Showing posts with label probability. Show all posts
Showing posts with label probability. Show all posts

Jun 27, 2011

Impact or Probability?

We all are more than familiar with the definition of Risk:

Risk = Probability × Impact= P×I

This way of measuring risk is a nice, simple, explainable and intuitive way of ordering risks in board or bath rooms, but unfortunately quite useless.

To demonstrate the limits of this kind of typical Risk definition, let's take a look at the next story:

The Risk of bicycling

You decided to start a 3 year math study at City University in London. From your brand new apartment in Southall, it's a 12.5 mile drive to the University at Southhampton Street.  As a passionate cyclist you consider the risk of cycling through London for the next three years.

Based on your googled " DFT's Reported Road Casualties 2009" research (resulting in a cycling death rate of 36 per billion vehicle miles), you first conclude that the probability of getting killed in a cycle accident during your three year study is relatively low : 0.1% (≈ 3[years] × 365[days] × 25[miles] × (36 [Killed]  ÷ 109[vehicle miles]).

Subjective probability
After this factfinding you start to realize it's YOU getting on the bike and it's YOUR 0.1% risk of DYING  in the next three years of your study....

Hmmmm...this comes closer; it makes things a little different, doesn't it? 

Its looks like 'subjective probability' - on reflection - is perhaps somewhat different from 'objective probability'.

While your left and right brain are still in a dormant paradoxical state of confusion, your left (logical) brain already starts to cope with the needs of the right (emotional) half that wants you on that bike at all costs!

Russian Roulette
Now your left brain tells you not to get emotional, after all it is 'only' an additional 0.1% risk. Already your left brain starts searching for reference material to legitimate the decision you're about to take.

Aha!.... Let's compare it with 'Russian Roulette', your left brain suggests. Instead of 6 chambers we have thousand chambers with one bullet. Heeee, that makes sense, you talk to yourself.

With such a 1000 chambers Russian gun against my head I would pull the trigger  without hesitating....  Or wouldn't I?..... No.., to be completely honest, 'I wouldn't risk it', my right brain tells me.

Hé... my left brain now tells me my right brain is inconsistent: It wants me on the bike but not to take part in a equal 'death probability game' of Russian roulette. Why not?

In Control
My left half concludes it must be the 'feeling' of my right side that makes me feel I'm 'in control' on my bike, but not in case of Russian Roulette. That makes sense, tells my left brain me. Of course! Problem solved! My right and left brain finally agree: It's only a small risk and it's I who can control the outcome of a healthy drive.  Besides, this way the health benefits of cycling massively outweigh the risks as well, my right brain convinces me superfluous.

A final check by my right brain tells me: If I can't trust myself, who can I?
This rhetorical question is the smashing argument in stepping on the bike and to enjoy a wonderful ride through London City.
As ever...,

After returning from my accidentless bike trip, I enjoy a drink with a colleague of mine, the  famous actuary Will Strike  [who doesn't know him? ;-)].

After telling him my 'bike decision story' he friendly criticizes me for my non-professional approach in this private decision problem. Will tells me that I should not only have analyzed the probability (P), but also the Impact (I) of my decision. Remember the equation: Risk=P×I?

Yes of course, Will is right. How could I forget? ..., the probability of getting a deathly accident was only 0.1%.

Yet, 'when' a car hits you full, the probability of meeting St. Petrus at heaven's gate is 100% and the Impact (I) is maximal (I=1; you're dead ...)


Risk[death on bike;25 miles/day; 3 years] =
Probability × Impact = 0.1% × 1=0.1%
From this outcome it's clear that, even though the Impact is maximal (1=100%) , on a '0% to 100% Risk scale' this 3 year 'London-Bike Risk Project' seems negligible and by no means a risk that would urge my full attention.

I'm finally relieved... it always makes a case stronger to have a taken decision verified by another method. In this case the Risk=P×I method confirmed my decision taken on basis of my left-right brain discussion.  Pff....

The next morning, after my subconscious brain washed the 'bike dishes' of the day before, I wake up with new insights. Suddenly I realize I tried to take my biking decision on the wrong variable: Probability, instead of Impact.

Actually, in both cases and without realizing, I took my decision finally on basis of the Impact and the possible 'Preventional Control' (not damage control !!!) I  could exert before and during my bike trip.

I had to conclude that in cases of high Impact (I>0.9), nor my left-right brain chat, nor the 'Risk=PxI' formula lead to a sound decision, because both are too much based on probability instead of Impact. In other words:

In case of high Impact, probability is irrelevant

In case of high Impact, only control counts

From now on this 'bike conclusion' will be engraved in my memory and I will apply it in my professional work as well.

P.S. for disbelievers, the tough ones!
If you're convinced you would take the risk of firing the 1000 chamber  Russian gun against your head, you probably valuate the fun of the bicycle trip higher than probability of the loss of your life or good health.

In this case, suppose someone would offer you an amount of money if you would take part in a 1000 chamber Russian roulette instead of a bicycle tour. At which amount would you settle?

Let's assume you would settle at € 10.000.000 (I wouldn't settle for less). In this case you really value your bicycle trip!!!! 

As we've seen in banking business as well : extreme low probabilities and high impact situations are tricky! That's why stress tests focus on impact and not on probability.

The different faces of Risk
Another issue when looking at risk is that risk is always conditional.
'The' probability of death or 'the' mortality rate doesn't exist. Mortality depends on a number of variables, such as age (the run down state of your DNA quality), the DNA-quality you where born with and lifestyle. Secondly mortality also depends on a number of uncertain events in your life.

To demonstrate this 'Chameleon property' of probability, lets take a look at the probability of a meteor hitting good old earth.

The initial probability of an asteroid devastating the earth within a 10 year time frame is around 0.1%. A typical case of low probability and high impact. Once we've become aware of a spotted meteor in our direction, the probability suddenly changes from a general probability in a time frame to a asteroid specific probability during his actual passage of the earth.
In case of  the asteroid '2011 MD'  that will pass the earth at a minimal distance of 11000 km on June 27, 2011, this specific probability turns out 0.11% (remember the Russian Gun...).

With a diameter of around 8 meter, this asteroid is no big threat to our civilization.

Here's a short impression what's coming flying in on us within the next decades (Source: Nasa; asteroid>50 meter or minimum distance< 100,000km):

Apart from some 'big asteroids' in the next decade, this picture puts our minds at rest. Yet we should keep in mind that most asteroids are discovered only a few weeks before a possibe collaps...

Risk Maps
A nice example of the limits of the Risk=P×I model in combination with a nice aleternative, is demonstrated by Fanton and Neil in in a document called: 'Measuring your Risks: Numbers that would make sense to Bruce Willis and his crew'.

In  their document they analyze the case of the film Armageddon, where an asteroid of the size of Texas is on a direct collision course with the earth and  Harry Stamper (alias Bruce  Willis) saves the world by blowing it up.

Trying to fill in the Risk=P×I model in this Armageddon case is useless.

In this case, Risk is defined as:

Risk =  [Probability of Impact]  × [Impact of asteroid striking the earth]
Fanton and Neil conclude:
  • We cannot get the Probability number.
    The probability number is a mix up. In general it makes no sense and it's too difficult for a risk manager to give the unconditional probability of every ‘risk’ irrespective of relevant controls, triggers and mitigants.
  • We  cannot  get  the  Impact  number. 
    Impact (on what?) can't be unconditional defined without considering also the possible mitigating events. 
  • Risk  score  is  meaningless.
    Even  if  we  could  get  round  the  two problems above, what exactly  does  the  resulting  number  mean?  
  • It  does  not  tell  us  what  we  really  need  to  know. 
    What  we  really  need  to  know is the probability, given our current state of knowledge, that there will be massive loss of life.

Instead of the Risk=P×I model,  Fanton and Neil propose (Measuring risks) the use of  causal models (risk maps) in which a risk is characterised by a set of uncertain events.

Each of these events has a set of  outcomes and the  ‘uncertainty’  associated  with  a  risk  is  not  a  separate  notion  (as  assumed  in  the  classic approach).
Every event  (and  hence  every  object  associated  with  risk)  has  uncertainty  that  is characterised by the event’s probability distribution.


The Initial risk of meteor strike
The probability of loss of life (meaning at least 80% of the world population) is about 77%:

In terms of the difference that Bruce Willis and his crew could make there are two scenarios: (1) the meteor is blown up and (2) where it is not.

Reading off the values for the probability of “loss of life” being false we find that we jump from 8.5% (meteor not blown up) to 82% (meteor blown up). This near tenfold increase in the probability of saving the world clearly explains why it merited an attempt.

Lessons learned
Use (Bayesian) Risk Maps rather than the Probability Impact Model or Risk Heat Maps, if you want to take decisions on facts instead of your intuition.

P.S. Many thanks to Benedict Escoto, who attended me on a wrong interpretation of the bicycle risk on bases of the Biomed info.
See document: Deaths of Cyclists in London: trends from 1992 to 2006
I rewrote this blog on information of DFT.

Related Links:
- DFT's Reported Road Casualties 2009
- Pedal cyclist casualties in reported road accidents: 2008 
- Is Cycling Dangerous?
- Cycling in London – How dangerous is it? (2011)
- Nasa: Small Asteroid to Whip Past Earth on June 27, 2011
- Nasa: Close (future) asteroid approaches...
- Nasa: Differences between Asteroid, Comet, Meteoroid, etc.
- Nasa: Search asteroid approaches in data base
- Nasa: Impact Probability of asteroids 
- Fanton & Neil: Measuring risks
- Fanton & Neil: Bayesian networks explained (pdf)
- Neil: Using Risk Maps!

Using Risk Maps

Deathly bike accedents in London

Jun 26, 2010

Death by Solvency

Risk Management can be a strange and deathly game. Normally one would expect that the more the demand of Probability of Insolvency (POI) is decreased:
  • the more Prevention- , Risk-reduction- and Damage-control-measures will be taken
  • the less actual Risk and corresponding Loss will actually occur
  • the higher the resulting average yearly profit
  • the lower the resulting yearly profit volatility

This appears to be true in situations where Risk Management is hardly developed and POI-Demands are relatively modest (5%-2.5%).

Increasing POI-Demands
However, depending on the type of risk, beyond certain POI-Demands (smaller than roughly 2.5%) , the costs of Risk management measures, maintenance and capital requirements become higher than the average expected Loss-reduction, resulting in - on average - lower profits.
Of course, these extra risk management investments and capital requirements can financed by raising consumers prices, but - on balance - this will result a smaller market corresponding with a lower profitability level.

The question can be asked if this still is what we, management and consumers, intended to achieve.......?

Next, in our passion to reduce Risk to an even more extreme low level, we can get carried away completely...

Excessive POI-Demands
When POI-Demands get to levels of 1% or less, a remarkable psychological effect enters the Risk management arena.

Management perceives that the Risk-level is now actually so low that they cannot fail anymore.
In their ambitious goal to outperform the profit level of their competitors, management gets overconfident and reckless. What would you attempt to do if you knew you could not fail?

When POI-Demands are set to levels of 0.5% or less (as they are mostly now in 2010) it becomes almost impossible to beat your competitors with an approach of 'taking more risk'. Even if one would try to manage or hedge these extra risks 'best' in the market. In the long-term, the price of this risk would equal or beat the expected loss.

In this situation some managers get desperate and instead of considering things 'right', they see only one option 'left'....

'Working Around (the) Rules"

WAR, Working Around the Rules, comprises actions like:
  • Taking (extra) risks on non-measurable or non-measured financial transactions, or or 'non-obligated-reporting risks'
  • Manipulating, disguising or mitigate risk information, or risk-control reports
  • Misuse legally allowed methods and accounting principles to create legally unintended financial effects or transactions

It's perhaps hard to admit, but as actual developments show, we've entered the final WAR phase. Some Examples: subprime, Madoff accounting, BP-deep horizon oil failure, bank multipliers, etc, etc.

In all these examples, managers (are pushed to) become over-creative by working around the rules to deliver what they've promised: more profit.

However this approach always results in
  1. More short-term profits
  2. Less long-term profits
  3. Sudden bankruptcy in the end

This development, resembles the 2010 situation in the Insurance and Banking industry where, after each financial debacle, POI-Demands where successively decreased to a 0.5% level  and have resulted in marginal profits and a highly volatile Profits or even losses. Pension Funds (NL: 2.5% POI-Demand) appear to be the next patient the operating table.....

The situation is out of control. Nothing really seems to help anymore....

Are there any solution to prevent this solvency meltdown process?
Yes, but that's for another blog as this one is getting too long...

Related links:
- Why excessive capital requirements harm consumers, insurers and...(2010)
- Presentation - Modelling of Long-Term Risk (2010)

Mar 21, 2010

Country Default Probability

National debts are growing worldwide. It seems we're drowning in a sea of debt. Who's gonna survive?

By experience we know that whenever our gut-feeling takes us for a ride, help of statistical models is necessary to rebalance and get sight at the real problem.

Sovereign Risk Monitor
In this case of 'national debt', the help of CMA's Sovereign Risk Monitor comes in. The CMA Sovereign Risk Monitor identifies and ranks the world’s most volatile sovereign debt issuers according to percentage changes in their 5 year CDS. CMA also calculates the Cumulative Probability of Default (CPD), the 5 year probability of a country being unable to honour its debt obligations.

Let's take a look at the world's most risky countries in Q4 2009:

Yet, the 'Default Landscape' is rapidly changing as becomes clear in CMA's interesting daily 19 March 2010 report showing Greece 'Cumulative Probability of Default' rising to 24.27%.

On the other hand we've got the world's best Countries, with Norway on top....

More actual information is available at CMA (registration required).

Let's hope for the best....

- CMA Sovereign Risk Report for Q4 2009
- Source: CMA
- Latest CMA Update

Jan 17, 2010

Once-in-a-Century Credit Tsunami

When will the next crisis happen and what magnitude will it be?
Investor or actuary, this question puzzles our mind, isn't it?

In the Financial Analysts Journal (January/February 2010) professors Guofu Zhou and Yingzi Zhu raised a similar key question:

Actually Zhou and Zhu did research on a 'October 2008 congress quote' by Alan Greenspan:

We are in the midst of a 'once in a century' credit tsunami


Zhou and Zhu Research
Given the fact that the Dow Jones Industrial Average (DJIA) dropped more than 50 percent, from 14,164 on 9 October 2007 to 6,547 on 9 March 2009, Zhou and Zhu answered the question whether a drop of 50% would be likely to occur once in a century.

Using daily data on the DJIA from 26 May 1896 to 19 June 2008, Zhou and Zhu estimated the long-term average DJIA-return (sample) at µ = 7.4% (excluding dividends) and the long-term volatility, known as the sample standard deviation, at s = 18.2% a year.

DowJones Industrial Average
May 1896 - June 2008
Average return: 7,4%
Standard deviation: 18.2%

On basis of the long-term data, Zhou and Zhu calculated the probability for the market to drop more than 50 percent from a high to a low over a 100-year horizon, considering two different models:
  1. Random Walk Model, excluding dividend
  2. Long Run Risks Model: complex dynamic simulation model, including consumption growth, dividend growth and asset prices

Here is the summarized outcome of their calculations:

As is clear on basis of the Long-Term Risks Model (LTR-Model), no matter what average return or standard deviation, the probability of a 50% draw down over a 100-year horizon is practically almost 100%.

50% draw down over an n-year horizon?
Given these results of Zhou and Zhu, we can now easily and (very) roughly approximate the probability, P(n), of a 50% draw down over an n-year horizon.

with r= P(1). We now roughly 'fit' P(n) to the results of the Long-Term Risks Model as follows:

It turns out the LTR-Models roughly corresponds with one year '50% draw downs probabilities' between r=4% and r=10% [r=P(1)].
As is clear from the table above, even over a 10-year period there's a substantial probability, somewhere between 33% and 65%, of a 50% market breakdown.

Also we can be more than 90% sure to become a witness of a market tsunami once in a lifetime......

The next market tsunami
Up to the next market tsunami, I would guess...., as tsunamis don't have a memory or allow themselves to fit into statistics or models like the ones mentioned above. Unlike natural sea-tsunamis, we - ourselves - are responsible for creating these 'financial tsunamis'.

Irrational Risk Attitude
But even if we are aware of the risk and are not responsible for creating the risk, we have an irrational risk attitude as human beings.
With the recent (2010) Haiti earthquake fresh in mind, let's take a look at the way we deal with the probability of an earthquake.

Los(t) Angeles .....................?
According to the 2007 Working Group on California Earthquake Probabilities (WGCEP 2007) the probability of a magnitude 6.7 or larger earthquake over the next 30 years striking the greater Los Angeles area is 67% (mark the similarity in our P(n) table!).

Yet we deny this reality and 'hope' for the better. Perhaps if every city would have to value the estimated fair value of this earthquake expectation in his balance sheet, things would change. However, I doubt.....

People act irrational with regard to Risk. If we can't manage it, we deny it. If we can manage it, we screw it up!

- Article Is the Recent Financial Crisis Really a ‘Once-in-a-Century’ Event?
- Wall Street Journal article, October 2008: Greenspan
- Credits: CFA Institute
- California Earthquake Probabilities
- Download Spreadsheet of tables used in this blog