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 13, 2011

Actuary Garfield

There's not a lot of 'Actuary Humor' on the Internet. Here's one...

Actuary Garfield explains how actuaries think...

Great and lots of humor, those Garfield cartoon strips, (especially those about actuaries....).

Original Sources:
- Garfield Snow
- Garfield Snowman

Jun 5, 2011

Short Term Longevity Risk

As well-born actuaries we all know the long term risks of longevity:

Lots of actuaries keep expending their energy on calculations of 50 years ahead mortality probabilities....  And indeed..., this is challenging....

Some research reports predict a decline in life expectation, others and more serious recent reports show a steady increase of life expectation.

Mission Impossible
Fact of actuarial life is that - although long term research is useful and educational - we are no Actuarial Magicians.

We should never suggest that we're able to value a bunch of complex and systemic risks  (liabilities, assets,mortality, costs, demographics, etc) into a reliable consistent model that predicts reality.

It's a farce!

What CAN we do?
Instead trying to compress a complex of long term risky cash flows into one representing unique value, we need to:
  1. Analyze and model the short term risks
  2. Develop a method (system) that enables boards of directors to manage and control their risky cash flows (profit share systems, experience rating, etc.).

Example: Short Term Longevity Risk
As a 2011 report of the National Research Council clearly shows:  The previous 50 years we've seen a 3 months yearly increase of lifespan every calendar year.

Instead of recalculating, checking and pondering this trend, let's take a look at the short term effects of this longevity increase trend.

Effect of 'one year life expectancy' increase 
First we take a look at the cost effect of the increase of 'one year of life expectancy' on a single-premium of a (deferred) life annuity (paid-up pensions)...
( Life table total population: United States, 2003 )

Depending on the discounting interest rate, a one year improvement of longevity for a 65 old person demands a 2,3% to 4,0% increase of the liabilities.

Of course the increase of the liabilities of a portfolio (of a pension fund) depends on the (liability weighted) age distrubution of the corresponding portfolio.

Here's a simple example:

This comes close to the rule of thumb as mentioned by AEGON:

10% mortality improvement adds one year to life expectancy, and one year of life expectancy adds 4% to the required value of a pension fund’s reserves

From the above presented visual sensitivity analysis we may conclude that for general (distributed) portfolio's a 'one year lifetime increase' will demand approximately 4-5% of the actual liabilities.

A three to four months yearly longevity-increase - as is still the actual trend - will therefore demand roughly a substantial 1,5% (yearly) of the liabilities.
This implies that in case your contribution is calculated at 4% and your average portfolio return is 7%, there's 3% left for financing longevity and indexation (=method). As 'longevity growth' in the near future will probably cost about 1,5%, there's  only 1,5% left for indexation on the long run.

Case closed

Related links:
Spreadsheet (xls) with data used in this blog
- Forecasting longevity of Dutch pension scheme members using postcodes
- Increasing life expectancy at pension funds (uvt;2011)
- Life Tables for the United States Social Security Area 1900-2100
- Valuing Pension Fund Liabilities on the Balance Sheet
- No limits to life expectancy?
- Broken Limits to Life Expectancy
- NRC: Explaining divergent levels of longevity (pdf;2011)
- Wolfram Alpha: Longevity U.S.
- AEGON: Longevity Rule of thumb