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Econophysics VII: Credits and the Instability of the Financial System - Thomas Guhr

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Let’s face complexity September 4-8, 2017

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Econophysics VII: Credits and the Instability of the Financial System - Thomas Guhr

  1. 1. Introduction Models Numerics RM Approach Conclusions Fakult¨at f¨ur Physik Econophysics VII: Credits and the Instability of the Financial System Thomas Guhr Let’s Face Complexity, Como, 2017 Como, September 2017
  2. 2. Introduction Models Numerics RM Approach Conclusions Outline Introduction — econophysics, credit risk Structural model and loss distribution Numerical simulations and random matrix approach Conclusions — general, present credit crisis Como, September 2017
  3. 3. Introduction Models Numerics RM Approach Conclusions Wrapping up Some Messages Como, September 2017
  4. 4. Introduction Models Numerics RM Approach Conclusions Old Connection Physics ←→ Economics Como, September 2017
  5. 5. Introduction Models Numerics RM Approach Conclusions Old Connection Physics ←→ Economics Einstein 1905 Bachelier 1900 Como, September 2017
  6. 6. Introduction Models Numerics RM Approach Conclusions Old Connection Physics ←→ Economics Einstein 1905 Bachelier 1900 Como, September 2017
  7. 7. Introduction Models Numerics RM Approach Conclusions Old Connection Physics ←→ Economics Einstein 1905 Bachelier 1900 Mandelbrot 60’s Como, September 2017
  8. 8. Introduction Models Numerics RM Approach Conclusions Growing Jobmarket for Physicists “Every tenth academic hired by Deutsche Bank is a natural scientist.” Como, September 2017
  9. 9. Introduction Models Numerics RM Approach Conclusions A New Interdisciplinary Direction in Basic Research Theoretical physics: construction and analysis of mathematical models based on experiments or empirical information physics −→ economics: much better economic data now, growing interest in complex systems Study economy as complex system in its own right economics −→ physics: risk managment, expertise in model building based on empirical data Como, September 2017
  10. 10. Introduction Models Numerics RM Approach Conclusions Economics — a Broad Range of Different Aspects Psychology Ethics Business Administration Laws and Regulations Politics ECONOMICS Quantitative Problems Como, September 2017
  11. 11. Introduction Models Numerics RM Approach Conclusions Return Distributions Revisited R∆t(t) = S(t + ∆t) − S(t) S(t) non–Gaussian, heavy tails! (Mantegna, Stanley, ..., 90’s) Como, September 2017
  12. 12. Introduction Models Numerics RM Approach Conclusions Diversification in a Stock Portfolio without Correlations empirical distribution of normalized returns (400 stocks) Como, September 2017
  13. 13. Introduction Models Numerics RM Approach Conclusions Diversification in a Stock Portfolio without Correlations empirical distribution of normalized returns (400 stocks) portfolio: superposition of stocks Como, September 2017
  14. 14. Introduction Models Numerics RM Approach Conclusions Diversification in a Stock Portfolio without Correlations empirical distribution of normalized returns (400 stocks) portfolio: superposition of stocks risk reduction by diversification (no correlations yet!): returns are more normally distributed, market risk reduced by approx. 50 percent Como, September 2017
  15. 15. Introduction Models Numerics RM Approach Conclusions Effect of Correlations stocks highly correlated to overall market risk reduction by diversification (with correlations): unsystematic risk can be removed, systematic risk (overall market) remains Como, September 2017
  16. 16. Introduction Models Numerics RM Approach Conclusions Introduction — Credit Risk Como, September 2017
  17. 17. Introduction Models Numerics RM Approach Conclusions Credits and Stability of the Economy credit crisis shakes economy −→ dramatic instability Como, September 2017
  18. 18. Introduction Models Numerics RM Approach Conclusions Credits and Stability of the Economy credit crisis shakes economy −→ dramatic instability claim: risk reduction by diversification Como, September 2017
  19. 19. Introduction Models Numerics RM Approach Conclusions Credits and Stability of the Economy credit crisis shakes economy −→ dramatic instability claim: risk reduction by diversification questioned with qualitative reasoning by several economists Como, September 2017
  20. 20. Introduction Models Numerics RM Approach Conclusions Credits and Stability of the Economy credit crisis shakes economy −→ dramatic instability claim: risk reduction by diversification questioned with qualitative reasoning by several economists I now present our quantitative study and answer Como, September 2017
  21. 21. Introduction Models Numerics RM Approach Conclusions Defaults and Losses default occurs if obligor fails to repay → loss possible losses have to be priced into credit contract correlations are important to evaluate risk of credit portfolio statistical model to estimate loss distribution Como, September 2017
  22. 22. Introduction Models Numerics RM Approach Conclusions Zero–Coupon Bond Creditor Obligort = 0 Principal Creditor Obligort = T Face value principal: borrowed amount face value F: borrowed amount + interest + risk compensation credit contract with simplest cash-flow credit portfolio comprises many such contracts Como, September 2017
  23. 23. Introduction Models Numerics RM Approach Conclusions Modeling Credit Risk Como, September 2017
  24. 24. Introduction Models Numerics RM Approach Conclusions Structural Models of Merton Type t Vk(t) F Vk(0) T microscopic approach for K companies economic state: risk elements Vk(t), k = 1, . . . , K maturity time T default occurs if Vk(T) falls below face value Fk then the (normalized) loss is Lk = Fk − Vk(T) Fk Como, September 2017
  25. 25. Introduction Models Numerics RM Approach Conclusions Geometric Brownian Motion with Jumps K companies, risk elements Vk(t), k = 1, . . . , K represent economic states, closely related to stock prices dVk(t) Vk(t) = µk dt + σkεk(t) √ dt + dJk(t) we include jumps ! drift term (deterministic) µk dt diffusion term (stochastic) σkεk(t) √ dt jump term (stochastic) dJk(t) parameters can be tuned to describe the empirical distributions Como, September 2017
  26. 26. Introduction Models Numerics RM Approach Conclusions Jump Process and Price or Return Distributions t Vk(t) jump Vk(0) with jumps without jumps jumps reproduce empirically found heavy tails Como, September 2017
  27. 27. Introduction Models Numerics RM Approach Conclusions Financial Correlations asset values Vk(t ), k = 1, . . . , K measured at t = 1, . . . , T returns Rk(t ) = dVk(t ) Vk(t ) Jan 2008 Feb 2008 Mar 2008 Apr 2008 May 2008 Jun 2008 Jul 2008 Aug 2008 Sep 2008 Oct 2008 Nov 2008 Dec 2008 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 S(t)/S(Jan2008) IBM MSFT normalization Mk(t ) = Rk(t ) − Rk(t ) R2 k (t ) − Rk(t ) 2 correlation Ckl = Mk(t )Ml (t ) , u(t ) = 1 T T t =1 u(t ) K × T data matrix M such that C = 1 T MM† Como, September 2017
  28. 28. Introduction Models Numerics RM Approach Conclusions Inclusion of Correlations in Risk Elements εi (t), i = 1, . . . , N set of iid time series K × N dependence matrix X correlated diffusion, uncorrelated drift, uncorrelated jumps dVk(t) Vk(t) = µk dt + σk N i=1 Xki εi (t) √ dt + dJk(t) correlation matrix is C = XX† covariance matrix is Σ = σCσ with σ = diag (σ1, . . . , σK ) Como, September 2017
  29. 29. Introduction Models Numerics RM Approach Conclusions Loss Distribution Como, September 2017
  30. 30. Introduction Models Numerics RM Approach Conclusions Individual Losses t Vk(t) F Vk(0) T normalized loss at maturity t = T Lk = Fk − Vk(T) Fk Θ(Fk −Vk(T)) if default occurs Como, September 2017
  31. 31. Introduction Models Numerics RM Approach Conclusions Portfolio Loss Distribution homogeneous portfolio portfolio loss L = 1 K K k=1 Lk stock prices at maturity V = (V1(T), . . . , VK (T)) distribution g(V |Σ) with Σ = σCσ want to calculate p(L) = d[V ] g(V |Σ) δ L − 1 K K k=1 Lk Como, September 2017
  32. 32. Introduction Models Numerics RM Approach Conclusions Large Portfolios Real portfolios comprise several hundred or more individual contracts −→ K is large. one is tempted to apply the Central Limit Theorem: For very large K, portfolio loss distribution p(L) should become Gaussian. does that really make sense ? if yes, under which conditions ? how large is “very large” ? Como, September 2017
  33. 33. Introduction Models Numerics RM Approach Conclusions Typical Portfolio Loss Distributions Unexpected loss Expected loss Economic capital α-quantile Loss in % of exposure Frequency highly asymmetric, heavy tails, rare but drastic events is the shape non–Gaussian because K is not large enough ? or are the heavy tails a feature that cannot be avoided ? Como, September 2017
  34. 34. Introduction Models Numerics RM Approach Conclusions Simplified Model — without Jumps, without Correlations analytical, good approximations slow convergence to Gaussian for large portfolio kurtosis excess of uncorrelated portfolios scales as 1/K Como, September 2017
  35. 35. Introduction Models Numerics RM Approach Conclusions Simplified Model — without Jumps, without Correlations analytical, good approximations slow convergence to Gaussian for large portfolio kurtosis excess of uncorrelated portfolios scales as 1/K diversification works slowly, but it works! Como, September 2017
  36. 36. Introduction Models Numerics RM Approach Conclusions Influence of Correlations, Numerical Simulations Como, September 2017
  37. 37. Introduction Models Numerics RM Approach Conclusions Numerics: with Correlations, without Jumps fixed correlation Ckl = c, k = l , and Ckk = 1 c = 0.2 c = 0.5 Como, September 2017
  38. 38. Introduction Models Numerics RM Approach Conclusions Kurtosis Excess versus Fixed Correlation γ2 = µ4 µ2 2 − 3 limiting tail behavior quickly reached −→ diversification does not work Sch¨afer, Sj¨olin, Sundin, Wolanski, Guhr (2007) Como, September 2017
  39. 39. Introduction Models Numerics RM Approach Conclusions Value at Risk versus Fixed Correlation 0.0 0.2 0.4 0.6 0.8 1.0 0.0100 0.0050 0.0020 0.0200 0.0030 0.0300 0.0150 0.0070 Correlation ValueatRisk VaR 0 p(L)dL = α here α = 0.99 K = 10, 100, 1000 99% quantile, portfolio losses are with probability 0.99 smaller than VaR, and with probability 0.01 larger than VaR diversification does not work, it does not reduce risk ! Como, September 2017
  40. 40. Introduction Models Numerics RM Approach Conclusions Numerics: with Correlations and with Jumps correlated jump–diffusion fixed correlation c = 0.5 jumps change picture only slightly tail behavior stays similar with increasing K Como, September 2017
  41. 41. Introduction Models Numerics RM Approach Conclusions Numerics: with Correlations and with Jumps correlated jump–diffusion fixed correlation c = 0.5 jumps change picture only slightly tail behavior stays similar with increasing K diversification does not work Como, September 2017
  42. 42. Introduction Models Numerics RM Approach Conclusions Random Matrix Approach Como, September 2017
  43. 43. Introduction Models Numerics RM Approach Conclusions Quantum Chaos result in statistical nuclear physics (Bohigas, Haq, Pandey, 80’s) resonances “regular” “chaotic” spacing distribution universal in a huge variety of systems: nuclei, atoms, molecules, disordered systems, lattice gauge quantum chromodynamics, elasticity, electrodynamics −→ quantum chaos −→ random matrix theory Como, September 2017
  44. 44. Introduction Models Numerics RM Approach Conclusions Search for Generic Features of Loss Distribution large portfolio → large K correlation matrix C is K × K “second ergodicity”: spectral average = ensemble average C = XX†, make dependence matrix X random goal: average over correlation matrices will drastically reduce number of degrees of freedom Como, September 2017
  45. 45. Introduction Models Numerics RM Approach Conclusions Search for Generic Features of Loss Distribution large portfolio → large K correlation matrix C is K × K “second ergodicity”: spectral average = ensemble average C = XX†, make dependence matrix X random additional motivation: correlations vary over time goal: average over correlation matrices will drastically reduce number of degrees of freedom Como, September 2017
  46. 46. Introduction Models Numerics RM Approach Conclusions Correlations Vary over Time fourth quarter 2005 first quarter 2006 S&P500 data Como, September 2017
  47. 47. Introduction Models Numerics RM Approach Conclusions Return Distribution returns R = (R1(t), . . . , RK (t)) at a time t, for example t = T assume Gaussian for constant Σ (correlations and variances) g(R|Σ) = 1 det (2πΣ) exp − 1 2 R† Σ−1 R 10−5 10−3 10−1 pdf -4 0 4 ˜r S&P500 data short time intervals 2 × 2 subspaces rotated and aggregated Gaussian is ok Como, September 2017
  48. 48. Introduction Models Numerics RM Approach Conclusions Random Correlation Matrices covariance matrix σXX†σ with X rectangular real K × N, N free parameter assume Wishart model for X with variance 1/N w(X|C0) = 1 det N/2 (2πC0) exp − N 2 tr X† C−1 0 X C0 is the mean, average correlation matrix XX† = C0 Como, September 2017
  49. 49. Introduction Models Numerics RM Approach Conclusions Average Return Distribution g (R|C0, N) = g(R|σXX† σ) w(X|C0) d[X] can be done in closed form g (R|C0, N) = 1 2N/2+1Γ(N/2) det(2πΣ0/N) × K(K−N)/2 NR†Σ−1 0 R NR†Σ−1 0 R (K−N)/2 only dependent on R†Σ−1 0 R with Σ0 = σC0σ similar to statistics of extreme events Schmitt, Chetalova, Sch¨afer, Guhr (2013) Como, September 2017
  50. 50. Introduction Models Numerics RM Approach Conclusions Heavy Tailed Average Return Distribution S&P500 data, rotated in eigenbasis of Σ0, aggregated 10−4 10−3 10−2 10−1 1 pdf -4 0 4 ˜r -4 0 4 ˜r daily returns, N = 5 monthly returns, N = 14 N smaller −→ stronger correlated −→ heavier tails Como, September 2017
  51. 51. Introduction Models Numerics RM Approach Conclusions Back to Credit Risk — Average Loss Distribution return distribution g (R|C0, N) easily transferred to risk element distribution g (V |C0, N) average loss distribution p (L) = d[V ] g (V |C0, N) δ L − 1 K K k=1 Lk can be cast into double integral if the average C0 is approximated as c0, that is C0,kl = c0, k = l no “fixed correlations”, the correlations fluctuate around C0 Como, September 2017
  52. 52. Introduction Models Numerics RM Approach Conclusions Average Loss Distribution Reaches Limiting Curve 10−2 10−1 1 10 102 p(L) 0.0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 L K = 10 K = 100 K → ∞ maturity time T = 1y, realistic input values from S&P500 data: µk = 0.015/y, σk = 0.25/ √ y, c0 = 0.25, N = 5 heavy tails remain, limiting curve −→ no diversification benefit Schmitt, Chetalova, Sch¨afer, Guhr (2014) Como, September 2017
  53. 53. Introduction Models Numerics RM Approach Conclusions General Conclusions uncorrelated portfolios: diversification works (slowly) unexpectedly strong impact of correlations due to peculiar shape of loss distribution correlations lead to extremely fat–tailed distribution fixed correlations: diversification does not work ensemble average reveals generic features of loss distributions heavy tails remain, no diversification benefit limiting curve for infinitely many obligors Como, September 2017
  54. 54. Introduction Models Numerics RM Approach Conclusions Conclusions in View of the Present Credit Crisis contracts with high default probability rating agencies rated way too high credit institutes resold the risk of credit portfolios, grouped by credit rating lower ratings → higher risk and higher potential return effect of correlations underestimated benefit of diversification vastly overestimated Como, September 2017
  55. 55. Introduction Models Numerics RM Approach Conclusions Conclusions in View of the Present Credit Crisis contracts with high default probability rating agencies rated way too high credit institutes resold the risk of credit portfolios, grouped by credit rating lower ratings → higher risk and higher potential return effect of correlations underestimated benefit of diversification vastly overestimated Como, September 2017
  56. 56. Introduction Models Numerics RM Approach Conclusions Conclusions in View of the Present Credit Crisis contracts with high default probability rating agencies rated way too high credit institutes resold the risk of credit portfolios, grouped by credit rating lower ratings → higher risk and higher potential return effect of correlations underestimated benefit of diversification vastly overestimated Como, September 2017

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