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...

Coastline Fair Value

Close your eyes and take a guess at the Australian coast length? Answer : 'Exactly' 25,760 km.

1. Right, according to Wikipedia
2. Wrong! Because the exact coast length depends on the length of your ruler!

If you would measure the Australian coastline with a 1-mm ruler, you would get a length of more than 100 .000 km!

This leads to the question:

Does a 'coastline fair value' exist?

After all, as the ruler gets diminishingly small, the coastline's length gets infinitely large.
This phenomenon is also known as the Richardson Effect (or the coastline paradox).

Coastline Formula?
In 1967 a document called "How long is the coast of Britain?" was published by the great mathematician Mandelbrot

In 1967 he revived the original formula, earlier developed by Richardson :

L(G) = F . G^(1-D)

with

L=length of the coastline as a function of G

G=Ruler length

F=positive constant factor,
D=constant (D>=1). D is a ‘‘characteristic’’ of a frontier, varying from D=1 for a straight frontier to D=1.25 for a very irregular coastline like Britain. It turns out that D = 1.13 for the Australian coast and D=1.02 for the very smooth South Africa coastline.

Fractals
The constant D also stands for 'Dimension' and in 1975 Mandelbrot develops this Dimension- idea to what is called the Fractal Dimension.

Fractals turn out to be the perfect (math) language for describing all kind of natural phenomenas like leaves, trees , etc.

Fractals are even used to describe the stock market, the credit crisis or the coastline of the law.


Coastline Formula & Valuation
What can we learn from this fractal coastline measurement with regard to valuations?

1. Stop changing the rules
   If accounting standards like IFRS , GAAP and IAS or legislation are constantly changing (e.g. amendments) and getting more and more specific, valuing a company becomes like measuring the coastline with different rulers.
   
   In this case management, supervisors, stake- and shareholders lack a sustainable view on their business. You can't justify the results and value of your company if you have to measure yourself with a dynamic ruler!
   
   
2. Stop digging
   More and more deep going risk research will eventually lead to an substantial increase or even 'infinite' Value at Risk.
   
   Therefore it's important to define portfolio-, market- and product-risk- limits and structures first, right from the companies (risk) strategy.
   
   These instruments reduce the needed depth of risk research and therefore increase the control- and efficiency-level of the company.

Try to think scale free and have fun by applying fractals in actuarial science!