Dec 3, 2012

Solvency II or Basel III ? Model Fallacy

Managing investment models - ALM models in particular - is a professional art. One of the most tricky risk management fallacies when dealing with these models, is that they are being used for identifying so called 'bad scenarios', which are then being 'hedged away'.

To illustrate what is happening, join me in a short every day ALM thought experiment...

Before that, I must warn you... this is going to be a long, technical, but hopefully interesting Blog. I'll try to keep the discussion on 'high school level'. Stay with me, Ill promise: it actuarially pays out in the end!

ALM Thought Experiment
  • Testing the asset Mix
    Suppose the board of our Insurance Company or Pension Fund is testing the current strategic asset mix with help of an ALM model in order to find out more about the future risk characteristics of the chosen portfolio.
     
  • Simulation
    The ALM model runs a 'thousands of scenarios simulation', to find out under which conditions and in which scenarios the 'required return' is met and to test if results are in line with the defined risk appetite.
     
  • Quantum Asset Return Space
    In order to stay as close to reality as possible, let's assume that the 'Quantum Asset Return Space' in which the asset mix has to deliver its required returns for a fixed chosen duration horizon N, consists of: 
    1. 999,900 scenarios with Positive Outcomes ( POs ),
      where the asset returns weigh up to the required return) and 
    2. 100 scenarios with Negative Outcomes ( NOs ),
      where the asset returns fail to weigh up to the required return.
       
    Choose 'N' virtual anywhere between 1 (fraction) of a year up to 50 years, in line with your liability duration.
     

  • Confidence (Base) Rate
    From the above example, we may conclude that the N-year confidence base rate of a positive scenario outcome (in short: assets meet liabilities) in reality is 99.99% and the N-year probability of a company default due to a lack of asset returns in reality is 0.01%.
     
  • From Quantum Space to Reality
    As the strategic asset mix 'performs' in a quantum reality, nobody - no board member or expert - can tell which of the quantum ('potential') scenarios will come true in the next N years or (even) what the exact individual quantum scenarios are.

    Nevertheless, these quantum scenarios all exist in "Quantum Asset Return Space" (QARS) and only one of those quantum scenarios will finally turn out as the one and only 'return reality'.

    Which one...(?), we can only tell after the scenario has manifested itself after N years.
     
  • Defining the ALM Model
    Now we start defining our ALM Model. As any model, our ALM model is an approach of reality (or more specific: the above defined 'quantum reality') in which we are forced to make simplifications, like: defining an 'average return', defining 'risk' as standard deviation, defining a 'normal' or other type of model as basis for drawing 'scenarios' for our ALM's simulation process.
    Therefore our ALM Model is and cannot be perfect.

    Now, because of the fact that our model isn't perfect, let's assume that our 'high quality' ALM Model has an overall Error Rate of 1% (ER=1%), more specific simplified defined as:
    1. The model generates Negative Scenario Outcomes (NSOs) (= required return not met) with an error rate of 1%. In other words: in 1% of the cases, the model generates a positive outcome scenario when it should have generated a negative outcome scenario
       
    2. The model generates Positive Scenario Outcomes (PSOs) (= required return met) with an error rate of 1%. In other words: in 1% of the cases, the model generates a negative outcome scenario when it should have generated a positive outcome scenario
       

The Key Question!
Now that we've set the our ALM model, we run it in a simulation with no matter how much runs. Here is the visual outcome:


As you may notice, the resulting ALM graph tells us more than a billion numbers....At once it's clear that one of the scenarios (the blue one) has a very negative unwanted outcome.
The investment advisor suggests to 'hedge this scenario away'. You as an actuary raise the key question:

What is the probability that a Negative Outcome (NO) scenario in the ALM model is indeed truly a negative outcome and not a false outcome due to the fact that the model is not perfect?

With this question, you hit the nail (right) on the head...
Do you know the answer? Is it 99% exactly, more or less?

Before reading further, try to answer the question and do not cheat by scrolling down.....

To help you prevent reading further by accident, I have inserted a pointful youtube movie:



Answer 
Now here is the answer: The probability that any of the NOs (Negative Outcomes) in the ALM study - and not only the very negative blue one - is a truly a NO and not a PO (Positive Outcome) and therefore false NO, is - fasten your seat belts  - 0.98%! (no misspelling here!)

Warning
So there's a 99.02% (=100%-0.98%) probability that any Negative Outcome from our model is totally wrong, Therefore one must be very cautious and careful with drawing conclusions and formulating risk management actions upon negative scenarios from ALM models in general.

Explanation
Here's the short Excel-like explanation, which is based on Bayes' Theorem.
You can download the Excel spreadsheet here.


There is MORE!
Now you might argue that the low probability (0.98%) of finding true Negative Outcomes is due to the high (99,99%) Positive Outcome rate and that 99,99% is unrealistic much higher than - for instance - the Basel III confidence level of 99,9%. Well..., you're absolutely right. As high positive outcome rates correspond one to one with high confidence levels, here are the results for other positive outcome rates that equal certain well known (future) standard confidence levels (N := 1 year):


What can we conclude from this graph?
If the relative part of positive outcomes and therefore the confidence levels rise, the probability that an identified Negative Output Scenario is true, decreases dramatically fast to zero. To put it in other words:

At high confidence levels (ALM) models can not identify negative scenarios anymore!!!


Higher Error Rates
Now keep in mind we calculated all this still with a high quality error rate of 1%. What about higher model error rates. Here's the outcome:


As expected, at higher error rates, the situation of non detectable negative scenarios gets worse as the model error rate increases......

U.S. Pension Funds
The 50% Confidence Level is added, because a lot of U.S. pension funds are in this confidence area. In this case we find - more or less regardless of the model error rate level - a substantial probability ( 80% - 90%) of finding true negative outcome scenarios. Problem here is, it's useless to define actions on individual negative scenarios. First priority should be to restructure and cut ambition in the current pension agreement, in order to realize a higher confidence level. It's useless to mop the kitchen when your house is flooded with water.....

Model Error Rate Determination
One might argue that the approach in this blog is too theoretical as it's impossible to determine the exact (future) error rate of a model. Yes, it's true that the exact model error rate is hard to determine. However, with help of backtesting the magnitude of the model error rate can be roughly estimated and that's good enough for drawing relevant conclusions.

A General Detectability Equation
The general equation for calculating the Detectability (Rate) of Negative Outcome Scenarios (DNOS) given the model error rate (ER)  and a trusted Confidence Level (CL) is:

DNOS = (1-ER) (1- CL) / ( 1- CL + 2 ER CL -ER )

Example
So a model error rate of 1%, combined with Basel III confidence level of 99.9% results in a low 9.02% [ =(1-0.01)*(1-0.999)/(1-0.999+2*0.01*0.999-0.01) ] detectability of Negative Outcome scenarios.

Detectability Rates
Here's a more complete oversight of detectability rates:


It would take (impossible?) super high quality model error rates of 0.1% or lower to regain detectability power in our (ALM) models, as is shown in the next table:



 Required  Model Confidence Level
If we define the Model Confidence Level as MCL = 1 - MER, the rate of Detectability of Negative Outcome Scenarios as DR= Detectability Rate = DNOS and the CL as CL=Positive Outcome Scenarios' Confidence Level, we can calculate an visualize the required  Model Confidence Levels (MCL) as follows:

From this graph it's at a glance clear that already modest Confidence Levels (>90%) in combination with a modest Detectability Rate of 90%, leads to unrealistic required Model Confidence Rates of around 99% or more. Let's not discuss the required Model Confidence Rates for Solvency II and/or Basel II/III.

Conclusions
  1. Current models lose power
    Due to the effect that (ALM) models are limited (model error rates 1%-5%) and confidence levels are increasing (above > 99%) because of more severe regulation, models significantly lose power an therefore become useless in detecting true negative outcome scenarios in a simulation. This implies that models lose their significance with respect to adequate risk management, because it's impossible to detect whether any negative outcome scenario is realistic.
     
  2. Current models not Solvency II and Basel II/III proof
    From (1) we can conclude in general that - despite our sound methods -our models probably are not Solvency II and Basel II/III proof. First action to take, is to get sight on the error rate of our models in high confidence environments...
     
  3. New models?
    The alternative and challenge for actuaries and investment modelers is to develop new models with substantial lower model error rates (< 0.1%).

    Key Question: Is that possible?

    If you are inclined to think it is, please keep in mind that human beings have an error rate of 1% and computer programs have an error rate of about 3%.......
     

Links & Sources:

By Maggid on 12/03/2012 11:32:00 PM 2 responses
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Nov 24, 2012

Dying Age Quiz

Ever heard of Club 27? It turns out that famous pop artists have a preferred age of dying: 27.

Among this 'club', with around 50 unlucky 'members' that all died at the age of 27, are well known names like Brian Jones, Jimi Hendrix, Janis Joplin, Jim Morrison and (lately, 2011) Amy Winehouse.

There's been a lot of (actuarial) discussion whether this club 27 phenomenon is a mortality anomaly or not.

In a statistical study from BMJ (British Journal of Medicine) called "Is 27 really a dangerous age for famous musicians? Retrospective cohort study", it's shown that  there's no peak in the risk of death for famous musicians at age 27.


Club 27, or its movie,  is therefore a nice opportunity to study some interesting artists who died young, but not based on any statistical relevance.

Quiz
Not only some top musicians died young, but also some 'historical' celebrities.

Now take the next quiz to test your knowledge on the dying age of the next famous people who changed the world, each on in his/her own way:





Links and sources:
- BMJ Statistical Study
- Dying Age Quiz of Famous People
- Death, Actuarial Science and Rock n’ Roll-the 27 Club

By Maggid on 11/24/2012 10:38:00 PM 0 responses

Nov 17, 2012

Pension for Contribution

People are lost if it comes down to their pension. A recent (2012) Friends Life survey found that 68% of Britons do not know the collective value of their pension funds.....

This result is in line with a Dutch 2011 survey, that concludes that 66% has no knowledge of their pension.

Pension illiteracy is clearly a worldwide phenomenon. Pensions are a 'low interest' product. Unfortunately - nowadays - in the double sense of the latter words.

As an actuary, people often ask me at a birthday party : I'm paying a 1000 bucks contribution each year for my pension, but does it pay out in the end? Can you tell me?

Unfortunately most actuaries, including myself, answer this question by telling that this is a difficult question to answer straightforward and that the pension outcome depends on topics like age, mortality, return, inflation, gender, indexation, investment scheme, asset mix, etc., etc.....

Simplifying
To make a breakthrough in this pension communication paradox, let's try to create more pension insight with a simple approach. But remember - as with everything in life - the word 'simple' implies that we can not be complete as well as consistent at the same time. After all, Kurt Gödel's incompleteness theorems clearly show that nothing in life can be both complete and consistent at the same time.

Thanks to God and Gödel, we can stay alive on this planet by simplifying everything in life to a level that our brains can comprise. We'll keep it that way in this blog as well.

How much pension Benefits for how much contribution?
First thing to do, is to give the average low pension interested person on this planet an overall hunch on what a yearly investment of a 1000 bucks(first simplification: S1)until the pension age of 65 year (S2) delivers in terms of a yearly pension as of age 65 in case of an average pension fund.

If we state 'bucks' here, we mean your local general currency. We denote 'bucks' here simply as $, or leave it out. So $ stands for €, ¥ , £ or even $ itself.

Now let's calculate for different pension contribution start ages (S3)what a yearly contribution of $ 1000 (payable in months at the beginning of each month; S4), pays back in terms of a yearly pension (payable in months at the end of each month; S5) on basis of a set of different constant return rates (S6). The calculation is on a net basis (so without costs; S7), a Dutch (2008) mortality table (S8) and without any inflation (S9), any pension indexation (S10), any contribution indexation (S11), or any tax influence (S12).

Here's the simple table we're looking for:

TABLE 1
Yearly Pension at age 65 on basis of 1000 yearly contribution
Pension Indexation=0%, Contribution Indexation=0%, Inflation: 0%
StartNet Yearly Return Rate
Age0%1%2%3%4%5%6%7%8%
252692366950196898952113192183362555335681
30234531134134550373429812131331760123612
3519992584333543045555717192571194915421
40165420842614327340925109637079339867
45131116101964238828963502422550856107
5097211631380163219212254263530723570
5563874486098911321291146516571868
60312355400448499554612674738

In a graphical view on a logarithmic pension benefits scale, it looks something like this:

Example
To illustrate what is happening, a simple example:
When you join your pension fund at age 40 and start saving $ 1000 a year (the first of every month: $ 83.33) until your 65, you'll receive a yearly pension benefit of $ 4092 yearly ( $ 341 at the end of every month) from age 65 of, as long as you live.

From this table, we can already draw some very basic conclusions:
  • To build up a substantial pension, it pays out if you start early in life
  • The pension outcome is heavily dependent on the yearly return of your pension fund
  • Most pension funds operate on basis of a 'general employee and/or employer contribution' instead of individual employee contributions.
    This implies that younger employees pay more than they should have paid on an individual basis and older employees less. In other words, younger employees subsidize older employees. How much more, you can derive from the tables above and by comparing the individual contributions to the general contribution level of the pension fund.


Pension Indexation
As we all want to protect our pension against inflation, let's calculate the outcome of a 'real pension' instead of a 'nominal pension'. As long term yearly inflation rates vary between 2% and 3%, we make the same calculation as above, but now the yearly pension outcome (as from age 65) will be indexed with 3% (fixed) at the end of every year and the yearly contribution paid, will also be yearly indexed with 3%.
Here's the outcome:

TABLE 2
Yearly Pension at age 65 on basis of 1000 yearly contribution
Pension Indexation=3%, Contribution Indexation=3%, Inflation: 0%
StartNet Yearly Return Rate
Age0%1%2%3%4%5%6%7%8%
2536874914656688141188916112219332997841124
302947385950516624870711468151382002126526
35230929683803487262417994102401311916809
401760221827813479434454136735836810382
45128815901949238128973515425151276168
5088210661277152318072134251229443440
555366337418629981149131615021706
60243281321364411461515573634

To get grip at the comparison between a real and a nominal pension, we express the real pension (3% Indexed Pensions and Contribution) as a percentage of the nominal pension:

TABLE 3
Yearly Pension at age 65 on basis of 1000 yearly contribution
'3% P&C-Indexed Pensions' as percentage '0% P&C-Indexed Pensions'
StartNet Yearly Return Rate
Age0%1%2%3%4%5%6%7%8%
25137%134%131%128%125%122%120%117%115%
30126%124%122%120%119%117%115%114%112%
35116%115%114%113%112%111%111%110%109%
40106%106%106%106%106%106%106%105%105%
4598%99%99%100%100%100%101%101%101%
5091%92%93%93%94%95%95%96%96%
5584%85%86%87%88%89%90%91%91%
6078%79%80%81%82%83%84%85%86%

From this last table we can conclude that if you start saving for your pension below the age of 40 your indexed savings weight up to the indexed pension. Above the age of 45 it is the other way around.

The above figures are the kind of figures (magnitude) you'll find on your benefits statements. You can compare in practice whether your benefit statement is in line with the above tables....

The Inflation Monster
In the last given example, pension is 3% inflation protected as from the moment of retirement.

However, if pension is not also yearly fully indexed (in this case: 3%) during the contribution period, there still is a major potential inflation erosion risk left.

In this case it's interesting to examine what the value of a 3% indexed pension in combination with a 3% indexed contribution is worth in terms of actual money, as inflation would continue at a constant 3% level each year. Here's the answer:

TABLE 4
Yearly Pension at age 65 on basis of 1000 yearly contribution
Pension Indexation=3%, Contribution Indexation=3%, Inflation: 3%
StartNet Yearly Return Rate
Age0%1%2%3%4%5%6%7%8%
251130150720132702364549396724919012607
30104713711795235430944076538071159427
3595112231567200725713293421954056925
4084110591328166220752586321739974959
457138801079131816041946235428393415
5056668482097711601370161218902208
5539947155264274285597911171269
60210242277314355398445494547

What we notice is a substantial inflation erosion effect as the pension fund participants get younger.
Let's zoom in on an example to see what we can achieve with these tables.

Example
  • From table 2 we can conclude that - at a 4% return rate - a 40 year old starting pension fund member, with a $ 1000 dollar yearly 3% indexed contribution will reach a 3% yearly indexed pension of $ 4344 yearly at age 65.
  • From table 4 we can subsequently conclude that, based on an inflation rate of 3%, this $ 4344 pension has a 'real' value of $ 2075, if it's expressed in the value money had when the participant was 40 years old (so, at the start).
  • From table 4 we can also conclude that in order to 'compensate' inflation erosion for this pension member, the pension fund has to achieve a return of around 7.4%.
    This follows from simple linear interpolation:
    7,4% = 7% + 1% * (4344-3997)/(4959-3997)

I'll leave other examples to your own imagination.

The effect of a constant inflation on a pension is devastating, as the next table shows

TABLE 5
Inflation Erosion
  • Pension indexation=3%
    as of age 65
  • Contribution indexation=3%
  • Inflation=3%
Start
Age		Inflation
Erosion
2569%
3064%
3559%
4052%
4545%
5036%
5526%
6014%
From table 5 it becomes clear that Inflation erosion is indeed substantial.
If you have a fully indexed pension from age 65 (who has?) of and you're N years away from your retirement, an inflation of i% will erode your pension with E%. In formula:
         
Example
Set inflation to 3%. If you're 40 years old and about to retire at 65, you've got 25 years (N=25=65-40) ahead of you.

If your pension of let's say $ 10,000 a year is not indexed during this period, you can buy with this $ 10,000 no more than you could buy today with $ 4,800.

Your pension is eroded due to inflation with 52% = 1- 1.03^-25. So only 48% is left.....

Finally
I trust these tables and examples contribute a little to your pension insight. Just dive into your pension, it's financially relevant and certainly will pay out!
Remember that all results and examples in this blog are approximations and simplifications on a net base (no costs or taxes are included). In practice pension funds or insurers have tot charge costs for administration, asset management, solvency, guarantees, mortality risk, etc. . This implies that in practice the results could differ strongly with the results as shown in this blog. The examples in this blog are therefore for learning and demonstration purposes only.

The above calculations were made in a few minutes with help of the Excel Pension Calculator that was developed in 2011 and updated in 2012.
With help of this pension planner you can calculate all kind of variations and set different variables, including different mortality tables (or even define your own mortality table).

You can download the pension calculator for free and make your own pension calculations.
More information about pension calculating with this simple pension calculator at:


Enjoy your pension, beware of inflation....

Links & Sources:
By Maggid on 11/17/2012 12:41:00 AM 0 responses
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