Report

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 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) 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 tz 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 XY 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 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 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 • • • • • 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 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 (?)