Financial crises, crisis spill-overs and the
business cycle
Stefan Straetmans, Maastricht university
School of Business and Economics
Risk Forum Paris, March 2015
Motivation and contribution




Tail behavior
- Univariate: fat tails (nonnormality)
- Multivariate: tail dependence
Features of the unconditional (time constant) df
This paper: do they exhibit regime dependence?
Extreme value analysis (EVT) presupposes a
stationary unconditional (“long-term”) df
→ reconcilable with short-term regimes/states
Motivation and contribution (2)



Tail indices, scalig constants, tail copula most likely
not invariant over time..
Regime dependence may shed light on the
determinants of tail fatness but careful: this is NOT
a causality analysis
Is there a way in between volatility and dependence
modelling based on conditional df (multivariate
GARCH, SV..) vs. “pure” EVT?
Motivation and contribution (3)

Simplest way to introduce regime dependence that
preserves unconditional stationarity of df:
- mixture of Pareto tails (univariate EVT)
- mixture of tail copula (multivariate EVT)

In this paper: (exogenous) regime = business cycle

We do find regime dependence for scaling constant,
tail index and strength of tail dependence
Anticipating our results....




Recession-based downside asset tail risk highest
for large variety of assets
Recession-based bank linkages highest
Stock-bond flight-to-quality dominates co-crashes
during recessions; asymmetry much smaller in
expansions
Minimum variance portfolios are regime
dependent; diversification meltdown in recessions
Univariate mixture model

Survivor function as mixture of survivor
functions:
1  F x   1  F1 x  1   1  F2 x

model survivor functions with Pareto tail:
PX  x  L1  x x 1  1   L2  x x  2
1   2  PX  x  L x x 
2
with L x   L1  x x 2 1  1   L2  x 
L x  also slowly var ying!
Bivariate mixture model

Survivor function as mixture:
1  F x, y    1  F1 x, y   1   1  F2 x, y 

Tail copula 1-1 with joint survivor function:
l u, v   lim t 0 t 1 1  F Q1 tu , Q2 tv

Mixture in survivor functions transfers to copula
and tail copula:
l u, v  l1 u, v  1   l2 u, v
Statistical relevance



Regime dependence in tail fatness provides
additional source of bias in tail index estimation
Why? Pareto mixture is equivalent to higher order
behavior which induces Hill bias
Same should hold for multivariate mixtures and
the strength of tail dependence (bias in extreme
linkage measures)
Mixtures and higher order behavior

n=2 states:
1   2

1  F  x   x  2 1    x 2 1


n=3 states:
1   2   3

1  F x   x  3 1  1  2  1 x 3 1  2 x 3  2
etc.

Economic relevance



Short-term horizon risk managers probably
interested to identify regime-dependent risk
...but even financial regulators with long-term
focus aim for countercyclical micro/macro
prudential risk indicators
Neglecting regime dependence potentially leads
to too conservative capital requirements
Application: trading limits for bank
traders




Return series
X i i  1,, n
s>0 : maximum loss such that pension fund, bank does
not become insolvent/illiquid
s depends on actual solvency/liquidity position
Maximum allowable investment I in risky position?
PrX  x p   1  F x p   p  PrI  X  I  x p   p  I  s / x p
x p  F 1 1  p 
 extreme quantile
estimator
Previous evidence on regime
dependence in tail behavior




Only structural break analyses
Univariate: breaks in tail index/tail quantiles?
(scant) evidence limited to (emerging) currency
returns (Koedijk et al. (1990, 1992), Straetmans et
al. (2013))
Multivariate: breaks in tail copula: EMS forex
return pairs (Straetmans (1998))
We adopt more complex subsample partitioning
Crisis and crisis spillover indicators





Downside tail risk: tail index/probs/quantiles
- individual assets and portfolios
Bivariate co-crash probability
- tail-β as bivariate co-movement indicator
- compare co-crash vs. Flight-to-Quality probs
Conditional expected number of co-crashes
Multivariate co-crash probability
Estimate on full, recession, expansion sample
Tail probs/tail quantiles

Use marginal quantile estimator that exploits fat
tails:
m

Pˆ  X  x    X m,n  x  , large x
n


Use Hill statistic for α
Inverse of estimator renders “extreme” Value-at-Risk
estimate for given p:
1/ 
 m 

xˆ p  X m ,n 
 pn 
,
p  n 1
Co-crash tail probability
CPXY  PX 1  Q1  p  X 2  Q2  p  
2 p  PX 1  Q1  p  or X 2  Q2  p 
PX 2  Q2  p 
 2  l 1,1
X 2  X M  CPXY is " tail -  "
l 1,1 : " stable tail dependence function" or " tail copula"
lˆ1,1 estimator following Huang (1992)
Alternative extreme comovement
measures

Conditional expected number of co-extremes and
(scaled) multivariate co-exceedance likelihood:
E   1 
Np
N

l  p,..., p  l 1,...,1


PN 1  P X 1  Q1  p ,..., X i  Qi  p ,..., X N  QN  p  X j  Q j  p 
PX 1  Q1  p ,..., X i  Qi  p ,..., X N  QN  p 

p
Note: co-crash probability as mixture
CPXY  2  l 1,1
 2 - l1 1,1  1   l2 1,1
  2  l1 1,1  1   2  l2 1,1
2
 CPX1  1   CPXY
Data


Monthly business cycle dummies (1985-2012)
- US: NBER dating committee
- GE, FR, UK, JP: Economic Cycle Research
Institute (ECRI)
DS Daily financial returns (1/2/1985-31/12/2012)
- 16 US bank stocks, G-5 stock and (10-year)
sovereign bond indices, commodities (oil, gold,
silver), exchange rates
Bank tail risk: representative results
(p=0.1%)
Banks
Recession
R
Expansion
qR
E
qE
Equality tests
R  E
q R  qE
BOA
1.75
0.76
2.73
0.12
-3.29
2.51
BB&T
2.37
0.30
3.13
0.09
-1.98
2.79
NYMellon
1.94
0.46
3.26
0.10
-3.89
2.60
Comerica
1.92
0.49
2.85
0.10
-2.91
2.62
Huntington
1.82
0.84
2.70
0.12
-2.89
2.67
JP Morgan
2.16
0.40
3.23
0.11
-2.96
2.70
Keycorp
1.80
0.69
2.81
0.11
-3.30
2.59
North. Trust
2.09
0.34
2.92
0.10
-2.42
2.52
Wells Fargo
2.04
0.45
3.29
0.09
-3.54
2.79
General stock and bond tail risk
(indices)
Recession
Stocks
R
Expansion
qR
E
Equality tests
qE
R  E
q R  qE
US
2.02
0.20
2.96
0.06
-2.80
2.40
GE
2.51
0.10
2.80
0.07
-0.89
1.53
UK
2.46
0.10
2.73
0.06
-0.83
1.99
FR
2.44
0.13
2.94
0.07
-1.36
2.00
JP
2.88
0.09
2.63
0.08
0.75
0.88
US
2.69
0.04
3.36
0.02
-1.55
2.02
GE
2.25
0.03
3.18
0.02
-2.91
1.43
UK
3.15
0.02
3.12
0.02
0.10
0.05
FR
2.63
0.02
3.08
0.02
-1.15
0.65
JP
2.18
0.03
2.27
0.03
-0.31
0.12
Bonds
Forex and commodities
Recession
Forex
R
Expansion
qR
E
Equality tests
qE
R  E
q R  qE
US$/uk£
2.47
0.06
3.16
0.03
-1.76
2.09
US$/JPY
2.18
0.11
3.35
0.04
-3.18
2.39
US$/SFR
2.21
0.08
3.46
0.03
-3.30
2.35
Oil
2.23
0.32
3.29
0.12
-2.85
2.43
Silver
2.03
0.24
2.54
0.13
-1.57
1.53
Gold
2.35
0.13
2.88
0.06
-1.44
2.24
Commodities
Bank linkages: MES and tail-β
Bank
Recession
R
Expansion
MES R ρ
Equality tests
E
MES E
ρ
R E
MESR  MESE
BOA
0.74
0.14
0.91
0.63
0.04
0.81
1.66
5.87
BB&T
0.62
0.08
0.83
0.46
0.03
0.62
3.17
3.10
NY Mellon
0.58
0.08
0.78
0.54
0.03
0.73
0.72
2.74
Comerica
0.66
0.09
0.85
0.56
0.03
0.72
2.11
4.78
Huntington
0.58
0.14
0.71
0.48
0.03
0.62
1.99
4.99
JP Morgan
0.73
0.09
0.89
0.58
0.04
0.80
2.75
5.33
Keycorp
0.69
0.13
0.80
0.54
0.03
0.74
3.08
3.42
North. Trust
0.62
0.07
0.78
0.51
0.03
0.65
2.06
4.21
Wells Fargo
0.74
0.11
0.92
0.51
0.03
0.75
3.88
4.61
Multivariate systemic risk results
N  16 banks
1  E  16
Full sample : E  2.558
P MUL  0.139
Recession : E R  3.676
PRMUL  0.319
Expansion : E E  2.053
PEMUL  0.109
Bank tail risk and systemic risk




Consistent with fundamentals-based banking crises
literature
Incidence of banking panics increases during recessions
cf. Gorton (1988)
How to make it countercyclical? Turn it around...
Macro prudential regulation: use recession (expansion) SR
indicators to assess true expansion (recession) SR
Co-crash vs. Flight-to-quality
S  stock return
B  bond return
3rd quadrant :
CPCO  PB  QB 1  p  S  QS 1  p 
2nd quadrant :
CPFTQ  PB  QB  p  S  QS 1  p 
p  0.05%
Stock/bond co-crashes vs. Flight-toquality probabilities
Full sample
Recession
ρ
Expansion
ρ
PFTQ
US
0.070
0.225
-0.096 0.068
0.295
-0.252 0.086
0.176
-0.051
GE
0.050
0.180
-0.107 0.039
0.195
-0.185 0.048
0.196
-0.080
UK
0.055
0.165
-0.066 0.042
0.366
-0.288 0.076
0.076
0.021
FR
0.060
0.100
0.001
0.083
0.188
-0.241 0.065
0.097
0.048
JP
0.060
0.090
-0.068 0.011
0.092
-0.151 0.106
0.099
-0.030
PCO
PFTQ
PCO
PFTQ
ρ
PCO
Minimum tail risk portfolios

Form 2-asset portfolio with pairs of DJ stocks
R p  wR1  1  wR2
 22  1 2
 2
 1   22  2 1 2

Minimum variance portfolio:

Minimum Value-at-Risk portfolio for low pvalue:
wMV
1/ 
 m 
ˆ

min q p w  Rm , n w
w
 pn 
DJ Correlations and the business cycle
DJ Stock pairs
Correlations
Full sample
recession
expansion
ExxonMobil/Wal Mart
0.33
0.43
0.32
Wall Mart/P&G
0.37
0.57
0.34
PG/J&J
0.47
0.63
0.45
JJ/GE
0.44
0.47
0.44
GE/JP Morgan
0.56
0.63
0.53
JP Morgan/Pfizer
0.34
0.45
0.32
Minimum Variance and tail risk
portfolios
DJ Stock pairs
Full sample
recession
expansion
Minimum Variance results
w
MV
w
MV
w
MV
Exxon/WalMart
0.6
0.013
0.41
0.018
0.64
0.013
Wall Mart/P&G
0.40
0.014
0.33
0.017
0.40
0.013
P&G/J&J
0.45
0.013
0.36
0.015
0.46
0.013
J&J/JP Morgan
0.83
0.014
1
0.016
0.74
0.013
Minimum tail risk results
w
q
w
q
w
q
Exxon/WalMart
0.59
0.078
0.41
0.095
0.63
0.074
Wall Mart/P&G
0.31
0.084
0.45
0.096
0.30
0.080
P&G/J&J
0.52
0.078
0.28
0.080
0.54
0.075
J&J/JP Morgan
0.78
0.093
0.93
0.089
0.74
0.089
Diversification of portfolio tail risk:
summary



Tactical asset allocation: if your aim is to avoid
extreme portfolio returns you have to invest
differently during recessions
Diversification meltdown during recessions:
higher correlation, higher (minimized) portfolio
variance and portfolio tail risk (VaR)
Portfolio allocation w for minimum tail risk
portfolios strongly differs from minimum
variance portfolios
Extensions/avenues for future
research




Regime-dependence of left-right tail asymmetry
→to be expected to be much more severe during
recessions
Allowing for thin and fat tails (Generalized EVD)
Alternatives for the business cycle? Borio’s
financial cycle
Endgenous regime determination, more thzn two
regimes...