### slides pptx - School of Mathematics

```Modeling Related Failures in
Finance
MFM Orientation, 2010
Outline
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Relationships and Related Events
Related Failures: Insurance, Survival, Reliability
Failures in Finance
Probability Structure
Default Correlation (w/example)
Copula Models
Applications of Copulas
References
Conclusion
Relationships and Related Events
• Old, old story…
• Relationships that do not matter (hypothesis of
independence)
• Relationships that do matter
• How to model relationships?
• Random variables – height or weight, personal
income, stock prices
• Random variables –length of life or age at death
Related Failures
• Insurance (mortality structure on associated
human lives)
• Survival (biological species)
• Reliability (connected components in complex
engineering systems)
• Finance (?)
Insurance
• Associated human lives (e.g., husbands and
wives)
• Common lifestyles
• Common disasters (accidents)
• Broken-heart syndrome
• Exclusions!
Survival
• Biological species within certain environment
(e.g., life on an island)
• Common environmental concerns
• Predator/prey interactions
• Symbiosis
Reliability
• Interaction of components of a complex
engineering system (e.g., power grid)
• Links in a chain (series or parallel)
• Climate and natural disasters
• Sayano-Shushenskaya HPS
Finance
• Bank failures, credit events, defaults on
mortgages
• Market situation
• Macroeconomic indicators
• Deficit of trust
• Chain reaction of failures
Probability Distributions
• Distribution function (d.f.; c.d.f)
F (t )  P  X  t 
• Survival function
S (t )  P  X  t   1  F (t )
• Distribution density function (d.d.f.)
f ( z) 
d
F (t )
dt
tz
Joint Distributions
• Joint distribution function
H (s, t )  P  X  s, Y  t 
• Joint survival function
K (s, t )  P  X  s, Y  t ) 
• Joint density
d2
h( z, w) 
H ( s, t )
dsdt
s  z ,t  w
Independence
• For any s, t
H (s, t )  F (s)G(t )  P( X  s) P(Y  t )
• For any s, t
K (s, t )  P( X  s) P(Y  t )
• For any s, t
h(s, t )  f (s) g (t )
• Joint functions are built from marginals
Pearson’s Moment Correlation
• Pearson’s moment correlation (correlation
coefficient) is defined as
Cov  X , Y 
EXY  EX  EY
  X ,Y  

 XY
Var  X  Var (Y )
• It is a good measure of linear dependence,
strongly connected with the first two
moments, and is known not to capture nonlinear dependence
Sample Pearson’s Correlation
• Given a paired (matched) sample
 x, y    x1 , y1  ,...,  xn , yn  ,
the sample correlation coefficient is defined as
n
ˆ  x, y  
n
1 n
xi yi   xi  yi

n i 1 i 1
i 1
 n 2 1  n 2  n 2 1  n 2 
  xi    xi    yi    yi  
 i 1
 i 1

n
n
i 1
i 1

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
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

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Default Correlation
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Time-to-default random variables
CDS (Credit Default Swaps)
CDO (Collateralized Debt Obligations)
Recent crisis
Problem: mathematical models failed to
accurately predict the risks
• Problems with default correlation
• Example: three-mortgage portfolio
Example (Absolutely Unrealistic)
• We underwrite three identical mortgages, each
with \$100K principal
• Term: 1 year
• Probability of default: 0.1 for each
• Annual payment is made in the beginning of the
year
• Interest rate of 11%
• Expected gain: \$1,000 per mortgage per year
• Problem: relatively high risk of a big loss
Losses
• We can lose as much as over \$250K while
making on the average \$3K!
• Expected gain = \$11,000 x 0.9 - \$89,000 x 0.1
= \$1,000
• Potential loss = \$89,000
• We collect (three mortgages) the interest of
\$33,000 = \$ 30,000 + \$3,000
• We bear the risk of losing the principal
3 x \$89,000 = \$267,000
Selling the Risk
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Is it possible to hedge the risks (sell the risks)?
CDO structure: how many defaults?
Senior tranche (safe)
Equity tranche (risky)
Find the buyers (investors): those who will
receive our cash flows and accept
responsibility for possible defaults
Default Probabilities - Independence
• P(all three defaults) = P(ABC) = 0.1 x 0.1 x 0.1
= 0.001
• P(at least two defaults) = 0.027 + 0.001 =
=0.028
• P(at least one default) = 0.243 + 0.027 + 0.001
= 0.271
Investors’ Side - Independence
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Assume independence of failures
Senior tranche: expected loss of \$100
Mezzanine tranche: expected loss of \$2,800
Equity tranche: expected loss of \$27,100
Expected losses of all tranches add up to
\$30,000
• For us: margin of \$3,000 and no risk!
• We might have to split the margin
Diagram 1 (Independence)
Correlation
• Assume that there is no independence and we
expect pair-wise correlations (Pearson’s
moment correlations) between the individual
defaults at 0.5
• That corresponds to joint probability of two
defaults being 0.055
probability of three defaults
• Different scenarios are possible
Calculation of the Multiple Default
Probabilities
  X ,Y  
EXY  EX  EY
Var  X Var (Y )

P ( AB )  P ( A) P ( B )

P ( A)(1  P ( A)) P ( B )(1  P ( B ))
P ( AB )  0.1 0.1

 0.5  P ( AB )  0.055
0.1 0.9
P ( ABC )  ?
Diagram X - Correlation
Diagram 2 (Extreme Scenario 2)
Default Probabilities – Scenario 2
• P(all three defaults) = 0.01
• P(at least two defaults) = 0.145
• P(at least one default) = 0.145
Investors’ Side – Scenario 2
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Assume default correlations of 0.5
Senior tranche: expected loss of \$1,000
Mezzanine tranche: expected loss of \$14,500
Equity tranche: expected loss of \$14,500
Expected losses of all tranches add up to
\$30,000
Diagram 3 (Extreme scenario 3)
Default Probabilities – Scenario 3
• P(all three defaults) = 0.055
• P(at least two defaults) = 0.055
• P(at least one default) = 0.19
Investors’ Side – Scenario 3
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Assume default correlations equal to 0.5
Senior tranche: expected loss of \$5,500
Mezzanine tranche: expected loss of \$5,500
Equity tranche: expected loss of \$19,000
Expected losses of all tranches add up to
\$30,000
What do we conclude?
• Correlation between the default variables is
important in order to estimate expected losses
(i.e., to price) the tranches
• Results are sensitive to the value of the
correlation coefficient
• Knowing pair-wise correlation coefficients is not
enough to price the tranches in case of more than
2 assets
• It would be enough under assumption of
normality
Definition of Copula Function
• A function C : I 2  0,1 [0,1]  I  0,1
is called a copula (copula function) if:
1. For any u, v  I
C (u,0)  C (0, v)  0
2. It is 2-monotone (quasi-monotone).
3. For any u, v  I
C (u,1)  u; C (1, v)  v
Frechet Bounds
• For any copula C (u , v)
the following inequalities (Frechet bounds)
hold:
W (u, v)  C (u, v)  M (u, v),
W (u, v)  max u  v  1, 0 ,
M (u, v)  min u, v
Maximum Copula
M (u, v)  min{u, v}
Maximum Copula
M (u, v)  min{u, v}
Minimum Copula
W (u, v)  max u  v 1,0
Minimum Copula
W (u, v)  max u  v 1,0
Product Copula
P(u, v)  uv
Product Copula
P(u, v)  uv
Sklar’s Theorem
• Theorem: 1. For any correctly defined joint
distribution function H ( x, y) with marginals
F ( x), G( y) , there exists such a copula function
that
H ( x, y)  C  F ( x), G( y) 
2. If the marginals are absolutely continuous,
then this representation is unique.
Applications of Copulas
• Going beyond correlation
• Extreme co-movements of currency exchange
rates
• Mutual dependence of international markets
• Evaluation of portfolio risks
• Pricing CDOs
References
• Joe Nelsen; An Introduction to Copulas,
Springer
• Umberto Cherubini, Elisa Luciano, Walter
Vecchiato; Copula Methods in Finance, Wiley
• Attilio Meucci; Computational Methods in
Decision-making, Kluwer
• Robert Engle et al.
• Paul Embrechts et al.
Conclusions
• Work in progress – the world is in search for
better models (?)
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