17jun10 measure of systemic risk m kritzman

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1© 2010 Windham Capital Management, LLC

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



■ 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



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.



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.



Three-dimensional scatter plot of asset returns



First eigenvector



Second eigenvector



‐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



‐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



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



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.



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%



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



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




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



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



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



Absorption ratio as a market timing signal

Absorption Ratio Stocks/Bonds

-1σ <= ∆AR <= + 1σ 50/50

∆AR > + 1σ 0/100

∆AR < - 1σ 100/0




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




$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



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


StockExposure

Stock Exposure




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



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


Return/Risk

AR Risk Adjusted Performance

Static

Dynamic



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



Herfindal Index versus sum

0%

2%

4%

6%

8%

10%

12%

14%


Total R

eturn

AR Global Performance

Static

Dynamic, Herfindal

Dynamic, Sum



Herfindal Index versus sum

0.00

0.20

0.40

0.60

0.80

1.00

1.20


Return/Risk

AR Risk Adjusted Performance

Static

Dynamic, Herfindal

Dynamic, Sum



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



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:




■ 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.



Financial turbulence throughout history

Global Financial Crisis

9/11

Tech Bubble

Russian Default

Gulf War

Black Monday

Stagflation



Strange connections

■ Skulls and financial turbulence

■ Missile guidance and portfolio rebalancing

■ Heat transfer and option pricing



Strange connections

■ Skulls and financial turbulence

■ Missile guidance and portfolio rebalancing

■ Heat transfer and option pricing

■ Italian toads and the absorption ratio



Italian toads predict earthquakes



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




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



Returns to risk



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



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



Absorption ratio and contagion



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.

17jun10 measure of systemic risk m kritzman

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