QRM, Extremes, Mathematics
and the Financial Crisis
Paul Embrechts
Department of Mathematics andDirector of RiskLab, ETH Zurich
Senior SFI Chairwww.math.ethz.ch/~embrechts

…
Embrechts, P., Resnick, S., Samorodnitsky, G.:
Living on the Edge. RISK, January 1998, 96-100.
Chavez-Demoulin, V., Embrechts, P.: Revisiting theedge, ten years on. To appear in Communications in
Statistics - Theory and Methods (2008/9)

About the title
I want to discuss with you four concepts:
Concept 1: QRM
Concept 2: Extremes (low probability events)
Concept 3: Mathematics
Concept 4: Financial Crisis
From a (somewhat personal) historical perspective
and in the light of the current economic crisis.
(Background: A.J. McNeil, R. Frey and P. Embrechts,
Quantitative Risk Management. Concepts, Techniques, Tools. Princeton UP, 2005)

Some relevant examples• February 1, 1953: Dutch dyke disaster• January 28, 1985: Challenger explosion• October 19, 1987: Black Monday• July 6, 1988: Piper Alpha• January 17, 1994: Northridge Earthquake• January 17, 1995: Barings and Kobe
• September 1998: LTCM hedge fund crisis• 2007-200x (x>9!): Credit crisis• and (unfortunately) many more …

Ex.1: 31. Jan. 1953 – 1. Feb. 1953*(February floading)
•
•
•
•
•

•

1836 people killed
72000 people evacuated
49000 houses and farms
floaded201000 cattle drowned
500 km coastal defenses
destroyed; more than 400
breaches of dykes
200000 ha land floaded

The Delta – Project••••

Coastal fload-protection
Requested dyke height at l: hd(l)Safety margin at l: MYSS(l) =Maximal Yearly Sea Surge at l:
Probability(MYSS(l) > hd(l)) should be „small“,
whereby „small“ is defined as:(Risk)– 1 / 10000 in the Randstad– 1 / 250 in the Deltaregion in the North– Similar requirements for rivers, but with 1/10 – 1/100•

For the Randstad (Amsterdam-Roterdam):
Dyke height = Normal-level (= NAP) + 5.14 m

PE

NAP

Guus Balkema

PE
Laurens de Haan

Guus Balkema

Ex.2: Northridge Earthquake: some loss ratio
numbers (%) to think about!

Fin-Ex.1: February 1995

The Great Hanshin (Kobe) earthquake ofJanuary 17, 1995

Prime example for Operational Risk,
external event (on top of all else)How Kobe earthquake and a straddle position
finally broke the back of Barings bank
Straddle = Short Call and Short Put on Common Strike

Volume of Nikkei Futures

Fin-Ex.2: The Black-Scholes Formula(s)Conditionsc = S0 N (d1 ) − K e−rT N (d2 )
p=Ke

−rT

N (−d2 ) − S0 N (−d1 )ln(S0 / K ) + (r + σ2 / 2)Twhere d1 =σ T
2ln(S0 / K ) + (r − σ / 2)Td2 == d1 − σ T
σ TFinancial Derivatives: where (*) it all started (*)• Black, F., and M. Scholes (1973): “The Pricing of
Options and Corporate Liabilities,” Journal of Political
Economy, 81, 637–654.
• Merton R. C. (1973): “Theory of Rational Option Pricing,”
Bell Journal of Economics and Management Science, 4,
141–183.
• However (*), L. Bachelier (1900), V. Bronzin (1908),
E.O. Thorp (1969)!

The Black-Scholes Model and Model Uncertainty

Just waiting for the storm to hit!

And the Perfect Storm came in
September 1998

Nobel

Prize 1997

Some vivid recollections of meetings/discussions
with RM and MS:• 1996: Meetings with RM and MS selling LTCM
investment to Swiss Private Banks• 1996-1998: Several discussions with MS on theuse of EVT to calculate (market) regulatory
capital: “Who is going to pay for the difference?”,i.e. Δ = (VaR-EVT) – (VaR-Normal) >> 0
• Cambridge Newton Institute Workshop on
Managing Uncertainty, 2001: “Insurance is (just)the other side of the coin (of finance)!” (MS)• Copula confusion

What about regulation?
• 1988: Basel I• 1994 – 2000:
Amendment to Basel I(Basel I ½), Value-atRisk (VaR) is born
• 2000 – 2009/10(?):
Basel II (Credit andOperational Risk)• Future: Basel III ???
• Later more on this!

What we should have learned from these andsimilar events:• One (past) example:- The Challenger explosion
and then more on the current issue:- The credit crisis …

Lesson 1: A (very) brief discussion on the
Challenger explosion

O-RingRichard Feynman

Some key modeling input:• Logistic regression: theory exists!
• Rare event prediction (31 degr. F)• Model Uncertainty• Statistical analysis: data matters!
• Statistical estimation of this uncertainty(95% confidence intervals)
These intervals are typically very wide forthe estimation of rare events

Lesson 2: A (very) brief discussion on the credit
crisisThe players (the agents, the components, the jigsaw pieces)(from Crouhy, Jarrow, and Turnbull (2008)):- Rating Agencies- Mortgage Brokers and Lenders- Special Investment Vehicles (SIVs)- Monolines- ABS Trust, CDS, CDO and CDO Squared Equity Holders- Financial Institutions- The Economy and Central Banks- Valuation Uncertainty … once again!!!!
- Transparency, or better… Opaqueness!
- Systemic Risk
- Politicians/the press/lawyers/accountants/the public …

All of these “components” need a careful and in depth discussion!

As examples of credit derivatives:

CDS = Credit Default Swap
A relatively simple instrument
CDO = Collateralized Dept Obligation
A rather complex instrument
A brief technical discussion of the latter and asomewhat more general discussion on the former:

Exhibit 2.9: The conventional wisdom – 2006 (!!!!!)“ There is growing recognition that the dispersion of credit
risk by banks to a broader and more diverse group ofinvestors, rather than warehousing such risk on theirbalance sheets, has helped make the banking and overall
financial system more resilient .

The improved resilience may be seen in fewer bank failures
and more consistent credit provision. Consequently thecommercial banks may be less vulnerable today to credit or
economic shocks ”
IMF Global Financial Stability Report, April 2006

A stylized Credit Default Swap Set-Up

1 bio USD

PF1

F-BB
10%/year
Insurance on
F-BB’s debt

1%/yearrating…
IC-AA
PF2/F2
PF3/F3… PFn/FnRA

rating

HF1
Betting on

HFk
default, no

link

CDOs
Complexity,
Opacity,
Distance,
Greed,
Economicand Political
Stupidity,
Regulatory
Blindness,
Academic Naivity,and Arrogance…
We are all to blame!

A sure road for disaster• In the previous picture, problems occur if several
corporations
default at the same time, inthat case the insurance companies
have topay, may loose their high rating causing thepension funds (investors)
more problems,
etc, etc … someone at some time will blame therating agencies• But what about the hedge funds … ?
• In the end all depends on default correlation …
enters the Gauss-copula.

The normal distribution

Extremes matter

(Synthetic)

Credit Default Swaps

Securitisation
construction

The investors

The waterfall principle

?
Eq.

Mez.

Sen.

An import critical voice:
Warren Buffet on Derivatives (Berkshire Hathaway annual report for 2002):

The derivatives genie is now well out of the bottle,
and these instruments will almost certainly multiply
in variety and number until some event
makes their toxicity clear. Central banks
and governments have so far found noeffective way to control, or even monitor,
the risks posed by these contracts. In
my view, derivatives are financial weapons of mass destruction, carrying dangers that, while now latent, are potentially lethal.

Some dimensions before we continue:
Thousand $ = 1 000 $Million $= 1 000 000 $Billion (US) $ = 1 000 000 000 $= 1 Milliard (UK) $Trillion (US) $ = 1 000 000 000 000 $= 1 Billion $ (UK)
= 1 Billion $ (Germany)
Trillion (UK) $ = 1 000 000 000 000 000 000 $Everything clear … I hope.

50 000 000 000 000 $ *• CDS is almost a brand new investment vehicle, but themarket is already 20 times its size in 2000. The principal
amount of CDS outstanding equals $50 trillion, or morethan three times the U.S. Gross Domestic Product andbigger than all the U.S. credit markets put together. And
the CDS has been a huge source of "financial
engineering" profits, both for Wall Street and the hedge
fund community over the last few years.
• World GDP is about $66 trillion.
• First CDS about 1995.
• Total nominal volume of OTC derivatives 550 Tri. $* 3.7 Tri. $ after nettingThe basic copula construction (d = 2)

Distribution Functions A-B

Distribution Functions C-D

David X. Li (2000) On Default Correlation: A Copula
Function Approach, Journal of Fixed Income 9:43-54••

This paper studies the problem of default
correlation. We first introduce a randomvariable called "time-until- default" to denotethe survival time of each defaultable entity or
financial instrument, and define the default
correlation between two credit risks as thecorrelation coefficient between their survival
times. Then we argue why a copula function
approach should be used to specify the joint
distribution of survival times after marginal
distributions of survival times are derived
from market information, such as risky bondprices or asset swap spreads. The definitionand some basic properties of copulafunctions are given. We show that thecurrent CreditMetrics approach to default
correlation through asset correlation isequivalent to using a normal copula function.
Finally, we give some numerical examples
to illustrate the use of copula functions in thevaluation of some credit derivatives, such as
credit default swaps and first-to-default
contracts.
April 1, 2000 (sic)(The Gauss-copula)David Li 8 years later

The Gauss-Copula (d=2)
In the two-dimensional case with correlation parameter ρ

And with density function

Let us call the Gauss-Copula the normal-copula!

Two results from the 1998 RiskLab report

Remark 1: See Figure 1 next page

A very early warning!

Remark 2: In the above paper it is shown that1959

Li - model(3)

Stress-model(12)

The Gauss-copula model caused a firststrong breeze:

September 12, 2005

How a Formula Ignited Market
That Burned Some Big Investors

The model Mr. Li devised helped estimate what returninvestors in certain credit derivatives should demand,
how much they have at risk and what strategies theyshould employ to minimize that risk. Big investors started
using the model to make trades that entailed giant bets
with little or none of their money tied up. Now, hundredsof billions of dollars ride on variations of the model everyday.
"David Li deserves recognition," says Darrell Duffie, aStanford University professor who consults for banks. He"brought that innovation into the markets [and] it has
facilitated dramatic growth of the credit-derivatives
markets."

David Li warned himself early on:
The problem: The scale's calibration isn't
foolproof. "The most dangerous part,"
Mr. Li himself says of the model, "is whenpeople believe everything coming out
of it." Investors who put too much trust in
it or don't understand all its subtleties may
think they've eliminated their risks when they haven't.
David X. Li in Wall Street Journal Article, 2005.

But then the Perfect Storm struck (again)!

And once more, the popular press blamed
the mathematicians, the quants, throughthe Gaussian-(normal-)copula, for having
blown up the economy!

Numerous newspaper articles:• Felix Salmon, 23 February, 2009, Wired
Magazine (a web-blog): Recipe for Disaster:
The Formula That Killed Wall Street• The Financial Times, Sam Jones (April 24,2009), On Couples and Copulas• Steve Lohr, September 12, 2009, NY Times,
Wall Street’s Math Wizards Forgot a Few
Variables• …

Recipe for Disaster: The Formula That Killed
Wall Street
By Felix Salmon 23 February, 2009
Wired Magazine

The popular press is full of statements like:• From risk-free return to return-free risk
• Mark-to-market, mark-to-model, mark-to-myth• Here’s what killed your 401(k)• Mea Copula• Anything that relies on correlation ischarlatanism (N.N.Taleb)• Double defeat for Wall Street and Mathematics• Rather than common sense, financial
mathematics was ruling• Etc …

Even the Financial Times joins in:
Of couples and copulas by Sam Jones (April 24, 2009)
In the autumn of 1987, the man who would
become the world’s most influential actuary
landed in Canada on a flight from China.
He could apply the broken hearts maths tobroken companies.
Li, it seemed, had found the final piece of a riskmanagement jigsaw that banks had been slowly piecing
together since quants arrived on Wall Street.
Why did no one notice the formula’s Achilles heel?

Johnny Cash and June Carter

Dear Sir
The article "Of couples and copulas", published on 24 April 2009,
suggests that David Li's formula is to blame for the current financial
crisis. For me, this is akin to blaming Einstein's E=mc² formula forthe destruction wreaked by the atomic bomb.
Feeling like a risk manager whose protestations of imminent danger
were ignored, I wish to make clear that many well-respected
academics have pointed out the limitations of the mathematical tools
used in the finance industry, including Li's formula. However, thesewarnings were either ignored or dismissed with a desultory
response: "It's academic".
We hope that we are listened to in the future, rather than being
made a convenient scapegoat.
Yours Faithfully,
Professor Paul Embrechts
Director of RiskLab
ETH ZurichSome personal recollections on the issue:
28 March 1999
Columbia-JAFEE Conference on the Mathematics of Finance,
Columbia University, New York.
10:00-10:45

P. EMBRECHTS (ETH, Zurich):"Insurance Analytics:
Actuarial Tools in Financial Risk-Management“
Why relevant?

1. Paper: P. Embrechts, A. McNeil, D. Straumann (1999)
Correlation and Dependence in Risk Management:
Properties and Pitfalls. Preprint RiskLab/ETH Zürich.
2. Coffee break: discussion with David Li.

If you are interested in my views on Copulas and
QRM:• Read my paper:“Copulas: A personal view”
Journal of Risk and Insurance, 2009.
See also my website:www.math.ethz.ch/~embrechts

There were however several early warnings (1)Economists’ Voice: www.bwpress.com/ev November, 2008

“I went on to explain how securitization can give rise to perverse incentives …
Has the growth in securitization been result of more efficient transactions technologies, or an unfounded reduction in concern about the importance of screening loan applications? … we should at least entertain the possibility that it is
the latter rather than the former.”At the very least, the banks have demonstrated
an ignorance of two very basic aspects of risk:(a) the importance of correlation, and(b) the possibility of price decline.

There were however several early warnings (2)Embrechts, P. et al. (2001): An academic response to Basel II.
Financial Markets Group, London School of Economics.
(Mailed to the Basel Committee)(Critical on VaR, procyclicality, systemic risk)

There were however several early warnings (3)Markopolos, H. (2005): The world’s largest
hedge fund is a fraud. (Mailed to the SEC)
(Madoff runs a Ponzi scheme)

Harry Markopolos

Bernard MadoffCharles Ponzi
1910

The Turner Review
A regulatory response to theglobal banking crisis
March 2009, FSA, London(126 pages)

1.1 (iv) Misplaced reliance on sophisticated maths
There are, however, fundamental questions about
The validity of VAR as a measure of risk (see Section
1.4 (ii) below). And the use of VAR measures basedon relatively short periods of historical observation(e.g. 12 months) introduced dangerous procyclicality into the assessment of tradingbook risk for the reasons set out in Box 1A (deficiencies of VAR).
The very complexity of the mathematics used to measure and manage risk, moreover,made it increasingly difficult for top management and boards to assess and exercise
judgement over the risks being taken. Mathematical sophistication ended up not containing risk, but providing false assurance that other prima facie indicators of increasing risk (e.g. rapid credit extension and balance sheet growth) could be safely ignored.
1.1 (v) Hard-wired procyclicality: …

1.4 (iii) Misplaced reliance on sophisticated maths: fixable
deficiencies or inherent limitations?
Four categories of problem can be distinguished:• Short observation periods• Non-normal distributions• Systemic versus idiosyncratic risk
• Non-independence of future events; distinguishing risk
and uncertainty
Frank H. Knight, 1921

This is the main reason why wemake a difference between
Model Risk and Model Uncertainty.
We very much stress the latter!

Supervisory guidance for assessing banks’ financial
instrument fair value practices
April 2009, Basel Committee on Banking Supervision• Principle 8: Supervisors expect bank valuation and risk measure-ment systems to systematically recognise and account for valuation
uncertainty. In particular, valuation processes and methodologies
should produce an explicit assessment of uncertainty related to theassignment of value for all instruments or portfolios. When appropriate this may simply be a statement that uncertainty for a particular
set of exposures is very small. While qualitative assessments are auseful starting point, it is desirable that banks develop methodologies that provide, to the extent possible, quantitative assessments.
These methodologies may gauge the sensitivity of value to the useof alternative models and modelling assumptions (when applicable),to the use of alternative values for key input parameters to thepricing process, and to alternative scenarios to the presumed
availability of counterparties. The extent of this analysis should becommensurate to the importance of the specific exposure for theoverall solvency of the institution.

Financial Mathematics and the Credit Crisis“If Financial Mathematicians have an
understanding of the derivative products at theroot of the credit crisis, can they offer any
insights on the current economic situation.
Specifically, there is a sense of gloom that “TheCity is over” and is there a more positive view.”(Question posed to researchers by Lord
Drayson, the UK Science and Innovation Minister)Some replies by researchers:• (L.C.G. Rogers) The problem is not thatmathematics was used by the banking industry,
the problem was that it was abused by thebanking industry. Quants were instructed to build
models which fitted the market prices. Now if themarket prices were way out of line, thecalibrated models would just faithfully reproduce
those wacky values, and the bad prices get
reinforced by an overlay of scientificrespectability!

And Rogers continues:• The standard models which were used for a longtime before being rightfully discredited by (some)academics and the more thoughtful practitioners
were from the start a complete fudge; so you
had garbage prices being underpinned by
garbage modelling.
• (M.H.A. Davis) The whole industry was stuck in
a classic positive feedback loop which no party
could (P.E. “wanted to”) walk away from.
Indeed only some!

EVT = Extreme Value Theory
EVT (first established around the 1920’s) offersa sound set of techniques for the understanding
and statistical estimation of rare events, beyond
the bell-curve world: it describes the statisticalbehavior of the largest observation, the biggest
loss, the worst case, rather than the averageobservation, the average loss, the average case.
For details: see the following textbooks!

EVT has become a standard
Some examples (but there are many more)A message for my students
New generations of students will have to use the tools
and techniques of QRM wisely in a world where therules of the game will have been changed.

Always be scientifically critical, as well as sociallyhonest, adhere to the highest ethical principles,
especially in the face of temptation … which will
come!

And on the boundedness of our knowledge:

There are more things in heaven and earth,
Horatio, than are dreamt of in yourphilosophy!
William Shakespeare(Hamlet I.v. 166)