Homo Actuarius Bayesianis

Bayesian fallacies are often the most trickiest.....

A classical example of a Bayesian fallacy is the so called "Prosecutor's fallacy" in case of DNA testing...

Multiple DNA testing (Source: Wikipedia)

A crime-scene DNA sample is compared against a database of 20,000 men.

A match is found, the corresponding man is accused and at his trial, it is testified that the probability that two DNA profiles match by chance is only 1 in 10,000.


Sounds logical, doesn't it?
Yes... 'Sounds'... As this does not mean the probability that the suspect is innocent is also 1 in 10,000. Since 20,000 men were tested, there were 20,000 opportunities to find a match by chance.

Even if none of the men in the database left the crime-scene DNA, a match by chance to an innocent is more likely than not. The chance of getting at least one match among the records is in this case:



So, this evidence alone is an uncompelling data dredging result. If the culprit was in the database then he and one or more other men would probably be matched; in either case, it would be a fallacy to ignore the number of records searched when weighing the evidence. "Cold hits" like this on DNA data-banks are now understood to require careful presentation as trial evidence.

In a similar (Dutch) case, an innocent nurse (Lucia de Berk) was at first wrongly accused (and convicted!) of murdering several of her patients.

Other Bayesian fallacies
Bayesian fallacies can come close to the actuarial profession and even be humorous, as the next two examples show:

1. Pension Fund Management
   
   
   It turns out that from all pension board members that were involved in a pension fund deficit, only 25% invested more than half in stocks.
   
   Therefore 75% of the pension fund board members with a pension fund deficit invested 50% or less in stocks.
   
   From this we may conclude that pension fund board members should have done en do better by investing more in stocks....
   
   
2. The Drunken Driver
   It turns out that of from all drivers involved in car crashes 41% were drunk and 59% sober.
   
   Therefore to limit the probability of a car crash it's better to drink...

It's often not easy to recognize the 'Bayesian Monster' in your models. If you doubt, always set up a 2 by 2 contingency table to check the conclusions....

Homo Actuarius
Let's  dive into the historical development of Asset Liability Management (ALM) to illustrate the different stages we as actuaries went through to finally cope with Bayesian stats. We do this by going (far) back to prehistoric actuarial times.
 

As we all know, the word actuary originated from the Latin word actuarius (the person who occupied this position kept the minutes at the sessions of the Senate in the Ancient Rome). This explains part of the name-giving of our species.

Going back further in time we recognize the following species of actuaries..

1. Homo Actuarius Apriorius
   This actuarial creature (we could hardly call him an actuary) establishes the probability of an hypothesis, no matter what data tell.
   
   ALM example: H₀: E(return)=4.0%. Contributions, liabilities and investments are all calculated at 4%. What the data tell is uninteresting.
   
   
2. Homo Actuarius Pragmaticus
   The more developed 'Homo Actuarius Pragamiticus' demonstrates he's only interested in the (results of the) data.
   ALM example: In my experiments I found x=4.0%, full stop.
   Therefore, let's calculate with this 4.0%.
   
   
3. Homo Actuarius Frequentistus
   In this stage, the 'Homo Actuarius Frequentistus' measures the probability of the data given a certain hypothesis.
   
   ALM example: If H₀: E(return)=4.0%, then the probability to get an observed value more different from the one I observed is given by an opportune expression. Don't ask myself if my observed value is near the true one, I can only tell you that if my observed value(s) is the true one, then the probability of observing data more extreme than mine is given by an opportune expression.
   In this stage the so called Monte Carlo Methods was developed...
   
   
4. Homo Actuarius Contemplatus
   The Homo Actuarius Contemplatus measures the probability of the data and of the hypothesis.
   
   ALM example:You decide to take over the (divided!) yearly advice of the 'Parameters Committee' to base your ALM on the maximum expected value for the return on fixed-income securities, which is at that moment  4.0%. Every year you measure the (deviation) of the real data as well and start contemplating on how the two might match...... (btw: they don't!)
   
   
5. Homo Actuarius Bayesianis
   The Homo Actuarius Bayesianis measures the probability of the hypothesis, given the data.  Was the  Frequentistus'  approach about 'modeling mechanisms' in the world, the Bayesian interpretations are more about 'modeling rational reasoning'.
   
   ALM example: Given the data of a certain period we test wetter the value of H₀: E(return)=4.0% is true : near 4.0% with a P% (P=99?) confidence level.

Knowledge: All probabilities are conditional
Knowledge is a strange  phenomenon...

When I was born I knew nothing about everything. When I grew up learned something about some thing. Now I've grown old I know everything about nothing. Joshua Maggid

The moment we become aware that ALL probabilities - even quantum probabilities - are in fact hidden conditional Bayesian probabilities, we (as actuaries) get enlightened (if you don't : don't worry, just fake it and read on)!

Simple Proof: P(A)=P(A|S), where S is the set of all possible outcomes.

From this moment on your probabilistic life will change.

To demonstrate this, examine the next simple example.

Tossing a coin

• When tossing a coin, we all know: P (heads)=0.5
• However, we implicitly assumed a 'fair coin', didn't we?
• So what we in fact stated was: P (heads|fair)=0.5
• 
  Now a small problem appears on the horizon: We all know a fair coin is hypothetical, it doesn't really exist in a real world as every 'real coin' has some physical properties and/or environmental circumstances that makes it more or less biased.
• We can not but conclude that the expression
  'P (heads|fair)=0.5'  is theoretical true, but has unfortunately no practical value.
• The only way out is to define fairness in a practical way is by stating something like:  0.4999≥P(heads|fair)≤0.5001
• Conclusion: Defining one point estimates in practice is practically  useless, always define estimate intervals (based on confidence levels).

From this beginners  example, let's move on to something more actuarial:

Estimating Interest Rates: A Multi Economic Approach

• Suppose you base your (ALM) Bond Returns (R) upon:
  μ= E(R)=4% and σ=2%
  
  
• Regardless what kind of brilliant interest- generating model (Monte Carlo or whatever) you developed, chances are your model is based upon several implicit assumptions like inflation or unemployment.
  
  The actual Return (R_t) on time (t) depends on many (correlated, mostly exogenous) variables like Inflation (I), Unemployment (U), GDP growth(G), Country (C) and last but not least  (R_[t-x]).
  
  A well defined Asset Liability Model should therefore define (R_t) more on basis of a 'Multi Economic Approach'  (MEA) in a form that looks more or less something like: R_t = F(I,U,G,σ,R_[t-1],R_[t-2],etc.)
  
  
• In discussing with the board which economic future scenarios will be most likely and can be used as strategic scenarios, we (actuaries) will be better able to advice with the help of MEA. This approach, based on new technical economic models and intensive discussions with the board, will guarantee  more realistic output and better underpinned decision taking.

Sources and related links:
I. Stats....
- Make your own car crash query
- Alcohol-Impaired Driving Fatalities (National Statistics)
- D r u n k D r i v i n g Fatalities in America (2009)
- Drunk Driving Facts (2006)

II. Humor, Cartoons, Inspiration...
- Jesse van Muylwijck Cartoons (The Judge)
- PHDCOMICS
- Interference : Evolution inspired by Mike West

III. Bayesian Math....
- New Conceptual Approach of the Interpretation of Clinical Tests (2004)
- The Bayesian logic of frequency-based conjunction fallacies (pdf,2011)
- The Bayesian Fallacy: Distinguishing Four Kinds of Beliefs (2008)
- Resource Material for Promoting the Bayesian View of Everything
- A Constructivist View of the Statistical Quantification of Evidence
- Conditional Probability and Conditional Expectation
- Getting fair results from a biased coin
- INTRODUCTION TO MATHEMATICAL FINANCE

Gold: Risk or Rescue?

For those of you who are still doubting...we live in a crazy world....

The Dutch Central Bank (DNB) has ordered (by court !) the glass-workers pension fund (SPVG) to decrease its 13% Gold allocation to less than 3% within two months.

DNB and Court arguments in short:

1. An investment of 13% is not in line with the Prudent Person Rule, which includes the principle that: assets must be invested in such a manner as to ensure the security, quality, liquidity and profitability of the portfolio as a whole.
   
   
2. Gold is a commodity and holding 13%  is classified as 'overweight' in comparison to the 2.7% average that Dutch pension funds have invested in commodities.
   
   
3. 15% allocation in Gold is a 'concentration risk' that could lead to a coverage shortage if the gold price imploded (volatility of Gold is relatively large).

At first, it seems unbelievable that important decisions, with substantial financial impact  - even in Court - are not based on financial facts, but on 'general principles' and the way the market 'used to do it'.

A decision based on an argument that refers to 'the average pension fund,' would more or less imply that pension funds would not be allowed to base their investment strategy on their own specific situation or a changing market outlook. Pension Fund Boards appear to be  'captured' by the market and a Supervisor who obviously has a hard time to develop 'own standards'....

Secondly, DNB actually takes over the investment responsibility of the pension Board. One could wander if DNB is (sufficiently) aware of the possibility that it can be hold financially responsible for the effect of a negative outcome if it turns out in the near future that SPVG has suffered a substantial financial loss, caused by this DNB-designation.

Is Gold really a risk?....  or a rescue?

Checking the facts.... 
Let's just check if DNB's and Court's arguments are valid.....

Yearly Return
We start by comparing the yearly returns of Gold, the S&P-500 Index and '10-Y Treasury Bonds' over the period 1971-2010.

To make Bonds risk-comparable with Gold and the S&P-500 Index, the yearly average Bond interest rate is translated into a yearly Market Value performance. This is done by assuming that each year, all '10-Y Bonds' bought in a specific year are valued, and sold at the average interest rate one year later (approximation).


Here is the result:

To bring some sense and order into this chart, we calculate the 'Moving Compound Annual Growth Rate' (MCAGR).
We start in 2010 and calculate the  compound average yearly return backwards moving up (year by year) to 1971. This is the result:

Now, this looks better... and a bit surprising as well!!! On the long term Gold (μ=9.2%) and the S&P-500 (μ=10.2%) are tending to a rough 9-10% yearly return......  A little bit Surprising is that Bonds (μ=7.6%) get along very well with their big risky brothers...
Take your time to 'absorb' the impact of this chart.....

Risk
Next, we take a look at Risk. We define Risk at first as the Standard Deviation (SD). We directly cut trough to the 'Moving Risk' (Moving SD).

We might conclude here that during recent years there was an increase of risk with regard to the S&P-500 (the 'red' crisis 'Mount K2' is clearly visible). Note that also for a longer period, i.c. the last 30 years, the S&P-500 Risk is substantial higher than the Risk of Gold and much higher than the Risk of Bonds. Only looking at a period of 40 years, Gold shows 'optical' up as more risky (SD=σ=25.8%) than the two other asset categories, Bonds (SD=σ=6.9%) and S&P-500 (SD=σ=18.1%).

However this way of presenting Risk is strongly discussable. Another view of Risk that comes closer to what we naturally 'perceive' as Risk, is to define Risk as only as the Downside Standard Deviation (look up : Sortino ratio ), where all positive yearly returns are eliminated (DSD) or set to zero (DSDZ).....
Let's have a look:

Now, these charts give us a quite a different sight on Risk-reality....
It shows that -on the long term -  not Gold (DSD=Dσ=7.5%) is the riskiest asset, but the S&P-500 (DSD=Dσ=10.6%). Bonds (DSD=Dσ=0.5%), as aspected, have the least volatility and are therefore less risky.

Perhaps the Risk of Bonds is a bit underestimated (very few observations) by the DSD-method (excluding positive yearly returns). In this case the downside deviation of yearly Bond-returns, replacing positive returns by zero, which generates a standard deviation of 3.2%, gives a better indication of a more likely standard deviation on the long run.


Why Gold? 
Although these simple calculations already put the DNB conclusions in a different light, let's get to the main point that should be addressed in defending why Gold should be a substantial part of any Pension Fund portfolio:
 

 Gold Reduces VaR

In a 2010 (october) publication the World Gold Council published a document called Gold: Hedging Against Tail Risk. This interesting report concludes:

1. Gold is first and foremost a consistent portfolio diversifier
2. Gold effectively helps to manage risk in a portfolio, not only by means of increasing risk-adjusted returns, but also by reducing expected losses incurred in extreme circumstances such tail-risk events (VaR).

Following this excellent WGC report, let's test the balancing and risk-reducing  power of Gold by analyzing (classical) Risk (SD) in combining Gold with different allocations (0% up to 100%)  in an asset mix with Bonds, respectively investments in S&P-500 stocks.

This chart clearly shows that Gold has the power to reduce the S&P-500 Risk (SD) from18.1% to 13,3% with an optimal asset location mix of  approximately 60% S&P-500 and 40% Gold. 

In case of Bonds the Risk (SD) is reduced from 6.9%  to 4.8% with an optimal mix of 80% Bonds and 20% Gold.

Asset Liability Model (ALM)
In practice it is necessary to optimize, by means of an adequate ALM study, the  allocation mix of stocks, Bonds and Gold. Just as a 'quick & dirty' excercise, let's take a look at the next asset-combination scenarios, based on data over the period 1971-2010:

Just some head line observations:

• From scenario M1 it becomes clear that even a 100% Bond scenario is't free from Risk. So diversification with other assets is a must.
  
• Looking at M2-M5 we find that the optimal mix, defined as the mix that best maximizes Return (Compound Annual Growth Rate)  and Sharpe Ratio (at a Risk free rate of 3% or 4%) and minimizes Risk (Standard deviation), is something something in the order of: 70% Bonds, 15% stock and 15% Gold.
  
• Scenarios M6-M8 and M9-M11 take todays most common (but strongly discussable!) practice as a starting point. Most pension funds have allocated around 50% or 40% to Bonds and 50% or 60% in more risky asset categories (stocks, etc.). It's clear that even in this situation Risk can be reduced and Return can be optimized, if Stocks are exchanged to Gold with a maximum allocation of 20% or 30%.

Notifier

Although this 'rule of thumb exercise' on this website provides some basic insights, please keep in mind that finding the optimal mix is work for professionals (actuaries).

A serious ALM Study is always necessary and should not only take into account a broad range of diversified asset categories, but should also focus and optimize on:

• The impact of the liabilities (duration) and coverage ratio volatility
• The Timing: Mean values and Standard Deviations are great, but the expected return highly depends on the actual moment of  investment or divestment in the market.
• Future expectations. In the current market situation (2011) the risk of interest rates going up and therefore Bond market value going strongly down, isn't hypothetical. Secondly, the stock market has been pumped up by trillions of 'investments' (?) in the US economy. Once this crisis-aid definitely stops, the question is if these 'cement investments' will be strong enough to keep stocks up. Personally I fear the worst...
  Not to mention a scenario with declining stock rates in combination with increasing interest rates and inflation......
  Who said the life of an actuary was easy???

Conclusion
We may conclude that:

• Investing in Gold up to a 10% to 15% allocation, reduces the Risk of a portfolio consisting of Bonds and S&P-500 Stocks substantially. 
• Gold is less Risky than investing in S&P-500 Stocks

Therefore the 'not with facts' underpinned intervention of DNB looks - to put it euphemistically -  at least strongly discussable....

A wise and modest underpinned allocation of Gold is no Risk, it's a Rescue!

Related Links:
- Spreadsheet with Data used in this Blog
- Prudent person Rule
- IPE: Dutch regulator orders pension scheme to dump gold
- GOLD: HEDGING AGAINST TAIL RISK
- Downside Risk:Sortino ratio
- Dutch Central Bank Orders Pension Fund To Sell Its Gold
- Pension Fund Benchmarking 
- Strategic Risk Managment and Risk Monitoring for Pension Funds

Bonus: Gold, Hedging against Tail Risk Video

Investment Strategy: The Price of Doubt

Most actuaries have seen it happen: A perfect designed investment strategy......., turning into a real nightmare. How could it come that far? What happened?

Life of an actuary...
Let's dive into a real life simplified actuarial case....:

As the actuary of your company, you've developed a perfect ALM study. Together with the head of the investment department, you've been able to convince your Board of the new developed 'Investment Strategy'. A consequent mix of 50% Bonds and 50% stocks, resulting in an average expected 6% return on the long term, turned out to be the best (optimal) investment mix given the risk appetite of your Board and the regulatory demands. All things are set for execution.

Now let's see how your strategic plan would develop (scenario I) and how it would probably be executed by the Board (scenario II) over the next ten years.

Although your investment strategy plan was designed on a rational basis and the execution of this plan was also intended to be a rational process, in practice they are not.....

Let's follow the discussion in the Board from year to year...

Year 1

The company's average portfolio return performs according plan (6%). Stocks: 8%, Bonds 4%, on average 6%. The Board concludes they have the right strategy. You, as an actuary, agree.

Year 2

Compliments from the Board. Stocks perform even higher (10%), leading to a 7% average return.
You sleep well that night.

Year 3

Another fabulous Stock performance year. A stock return of 20%, leading to an average return of 12%! Some Board members start to doubt and question your ALM-model. They are arguing that if stock prices are that high three years in a row, they would like to profit more from this development. They suggest to adjust the asset mix in favor of stocks. Your ALM model should me more flexible.

You are defending your Asset Liability Model to the grave, but after extensive discussions all board members agree that a slight 'temporary' adjustment to 70% stocks and 30% bonds would be 'worth the risk' to profit from this high stock return. With great reluctance, you agree....

Year 4

Although the performance of stocks is not as high as the year before, it's still relatively high (15%) and leads to an average return of 11.7%, which is 2.2% (!) higher than the 9.5% return that would have been achieved with a 50/50% mix.  The Board concludes that it took the right decision last year, to adjust the asset mix to 70/30%.

You - as the responsible actuary - warn again, but the facts are against you. Disappointed and misunderstood you return to your office as the President of the Board tries to cheer you up by thanking you for your 'constructive response' in the board meeting. You abstain from joining the festive Board Party that evening.

Year 5

Stocks are dramatically down to 0%, leading to an average mixed return of 1.2% this year.
The board meeting this year is chaotic. Some members support you as the 'responsible actuary' to readjust the asset mix to the original mix of 50/50%. Others argue that this stock dip is only temporary and that this year's average return is only 0.8% lower than would have been achieved with a 50/50% mix. On top of, most members strain that this year's 0.8% negative return is still lower than the 2.2% positive difference of last year. After two stressful board meetings, the Board decides to stick to their 70/30% investment mix.
The board president's eye fails to meet you, as you leave the board room that night.

Year 6

What was most feared, has become true.. A negative stock return of 10%, leading to an average return of -5.8% ....   When you walk into the board room that night, all eyes are on you as the 'responsible actuary'. You hold your breath, just like all other board members. After a short moment of silence the board president states that he proposes to bring back the asset mix to the original 50/50% mix. Without further discussion this proposal is accepted. There's no board party this year.

Year 7

Negative stock returns have increased to 15%, leading to an average return of -5.5% this year.
Some Board members fear that if stock prices will be down for another few years, the average 'needed' return of 6% will not be met. They doubt the current strategy.

Also the Regulator and some Rating Agencies insist on higher confidence and solvency levels with corresponding measures to be taken. Both are not positive and doubt the outlook on stock returns on the long term...

After a long meeting that night, the Board chooses for reasons of 'savety' (!) to adjust the asset mix to 30/70% in favor of the still 4% stable performing Bonds (Better something than nothing (!) ).

Again... you explain that night, that changing the asset mix following actual market performance, is the worst thing a company can do....  But again, you lose the debate.

The power of emotion is greater than the power of rationality. Now not only the Board seems against you, but the Regulator as well. Who wants to fight that! After all, 'ethical' rule number one is 'complying with the Regulator'. That evening you brainwash yourself and reprogram your attitude to 'actuarial follower' instead of 'actuarial leader'.

After two Johnnie Walkers you see the future bright again and seem ready for the new year.


Year 8
To everybody's surprise stocks performed extremely well at 25% this year. As a result the average return reaches a satisfying performance of 10.3%. With 'mixed feelings' board members take notice of the results. What nobody dears to say and everybody seems to think is: 'Had we stuck to our 70/30% asset mix, the performance would have been: 18.7% (!)......'

The Board President cautiously concludes that the Board took the right decision last year, leading to a proud 10.3% return this year. Compliments to everyone, including the actuary! Supported by your 'converted' mind, the 30/70% asset mix is continued. That evening you accept the invitation to the board party. Lots of Johnnie Walkers help you that night to cope with the decisions taken.

Year 9

Stocks perform at 20%, leading to an 8.8% average mixed return. No Board member dears to raise questions about the possibility of readjusting the asset mix to a 'more risky' (what's that?) one. After all, the overall performance is still higher than the needed 6%. So who may complain or doubt the new 'On the Fly Strategy'? Who cares or who dears? You go to bed early that night.

Year 10
Stocks returns have come down to a more 'realistic' level of 7%. As a consequence the average return is down to 4.9%, way down beneath the critical level of 6%. Board members have to strike a balance. Some of them doubt again. Continuing the 30/70% asset mix will not bring them the needed long term 6% objective return. Adjusting to a 50/50% mix probably will, but is more risky. What to do?

All eyes are on you as the 'final advising actuary'. With restrained pride you state: "Dear colleagues, what about our good friend, the original '50/50% asset mix'. Can we confirm on that?" Without anyone answering, the President takes a look around.... His gavel hits the table and the decision seems to have been taken.

AftermathEmatics.......

That night you decide to change Johnnie Walker for a well deserved glass of 'actuarial wine': a simple  'Mouton Rothschild 1945' (at the expense of the Board of course). You enjoy the moment and the pleasure of being an actuary. Even after the Rothschild you realize that the decade price of doubt was high: 0.9% p.a. ...

When you go to bed for a good night sleep, you smile...., as some little voice in your head tells you that next year this madness decade-cycle will probably start again...

Pension Fund Death Spiral

In a very simplified model (Pensions Dynamics, PPT), professor of investment strategy, Alan White, concludes that defined benefit pension plans probably cannot succeed on the long term.

Death Spiral
White shows that every pension fund with a non risk-free asset approach, will eventually encounter a “Death Spiral” which will lead to the collapse of the fund. The only solutions are:

• Raising contribution rates
• Lowering promised pension benefits.

Assumptions
All conclusions are based on the next summarized main assumptions:

• Compensation growth: 2% per year
• Pension contribution: 15% of yearly compensation
• Yearly retirement income objective: 70% of his final salary
• Risk-free rate of interest is 3%;Risk premium on the risky assets: 3%
• Annual volatility of the risky assets: 15%
• Time horizon: 100-year
• Risky Assets investment part : 60% of the portfolio
• Corresponding final pay pension defined as 20 year annuity
• Required minimum average Pension Fund asset value in steady state
  - at 3% return: €/$ 47,200
  - at 6% return: €/$ 23,600

Frequency Distribution Outcome
One of the most striking outcomes of this study is the fact that as we look farther in to the future of the simulated pension fund, the amplitude of the frequency distribution of asset values appears to be dropping to zero. The chance that (average) asset values will be between $10,000 and $100,000 gets smaller and smaller.

The reason for this is that the probability of very high asset values and the probability of entering a collapsed state (the collapsed funds are not shown in the next figure) both increase as we expand out time horizon. As a result the probability that assets remain in the intermediate interval, is reduced.

Another interesting facts are:

• Asset values appear to become more sustainable as the part 'risky assets' increases
• Collapse rates for growing pension funds are, (almost) independently of the asset mix, negligible.
• Collapse rates for more mature (steady state) pension funds are substantial and increase to deadly percentages as the time horizon increases from 50 to 100 years.

Remarks
Although Whites model is perhaps oversimplified and can be easily criticized, it clearly shows the essential principles of running a pension fund.

In a commentary, Rob Bauer (ABP, University of Maastricht) argues White's conclusions. Nevertheless, interesting stuff, that stimulates actuarial insight.

Links
Interesting corresponding links:

• Longer Horizons Can Handle Riskier Holdings
• Stochastic Interest rate models (2008)
• The calculus of retirement income