17jun10 measure of systemic risk m kritzman
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April 20101
Principal Components as a Measure of Systemic Risk
Mark Kritzman, Windham Capital Management and MITYuanzhen Li, Windham Capital ManagementSebastien Page, SSARoberto Rigobon, MIT and NBER
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■ The challenge
■ The absorption ratio defined
■ The absorption ratio and asset prices
■ The absorption ratio and financial turbulence
■ The global absorption ratio and financial crises
■ Summary
Principal components as a measure of systemic risk
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Challenge
■ Securitization obscures connections among stakeholders.
■ Private transacting leads to opacity.
■ Complexity reduces clarity (Lehman Brothers had 900,000
derivative contracts on its books when it defaulted).
■ “Flexible” accounting also hides financial linkages.
It is unlikely that we can directly observe the explicit linkages
of financial institutions.
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The absorption ratio as a measure of implied systemic risk
■ The absorption ratio equals the fraction of the total variance of a set of assets explained or “absorbed” by a finite number of eigenvectors.
■ A high absorption ratio implies that markets are compact or tightly coupled.
■ Compact markets are relatively fragile in that shocks propagate more quickly and broadly than when markets are loosely linked.
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Three-dimensional scatter plot of asset returns
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First eigenvector
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First eigenvector
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Second eigenvector
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‐1 ‐0.8 ‐0.6 ‐0.4 ‐0.2 0 0.2 0.4 0.6 0.8 1
factor 1
factor 2
Toy example
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‐1 ‐0.8 ‐0.6 ‐0.4 ‐0.2 0 0.2 0.4 0.6 0.8 1
Bordeaux 85
Burgundy 85
US Cabernet 85
Bordeaux 94
Burgundy 94
US Cabernet 94
US Pinot 94
Bordeaux 97
Burgundy 97
US Cabernet 97
US Pinot 97factor 1: vintage
factor 2: varietal
Toy example
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The absorption ratio
where,
AR: Absorption Ratio
N: number of assets
n: number of eigenvectors used to calculate AR
: variance of the i-th eigenvector, sometimes called eigenportfolio
: variance of the j-th asset
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The absorption ratio
0.00
0.25
0.50
0.75
1.00
500 475 450 425 400 375 350 325 300 275 250 225 200 175 150 125 100 75 50 25 0Day
Memory LossWeight
Variances are estimated from exponentially decaying returns.
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Absorption ratio versus average correlation
Absorption Ratio versus Average Correlation
Period 1 Correlations Standard
Assets 1 2 3 4 Deviations
1 1.00 0.12 ‐0.01 0.01 35.16%
2 0.12 1.00 ‐0.04 ‐0.03 35.07%
3 ‐0.01 ‐0.04 1.00 0.82 4.95%
4 0.01 ‐0.03 0.82 1.00 5.02%
Period 2 Correlations Standard
Assets 1 2 3 4 Deviations
1 1.00 0.64 ‐0.05 ‐0.01 34.46%
2 0.64 1.00 ‐0.05 ‐0.03 34.04%
3 ‐0.05 ‐0.05 1.00 0.03 4.92%
4 ‐0.01 ‐0.03 0.03 1.00 4.88%
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Absorption ratio versus average correlation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Period 1 Period 2
AC
AR
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Absorption ratio and U.S. stocks
Covariance matrix and eigenvectors are estimated from daily returns over prior 500 days.
Absorption ratio is estimated from first 10 eigenvectors.
0
500
1,000
1,500
0.5
1
1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
MSCI USA Price Index
Absorption Ratio
Absorption Ratio
MSCI USA Price Index
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1% Worst 2% Worst 5% Worst
1 Day 84.85% 87.69% 70.81%
1 Week 84.85% 83.08% 75.78%
1 Month 100.00% 98.46% 89.44%
Fraction of drawdowns preceded by spike in AR
1 standard deviation, 15 days / 1 year
1/1/1998 through 5/10/2010
Absorption ratio and drawdowns
Spike = 1 standard deviation outlier of (15-day moving average of AR minus
1-year moving average of AR) divided by standard deviation of 1-year AR
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1 Sigma Increase
1 Sigma Decrease Difference
1 Day -8.28% 9.27% -17.56%
1 Week -8.44% 10.06% -18.50%
1 Month -5.86% 12.16% -18.02%
Absorption ratio and subsequent returns
Annualized return after extreme AR
1 standard deviation, 15 days / 1 year
1/1/1998 through 5/10/2010
Spike = 1 standard deviation outlier of (15-day moving average of AR minus
1-year moving average of AR) divided by standard deviation of 1-year AR
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Absorption ratio and subsequent returns
‐10.00%
‐5.00%
0.00%
5.00%
10.00%
15.00%
20.00%
1 Day 1 Week 1 Month
Annualized Return after Spikes and Declines
Spike
Sharp Decline
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Absorption ratio as a market timing signal
Absorption Ratio Stocks/Bonds
-1σ <= ∆AR <= + 1σ 50/50
∆AR > + 1σ 0/100
∆AR < - 1σ 100/0
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Absorption ratio as a market timing signal
Performance: 100/0 versus 0/100
Dynamic 50/50
Return 9.58% 5.08%
Risk 11.50% 10.89%
Return/Risk 0.83 0.47
Turnover 86.01%
Number of Trades 1.72
1/1/1998 through 5/10/2010
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Absorption ratio as a market timing signal
$0
$1
$2
$3
$4
1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
Cumulative Wealth
50‐50
All‐stock
Dynamic
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Absorption ratio stock exposure
0
500
1,000
1,500
0%
100%
200%
300%
1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
MSCI USA Price Index
StockExposure
Stock Exposure
MSCI USA Price Index
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Absorption ratio as a market timing signal Non-U.S. markets
0%
2%
4%
6%
8%
10%
12%
14%
U.S. U.K. Germany Canada Japan
Total R
eturn
AR Global Performance
Static
Dynamic
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Absorption ratio as a market timing signal Non-U.S. markets
0.00
0.20
0.40
0.60
0.80
1.00
1.20
U.S. U.K. Germany Canada Japan
Return/Risk
AR Risk Adjusted Performance
Static
Dynamic
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Major stock market crashes
‐80%
‐60%
‐40%
‐20%
0%
20%
40%
60%
80%
100%
120%
1987 Crash Dot‐Com Meltdown
Global Financial Crisis
Absorption Ratio and Major Crashes
Stock Market Drawdown
AR Filtered Return
% Drawdown Avoided
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Herfindal Index versus sum
0%
2%
4%
6%
8%
10%
12%
14%
U.S. U.K. Germany Canada Japan
Total R
eturn
AR Global Performance
Static
Dynamic, Herfindal
Dynamic, Sum
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Herfindal Index versus sum
0.00
0.20
0.40
0.60
0.80
1.00
1.20
U.S. U.K. Germany Canada Japan
Return/Risk
AR Risk Adjusted Performance
Static
Dynamic, Herfindal
Dynamic, Sum
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Absorption ratio and the national housing bubble
0.00
50.00
100.00
150.00
200.00
250.00
0.6
1
1.4
Jan‐92
Oct‐92
Jul‐93
Apr‐94
Jan‐95
Oct‐95
Jul‐96
Apr‐97
Jan‐98
Oct‐98
Jul‐99
Apr‐00
Jan‐01
Oct‐01
Jul‐02
Apr‐03
Jan‐04
Oct‐04
Jul‐05
Apr‐06
Jan‐07
Oct‐07
Jul‐08
Apr‐09
Absorption Ratio
Case‐Shiller Composite Home Price Index
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Absorption ratio and financial turbulence
dt = vector distance from multivariate averageyt = return series = mean vector of return series ytΣ = covariance matrix of return series yt
dt = (yt - µ)Σ-1(yt - µ)′
How we define financial turbulence:
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Absorption ratio and financial turbulence
■ We measure financial turbulence as a condition in which asset prices behave in an
uncharacteristic fashion, given their historical pattern of behavior.
■ Extreme price moves
■ Decoupling of correlated assets
■ Convergence of uncorrelated assets
■ Differences from historical averages capture extent to which one or more return was
unusually high or low.
■ Multiplying by inverse of covariance matrix makes the measure scale independent and
captures interaction of assets.
■ Post multiplying by transpose of differences converts vector to a single number.
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Financial turbulence throughout history
Global Financial Crisis
9/11
Tech Bubble
Russian Default
Gulf War
Black Monday
Stagflation
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Strange connections
■ Skulls and financial turbulence
■ Missile guidance and portfolio rebalancing
■ Heat transfer and option pricing
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Strange connections
■ Skulls and financial turbulence
■ Missile guidance and portfolio rebalancing
■ Heat transfer and option pricing
■ Italian toads and the absorption ratio
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Italian toads predict earthquakes
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Common toads appear to be able to sense an impending earthquake and will flee their colony days before the seismic activity strikes.
The evidence comes from a population of toads which left their breeding colony three days before an earthquake that struck L'Aquila in Italy in 2009.
How toads sensed the quake is unclear, but most breeding pairs and males fled.
They reacted despite the colony being 74km from the quake's epicentre, say biologists in the Journal of Zoology.
BBC Earth News
Italian toads predict earthquakes
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Absorption ratio and financial turbulence
1
1.2
1.4
1.6
1.8
2
2.2
2.4
‐90 ‐80 ‐70 ‐60 ‐50 ‐40 ‐30 ‐20 ‐10 0 0 0 0 10 20 30 40 50 60
AR Standardized Shift, Median
Time
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Returns to risk
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Absorption ratio and financial crises
Asian Financial Crisis
Russian & LTCM Crisis Housing Bubble
Lehman Default
0.55
0.65
0.75
0.85
Jan‐97
Jan‐98
Jan‐99
Jan‐00
Jan‐01
Jan‐02
Jan‐03
Jan‐04
Jan‐05
Jan‐06
Jan‐07
Jan‐08
Absorption RatioEvents
9/11Dot com
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Absorption ratio and contagion
Asian Financial Crisis
Russian & LTCM Crisis Housing Bubble
Lehman Default
0.55
0.65
0.75
0.85
Jan‐97
Jan‐98
Jan‐99
Jan‐00
Jan‐01
Jan‐02
Jan‐03
Jan‐04
Jan‐05
Jan‐06
Jan‐07
Jan‐08
Absorption RatioEvents
9/11Dot com
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Absorption ratio and contagion
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Summary
■ We present a method for inferring systemic risk from asset prices, which we call the absorption ratio.
■ The absorption ratio equals the fraction of a set of assets’ total variance explained or absorbed by a finite number of eigenvectors.
■ A high absorption ratio implies that markets are relatively compact.
■ Compact markets are fragile, because shocks propagate more quickly and broadly.
■ Most significant stock price drawdowns were preceded by spikes in the absorption ratio.
■ Stock returns are lower, on average, following spikes in the absorption ratio than in the wake of sharp declines.
■ Investors could have profited by varying equity exposure following significant changes in the absorption ratio.
■ The absorption ratio provided early warning signs of the U.S. housing bubble.
■ The absorption ratio anticipated episodes of financial turbulence.
■ Shifts in the global absorption ratio coincided with many global financial crises.
■ We suggest that regulators and investors consider using the absorption ratio as an early warning signal of potential asset price depreciation and financial turbulence.