Life is Nonlinear, so is Risk!

From the day we were born, we've learned to survive in a complex world by applying linear mechanisms in life:

• On a short time scale things don't change much
• The future can be predicted by extrapolation of the past
• Every event now, must have a cause in the past
• Results are a (linear) combination of events in the past

Linear Thinking
In line with our linear culture, we - actuaries, (risk) managers, investment consultants or asset managers, etc. - have applied this way of linear thinking in our professional field:

• Mean reversion: Returns continue to go back to an average value over time
• Volatilities are more or less constant in time
• Increasing volatility is a good predictor of an upcoming financial crisis
• Standard deviation is similar to risk or volatility
• If a distribution is complex, a normal distribution nevertheless will do fine
• Tail risks are not really interesting or can't be modelled anyway

More detailed psychological linear thinking in the Risk area...

• Peer Risk: If all other professionals (institutions) are using a certain method or investment strategy, why should I take the risk of developing a new one?
  
• First Mover Risk: Why should I act first and carry all research investments?
  
• Supervisory Compliant:If the regulator prescribes new regulations, I'll apply those regulations as if it is my own risk appetite.
  
• Big Brother Hedge Risk: I base my investment strategy at a save distance on the biggest leader in the market. Might trouble arise, the Regulator first has to deal with my Big Brother.
  
• Regulation Risk: Regulation (change) is perceived as a given fact and not viewed or managed as a kind of risk
  
• Risk of Free Rate Risk: There must be some kind of risk free interest rate.

Thinking long term and two steps deeper, it's obvious that applying any of the above mentioned linear thinking methods will likely be the nail in the coffin of any financial institution.

Linear thinking and modelling make our daily life more simple. Unfortunately, 'too simple' to cope with financial markets reality on a long term. 

Metamorphosis by Escher...
If we are lucky, (market) circumstances only change slowly and we're able to adapt the value of the variables in our linear models gradually, while  keeping our traditional way of linear thinking and modeling.  We act just like the famous graphic artist Escher shows us in Metamorphosis...

If we are less 'lucky' (as we are in2012), our linear models all of a sudden seem to fail. Covariances and volatility increase. Systemic risk shows up everywhere and a 'risk free rate' turns out to be an illusion. Our risk dashboard is on fire and we'll have to admit: our linear MPT models are failing.

Navigation Risk Parable
Why is it so so hard to admit that our linear models fail?

Suppose you developed a 2D linear (x,y)-navigation app in your Florida flatland office. Your app works fine for years. Than you decide to visit Black Hills & Badlands of South Dakota. Suddenly your app seems to fail in the mountains. Travel times and distances on your display suddenly seem wrong.

You realise you urgently need to develop a nonlinear 3D (x,y,z)-navigation app.... However, you don't do it.

Why not?

Well, first of all your old linear 2D app worked fine for years and on short trips the app still works (approximately) fine.

Besides, nobody of your Californian friends uses a 3D app and developing a new nonlinear app is very expensive.

Well, it's time we realise that most developments in life are in fact nonlinear.
If the stakes are high, like in the investment business, linear models will eventually lead us to a disaster.

Summarised, we might conclude:

If life is Nonlinear, so why aren't our models?

New Solutions

What alternatives do we have for our old linear model?

Although there are many nonlinear models, I'll emphasize on two interesting nonlinear based models in this blog.

I. Predicting economic market crises using measures of collective panic

Is there an adequate predictor of a market crisis?

Using new statistical analysis tools based on complexity theory, the New England Complex Systems Institute (NECSI) performed a new research on predicting market crashes.

As we know volatility is a measure of risk. So one would expect an increase of volatility also to be an adequate predictor of a financial market crash. Unfortunately this is not the case, as is shown in the recent NECSI study. While volatility increases at the beginning of a crisis, it is unreliable as an adequate indicator of a nearby market crash.

What also is not true, is that a market crash is often triggered by market panic justified, or not justified, by external (bad) news.

The NECSI research indicates that it's the internal structure of the market and not an external crises, that's primarily responsible for a market crash.

It turns out that the number of different stocks that move up (U) or down (D) together is an indicator of the mimicry ( 'collective flight'; herding) within the market. When mimicry is high, many stocks follow each other's movements.

This "co-movement" of stocks  is an indicator of a nervous market that is ripe for panic. The existence of a large probability of co-movement of stocks on any given day, is a measure of systemic risk and vulnerability to self-induced panic.

So, rather than measuring volatility or correlation, the fraction of stocks that move in the same direction turns out to be a successful predictor of a market crash..

NECSI researchers showed that a dramatic increase in market mimicry occurred during the entire year before each market crash of the past 25 years, including the recent financial crisis.

II. Worst-Case Value-at-Risk of Non-Linear Portfolios 
We all know that VaR lacks some desirable theoretical properties:
- Not a coherent risk measure.
- Precise knowledge of the distribution function is critical
- Non-convex function of w → VaR minimization intractable
- To optimize VaR we have to resort to VaR approximations
- Normality assumption is unrealistic → may underestimate the actual VaR.

Zymler, Kuhn & Rustem of the Department of Computing Imperial College London now developed a nonlinear alternative for VAR, called



Worst Case Var

Two variations on WCVar lead to practical applications:

1. Worst-Case Polyhedral VaR (WCPVaR)
   A polyhedral VaR approximation for portfolios containing long positions in European options expiring at the end of the investment horizon
   
   
2. Worst-Case Quadratic VaR (WCQVaR)
   A suitable VaR approximation for portfolios containing long and/or short positions in European and/or exotic options expiring beyond the investment horizon.

Here's an example of WCQVar's results against  WCVar (plain) and the good old 'Monte Carlo Var' we mostly use in linear modeling. This graph needs no further comment.....

Using the WCQVar leads to more realistic modeling results. WCQVar-techniques can also be used for for index tracking leads to spectacular results (see pdf).

Worst-Case Value-at-Risk of Non-Linear Portfolios

Finally
After so many years of relative successfully using linear models, it's hard to recognize that we need new models based on new nonlinear approaches.
Therefore we need 'first movers'. Who's willing to take the risk and jump into the nonlinear deep-sea?



Be confident and stay on your happy feet.. after a successful jump, 'herding theory' tells us others will follow...


Sources & Related Links
- Predicting economic market crises using measures of collective panic (PDF)
-  NECSI Research (2011)
- Self-Induced Panic And The Financial Crisis 
- Worst Case Var Document (PDF; 2011)
- Worst Case Var Document (PDF;2009)
- Order Happy Feet Video