclosed graph shows these life spans down to Moses, of whom Deuteronomy 34:7 states:

“And Moses was a hundred and twenty years old when he died: his eye was not dim, nor his natural force abated.”

The curve which has been fitted to the data is of the form y = A + B-C^-X. With “A” set equal to 120, the R- squared of the fit is approximately 92% (the R-squared can be increased slightly using a lower value of “A”). This is a remarkably good fit to biological data.

I am not sure what all this means, except that, as always, there is more to the Bible than meets the eye. I welcome the comments of other readers.

Mark W. Campbell

In his original article Campbell doesn't mention the values of the variables A,B and C. However, in the following magazine of The Actuary (nr. 7), Samuel L. Tucker, defines those variables in an equation that 'fits the Campbell curve quite well' :

y=120+830*1.407 ^-x

the variable 'x' stands for the 'xth generation'.

In the same article, nr. 7, Tucker concludes that the Campbell equation overestimates the lifespan and therefore fails in case of earlier great-great grandfathers, back until Adam. He challenges Campbell to develop an integral equation regarding all 26 generation.

26 Generation Equation
Well, here it is. The formula, a logistic equation fitted at ZunZun, is now expressed into our modern western time line (t):
With a= 792.40, b= 1307204394.9 and
c= -0,00881292

In graphics:

The simple formula and good fit undoubtedly prove that:

God must be an Actuary! ;-)

The results in table form:


De equation is again modelled with an age limit of 120, as it appears that, although longevity in modern times is increasing, the 'omega age' (120) seems hard to beat.

More information about our great-great grandfathers at:

Noahs Age

For those who are interested, please download the corresponding spreadsheet.

Have fun in combining actuarial math and the bible.......

Swine Flu Counter update Sept 2009

Here you'll find the September 2009 update of the

Global Swine Flu Counter

Although there is still an increasing risk of underreporting, the counter has been renewed on basis of the latest available global reports as provided by Wikipedia/ECDC.

Swine Flu under Control?
The September 2009 developments suggest the Swine Flu development is under control, as the reported infections changed from a exponential growth recent months, to more linear growth in August 2009. In September the increase of infections was already declining.

New Model
The above developments are the main reason why data in the Swine Flu calculator have now been modelled by a logistic function.
Well considered curve fitting at ZunZun, showed a Gompertz function (with offset) resulted in a satisfying approximation :



Life actuaries will be familiar with good old Gompertz. The Gompertz equations are - by the way - also used to model Plant Desease Progres.

The number of death have now been modelled ruffly as 1.8% of the infected people a month earlier [Death=0.018*I(t-30)]

Results update
The results the new approximation show that the number of reported infections increases asymptotically towards a limit of about 323,000.

Correspondingly, the number of death, , increases to a limit of ruffly 6000.

All provided the actual controlled development continues and no new mutation of the H1N1 will develop in the next months.....

Risk
The risk of underreporting is not negligible . Modeling on basis of excluding the September data would result in a limit of 528,000 infects and about 9500 deaths. We'll just have to wait how H1N1 develops.....
But as becomes clear, the explosion of swine flue cases looks under control.

If necessary, the counter will be updated again on a on a regular basis. The latest data you'll find in this XLS spreadsheet.

Install Swine Flu Counter
How to implement this Swine Flu Counter on your web site?

• Put the next HTML-script (without the outer quotes) just before the end of the body tag:' <script language="javascript" type="text/javascript" src="http://sites.google.com/site/boooming/actuary/swine-flu-2009-update1.js"> </script>'
  


• Put the next HTML-line (without the outer quotes) where you want the Swine Flu table to appear on your site :
  ' <div id="swineflutable"></div> '
  
  
• Ready!