Let’s face complexity September 4-8, 2017

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. 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
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3. Introduction Models Numerics RM Approach Conclusions
Wrapping up Some Messages
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4. Introduction Models Numerics RM Approach Conclusions
Old Connection Physics ←→ Economics
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5. Introduction Models Numerics RM Approach Conclusions
Old Connection Physics ←→ Economics
Einstein 1905 Bachelier 1900
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6. Introduction Models Numerics RM Approach Conclusions
Old Connection Physics ←→ Economics
Einstein 1905 Bachelier 1900
Como, September 2017

7. Introduction Models Numerics RM Approach Conclusions
Old Connection Physics ←→ Economics
Einstein 1905 Bachelier 1900
Mandelbrot 60’s
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8. Introduction Models Numerics RM Approach Conclusions
Growing Jobmarket for Physicists
“Every tenth academic hired by Deutsche Bank is a natural scientist.”
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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. 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. 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)
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12. Introduction Models Numerics RM Approach Conclusions
Diversification in a Stock Portfolio without Correlations
empirical distribution of normalized returns (400 stocks)
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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
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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. 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
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16. Introduction Models Numerics RM Approach Conclusions
Introduction — Credit Risk
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17. Introduction Models Numerics RM Approach Conclusions
Credits and Stability of the Economy
credit crisis shakes economy −→ dramatic instability
Como, September 2017

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. 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. 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. 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
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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
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23. Introduction Models Numerics RM Approach Conclusions
Modeling Credit Risk
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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. 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
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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
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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. 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 )
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29. Introduction Models Numerics RM Approach Conclusions
Loss Distribution
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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
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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. 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. 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. 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. 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. Introduction Models Numerics RM Approach Conclusions
Influence of Correlations,
Numerical Simulations
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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
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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. 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. 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. 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
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42. Introduction Models Numerics RM Approach Conclusions
Random Matrix Approach
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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
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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. 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. Introduction Models Numerics RM Approach Conclusions
Correlations Vary over Time
fourth quarter 2005 first quarter 2006
S&P500 data
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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. 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
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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)
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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. 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. 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. 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. 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. 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. 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