The Recent Financial Turmoil and Related Financial
Engineering Research Problems1
Steven Kou
Columbia University
1 Based
on Two Papers:
Heyde, Kou, and Peng (2007). What Is a Good External Risk Measure: Bridging the
Gaps between Robustness, Subadditivity, Prospect Theory, and Insurance Risk Measures.
Peng and Kou (2008). Default Clustering and Pricing of CDO’s
Steven Kou Columbia University ()
1 / 49
1
Overview of Financial Engineering
2
The Recent Financial Turmoil
3
A New Model for CDO’s
What is a CDO?
Current Portfolio Credit Risk Models
The New Conditional Survival (CS) Model
CDO Pricing under the CS Model
4
What are Good External Risk Measures?
Review of Risk Measures
Motivation and Examples
Reasons to Relax Subadditivity
New Axioms and Characterization of Natural Risk Statistic
5
Summary
Steven Kou Columbia University ()
2 / 49
Outline
1
Overview of Financial Engineering
2
The Recent Financial Turmoil
3
A New Model for CDO’s
What is a CDO?
Current Portfolio Credit Risk Models
The New Conditional Survival (CS) Model
CDO Pricing under the CS Model
4
What are Good External Risk Measures?
Review of Risk Measures
Motivation and Examples
Reasons to Relax Subadditivity
New Axioms and Characterization of Natural Risk Statistic
5
Summary
Steven Kou Columbia University ()
2 / 49
Financial Engineering
Apply engineering modeling and tools to build realistic (i.e.
sophisticated) …nancial models, and to get simple (analytical or
approximate) solutions.
Steven Kou Columbia University ()
3 / 49
Financial Engineering
Apply engineering modeling and tools to build realistic (i.e.
sophisticated) …nancial models, and to get simple (analytical or
approximate) solutions.
Apply …nancial methods to solve engineering related problems.
Electricity Options
Revenue Management (arbitrage and airline revenue management)
Paper in Management Science 2008.
Steven Kou Columbia University ()
3 / 49
Outline
1
Overview of Financial Engineering
2
The Recent Financial Turmoil
3
A New Model for CDO’s
What is a CDO?
Current Portfolio Credit Risk Models
The New Conditional Survival (CS) Model
CDO Pricing under the CS Model
4
What are Good External Risk Measures?
Review of Risk Measures
Motivation and Examples
Reasons to Relax Subadditivity
New Axioms and Characterization of Natural Risk Statistic
5
Summary
Steven Kou Columbia University ()
4 / 49
The Recent Financial Turmoil
The crisis in subprime credit markets
Clustering defaults across time (time series correlation, Das et al.
2007) and cross-sectionally (contagion e¤ect, Longsta¤ and Rajan
2007)
Di¢ culties in modeling collateralized debt obligations (CDOs)
Risk Measures and BASEL Accord regulation
Steven Kou Columbia University ()
5 / 49
Outline
1
Overview of Financial Engineering
2
The Recent Financial Turmoil
3
A New Model for CDO’s
What is a CDO?
Current Portfolio Credit Risk Models
The New Conditional Survival (CS) Model
CDO Pricing under the CS Model
4
What are Good External Risk Measures?
Review of Risk Measures
Motivation and Examples
Reasons to Relax Subadditivity
New Axioms and Characterization of Natural Risk Statistic
5
Summary
Steven Kou Columbia University ()
6 / 49
What is a CDO
What is a CDO (Collateralized Debt Obligation)?
CDO is a security constructed from a portfolio of …xed-income
securities or credit derivatives.
CDO provides a way to create high quality debt out of a portfolio of
low quality debt.
Steven Kou Columbia University ()
7 / 49
Steven Kou Columbia University ()
8 / 49
CDO and Current Subprime Mortgage Crisis
The impact of credit crunch on iTraxx 5Y index tranche spreads
Tranches(%)
09/20/07
03/14/08
09/12/08
Steven Kou Columbia University ()
0-3
1812
5150
4011
3-6
84
649
475
6-9
37
401
277
9-12
23
255
152
12-22
15
143
68
22-100
7
70
35
8 / 49
CDO and Current Subprime Mortgage Crisis
The impact of credit crunch on iTraxx 5Y index tranche spreads
Tranches(%)
09/20/07
03/14/08
09/12/08
0-3
1812
5150
4011
3-6
84
649
475
6-9
37
401
277
9-12
23
255
152
12-22
15
143
68
22-100
7
70
35
CDO played an important role in the subprime mortgage crisis
Rating Agencies
Banks
Steven Kou Columbia University ()
8 / 49
CDO and Current Subprime Mortgage Crisis
The impact of credit crunch on iTraxx 5Y index tranche spreads
Tranches(%)
09/20/07
03/14/08
09/12/08
0-3
1812
5150
4011
3-6
84
649
475
6-9
37
401
277
9-12
23
255
152
12-22
15
143
68
22-100
7
70
35
CDO played an important role in the subprime mortgage crisis
Rating Agencies
Banks
Mispricing and improper credit rating of CDO have been criticized.
CDO itself should not be blamed for the …nancial crisis: it is a good
instrument as long as one knows how to price it.
Steven Kou Columbia University ()
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The Market Standard — Gaussian Copula Models
Idea: using copula functions to model default time correlation
Literature: Gaussian copula model (Li, 2000)
What is wrong with Gaussian copula?
Gaussian copula cannot generate tail dependence
lim P (Y > VaRq (Y )jX > VaRq (X )) = 0
q !1
Gaussian copula does not work during crisis, when the default
correlation is strong.
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Implied Copula Correlation of iTraxx 5Y CDO on 03/14/08
implied copula correlation
1
0.8
0.6
0.4
0.2
0
0-3%
3-6%
Steven Kou Columbia University ()
6-9%
9-12%
Tranche
12-22%
22-100%
10 / 49
Other Models
Top-down approach builds models for the portfolio cumulative loss
process directly
Good …t for standard CDO portfolios, but with no connection to
underlying single names
Longsta¤ and Rajan, 2007, Giesecke, et al., 2007, Halperin, 2007, Cont
and Minca, 2007
Bottom-up approach builds models for single name default times
Consistent with single name CDS spreads, but is hard to price and
calibrate CDO tranches
Examples: Static bottom-up models, e.g. copula models, dynamic
intensity models
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Dynamic Bottom-up Models
General idea:
Instantaneous default intensity of a single name
P (τ i
t + ∆t jτ i > t ) = λi (t )∆t + o (∆t )
Building correlation among instantaneous intensities λ1 (t ), . . . , λn (t )
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12 / 49
Dynamic Bottom-up Models
General idea:
Instantaneous default intensity of a single name
P (τ i
t + ∆t jτ i > t ) = λi (t )∆t + o (∆t )
Building correlation among instantaneous intensities λ1 (t ), . . . , λn (t )
Similar to survivial analysis in biostatistics
Steven Kou Columbia University ()
12 / 49
Dynamic Bottom-up Models
General idea:
Instantaneous default intensity of a single name
P (τ i
t + ∆t jτ i > t ) = λi (t )∆t + o (∆t )
Building correlation among instantaneous intensities λ1 (t ), . . . , λn (t )
Similar to survivial analysis in biostatistics
Factor instantaneous intensity models(Du¢ e and Gârleanu, 2001;
Mortensen, 2006)
λi (t ) = ai λM (t ) + λid
i (t )
M
id
λ (t ) and λi (t ) are independent a¢ ne jump di¤usion processes
Other papers: Papageorgiou and Sircar (2007), Joshi and Stacey
(2006), Schönbucher (2007)
Steven Kou Columbia University ()
12 / 49
Motivation
A good model should be able to produce clustering defaults, both
crosstionally and across the time.
Traditional intensity models
τ i = inf ft
Λi (t ) =
Z t
0
0 : Λi (t )
λi (s )ds
Ei g , Ei
d
exp(1), i.i.d.
(Λi (t ) is continuous!)
The continuity does not allow simultaneous defaults of several names.
Steven Kou Columbia University ()
13 / 49
The New Model: Conditional Survival (CS) Model
Our new model: conditional survival (CS) model is based on cumulative
intensity
Steven Kou Columbia University ()
14 / 49
The New Model: Conditional Survival (CS) Model
Our new model: conditional survival (CS) model is based on cumulative
intensity
Steven Kou Columbia University ()
14 / 49
The New Model: Conditional Survival (CS) Model
Our new model: conditional survival (CS) model is based on cumulative
intensity
τ i = inf ft
Λi (t ) =
J
0 : Λi (t )
Ei g , Ei
d
exp(1), i.i.d.
∑ ai ,j Mj (t ) + Xiid (t ), i = 1, . . . , n.
j =1
M (t ) = (M1 (t ), . . . , MJ (t )) 0 are market factor processes:
increasing and right continuous (possibly having jumps).
Xiid (t ) 0 is the idiosyncratic part of the cumulative default
intensity of the i-th name.
Steven Kou Columbia University ()
14 / 49
The New Model: Conditional Survival (CS) Model
The new model
Can produce simultaneous defaults of several names which leads to
stronger correlation.
Can lead to clustering default across the time.
No dynamics of idiosyncratic cumulative intensity is needed
The model provides automatic and exact calibration to single name
CDS data
The sensitivities w.r.t. each of the n single CDS’s can be obtained
concurrently with the CDO pricing
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15 / 49
Conditional Survival Probability
Conditional survival probability
h
qic (t ) , P ( τ i > t j M (t )) = E e
X iid (t )
Survival probability
h
qi (t ) , P (τ i > t ) = E e
Steven Kou Columbia University ()
X iid (t )
i
i
h
E e
e
∑Jj=1 a i ,j M j (t )
∑Jj=1 a i ,j M j (t )
i
16 / 49
Conditional Survival Probability
Conditional survival probability
h
qic (t ) , P ( τ i > t j M (t )) = E e
X iid (t )
Survival probability
h
qi (t ) , P (τ i > t ) = E e
X iid (t )
i
i
h
E e
e
∑Jj=1 a i ,j M j (t )
∑Jj=1 a i ,j M j (t )
i
Conditional survival probability is the building block
qic (t ) = qi (t )
Steven Kou Columbia University ()
e
h
E e
∑Jj=1 a i ,j M j (t )
∑Jj=1 a i ,j M j (t )
i
16 / 49
Conditional Survival Probability
Conditional survival probability
h
qic (t ) , P ( τ i > t j M (t )) = E e
X iid (t )
Survival probability
h
qi (t ) , P (τ i > t ) = E e
X iid (t )
i
i
h
E e
e
∑Jj=1 a i ,j M j (t )
∑Jj=1 a i ,j M j (t )
i
Conditional survival probability is the building block
qic (t ) = qi (t )
e
h
E e
∑Jj=1 a i ,j M j (t )
∑Jj=1 a i ,j M j (t )
i
No dynamics of idiosyncratic cumulative intensity Xiid (t )
Using survival probability qi (t ) as input, the model provides
automatic and exact calibration to single name CDS data.
Steven Kou Columbia University ()
16 / 49
Specifying Dynamics of Market Factors
A choice for M (t ) is Polya process
A Polya process is a mixed Poisson process with the rate λ having
Gamma (α, β) distribution
Clustering jumps: A Polya process has positive correlation between
increments, so the arrival of one event tends to trigger more events.
Polya process with parameter (α, β) has a negative binomial
distribution
P (M (t ) = i ) =
α+i
i
1
1
1 + βt
α
βt
1 + βt
i
E [exp( aM (t ))] has closed form.
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17 / 49
CDO Pricing by Exact Simulation
Key fact: conditional on M (t ), default indicators
Ii = 1fτi t g , i = 1, . . . , n are independent Bernoulli (1
Steven Kou Columbia University ()
qic (t )) r.v.
18 / 49
CDO Pricing by Exact Simulation
Key fact: conditional on M (t ), default indicators
Ii = 1fτi t g , i = 1, . . . , n are independent Bernoulli (1
qic (t )) r.v.
Exact simulation of Lt at given time t:
1 Generate market factors M (t ), M (t ), . . . , M (t ).
1
2
J
2
Calculate the conditional survival probability analytically
qic (t )
3
4
e
h
= qi (t )
E e
d
Generate independent Ii s Bernoulli(1
Calculate Lt = ∑ni=1 (1 Ri )Ni Ii .
Steven Kou Columbia University ()
∑Jj=1 a i ,j M j (t )
∑Jj=1 a i ,j M j (t )
i
qic (t )), i = 1, . . . , n.
18 / 49
CDO Pricing by Exact Simulation
Key fact: conditional on M (t ), default indicators
Ii = 1fτi t g , i = 1, . . . , n are independent Bernoulli (1
qic (t )) r.v.
Exact simulation of Lt at given time t:
1 Generate market factors M (t ), M (t ), . . . , M (t ).
1
2
J
2
Calculate the conditional survival probability analytically
qic (t )
3
4
e
h
= qi (t )
E e
d
Generate independent Ii s Bernoulli(1
Calculate Lt = ∑ni=1 (1 Ri )Ni Ii .
∑Jj=1 a i ,j M j (t )
∑Jj=1 a i ,j M j (t )
i
qic (t )), i = 1, . . . , n.
It takes 20 seconds to carry out 100,000 replication, using C++
program and a desktop with an Intel 2.33G cpu.
Steven Kou Columbia University ()
18 / 49
CDO Pricing by Exact Simulation
Key fact: conditional on M (t ), default indicators
Ii = 1fτi t g , i = 1, . . . , n are independent Bernoulli (1
qic (t )) r.v.
Exact simulation of Lt at given time t:
1 Generate market factors M (t ), M (t ), . . . , M (t ).
1
2
J
2
Calculate the conditional survival probability analytically
qic (t )
3
4
e
h
= qi (t )
E e
d
Generate independent Ii s Bernoulli(1
Calculate Lt = ∑ni=1 (1 Ri )Ni Ii .
∑Jj=1 a i ,j M j (t )
∑Jj=1 a i ,j M j (t )
i
qic (t )), i = 1, . . . , n.
It takes 20 seconds to carry out 100,000 replication, using C++
program and a desktop with an Intel 2.33G cpu.
Variance reduction by using control variant
E [Lt ] = ∑ni=1 (1 Ri )Ni (1 qi (t ))
Steven Kou Columbia University ()
18 / 49
Sensitivity w.r.t. Single Name CDS
The sensitivity w.r.t. to an individual CDS can be reduced to the
sensitivity w.r.t. to single name survival probability
In CS model, expected tranche loss is linear w.r.t. single name
survival probability
E [(Lt
a)+ ] = E fA(t )qi (t ) + B (t )g ,
e
A(t ) = h
E e
∑Jj=1 a i ,j M j (t )
∑Jj=1 a i ,j M j (t )
E [(Lt
i
i
a)+
(Lt i + (1
∂E [(Lt a)+ ]
= E [A(t )]
∂qi (t )
Ri )Ni
a)+ jM (t )]
where Lt i is the portfolio loss excluding the i-th name.
A(t ) can be easily calculated in each simulation trial for pricing.
The sensitivities w.r.t. each of the n single name CDS are obtained
concurrently with CDO tranche pricing.
Steven Kou Columbia University ()
19 / 49
iTraxx 5Y CDO Tranche Spread Calibration Results
CDO and CDS data on March 14, 2008, right before the collapse of
Bear Stern, whose stock dropped to $2.84 on March 17, 2008.
Steven Kou Columbia University ()
20 / 49
iTraxx 5Y CDO Tranche Spread Calibration Results
CDO and CDS data on March 14, 2008, right before the collapse of
Bear Stern, whose stock dropped to $2.84 on March 17, 2008.
We use single name 5Y CDS spread to obtain marginal survival
probability qi (t ), i = 1, . . . , 125
Steven Kou Columbia University ()
20 / 49
iTraxx 5Y CDO Tranche Spread Calibration Results
CDO and CDS data on March 14, 2008, right before the collapse of
Bear Stern, whose stock dropped to $2.84 on March 17, 2008.
We use single name 5Y CDS spread to obtain marginal survival
probability qi (t ), i = 1, . . . , 125
Calibration results by using 1 Polya process and 2 analog to integral
of CIR processes
Tranches(%)
Market
Model
B-A spread
Steven Kou Columbia University ()
0-3
5150
5230
158
3-6
649
665
24
6-9
401
374
25
9-12
255
250
20
12-22
143
165
12
22-100
70
67
3
20 / 49
iTraxx 5Y CDO Tranche Spread Calibration Results
CDO and CDS data on March 14, 2008, right before the collapse of
Bear Stern, whose stock dropped to $2.84 on March 17, 2008.
We use single name 5Y CDS spread to obtain marginal survival
probability qi (t ), i = 1, . . . , 125
Calibration results by using 1 Polya process and 2 analog to integral
of CIR processes
Tranches(%)
Market
Model
B-A spread
0-3
5150
5230
158
3-6
649
665
24
6-9
401
374
25
9-12
255
250
20
12-22
143
165
12
22-100
70
67
3
Pricing error: Chi-square = 6.7(p-value = 0.24), RMSE = 1.01,
v
!2
u
6
o
2
o
u1 6
s
s
( sk s k )
k
k
CHISQ = ∑
, RMSE = t ∑
o,a
o,b
s
6
s
s
k
k =1
k =1
k
k
Steven Kou Columbia University ()
20 / 49
Model and Market Implied Correlation
Implied Copula Correlation of iTraxx 5Y C DO Tranches Spread on 3/14/08
1
Market
Model
0.9
implied copula correlation
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0-3%
3-6%
6-9%
9-12%
12-22%
22-100%
tranche
Steven Kou Columbia University ()
21 / 49
Outline
1
Overview of Financial Engineering
2
The Recent Financial Turmoil
3
A New Model for CDO’s
What is a CDO?
Current Portfolio Credit Risk Models
The New Conditional Survival (CS) Model
CDO Pricing under the CS Model
4
What are Good External Risk Measures?
Review of Risk Measures
Motivation and Examples
Reasons to Relax Subadditivity
New Axioms and Characterization of Natural Risk Statistic
5
Summary
Steven Kou Columbia University ()
22 / 49
Measure of Risk Maps a R.V. to a Real Number
Risk: future loss of a position–random variable X
Measure of risk: mapping from a set of risks to the real line
X ! ρ (X )
Examples:
margin requirement for …nancial trading
capital requirement against market risk
insurance risk premium
Steven Kou Columbia University ()
23 / 49
Coherent Measure of Risk
Coherent Measure of Risk (Artzner et al. 1999, Huber 1981)
Translation invariance: ρ(X + a) = ρ(X ) + a
Positive homogeneity: ρ(λX ) = λρ(X ), λ
Monotonicity: ρ(X )
ρ(Y ), if X
Subadditivity: ρ(X + Y )
Steven Kou Columbia University ()
0
Y
ρ (X ) + ρ (Y )
24 / 49
Coherent Measure of Risk
Coherent Measure of Risk (Artzner et al. 1999, Huber 1981)
Translation invariance: ρ(X + a) = ρ(X ) + a
Positive homogeneity: ρ(λX ) = λρ(X ), λ
Monotonicity: ρ(X )
ρ(Y ), if X
Subadditivity: ρ(X + Y )
0
Y
ρ (X ) + ρ (Y )
Characterization of coherent measure of risk
Maximum expected loss
ρ(X ) = sup E P [X ]
P 2P
Acceptance sets
ρ(X ) = minfm : X
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m 2 acceptance setg
24 / 49
Insurance Measure of Risk
X and Y are comonotonic if and only if
8 ω 1 , ω 2 2 Ω, (X (ω 1 )
Steven Kou Columbia University ()
X (ω 2 ))(Y (ω 1 )
Y (ω 2 ))
0
25 / 49
Insurance Measure of Risk
X and Y are comonotonic if and only if
8 ω 1 , ω 2 2 Ω, (X (ω 1 )
X (ω 2 ))(Y (ω 1 )
Y (ω 2 ))
0
Insurance Measure of Risk (Wang et al. 1997)
Law invariance: ρ(X ) = ρ(Y ), if X and Y have the same distribution
Monotonicity: ρ(X )
ρ(Y ), if X
Y
Comonotonic additivity:
ρ(X + Y ) = ρ(X ) + ρ(Y ), if X and Y are comonotonic
Continuity:
lim ρ(max(X
d !0
d, 0)) = ρ(X + ), lim ρ(min(X , d )) = ρ(X )
lim ρ(max(X , d )) = ρ(X )
d !∞
d! ∞
Scale normalization: ρ(1) = 1
Steven Kou Columbia University ()
25 / 49
Insurance Measure of Risk (Continued)
It does not require subadditivity
ρ( ) is an insurance measure of risk if and only if ρ( ) has a Choquet
integral representation:
ρ (X ) =
=
Z
Xd (g
Z 0
∞
P)
(g (P (X > t ))
1)dt +
Z ∞
g (P (X > t ))dt
0
where g ( ) is increasing, g (0) = 0, g (1) = 1
It does not generate scenarios, unlike the coherent risk measure
Steven Kou Columbia University ()
26 / 49
What Is an Axiom
De…nition: a statement or principle which is generally accepted to be
true, but is not necessarily so. (Cambridge English Dictionary)
Axioms are subject to debate and change.
It is useful to provide alternative axioms.
Steven Kou Columbia University ()
27 / 49
What Is an Axiom
De…nition: a statement or principle which is generally accepted to be
true, but is not necessarily so. (Cambridge English Dictionary)
Axioms are subject to debate and change.
It is useful to provide alternative axioms.
Whether a risk measure is good or not depends on the objective of
proposing the risk measure
A risk measure may be suitable for internal risk management, but not
for external regulatory agencies, and vice versa.
We focus on external risk measures from the viewpoint of
governmental/regulatory agencies
Steven Kou Columbia University ()
27 / 49
Legal Realism and Legal Realism
Legal Realism
Legal Realism: The legal decision of a court is determined by the
actual practices of judges, rather than the law set forth in statutes or
precedents (Hart, 1994).
A law is only a guideline for judges and enforcement o¢ cers
The law should be robust.
Coherent risk measure generally lack robustness
Steven Kou Columbia University ()
28 / 49
Legal Realism and Legal Realism
Legal Realism
Legal Realism: The legal decision of a court is determined by the
actual practices of judges, rather than the law set forth in statutes or
precedents (Hart, 1994).
A law is only a guideline for judges and enforcement o¢ cers
The law should be robust.
Coherent risk measure generally lack robustness
Legal Positivism
Legal Positivism: the existence and content of law depend on social
norms and not on their merits (Hart, 1994).
If a system of rules are to be imposed by force, there must be a
su¢ cient number of people who accept it voluntarily.
A law should re‡ect the social norms.
In face of …nancial risk, people’s decision may violate the subadditivity
Axiom. (Kahneman and Tversky, 1979, 1992)
Steven Kou Columbia University ()
28 / 49
An Example of Tra¢ c Speed Limit
National Maximum Speed Law was repealed in 1995 because of
notoriously low compliance.
Not by studying geometrical or architectural features of road
“85th percentile rule”
The 85th percentile speed manifests:
A law should be robust
A law should be consistent to social behaviors
Steven Kou Columbia University ()
29 / 49
Example: Value-at-risk (VaR)
Value-at-risk (VaR) at level α:
VaRα (X ) = α-quantile of X , α 2 [95%, 99.9%]
Value-at-risk satis…es axioms for the insurance risk measure
Value-at-risk does not satisfy subadditivity
VaRα (X + Y )
VaRα (X ) + VaRα (Y ), in general
Inconsistency: 99% VaR is the primary measure of risk imposed by
Basel Accord, but it is excluded by coherent measure of risk
Steven Kou Columbia University ()
30 / 49
Example: Value-at-risk (VaR)
Value-at-risk (VaR) at level α:
VaRα (X ) = α-quantile of X , α 2 [95%, 99.9%]
Value-at-risk satis…es axioms for the insurance risk measure
Value-at-risk does not satisfy subadditivity
VaRα (X + Y )
VaRα (X ) + VaRα (Y ), in general
Inconsistency: 99% VaR is the primary measure of risk imposed by
Basel Accord, but it is excluded by coherent measure of risk
Steven Kou Columbia University ()
30 / 49
Example: Tail Conditional Expectation (TCE)
Tail conditional expectation (TCE) at level α (Artzner et.al,
1999):
TCEα (X ) = E [X jX
VaRα (X )]
Tail conditional expectation is also called conditional value-at-risk
(Rockafellar & Uryasev, 2002) and expected shortfall
(Acerbi, 2002)
Tail conditional expectation is subadditive.
Tail conditional expectation is not robust.
Steven Kou Columbia University ()
31 / 49
Tail conditional median (TCM)
A more robust risk measure:
Tail conditional median (TCM) at level α:
TCMα (X ) = median [X jX
Steven Kou Columbia University ()
VaRα (X )]
32 / 49
Tail conditional median (TCM)
A more robust risk measure:
Tail conditional median (TCM) at level α:
TCMα (X ) = median [X jX
VaRα (X )] = VaR 1+α (X )
2
Tail conditional median does NOT satisfy subadditivity
The criticism that VaR is not informative is inappropriate.
TCM satis…es subadditivity in examples in which VaR violates
subadditivity.
Steven Kou Columbia University ()
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TCE
TCM
9
9
Laplace
T-3
T-5
T-12
8
8
7
α
1.44
6
TCM
TCE
α
7
5
4
4
3
3
-6.5
-6
-5.5
log(1- α)
Steven Kou Columbia University ()
-5
-4.5
0.75
6
5
-7
Laplace
T-3
T-5
T-12
-7
-6.5
-6
-5.5
log(1- α)
-5
-4.5
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Regulator’s Viewpoint: Robustness Is Crucial
When a regulator impose a risk measure:
Each …rm calculates the level of risk and reports to the regulator
Each …rm has the freedom to choose its own internal models
The risk measure should be robust with respect to underlying models.
Steven Kou Columbia University ()
33 / 49
Regulator’s Viewpoint: Robustness Is Crucial
When a regulator impose a risk measure:
Each …rm calculates the level of risk and reports to the regulator
Each …rm has the freedom to choose its own internal models
The risk measure should be robust with respect to underlying models.
Heaviness of tail distribution is hard to identify
With 5,000 observations one can not distinguish between the Laplace
distribution and the T-distributions. (Heyde & Kou, 2004)
Steven Kou Columbia University ()
33 / 49
Axioms for Natural Risk Statistic
Data x̃ = (x1 , . . . , xn ) 2 Rn
Risk statistic: a mapping from the data set to the real line:
ρ̂ : Rn ! R
x̃ 7! ρ̂(x̃ )
Axioms for natural risk statistic:
Positive homogeneity and translation invariance:
ρ̂(ax̃ + b ) = aρ̂(x̃ ) + b, 8x̃ 2 Rn , a 0, b 2 R
Monotonicity: ρ̂(x̃ )
ρ̂(ỹ ), if x̃
ỹ , i.e., xi
yi , i = 1, . . . , n.
Comonotonic subadditivity: ρ̂(x̃ + ỹ ) ρ̂(x̃ ) + ρ̂(ỹ ), if x̃ and ỹ are
comonotonic, i.e., (xi xj )(yi yj ) 0 for any i 6= j.
Empirical law invariance:
ρ̂((x1 , . . . , xn )) = ρ̂((xi1 , . . . , xin )), for any permutation (i1 , . . . , in ).
Steven Kou Columbia University ()
34 / 49
Representation Theorem for Natural Risk Statistic
Theorem
Let x(1 ) , ..., x(n ) be the order statistics of the data x̃ = (x1 , ..., xn ) with
x(n ) being the largest. Then ρ̂ is a natural risk statistic if and only if there
exists a set of weights W = fw̃ = (w1 , . . . , wn )g Rn with each w̃ 2 W
satisfying ∑ni=1 wi = 1 and wi
0, 81 i n, such that
n
ρ̂(x̃ ) = sup f ∑ wi x(i ) g, 8x̃ 2 Rn .
w̃ 2W i =1
Natural risk statistic includes VaR along with scenario analysis and TCM
Steven Kou Columbia University ()
35 / 49
Representation Theorem for Natural Risk Statistic
Theorem
Let x(1 ) , ..., x(n ) be the order statistics of the data x̃ = (x1 , ..., xn ) with
x(n ) being the largest. Then ρ̂ is a natural risk statistic if and only if there
exists a set of weights W = fw̃ = (w1 , . . . , wn )g Rn with each w̃ 2 W
satisfying ∑ni=1 wi = 1 and wi
0, 81 i n, such that
n
ρ̂(x̃ ) = sup f ∑ wi x(i ) g, 8x̃ 2 Rn .
w̃ 2W i =1
Natural risk statistic includes VaR along with scenario analysis and TCM
The proof does not follow easily from Huber’s result in robust
statistics.
Subadditivity only holds for comonotonic sets, which are not open
sets.
Boundary points do not have as nice properties as interior points.
Steven Kou Columbia University ()
35 / 49
Outline
1
Overview of Financial Engineering
2
The Recent Financial Turmoil
3
A New Model for CDO’s
What is a CDO?
Current Portfolio Credit Risk Models
The New Conditional Survival (CS) Model
CDO Pricing under the CS Model
4
What are Good External Risk Measures?
Review of Risk Measures
Motivation and Examples
Reasons to Relax Subadditivity
New Axioms and Characterization of Natural Risk Statistic
5
Summary
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Summary
Steven Kou Columbia University ()
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Summary
Steven Kou Columbia University ()
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Summary
Steven Kou Columbia University ()
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The Future of Financial Engineering after the Recent
Financial Turmoil
Steven Kou Columbia University ()
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The Future of Financial Engineering after the Recent
Financial Turmoil
Olympic Slogan: “Citius, Altius, Fortius”– “Faster, Higher, Stronger”
Steven Kou Columbia University ()
38 / 49
The Future of Financial Engineering after the Recent
Financial Turmoil
Olympic Slogan: “Citius, Altius, Fortius”– “Faster, Higher, Stronger”
Faster: Analytical Solutions and Approximations
Steven Kou Columbia University ()
38 / 49
The Future of Financial Engineering after the Recent
Financial Turmoil
Olympic Slogan: “Citius, Altius, Fortius”– “Faster, Higher, Stronger”
Faster: Analytical Solutions and Approximations
Higher: High dimension modeling in option pricing, credit risks, VaR,
MBS
Steven Kou Columbia University ()
38 / 49
The Future of Financial Engineering after the Recent
Financial Turmoil
Olympic Slogan: “Citius, Altius, Fortius”– “Faster, Higher, Stronger”
Faster: Analytical Solutions and Approximations
Higher: High dimension modeling in option pricing, credit risks, VaR,
MBS
Stronger: better and more ‡exible models for stocks, interest rates,
energy prices.
Steven Kou Columbia University ()
38 / 49
The Future of Financial Engineering after the Recent
Financial Turmoil
Olympic Slogan: “Citius, Altius, Fortius”– “Faster, Higher, Stronger”
Faster: Analytical Solutions and Approximations
Higher: High dimension modeling in option pricing, credit risks, VaR,
MBS
Stronger: better and more ‡exible models for stocks, interest rates,
energy prices.
Going from the Wall Street to the Main Street.
Steven Kou Columbia University ()
38 / 49
Thank you!
Steven Kou Columbia University ()
39 / 49
Implicit Constraints on Model Parameters
The idiosyncratic cumulative intensities Xiid (t )
h
i
h
i
h
id
id
E e X i (T m )
E e X i (T m 1 )
E e
Recall that
h
E e
X iid (t )
i
h
=
E e
0 and increasing
i
X iid (T 1 )
1, 81 i
qi (t )
∑Jj=1 a i ,j M j (t )
This imposes parameter constraints:
h
E e
qi (Tm )
∑Jj=1
a i ,j M j (T m )
Steven Kou Columbia University ()
i
h
E e
qi (T1 )
∑Jj=1 a i ,j M j (T 1 )
i
i
1, 81
i
n
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n
Specifying Dynamics of Market Factors (Continued)
Choice B: model M (t ) by an anology to
M (t ) =
Rt
0
V (s )ds, V (t ) is a CIR process:
K 1
1
1
V (t0 )h + ∑ V (ti )h + V (tK )h, 0 = t0 < t1 <
2
2
i =1
< tK = t
h is the discretization step size, e.g. h = 1/32.
Rt
The exact simulation of 0 V (s )ds is time-consuming. The exact
simulation of M (t ) is reasonably fast: only involves simulation of CIR
process.
E [exp( aM (t ))] has closed form.
Steven Kou Columbia University ()
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Calibration Algorithm
1
2
Initialization: set market factor parameter Θ0 , and set s = 0.
Iteration: s ! s + 1
For given Θs , determine loading coe¢ cients ai by solving a constrained
optimization problem:
i
h
J
min E e ∑j =1 ai ,j N j (T m ) /qi (Tm ) 1
s.t.
E e
0
q i (T m )
∑Jj=1 a i ,j N j (T m )
E e
q i (T 1 )
∑Jj=1 a i ,j N j (T 1 )
1
ai ,j
These problems can be solved quickly by e.g., sequential quadratic
programming(we used the package CFSQP developed by Lawence
et.al).
Calculate tranche spreads using the market parameter Θs and loading
coe¢ cients ai obtained above.
Update the market factor parameter Θs ! Θs +1 by using some
optimization routine, e.g. Powell’s direction-set algorithm(Powell,
1964)
Repeat
Steven Kou Columbia University ()
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iTraxx 5Y CDO Tranche Spread Calibration Results
Fitted model parameters:
1
2
3
Polya process: α = 2.64516, β = 0.00583919
1st CIR process:
θ 1 = 0.1, κ 1 = 0.12792, σ1 = 1.34701, V1 (0) = 1.1411; h = 1/32
2nd CIR process: θ 2 = 0.1, κ 2 = 0.00100369, σ2 = 7.40483, V2 (0) =
0.106509; h = 1/32
Steven Kou Columbia University ()
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A merger may create extra risk
X and Y : the net payo¤ of two …rms before a merger.
Because of bankruptcy protection, the actual net payo¤ of the two
…rms would be X + = max(X , 0) and Y + = max(Y , 0)
After the merger, the net payo¤ of the joint …rm would be X + Y ,
and the actual net payo¤ would be (X + Y )+ , due to bankruptcy
protection.
(X + Y ) +
Steven Kou Columbia University ()
X + + Y +!
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Comonotonic Additivity Does Not Hold in Two Scenarios
Probability space: (Ω = fω 1 , ω 2 , ω 3 g, F , P ), P (ω i ) =
1
3
Two comonotonic random variables
(Z (ω 1 ), Z (ω 2 ), Z (ω 3 )) = (3, 2, 4), Y = Z 2
Two distortion function
1
2
g1 ( ) = 0.5, g1 ( ) = 1.0;
3
3
1
2
g2 ( ) = 0.72, g2 ( ) = 0.8
3
3
Risk measureRincorporating two
R scenarios:
ρ(X ) = max X d (g1 P ), X d (g2 P )
Only strict comonotonic subadditivity holds!
ρ(Z + Y ) = 9.28 < ρ(Z ) + ρ(Y ) = 2.5 + 6.8 = 9.3
Steven Kou Columbia University ()
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1. Diversi…cation and Tail subadditivity of VaR
For risks with …nite second moment, diversi…cation is preferable
because
SD (X + Y ) SD (X ) + SD (Y )
Diversi…ed portfolio has larger expected utility than a single risk.
(Samuelson, 1967)
Diversi…cation is not preferable if the risks have extremely heavy tails.
(Ibragimov and Walden, 2004)
VaR has tail subadditivity(Daníelsson et al. 2005): If the joint
distribution of X and Y does not have extremely heavy tail, then
there exists α0 2 (0, 1), such that
VaRα (X + Y )
Diversi…cation reduces risk?
VaR satis…es subadditivity?
Steven Kou Columbia University ()
VaRα (X ) + VaRα (Y ), 8α 2 (α0 , 1)
Not very heavy tails
Yes
Yes
Heavy tails
No
No
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Other Reasons to Relax Subadditivity
Merger of two …rms removes the bankruptcy protection between them
and hence increase the risk of complete breakdown of the joint …rm.
(Dhaene et al. 2005)
Prospect theory (Kahneman & Tversky, 1979, 1992, 1993) is
able to explain the major violations of expected utility theory
Non-comonotonic random loss can violate subadditivity
Steven Kou Columbia University ()
47 / 49
An Example Violating Subadditivity
A fair coin is ‡ipped
Three positions:
H: losing $10,000 if it comes up “head"
T: losing $10,000 if it comes up “tail"
S: losing $5,000 for sure
ρ(S ) = 5, 000, ρ(H ) = ρ(T )
ρ(H ) < ρ(S ) = 5, 000 because people are risk seeking in face of loss
ρ(H + T ) = ρ(sure loss of 10,000) = 10, 000
Subadditivity does not hold:
ρ(H ) + ρ(T ) < 10, 000 = ρ(H + T )!
Steven Kou Columbia University ()
48 / 49
Representation Theorem for Law Invariant Coherent Risk
Statistic
Theorem
ρ̂ is a law invariant coherent risk statistic if and only if there exists a set of
weights W = fw̃ = (w1 , . . . , wn )g Rn with each w̃ 2 W satisfying
0, 81 i n and w1 w2 . . . wn , such that
∑ni=1 wi = 1 and wi
n
ρ̂(x̃ ) = sup f ∑ wi x(i ) g, 8x̃ 2 Rn .
w̃ 2W i =1
Steven Kou Columbia University ()
49 / 49
Representation Theorem for Law Invariant Coherent Risk
Statistic
Theorem
ρ̂ is a law invariant coherent risk statistic if and only if there exists a set of
weights W = fw̃ = (w1 , . . . , wn )g Rn with each w̃ 2 W satisfying
0, 81 i n and w1 w2 . . . wn , such that
∑ni=1 wi = 1 and wi
n
ρ̂(x̃ ) = sup f ∑ wi x(i ) g, 8x̃ 2 Rn .
w̃ 2W i =1
Assigning larger weights to larger observations leads to less robust risk
statistics
Coherent risk statistic is generally not robust!
Steven Kou Columbia University ()
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