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Modeling Related Failures in Finance

Modeling Related Failures in Finance. Arkady Shemyakin MFM 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

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Modeling Related Failures in Finance

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  1. Modeling Related Failures in Finance Arkady Shemyakin MFM Orientation, 2010

  2. 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

  3. 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

  4. Related Failures • Insurance (mortality structure on associated human lives) • Survival (biological species) • Reliability (connected components in complex engineering systems) • Finance (?)

  5. Insurance • Associated human lives (e.g., husbands and wives) • Common lifestyles • Common disasters (accidents) • Broken-heart syndrome • Exclusions!

  6. Survival • Biological species within certain environment (e.g., life on an island) • Common environmental concerns • Predator/prey interactions • Symbiosis

  7. 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

  8. Finance • Bank failures, credit events, defaults on mortgages • Market situation • Macroeconomic indicators • Deficit of trust • Chain reaction of failures

  9. Probability Distributions • Distribution function (d.f.; c.d.f) • Survival function • Distribution density function (d.d.f.)

  10. Joint Distributions • Joint distribution function • Joint survival function • Joint density

  11. Independence • For any • For any • For any • Joint functions are built from marginals

  12. 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

  13. Sample Pearson’s Correlation • Given a paired (matched) sample the sample correlation coefficient is defined as

  14. 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 defaultcorrelation • Example: three-mortgage portfolio

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. Diagram 1 (Independence)

  21. 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

  22. Calculation of the Multiple Default Probabilities

  23. Diagram X - Correlation

  24. Diagram 2 (Extreme Scenario 2)

  25. Default Probabilities – Scenario 2 • P(all three defaults) = 0.01 • P(at least two defaults) = 0.145 • P(at least one default) = 0.145

  26. 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

  27. Diagram 3 (Extreme scenario 3)

  28. Default Probabilities – Scenario 3 • P(all three defaults) = 0.055 • P(at least two defaults) = 0.055 • P(at least one default) = 0.19

  29. 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

  30. 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

  31. Definition of Copula Function • A function is called a copula (copula function) if: • For any • It is 2-monotone (quasi-monotone). • For any

  32. Frechet Bounds • For any copula the following inequalities (Frechet bounds) hold:

  33. Maximum Copula

  34. Maximum Copula

  35. Minimum Copula

  36. Minimum Copula

  37. Product Copula

  38. Product Copula

  39. 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.

  40. Applications of Copulas • Going beyond correlation • Extreme co-movements of currency exchange rates • Mutual dependence of international markets • Evaluation of portfolio risks • Pricing CDOs

  41. References • Joe Nelsen; An Introduction to Copulas, Springer • Umberto Cherubini, Elisa Luciano, Walter Vecchiato; Copula Methods in Finance, Wiley • AttilioMeucci; Computational Methods in Decision-making, Kluwer • Robert Engle et al. • Paul Embrechts et al.

  42. Conclusions • Work in progress – the world is in search for better models (?)

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