Credits and the Instability of the Financial System: a Physicists - - PowerPoint PPT Presentation
Introduction Models Numerics RM Approach Conclusions Fakult at f ur Physik Credits and the Instability of the Financial System: a Physicists Point of View Thomas Guhr Spectral Properties of Complex Networks ECT* Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Fakult¨ at f¨ ur Physik
Thomas Guhr Spectral Properties of Complex Networks ECT* Trento, July 2012
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
◮ Introduction — econophysics, credit risk ◮ Structural model and loss distribution ◮ Numerical simulations and random matrix approach ◮ Conclusions — general, present credit crisis
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Einstein 1905 Bachelier 1900
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Einstein 1905 Bachelier 1900
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Einstein 1905 Bachelier 1900 Mandelbrot 60’s
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
“Every tenth academic hired by Deutsche Bank is a natural scientist.”
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
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
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Psychology Ethics Business Administration Laws and Regulations Politics
Quantitative Problems
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
R∆t(t) = S(t + ∆t) − S(t) S(t) non–Gaussian, heavy tails! (Mantegna, Stanley, ..., 90’s)
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
◮ credit crisis shakes economy −
→ dramatic instability
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
◮ credit crisis shakes economy −
→ dramatic instability
◮ claim: risk reduction by diversification
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
◮ credit crisis shakes economy −
→ dramatic instability
◮ claim: risk reduction by diversification ◮ questioned with qualitative reasoning by several economists
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
◮ 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
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
◮ 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
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Creditor Obligor t = 0
Principal
Creditor Obligor t = 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
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
t Vk(t) F Vk(0) T ◮ microscopic approach for K companies ◮ economic state: risk elements Vk(t), k = 1, . . . , K ◮ default occurs if Vk(T) falls below face value Fk ◮ then the (normalized) loss is Lk = Fk − Vk(T)
Fk
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
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
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Introduction Models Numerics RM Approach Conclusions
t Vk(t) jump Vk(0) with jumps without jumps
jumps reproduce empirically found heavy tails
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
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(Jan 2008)
IBM MSFT
normalization Mk(t′) = Rk(t′) − Rk(t′)
k(t′) − Rk(t′)2
correlation Ckl = Mk(t′)Ml(t′) , u(t′) = 1 T ′
T ′
u(t′) K × T ′ data matrix M such that C = 1 T ′ MM†
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
◮ εi(t), i = 1, . . . , I set of random variables ◮ K × I structure matrix A ◮ correlated diffusion, uncorrelated drift, uncorrelated jumps
dVk(t) Vk(t) = µk dt + σk
I
Akiεi(t) √ dt + dJk(t) for T → ∞ correlation matrix is C = AA† covariance matrix is Σ = σCσ with σ = diag (σ1, . . . , σK)
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
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
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Introduction Models Numerics RM Approach Conclusions
◮ homogeneous portfolio ◮ portfolio loss L = 1
K
K
Lk
◮ stock prices at maturity V = (V1(T), . . . , VK(T)) ◮ distribution p(mv)(V , Σ) with Σ = σCσ
want to calculate p(L) =
K
K
Lk
Introduction Models Numerics RM Approach Conclusions
Real portfolios comprise several hundred or more individual contracts − → K is large. Central Limit Theorem: For very large K, portfolio loss distribution p(L) must become Gaussian. Question: how large is “very large” ?
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Introduction Models Numerics RM Approach Conclusions
Unexpected loss Expected loss Economic capital α-quantile Loss in %
Frequency
◮ highly asymetric, heavy tails, rare but drastic events ◮ mean of loss distribution is called expected loss (EL) ◮ standard deviation is called unexpected loss (UL)
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Introduction Models Numerics RM Approach Conclusions
◮ analytical, good
approximations
◮ slow convergence to
Gaussian for large portfolio
◮ kurtosis excess of
uncorrelated portfolios scales as 1/K
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
◮ 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!
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
fixed correlation Ckl = c, k = l , and Ckk = 1 c = 0.2 c = 0.5
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Introduction Models Numerics RM Approach Conclusions
γ2 = µ4 µ2
2
− 3 limiting tail behavior quickly reached − → diversification does not work
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Introduction Models Numerics RM Approach Conclusions
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 Value at Risk
VaR
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 !
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Introduction Models Numerics RM Approach Conclusions
◮ correlated jump–diffusion ◮ fixed correlation c = 0.5 ◮ jumps change picture only
slightly
◮ tail behavior stays similar
with increasing K
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
◮ correlated jump–diffusion ◮ fixed correlation c = 0.5 ◮ jumps change picture only
slightly
◮ tail behavior stays similar
with increasing K
◮ diversification does not work
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
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
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Introduction Models Numerics RM Approach Conclusions
◮ large portfolio → large K ◮ correlation matrix C is K × K ◮ “second ergodicity”: spectral average = ensemble average ◮ set C = WW † and choose W as random matrix
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
◮ large portfolio → large K ◮ correlation matrix C is K × K ◮ “second ergodicity”: spectral average = ensemble average ◮ set C = WW † and choose W as random matrix ◮ additional motivation: correlations vary over time
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
Brownian motion, V = (V1(T), . . . , VK(T)), price distribution p(mv)(V , Σ) = 1 √ 2πT
K
1 √ det Σ exp
2T (V − µT)†Σ−1(V − µT)
N free parameter, such that Σ = σWW †σ assume Gaussian distribution for W with variance 1/N p(corr)(W ) =
2π
KN
exp
2 tr W †W
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
p(mv)(ρ) =
=
2πT
K 21− N
2
Γ(N/2)ρ
N+K−1 2
T
N−K 2
K N−K
2
T
V 2
k (T)
σ2
k
similar to statistics of extreme events easily transferred to geometric Brownian motion
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
about K = 400 stocks with complete time series from S&P500
20 40 60 80 1015 1012 109 106 0.001 1 Ρ pΡ 10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Ρ pΡ
N = 5, 10, 20, 30 (theory) N = 14 (fit to data) N smaller − → stronger correlated − → heavier tails
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
p(L) =
K
K
Lk
0.005 0.010 0.015 0.020 1 10 100 L pL
Ckl = 0 , k = l N = 5 → std (Ckl) = 0.45 K = 10, 100, 1000, 10000 best case scenario, but heavy tails remain − → little diversification benefit
Trento, July 2012
Introduction Models Numerics RM Approach 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 ◮ average correlation zero (best case scenario), but still: heavy
tails remain, little diversification benefit
◮ non–zero average correlation: work in progress
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
◮ 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
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
◮ 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
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
◮ 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
Trento, July 2012
Introduction Models Numerics RM Approach Conclusions
afer, M. Sj¨
Credit Risk - A Structural Model with Jumps and Correlations, Physica A383 (2007) 533 M.C. M¨ unnix, R. Sch¨ afer and T. Guhr, A Random Matrix Approach to Credit Risk, arXiv:1102.3900 both ranked for several months among the top–ten new credit risk papers on www.defaultrisk.com
afer, A. Koivusalo and T. Guhr, Credit Portfolio Risk and Diversification, invited contribution to “Credit Portfolio Securitizations and Derivatives”,
Trento, July 2012