# Thinking Coherently

📅 Wednesday, October 26, 2011

Niels Bohr once said that if all humanity would be lost and we were to pass only one sentence to the next generation that we should pass the information that matter is composed of atoms. Now humanity has built up almost hundred years more history, and with that hindsight I would prefer another sentence to be passed on. I would say “When thinking about risk, do it in a mathematically coherent way, that is only use convex risk measures that are related to the downside of the profit distribution”. I believe that our ability to start thinking coherently about risk will determine whether or not the capitalist system will survive and whether or not we leave a free world to our children or not.

Since 1638 expected value is was used as a measure for risk, during last century utility has been linked to risk and various risk measure have been proposed such as volatility. Back in 1997, Philippe Artzner and Freddy Delbaen and Jean-Marc Eber and David Heath, published a paper ‘Thinking Coherently’. With financial risks in mind, but with a deep and universal approach, they proposed four axioms that risk measures should satisfy in order to be “coherent”. To be honest, there might be more coherent sets of axioms, just as there are many coherent alternatives for Euclid’s axioms. Just as these alternatives describe another reality than the geometry on the plane (for example geometry on a sphere), it is also in finance possible to find other sets of axioms that are coherent. What is coherence in finance?

Different people might define coherence different, but few people would disagree that a coherent risk measure should satisfy the following conditions:

- is an investment X has always worse outcomes than Y (in every observation), then X is more risky—in other words, a risk measure should be related to the downside of the investments (or the left tail of the return distribution).
- If we invest more, our risk increases—Artzner asks a direct connection: twice the investment, then twice the risk.
- If we diversify, the risk must decrease.
- If we hedge our position, then the risk decreases with that amount—Artzner asks that if we add an amount of cash, that the risk should be decrease with the same amount.

The above statements seem to be acceptable, but are by far not the only possible way to define coherence. This definition of coherence seems to be fine for the investor that cannot influence prices, is not allowed to think in terms such as “once I default it does not matter any more”. The investor in Artzner’s approach cannot influence price and while an individual would not care if he has twice or ten times a debt that he can never repay, for the society it does matter. For example, once a bank defaults its shareholders lost their investment, and that’s it. They might not care if the bank defaulted with a net liability of one dollar or one billion dollar. The society that has to cope with the fallout (maybe a bailout) should care.

The most prominent issue for internal coherence might be the fact that a risk measure has only one minimum. Otherwise behaviour is erratic and has nothing to do with risk. For example, we would expect that if a portfolio is diversified more and more that the risk decreases more and more. Mathematicians would say that the risk measure needs to be “coherent”. This property ensures that We should only find one optimum, when we find local optima this means that our risk measure is internally incoherent and cannot be used to optimize portfolios.