modeling related failures in finance arkady shemyakin mfm orientation, 2010
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Modeling Related Failures in Finance
Arkady ShemyakinMFM Orientation, 2010
Outline
• 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)• High-load periods• Climate and natural disasters• Overloads• 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)
• Survival function
• Distribution density function (d.d.f.)
( )F t P X t
( ) 1 ( )S t P X t F t
( ) ( ) t zd
f z F tdt
Joint Distributions
• Joint distribution function
• Joint survival function
• Joint density
( , ) ,H s t P X s Y t
( , ) , )K s t P X s Y t
2
,( , ) ( , ) s z t wd
h z w H s tdsdt
Independence
• For any
• For any
• For any
• Joint functions are built from marginals
, ( , ) ( ) ( ) ( ) ( )s t H s t F s G t P X s P Y t
, ( , ) ( ) ( )s t K s t P X s P Y t
, ( , ) ( ) ( )s t h s t f s g t
Pearson’s Moment Correlation
• Pearson’s moment correlation (correlation coefficient) is defined as
• It is a good measure of linear dependence, strongly connected with the first two moments, and is known not to capture non-linear dependence
,,
( )X Y
Cov X Y EXY EX EYX Y
Var X Var Y
Sample Pearson’s Correlation
• Given a paired (matched) sample
the sample correlation coefficient is defined as
1 1, ,..., , ,n nx y x yx,y
1 1 1
2 2
2 2
1 1 1 1
1
ˆ1 1
n n n
i i i ii i i
n n n n
i i i ii i i i
x y x yn
x x y yn n
x,y
Default Correlation
• 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
• Is it possible to hedge the risks (sell the risks)?• CDO structure: how many defaults?• Senior tranche (safe)• Mezzanine tranche (middle-of-the-road)• 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
• 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
• Sadly, it says next to nothing about the joint probability of three defaults
• Different scenarios are possible
Calculation of the Multiple Default Probabilities
( ) ( ) ( ),
( )(1 ( )) ( )(1 ( ))( )
( ) 0.1 0.10.5 ( ) 0.055
0.1 0.9
( ) ?
EXY EX EY P AB P A P BX Y
P A P A P B P BVar X Var Y
P ABP AB
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
• 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
• 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
is called a copula (copula function) if:
1. For any
2. It is 2-monotone (quasi-monotone).
3. For any
2: 0,1 [0,1] 0,1C I I
, ( ,1) ; (1, )u v I C u u C v v
, ( ,0) (0, ) 0u v I C u C v
Frechet Bounds
• For any copula the following inequalities (Frechet bounds) hold:
( , )C u v
( , ) ( , ) ( , ),
( , ) max 1,0 ,
( , ) min ,
W u v C u v M u v
W u v u v
M u v u v
Maximum Copula ( , ) min{ , }M u v u v
Maximum Copula ( , ) min{ , }M u v u v
Minimum Copula ( , ) max 1,0W u v u v
Minimum Copula ( , ) max 1,0W u v u v
Product Copula ( , )P u v uv
Product Copula ( , )P u v uv
Sklar’s Theorem
• Theorem: 1. For any correctly defined joint distribution function with marginals
, there exists such a copula function that
2. If the marginals are absolutely continuous, then this representation is unique.
( , )H x y( ), ( )F x G y
( , ) ( ), ( )H x y C F x G y
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 (?)