On The Probabilistic and The Correlation Conundrum
We think of most, but not all, variables and experiences, in life as forms of the Standard Normal Probability Density function - the well known Gaussian Bell Curve. Variables can be from the mundane - distribution of height in a sample population, to the esoteric - daily percentage change in asset values (more on this later, but for now we assume a Normal distribution). Experiences, for this purpose, are behaviors relative to expectations of what is considered normal, in any given culture.
Every Normal distribution is defined by two inherent parameters: its 'Mean' and 'Variance'. For practical purposes, Mean is the value of the variable (or, experience) that is the average of all observed data, and Variance represents the degree of dispersion from Mean. As a matter of practice, these distributions are described using the Standard Deviation (square root of the Variance) instead. Standard deviation, represented by the Greek symbol for the letter Sigma, has become synonymous with "Sigma" itself. So when someone says, for example, that something is a 3 Sigma event, they are actually talking about the probabilistic nature of the event:
A Standard Normal Probability Distribution has a Mean of 0.0, and a Standard Deviation of 1. The distribution is symmetric around its mean - deviations from mean can be positive, or negative. Without going into some boring mathematics, the following list is the probability of an event given how many standard deviations the event is from its mean:
1 Sigma ~ 68.2%
2 Sigma ~ 95%
3 Sigma ~ 99.7%
Now, lets try and put this into context. The probability of an event, that is 3 or More Sigmas away from the Mean, is 0.3%. That is 3 in every 1,000 data points (or observations).
Folks speak of 3 Sigma events as something that is very rare. When you hear someone talk of a 3 Sigma event, it is that rarity they are talking about. As if, they are surprised, or would be shocked, by the event happening at all!
Continuing with the list:
4 Sigma ~ 0.0063% [1 in 15 million]
5 Sigma ~ 0.0000573% [1 in 1.744 billion]
6 Sigma ~ 0.0000001973% [1 in 506 billion]
5, and 6 Sigma events begin to truly represent the "rare". A Six Sigma event, for example, has a 1 in 506 billion chance of happening. In criminal forensic DNA sciences, for example, the accused typically get convicted if DNA evidence can show 1 in a few billion chance (2 to 4 billion) that the DNA tested did not come from the accused. Identical twins, excluded of course.
For those with some interest in US stock markets - the S&P500 (SP500) is a benchmark index of the market. It comprises 500 of the largest, and diversified industry stocks, representing 70% of the total US stock market capitalization, or about $10 Trillion. Stock market Index's Mean returns (average % change) and Standard Deviation of Return (Volatility of the Index) can be measured over any time period - 1 year, 3 years, since inception, etc. Looking back at the last 3 years, the Index has a Mean of 16.40% and Volatilty of 16.08%. These are annualized numbers - on average, the index returned (gain) of 16.4%, with a variability (standard deviation or volatility of 16.08%. With the SP500 currently at 1356, investors can expect a 16.40%, or 222 point annualized gain, and a 1 Sigma daily index variability of about 1.005% (16.08% annualized volatility converted into a daily volatility number), or about 13.5 index point a day.
A Six Sigma event, for the SP500 Index (1 in 506 billion chance, remember?) would be a 81 point change, in either direction (up, or down), on any given day.
Looking back upon the days where the SP500 Index experienced the largest changes:
Losing Days, Change in Index points:
1. on 2008-09-29 Change: -102.65
2. on 2008-10-15 Change: -86.76
3. on 2000-04-14 Change: -83.95
Winning Days, Change in Index points:
1. on 2008-10-28 Change: +91.59
2. on 2008-10-13 Change: +90.6
3. on 2000-03-16 Change: +66.32
So, within the last 12 years (approximately 3,072 trading days), we have seen 5 instances of a SIX SIGMA event. 1 in 506 billion is a myth. While mathematically correct, the fallacy lies in market participants assuming that the market returns are normally distributed. They are not, even on normal good days. Factor in EVENT RISK, and all notions of 6 SIGMA being an extremely rare event go out the window.
The second Monster, lurking in most every Investment Banks' trading books, is Correlation risk - both at a trading/model level, and in the tools/models used to estimate Value at Risk (VaR). Correlation estimates the degree to which 2 or more asset prices/returns move in tandem. Banks, particularly large global players, like using correlation matrices across large number of assets classes, to:
1. Create "Basket" trades, and price/hedge such trades - asset return correlation is a significant input into modeling, pricing, and hedging such transactions. Small changes in correlation, as an input parameter, can create large swings in pricing. Such correlation based models also guide/predict the amount of hedges across each of the component asset classes, but typically the traders have discretion in how to execute their hedges. What is quite important here is to note, that correlation input can create large price changes. So, at inception, a trader may, for example, buy at a few correlation points below fair value thereby creating a large profit (albeit funny money) upfront. This "funny money" profit is realized over time - either the hedges behave as they are supposed to, and re-balancing the hedges captures the upfront profits over time, or the trader is able to sell the exact same offsetting risk to a 3rd party, at a few correlation points above fair value (the Bid-Offer spread). Correlation trading, in theory, is a very profitable line of business.
2. Middle Office Risk Management VaR reporting - VaR, no matter how sophisticated the models and tool, is an estimate for projected 3 SIGMA (or, for some banks, 2 SIGMA) losses over a chosen time horizon. One can calculate 1 day VaR, or 1 Year VaR, and everything in between. Most VaR models simulate global changes in asset prices using a two step process:
a. Simulate non-Correlated Random Numbers, assuming some probability density distributions.
b. From these non-Correlated random numbers, generate Correlated random numbers - I know, "Correlated Random Numbers" sounds like an oxymoron, but it isn't.
Cholesky Decomposition is one popular technique used to go from (a) to (b).
Once the VaR models have correlated random numbers - and this may be based on 1000 X 1,000 correlation matrix, or larger, depending on the number of asset classes, and level of granularity desired (1 year vs. 2, 3, 5, 10, etc., on each yield curve, as an example) - calculating VaR is a simple matter of: either repricing every position by applying asset price changes based on the correlated random numbers, or by approximating the P&L for each simulation run by applying time series expansion techniques, given "delta" (first derivative) and "gamma" (second derivative) for each position, and the implied or historical price volatility for each asset class.
VaR estimates are generally used by senior management - to track portfolio risks and exposures (in P&L terms) in an abstract, 30k foot level, and to track positions against policy limits.
The problem with Correlation though is that it is not static. Even for a moment. Measuring and updating correlation estimates in real time is not for the faint of heart. Re-balancing and "Reserves" based pricing, techniques from the world of options pricing theory, is meant to capture correlation changes. But when Shit Hits The Fan, as it does under any financial, or political crisis, all correlation estimates (historical, or otherwise) go out the window, and correlation trading books are left naked, and at the whim of market gyrations. Correlation is the most profitable, yet most toxic of products a bank can trade either for itself, or with its customers. There is no known cure, for when correlation assumptions go astray, other than recognizing and absorbing large losses.
In crisis scenarios, trades in correlation trading books will produce completely unexpected day to day P&L. And these swings can be large in either direction. This creates a disincentive for the trader to update the marks on their positions when the losses keep rolling in - taking deep daily losses can be catastrophic to bonus expectations and to their trading careers. I am by no means suggesting all traders are dishonest. There are plenty of honest ones out there. The problem usually is either a dishonest trader who tries to cover-up losses in their book, hoping the market will turnaround, or an honest trader, who marks their book to market as they should, everyday, but one that is in "love" with their own positions, and can talk management out of being forced to liquidate such positions.
By the time management realizes the trades are not coming back, it is usually too late. The damages have been done. Unwinding such basket correlation trades is no trivial matter. Market participants will not let a JP Morgan, or any other bank for that matter, out from under these risks without first extracting their own pound of flesh. That is just Wall Street being Wall Street. This adds to the pain of unwinding them rotten correlation positions.
For "Senior Management" used to seeing comfortable 4:30PM daily VaR reports, daily P&L, assuming correlation marks are marked to market, no longer makes sense, for it is no longer in line with VaR estimates. Very few bank CEOs understand why, and the normal reaction is to reach out to the head of the trading unit for answers. The answers the CEO gets, is of course a function of the culture of the bank, and that of the trading unit.
An insulated CIOs office, thousands of miles and several time zones away from the CEOs desk, one that has enjoyed vast historical success, and that is now sitting on very large correlation risks gone bad - is a perfect recipe for disaster.
We understand why JP Morgan lost $6 BILLION on one single trade, the so called "WHALE" trade.
We think of most, but not all, variables and experiences, in life as forms of the Standard Normal Probability Density function - the well known Gaussian Bell Curve. Variables can be from the mundane - distribution of height in a sample population, to the esoteric - daily percentage change in asset values (more on this later, but for now we assume a Normal distribution). Experiences, for this purpose, are behaviors relative to expectations of what is considered normal, in any given culture.
Every Normal distribution is defined by two inherent parameters: its 'Mean' and 'Variance'. For practical purposes, Mean is the value of the variable (or, experience) that is the average of all observed data, and Variance represents the degree of dispersion from Mean. As a matter of practice, these distributions are described using the Standard Deviation (square root of the Variance) instead. Standard deviation, represented by the Greek symbol for the letter Sigma, has become synonymous with "Sigma" itself. So when someone says, for example, that something is a 3 Sigma event, they are actually talking about the probabilistic nature of the event:
A Standard Normal Probability Distribution has a Mean of 0.0, and a Standard Deviation of 1. The distribution is symmetric around its mean - deviations from mean can be positive, or negative. Without going into some boring mathematics, the following list is the probability of an event given how many standard deviations the event is from its mean:
1 Sigma ~ 68.2%
2 Sigma ~ 95%
3 Sigma ~ 99.7%
Now, lets try and put this into context. The probability of an event, that is 3 or More Sigmas away from the Mean, is 0.3%. That is 3 in every 1,000 data points (or observations).
Folks speak of 3 Sigma events as something that is very rare. When you hear someone talk of a 3 Sigma event, it is that rarity they are talking about. As if, they are surprised, or would be shocked, by the event happening at all!
Continuing with the list:
4 Sigma ~ 0.0063% [1 in 15 million]
5 Sigma ~ 0.0000573% [1 in 1.744 billion]
6 Sigma ~ 0.0000001973% [1 in 506 billion]
5, and 6 Sigma events begin to truly represent the "rare". A Six Sigma event, for example, has a 1 in 506 billion chance of happening. In criminal forensic DNA sciences, for example, the accused typically get convicted if DNA evidence can show 1 in a few billion chance (2 to 4 billion) that the DNA tested did not come from the accused. Identical twins, excluded of course.
For those with some interest in US stock markets - the S&P500 (SP500) is a benchmark index of the market. It comprises 500 of the largest, and diversified industry stocks, representing 70% of the total US stock market capitalization, or about $10 Trillion. Stock market Index's Mean returns (average % change) and Standard Deviation of Return (Volatility of the Index) can be measured over any time period - 1 year, 3 years, since inception, etc. Looking back at the last 3 years, the Index has a Mean of 16.40% and Volatilty of 16.08%. These are annualized numbers - on average, the index returned (gain) of 16.4%, with a variability (standard deviation or volatility of 16.08%. With the SP500 currently at 1356, investors can expect a 16.40%, or 222 point annualized gain, and a 1 Sigma daily index variability of about 1.005% (16.08% annualized volatility converted into a daily volatility number), or about 13.5 index point a day.
A Six Sigma event, for the SP500 Index (1 in 506 billion chance, remember?) would be a 81 point change, in either direction (up, or down), on any given day.
Looking back upon the days where the SP500 Index experienced the largest changes:
Losing Days, Change in Index points:
1. on 2008-09-29 Change: -102.65
2. on 2008-10-15 Change: -86.76
3. on 2000-04-14 Change: -83.95
Winning Days, Change in Index points:
1. on 2008-10-28 Change: +91.59
2. on 2008-10-13 Change: +90.6
3. on 2000-03-16 Change: +66.32
So, within the last 12 years (approximately 3,072 trading days), we have seen 5 instances of a SIX SIGMA event. 1 in 506 billion is a myth. While mathematically correct, the fallacy lies in market participants assuming that the market returns are normally distributed. They are not, even on normal good days. Factor in EVENT RISK, and all notions of 6 SIGMA being an extremely rare event go out the window.
The second Monster, lurking in most every Investment Banks' trading books, is Correlation risk - both at a trading/model level, and in the tools/models used to estimate Value at Risk (VaR). Correlation estimates the degree to which 2 or more asset prices/returns move in tandem. Banks, particularly large global players, like using correlation matrices across large number of assets classes, to:
1. Create "Basket" trades, and price/hedge such trades - asset return correlation is a significant input into modeling, pricing, and hedging such transactions. Small changes in correlation, as an input parameter, can create large swings in pricing. Such correlation based models also guide/predict the amount of hedges across each of the component asset classes, but typically the traders have discretion in how to execute their hedges. What is quite important here is to note, that correlation input can create large price changes. So, at inception, a trader may, for example, buy at a few correlation points below fair value thereby creating a large profit (albeit funny money) upfront. This "funny money" profit is realized over time - either the hedges behave as they are supposed to, and re-balancing the hedges captures the upfront profits over time, or the trader is able to sell the exact same offsetting risk to a 3rd party, at a few correlation points above fair value (the Bid-Offer spread). Correlation trading, in theory, is a very profitable line of business.
2. Middle Office Risk Management VaR reporting - VaR, no matter how sophisticated the models and tool, is an estimate for projected 3 SIGMA (or, for some banks, 2 SIGMA) losses over a chosen time horizon. One can calculate 1 day VaR, or 1 Year VaR, and everything in between. Most VaR models simulate global changes in asset prices using a two step process:
a. Simulate non-Correlated Random Numbers, assuming some probability density distributions.
b. From these non-Correlated random numbers, generate Correlated random numbers - I know, "Correlated Random Numbers" sounds like an oxymoron, but it isn't.
Cholesky Decomposition is one popular technique used to go from (a) to (b).
Once the VaR models have correlated random numbers - and this may be based on 1000 X 1,000 correlation matrix, or larger, depending on the number of asset classes, and level of granularity desired (1 year vs. 2, 3, 5, 10, etc., on each yield curve, as an example) - calculating VaR is a simple matter of: either repricing every position by applying asset price changes based on the correlated random numbers, or by approximating the P&L for each simulation run by applying time series expansion techniques, given "delta" (first derivative) and "gamma" (second derivative) for each position, and the implied or historical price volatility for each asset class.
VaR estimates are generally used by senior management - to track portfolio risks and exposures (in P&L terms) in an abstract, 30k foot level, and to track positions against policy limits.
The problem with Correlation though is that it is not static. Even for a moment. Measuring and updating correlation estimates in real time is not for the faint of heart. Re-balancing and "Reserves" based pricing, techniques from the world of options pricing theory, is meant to capture correlation changes. But when Shit Hits The Fan, as it does under any financial, or political crisis, all correlation estimates (historical, or otherwise) go out the window, and correlation trading books are left naked, and at the whim of market gyrations. Correlation is the most profitable, yet most toxic of products a bank can trade either for itself, or with its customers. There is no known cure, for when correlation assumptions go astray, other than recognizing and absorbing large losses.
In crisis scenarios, trades in correlation trading books will produce completely unexpected day to day P&L. And these swings can be large in either direction. This creates a disincentive for the trader to update the marks on their positions when the losses keep rolling in - taking deep daily losses can be catastrophic to bonus expectations and to their trading careers. I am by no means suggesting all traders are dishonest. There are plenty of honest ones out there. The problem usually is either a dishonest trader who tries to cover-up losses in their book, hoping the market will turnaround, or an honest trader, who marks their book to market as they should, everyday, but one that is in "love" with their own positions, and can talk management out of being forced to liquidate such positions.
By the time management realizes the trades are not coming back, it is usually too late. The damages have been done. Unwinding such basket correlation trades is no trivial matter. Market participants will not let a JP Morgan, or any other bank for that matter, out from under these risks without first extracting their own pound of flesh. That is just Wall Street being Wall Street. This adds to the pain of unwinding them rotten correlation positions.
For "Senior Management" used to seeing comfortable 4:30PM daily VaR reports, daily P&L, assuming correlation marks are marked to market, no longer makes sense, for it is no longer in line with VaR estimates. Very few bank CEOs understand why, and the normal reaction is to reach out to the head of the trading unit for answers. The answers the CEO gets, is of course a function of the culture of the bank, and that of the trading unit.
An insulated CIOs office, thousands of miles and several time zones away from the CEOs desk, one that has enjoyed vast historical success, and that is now sitting on very large correlation risks gone bad - is a perfect recipe for disaster.
We understand why JP Morgan lost $6 BILLION on one single trade, the so called "WHALE" trade.
