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!