Chris Humphrey

John Cotter

Kevin Dowd

Margaret Woods

English

Research Papers in Economics

 
UCD GEARY INSTITUTE 
DISCUSSION PAPER SERIES 
 
 
 
 
How Unlucky is 25-Sigma? 
 
By 
Kevin Dowd, John Cotter, Chris Humphrey and Margaret Woods
§ 
 
 
 
 
March 24, 2008 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
___________________ 
§
 Dowd and Woods: Centre for Risk and Insurance Studies, Nottingham University Business School, 
Jubilee Campus, Nottingham NG8 1BB, UK. Cotter: Centre for Financial Markets, School of Business, 
University  College  Dublin,  Carysfort  Avenue,  Blackrock,  Co.  Dublin,  Ireland.  Humphrey:  School  of 
Accounting and Finance, University of Manchester, Crawford House, Oxford Road, Manchester M13 
9PL, UK. Corresponding author: Kevin.Dowd@nottingham.ac.uk. How Unlucky is 25-Sigma? 
By  
Kevin Dowd, John Cotter, Chris Humphrey and Margaret Woods
♣ 
 
March 24 2008 
  
 
1. Introduction 
 
One of the more memorable moments of last summer’s credit crunch came when the 
CFO of Goldman Sachs, David Viniar, announced in August that Goldman’s flagship 
GEO hedge fund had lost 27% of its value since the start of the year. As Mr. Viniar 
explained, “We were seeing things that were 25-standard deviation moves, several 
days in a row.”
1 One commentator wryly noted: 
 
That Viniar. What a comic. According to Goldman’s mathematical models, 
August,  Year  of  Our  Lord  2007,  was  a  very  special  month.  Things  were 
happening that were only supposed to happen once in every 100,000 years. 
Either that … or Goldman’s models were wrong (Bonner, 2007b).  
 
But sadly Goldman were not alone. In 2007 alone, massive losses were announced by 
Bear  Stearns,  UBS,  Merrill  Lynch  and  Citigroup,  and  then  there  were  the  earlier 
financial disasters – 1987, Daiwa, Barings, Long-Term Capital, the dotcoms, Russia, 
East Asia, and so on – and afterwards Société Générale and Bear Stearns again in 
early 2008, with rumours of more yet to come. 
 
Citi’s case was particularly interesting. To quote from the same commentator: 
 
Gary Crittenden, Citi’s chief financial officer, claimed … that the firm was 
simply  a  victim  of  unforeseen  events.  …  No  mention  was  made  of  the 
previous five years, when Citi was busily consolidating mortgage debt from 
people who weren’t going to repay … pronouncing it ‘investment grade’ … 
mongering it to its clients … and stuffing it into its own portfolio … while 
paying itself billions in fees and bonuses. No, according to the masters of the 
universe, downgrades by Moody’s and Fitch’s were completely unexpected … 
like  the  eruption  of  Vesuvius;  even  the  gods  were  caught  off  guard. 
Apparently, as of September 30
th, Citigroup’s subprime portfolio was worth 
every penny of the $55 billion that Citi’s models said it was worth. Then, 
whoa, in came one of those 25-sigma events. Citi was whacked by a once-in-
a-blue-moon fat tail.  
 
Who could have seen that coming? (Bonner, 2007c).  
 
                                                
♣ Dowd and Woods: Centre for Risk and Insurance Studies, Nottingham University Business School, 
Jubilee Campus, Nottingham NG8 1BB, UK. Cotter: Centre for Financial Markets, School of Business, 
University College Dublin, Carysfort Avenue, Blackrock, Co. Dublin, Ireland. Humphrey: School of 
Accounting and Finance, University of Manchester, Crawford House, Oxford Road, Manchester M13 
9PL, UK. Corresponding author: Kevin.Dowd@nottingham.ac.uk. 
1 Reported in the Financial Times, August 13, 2007.  Be all this as it may, one thing is for sure: there are certainly a lot of very unlucky 
financial institutions around.  
 
 
2. How Unlikely is a 25 sigma event? 
 
The once-in-a-100,000 year figure was quoted in a number of places,
2 and suggests 
that Goldman, Citi and so on must have been very unlucky indeed. But exactly how 
unlikely is a 25-sigma shock?
3  
 
To start with, lets assume that losses are normally distributed – assume that losses 
obey the classic bell curve – and ask the question: what is the probability of a loss that 
is, say, 2 standard deviations or more away from the mean, i.e., what is the probability 
of a 2-sigma loss event? 
 
The answer is given in Figure 1: the probability associated with a 2 sigma event is 
equal to the mass of the right-hand tail of the distribution demarcated at the point 
where the number of standard deviations from the mean is equal to 2, and this is equal 
to 2.275%.
4 We might therefore expect to see a 2-sigma loss event on one trading day 
out of 1/2.275%=43.956 days, i.e., on approximately 1 day out of 44 days. A 2-sigma 
event is unlikely to occur on any given day, but we would expect to see a few of them 
in any given year.
5 
 
Figure: Probability of a 2-Sigma Event 
                                                
2  See,  e.g.,  Bonner  (2007b,c)  Randomwalk  (2007)  or  Hutchinson  (2008).  A  more  colourful 
interpretation, though sadly one that we are unable to verify – was that a 25-sigma event was as likely 
as Hell freezing over (Bonner, 2007a).  Another commentator, Seth Jayson, suggested that a 25-sigma 
event is as likely as catching an asteroid in the hand (Jason, 2007). Mr. Jayson also made another 
entertaining analogy, but one that we are sadly unable to repeat.  
3 A straw poll of students and colleagues at Nottingham University Business School carried out by one 
of the authors indicated that not a single person had the remotest idea of the true probability of a 25 
sigma event, and most seemed to think it was likely to occur more often than once in 100,000 years. 
4  For  readers  who  wish  to  verify  it,  this  number  can  be  obtained  using  the  command  ‘=1-
NORMSDIST(2)’ in Excel. 
5 Again, in the interests of verifiability: if we go by the common market convention of there being 250 
trading days in a year, then we would expect to see about 250/44 or approximately 5.68 such events in 
any given year. -4 -3 -2 -1 0 1 2 3 4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Number of standard deviations from mean
P
r
o
b
a
b
i
l
i
t
y
 
d
e
n
s
i
t
y
Probability = 2.275% 
 
 
 
We  now  use  the  same approach  to  estimate  the  probabilities  of  losses  that  are  3 
sigmas away from the mean, 4 sigmas away from the mean, and so on. The results of 
this exercise for 3, 4, 5, 6 and 7 sigma events are given in Table 1. 
 
Table 1: Probabilities of k-sigma events: k=3, 4, 5, 6 and 7 
k  Probability in any given day  Expected occurrence: once in every  
3  0.135%  740.8  days 
4  0.00317%  31559.6  days 
5  0.000029%  3,483,046.3  days 
6  0.000000099%  1,009,976,678  days 
7  0.000000000129%  7.76e+11  days 
 
 
The reader will note that as k gets bigger the probabilities of a k-sigma event fall 
extremely rapidly:  
•  a 3-sigma event is to be expected about every 741 days or about 1 trading day 
in every three years; 
•  a 4-sigma event is to be expected about every 31,560 days or about 1 trading 
day in 126 years (!);  
•  a 5-sigma event is to be expected every 3,483,046 days or about 1 day every 
13,932 years(!!) 
•  a 6-sigma event is to be expected every 1,009,976,678 days or about 1 day 
every 4,039,906 years; 
•  a 7-sigma event is to be expected every 7.76e+11 days – the number of zero 
digits is so large that Excel now reports the number of days using scientific 
notation, and this number is to be interpreted as 7.76 days with decimal point 
pushed back 11 places. This frequency corresponds to 1 day in 3,105,395,365 
years.  
 These  results  are  breathtaking.  To  give  them  some  perspective:  a  5-sigma  event 
corresponds to an expected occurrence of less than just one day in the entire period 
since  the  end  of  the  last  Ice  Age;  a  6-sigma  event  corresponds  to  an  expected 
occurrence of less than one day in the entire period since our species, Homo Sapiens, 
evolved  from  earlier  primates;  and  a  7-sigma  event  corresponds  to  an  expected 
occurrence of just once in a period approximately five times the length of time that 
has elapsed since multicellular life first evolved on this planet.
6  
 
At this point, there are so many decimal points in the numbers involved that Excel is 
unable to handle values of k  bigger than 7, and we are still a very long way from the 
25 sigma events that Goldman and Co. experienced. To get around the limitations of 
Excel,  we  now  switch  over  to  MATLAB  and  use  the  MATLAB  command  ‘1-
normcdf(8,0,1)’ to estimate the corresponding probability for an 8-sigma event, and 
this corresponds to an expected occurrence once every 6.429e+012 years. To put this 
into perspective, this is a period (considerably) longer than the entire period that has 
elapsed since Big Bang.
7 If we observe a profit or loss once a day, then a mere 8-
sigma event should occur less than once in the entire history of the universe.  
 
We then moved to 9 sigma events, but ran into a problem because the MATLAB 
‘normcdf’  function  gives  9  sigma  and  larger  events  a  flat  value  of  zero:  the 
probabilities are so small they now fall under the function’s radar. To get around this 
problem  we  wrote  a  specially  designed  MATLAB  function  to  estimate  the 
probabilities and expected occurrence periods associated with larger losses, and the 
details involved are reported in the Appendix. We then used this function to produce 
the high-sigma results shown in Table 2: 
 
Table 2: Probabilities of High Sigma Events 
k  Probability in any given day  Expected occurrence: once in every  
10  7.620e-022 %  5.249e+020  years 
15  3.671e-049 %  1.090e+048  years 
20  2.754e-087 %  1.453e+086  years 
25  3.0570e-136 %  1.309e+135  years 
 
These numbers are on truly cosmological scales, and a natural comparison is with the 
number of particles in the Universe, which is believed to be between 1.0e+73 and 
1.0e+85 (Clair, 2001). Thus, a 20-event corresponds to an expected occurrence period 
measured  in  years  that  is  10  times  larger  than  the  higher  of  the estimates  of  the 
number of particles in the Universe. For its part, a 25-sigma event corresponds to an 
expected occurrence period that is equal to the higher of these estimates but with the 
decimal point moved 52 places to the left!  
 
To give a more down to earth comparison, on February 29 2008, the UK National 
Lottery is currently was offering a prize of £2.5m for a ticket costing £1. Assuming it 
to be a fair bet, the probability of winning the lottery on any given attempt is therefore 
0.0000004.  The  probability  of  winning  the  lottery  n  times  in  a  row  is  therefore 
                                                
6 The end of the last Ice Age was about 10,000 years ago; Homo Sapiens evolved within the last 
million years; and multicellular life on Earth originated about 600 million years ago.  
7 The NASA website reports that Big Bang occurred between 12 and 14 billion years ago.  0.0000004
n , and the probability of a 25 sigma event is comparable to the probability 
of winning the lottery 21 or 22 times in a row.
8  
 
And we should not forget Goldman’s losing streak – Goldman did not just experience 
a  single  25-sigma  event,  but  experienced  several  in  a  row  –  or  forget  that  other 
institutions also experienced 25-sigma events. If the probability of a single 25-sigma 
event is low, the odds of two or more such events are truly infinitesimal. For example, 
the odds of two 25-sigma events on consecutive days are equal to 3.057e-136 squared, 
which is 9.3450e-272. This is as likely as winning the lottery about 42 times in a row. 
The corresponding expected occurrence period is the square of 1.309e+135 years – 
that  is,  1.713e+270  years  –  a  number  so  vast  that  it  dwarves  even  cosmological 
figures. As Oscar Wild might have put it: to experience a single 25-sigma event might 
be  regarded  as  a  misfortune,  but  to  experience  more  than  one  does  look  like 
carelessness. 
 
It is pretty clear by now that a 25-sigma event is much, much, much less likely than is 
suggested by an expected occurrence every 100,000 years. In fact, if we take the true 
figure (1.309e+135) and divide it by 100,000 we get a figure equal to 1.309e+130, 
and this tells us that the estimate of 100,000 is out by 130 decimal points: it is not 
even close to within 100 decimal points. We suspect, on the other hand, that the 
estimate of a 25-sigma event being on a par with Hell freezing over is probably about 
right. 
 
 
3. Bad luck or just incompetence? 
 
However low the probabilities, and however frequently 25-sigma or similar events 
actually  occur,  it  is  always  possible  that  Goldman’s  and  other  institutions  that 
experienced such losses were just unlucky – albeit to an extent that strains credibility.
9 
But if these institutions are really that unlucky, then perhaps they shouldn’t be in the 
business of minding other people’s money. Of course, those who are more cynical 
than us might suggest an alternative explanation – namely, that Goldmans and their 
Ilk are simply not competent at their job. Heaven forbid!  
 
All  this  poses  an  interesting  dilemma  for  investors:  would  you  prefer  the  people 
looking after your money to be incredibly unlucky or just plain incompetent? There 
again,  maybe  there  is  truth  in  both  explanations.  Benjamin  Franklin  once  aptly 
observed that “Diligence is the mother of good luck”, and it would stand to reason 
that their opposites, bad luck and incompetence, might also be related. 
 
As for Mr. Viniar: he promised that Goldman would take a more “robust approach” in 
future, and this will no doubt be welcome news to hard-hit investors and the millions 
                                                
8 Note that 
21 0.0000004 4.3980e-135 =  and 
22 0.0000004 1.7592e-141 =  so the probability  of a 25-
sigma event lies somewhere between the two. 
9 Defenders of the ‘bad luck’ scenario might argue that the probabilities are only as low as they are 
because we have assumed normality, whereas real world loss distributions are fat-tailed and fat-tailed 
distributions lead to much higher probabilities of extreme events. This is true but irrelevant: so what if 
the true probability of a 25-sigma event is 3.057e-36 rather than 3.057e-136? Moving the decimal point 
a hundred places or more makes no practical difference. We would invite anyone who disagrees to do 
the calculations for themselves. facing foreclosure on their mortgages. He went on to acknowledge that the experience 
“makes you reassess how big the extreme moves can be”.
10 Indeed.  
 
Funny things these 25-sigma events. And surprisingly common too. 
 
 
References  
 
Bonner, B. “Goldman Sachs fund loses 30%, Wall Street math fails to predict future.” 
August  16,  2007.  Available  at  www.dailyreckoning.com.au/wall-street-
math/2007/08/16. (a) 
Bonner, B. “25 standard deviations in a blue moon.” November 11, 2007. Available at 
www.moneyweek.com/file/37640/25-standard-deviations-in-a-blue-moon.html. (b) 
Bonner, B. “Goldman Sachs was wrong and 2 million families may lose their homes.” 
November  14,  2007.  Available  at  www.dailyreckoning.com.au/goldman-sachs-
3/2007/11/14. (c) 
Clair, B. “The biggest numbers in the Universe.” Strange Horizons, April 2, 2001. 
Available  on  the  web  at 
http://www.strangehorizons.com/2001/20010402/biggest_numbers.shtml. 
Hutchinson, M. “Scrap heap for financial models.” The Bear’s Lair, February 11, 
2008. Published on the Prudent Bear website, www.prudentbear.com. 
Jayson, S. “Goldman’s CFO: short truth or long ego.” The Motley Fool. Available on 
the web at http:/www.fool.com/investing/value/2007/08/15/goldmans-cfo-short-truth-
or-long-ego.aspx. August 15, 2007. 
Larsen, P. T. “Goldman pays the price of being big.” Financial Times, August 13, 
2007. 
Randomwalk “The day the gammas lost sight of the betas and alphas.” September 30, 
2007.  Available  on  the  web  at 
www.scribblesheet.co.uk/article/the_day_the_gammas_lost_sight_of_the_betas_and_
alphas. 
Van Zeghbroeck, B. (2007) Principles of Semiconductor Devices. Published on the 
web at http://ece-www.colorado.edu/~bart/book/book/index.html. 
 
 
Appendix: Estimating the Probabilities of High Sigma Events 
 
Following Zeghbroeck (2007), let the Gaussian probability density be  ( ) G x  and the 
error function be  ( ) erf x . These are related via 
 
(1)                                                ( )
2
x
x
x
G x dx erf
σ −
 
=  
  ∫  
 
If we set  1 σ =  then  x is the number of standard deviations away from the mean. 
Hence,  
 
(2)                                                 ( )
2
x
x
x
G x dx erf
−
 
=  
  ∫  
                                                
10 Financial Times, August 13, 2007.  
Let  p   be  the  tail  probability  associated  with  some  value  x.  It  follows  from 
rearranging (2) that 
 
(3)                                              0.5 0.5
2
x
p erf
 
= − ×  
 
 
However, it is also known that for large  y ,  
 
(4)                         
2
2 2 2 2 3
1 1 3 1 3 5
( ) 1 1 ...
2 (2 ) (2 )
y e
erf y
y y y y π
−   × × ×
≈ − − + + +  
 
 
 
It then follows from further rearranging that 
 
(5)                                 
2
2 2 2 2 3
1 1 3 1 3 5
1
2 (2 ) (2 ) 2
y e
p
y y y y π
−   × × ×
≈ − + +  
 
 
where  / 2 y x = .  
 
The MATLAB function used to produce the high-sigma results reported in the paper 
is reproduced below: 
 
function p=high_sigma_prob(x) 
% Function estimates p, the probability of a high-sigma loss event. 
% 
% Input: 
%  - x: size of loss in terms of number of sigmas away from mean 
% 
% Written by Kevin Dowd March 1, 2008. 
********************************************************************* 
y=x/sqrt(2); 
p=0.5*(exp(-y^2)/(sqrt(pi)*y))*... 
(1-1/(2*y^2)+1*3/(2*y^2)^2+1*3*5/(2*y^2)^3); 
 
 
 
 

Goldman pays the price of being big.” Financial Times,

Goldman Sachs fund loses 30%, Wall Street math fails to predict future.”

Goldman’s CFO: short truth or long ego.” The Motley Fool. Available on the web at http:/www.fool.com/investing/value/2007/08/15/goldmans-cfo-short-truthor-long-ego.aspx.

How Unlucky is 25-Sigma?

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