Paper Principal Components - JPM 2011

8/11/2019 Paper Principal Components - JPM 2011

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VOLUME 37 NUMBER 4 SUMMER 2011www.iijpm.com

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Principal Components as aMeasure of Systemic RiskM ARK K RITZMAN , Y UANZHEN L I , S EBASTIEN PAGE ,AND R OBERTO R IGOBON


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PRINCIPAL C OMPONENTS AS A M EASURE OF S YSTEMIC R ISK SUMMER 2011

Principal Components as a

Measure of Systemic RiskM ARK KRITZMAN , YUANZHEN LI , SEBASTIEN PAGE ,AND ROBERTO R IGOBON

M ARK K RITZMAN

is the president and CEO

of Windham CapitalManagement, LLC, and asenior lecturer at the SloanSchool of Management,MIT, in Boston, [email protected]

Y UANZHEN L I

is a research associateat Windham CapitalManagement, LLC,in Boston, [email protected]

S EBASTIEN P AGE

is an executive vice presi-dent at PIMCO in New-port Beach, [email protected]

R OBERTO R IGOBON

is a professor at the SloanSchool of Management,MIT, in Boston, MA,and a research associateof the National Bureauof Economic Research in

Cambridge, [email protected]

The U.S. governments failure toprovide adequate oversight andprudent regulation of the finan-cial markets, together with exces-

sive risk taking by some financial institutions,pushed the world financial system to the brinkof systemic failure in 2008. As a consequenceof this near catastrophe, both regulators andinvestors have become keenly interested indeveloping tools for monitoring systemic risk.But this is easier said than done. Securitization,private transacting, complexity, and flexibleaccounting 1 prevent us from directly observing

the many explicit linkages of f inancial institu-tions. As an alternative, we introduce a measureof implied systemic risk called the absorptionratio, which equals the fraction of the totalvariance of a set of asset returns explained orabsorbed by a f ixed number of eigenvectors. 2 The absorption ratio captures the extent towhich markets are unified or tightly coupled.When markets are tightly coupled, they aremore fragile in the sense that negative shockspropagate more quickly and broadly than whenmarkets are loosely linked.

We offer persuasive evidence that theabsorption ratio effectively captures marketfragility. We show that

1. Most significant U.S. stock marketdrawdowns were preceded by spikes inthe absorption ratio.

2. Stock prices, on average, depreciatedsignificantly following spikes in theabsorption ratio and, on average, appre-ciated significantly in the wake of sharpdeclines in the absorption ratio.

3. The absorption ratio provided earlysigns of the consolidation of the U.S.housing market.

4. The absorption ratio systematically rosein advance of market turbulence.

5. Most global financial crises coincidedwith positive shifts in the absorptionratio.

6. The absorption ratio captures a largefraction of the information of morecomplex and computationally inten-sive structural models of financialcontagion.

We proceed as follows. We begin with aliterature review of systemic risk and relatedtopics. Then we provide a formal descrip-tion of the absorpt ion ratio. We next presenthistorical estimates of the absorption ratio fora variety of asset markets, and we show howit relates to asset prices, financial turbulence,the global financial crisis, and financial con-tagion. We conclude with a summary andsuggestions about how regulators and inves-tors might use the absorption ratio as an earlywarning signal of market stress.


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THE JOURNAL OF PORTFOLIO M ANAGEMENT SUMMER 2011

LITERATURE REVIEW

De Bandt and Hartmann [2000] provided anextensive review of the literature on systemic risk. Moststudies in this review focus on contagion and finan-

cial fragility, and the literature on contagion itselfis quite rich (see, for example, Pavlova and Rigobon[2008]). Recently, the IMFs Global Financial StabilityReport [2009] included a chapter on detecting systemicrisk in which it stated, The current crisis demonstratesthe need for tools to detect systemic risk (p. 111), andBeing able to identify systemic events at an early stageenhances policymakers ability to take necessary excep-tional steps to contain the crisis (p. 111). As a simplestarting point, the IMF report suggests monitoring con-ditional (stress) correlations.

In a related study, Billio et al. [2010] showedthat correlations increase during market crashes. Priorstudies have shown that exposure to different countryequity markets offers less diversification in down mar-kets than in up markets. 3 The same is true for globalindustry returns (Ferreira and Gama [2004]), individualstock returns (Ang, Chen, and Xing [2002], Ang andChen [2002], and Hong, Tu, and Zhou [2007]), hedgefund returns (Van Royen [2002a]), and internationalbond market returns (Cappiello, Engle, and Sheppard[2006]).

Both the IMFs Global Financial Stability Report [2009] and Bill io et al. [2010] suggested that an impor-tant symptom of systemic risk is the presence of suddenregime shifts. Investors have long recognized that eco-nomic conditions frequently undergo abrupt changes.The economy typically oscillates between

a steady, low-volatility state characterized by eco-nomic growth, and

a panic-driven, high-volatility state characterizedby economic contraction.

Evidence of such regimes has been documented in

short-term interest rates (Gray [1996], Ang and Bekaert[2002], and Smith [2002]), GDP or GNP (Hamilton[1989], Goodwin [1993], Luginbuhl and de Vos [1999],and Lam [2004]), inf lation (Kim [1993] and Kumar andOkimoto [2007]), and market turbulence (Chow et al.[1999], Kritzman, Lowry, and Van Royen [2001], andKritzman and Li [2010]).

Bill io et al. [2010] independently applied principalcomponents analysis to determine the extent to whichseveral f inancial industries became more unif ied acrosstwo separate regimes. They found that the percentage ofthe total variance of these industries explained by a single

factor increased from 77% during the 19942000 periodto 83% during the 20012008 period. We instead applyprincipal components analysis to several broad marketsand estimate on a rolling basis throughout history thefraction of total market variance explained by a finitenumber of factors. We call this measure the absorp-tion ratio. We also introduce a standardized measureof shifts in the absorption ratio, and we analyze howthese shifts relate to changes in asset prices and finan-cial turbulence. By applying a moving window in ourestimation process, we account for potential changes inthe risk factors over time. Because Billio et al. dividedhistory into only two periods, they assumed implicitlythat these periods are distinct regimes and are stationarywithin themselves.

Pukthuanthong and Roll [2009] independentlydeveloped a measure similar to the absorption ratioto measure global market integration. They regressedcountry returns on principal components estimatedfrom a prior period and averaged the R 2s to producea multiple R 2. They demonstrated that this measure ofintegration is more reliable than correlation and some-times contradictory to correlation. We make a similar

argument that the absorption ratio is a better measureof systemic than average correlation, but our main focusis to demonstrate that the absorption ratio is a reliableindicator of market fragility.

THE ABSORPTION RATIO

Consider a covariance matrix of asset returns esti-mated over a particular time period. The first eigen-vector is a linear combination of asset weights thatexplains the greatest fraction of the assets total vari-ance. The second eigenvector is a linear combinationof asset weights orthogonal to the f irst eigenvector thatexplains the greatest fraction of leftover asset variance;that is, variance not yet explained or absorbed by thefirst eigenvector. The third eigenvector and beyond areidentified the same way. They absorb the greatest f rac-tion of leftover variance and are orthogonal to precedingeigenvectors.


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It is perhaps more intuitive to visualizeeigenvectors. The left panel of Exhibit 1shows a three-dimensional scatter plot ofasset returns with a vector piercing theobservations. Each observation is the inter-

section of returns of three assets for a givenperiod, which might be a day, a month,or a year. This vector represents a linearcombination of the assets and is a potentialeigenvector. The right panel of Exhibit 1shows the same scatter plot of asset returnsbut with a different vector piercing theobservations.

Of all the potential vectors piercingthe scatter plot, we determine the firsteigenvector by perpendicularly projectingthe observations onto each potential eigen-vector, shown by Exhibit 2.

The first eigenvector is the one withthe greatest variance of the projected obser-vations, as shown in Exhibit 3. 4

In order to identify the second eigen-vector, we first consider a plane passingthrough the scatter plot that is orthogonal tothe f irst eigenvector, shown by Exhibit 4.

The second eigenvector must lie onthis orthogonal plane, thereby limiting oursearch. It is the vector that yields the second

highest variance of projected observations.We find the third eigenvector in the samefashion. It is the vector that yields the thirdgreatest variance and is orthogonal to thefirst two vectors. These three eigenvectorstogether explain the total variance of theassets.5

We may or may not be able to asso-ciate these eigenvectors with observable economic orfinancial variables. In some cases the asset weights ofthe eigenvector may suggest an obvious factor. Forexample, if we were to observe short exposures to theairline industry and other industries that consume fueland assume long exposures to the oil industry and otherindustries that profit from r ising oil prices, we mightconclude that this eigenvector is a proxy for the price ofoil. Alternatively, an eigenvector may ref lect a combi-nation of several inf luences that came together in a par-ticular way that is unique to the chosen sample of assets,in which case the factor may not be definable other

than as a statist ical art ifact. Moreover, the compositionof eigenvectors may not persist through time. Sourcesof risk are likely to change from period to period.

In some applications this ar tif iciality or nonsta-

tionarity would be problematic. If our intent were toconstruct portfolios that were sensitive to a particularsource of risk, then we would like to be able to iden-tify it and have some confidence of its persistence asan important risk factor. But here our interest is not tointerpret sources of risk. Instead, we seek to measurethe extent to which sources of risk are becoming moreor less compact.

E X H I B I T 1Three-Dimensional Scatter Plots of Asset Returns

E X H I B I T 2Projection of Observations onto Vectors


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The particular measure we use as an indicator ofsystemic risk is the absorption ratio, which we defineas the fraction of the total variance of a set of assetsexplained or absorbed by a f inite set of eigenvectors, asshown in the following equation:

i N

Ei

j N

Aj

=

=

=

12

12

(1)

where,AR = absorption ratioN = number of assetsn = number of eigenvectors in numer-

ator of absorption ratio

Ei 2 = variance of the i th eigenvector Aj

2 = variance of the j th asset

A high value for the absorption ratiocorresponds to a high level of systemicrisk because it implies that the sources ofrisk are more unified. A low absorptionratio indicates less systemic risk because itimplies that the sources of risk are moredisparate. We should not expect highsystemic risk to necessarily lead to assetdepreciation or financial turbulence. It issimply an indication of market fragilityin the sense that a shock is more likelyto propagate quickly and broadly whensources of risk are t ightly coupled.

The Absorption Ratio versusAverage Correlation

One might suspect that the average correlation ofthe assets used to estimate the absorption ratio providesthe same indication of market unity, but it does not.

Unlike the absorption ratio, the average correlation failsto account for the relevance of the asset correlations thatmake up the average.

Exhibit 5 shows an increase in the correlation oftwo hypothetical assets with relatively high volatility anda decrease in the correlation of two hypothetical assetswith relatively low volatility. It turns out that although theaverage correlation decreases slightly from 0.36 to 0.32,the absorption ratio increases sharply from 0.55 to 0.80.The key distinction is that the absorption ratio accountsfor the relative importance of each assets contribution tosystemic risk, whereas the average correlation does not. 6

EMPIRICAL ANALYSIS: THE ABSORPTIONRATIO

We present empirical evidence that the absorp-tion ratio effectively captures market fragility. We showthat stock prices, on average, depreciate significantlyfollowing spikes in the absorption ratio. We demonstrate

E X H I B I T 3First Eigenvector

E X H I B I T 4Second Eigenvector


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steep decline in stock prices, and that although stockprices have partial ly recovered as of the second quarterof 2010, the absorption ratio has fallen only slightly. Thiscontinued high level for the absorption ratio, while per-haps worrisome, does not necessarily foretell a renewed

sell-of f in stocks. It does suggest, however, that the U.S.stock market remains extremely fragile and thereforehighly vulnerable to negative shocks.

A casual review of the absorption ratio alongsidestock prices suggests a coincident relationship, whichperhaps casts doubt on the notion that the absorptionratio might be useful as a signal of impending trouble.Exhibit 7 sheds some light on this question. It shows thefraction of signif icant drawdowns preceded by a spike inthe absorption ratio. We first compute the moving averageof the absorption ratio over 15 days and subtract it fromthe moving average of the absorption rat io over one year.We then divide this difference by the standard deviationof the one-year absorption ratio, as shown. We call thismeasure the standardized shift in the absorption ratio,

AR = ( AR 15 Day AR 1 Year )/ (2)

where,AR = standardized shift in absorption ratio

AR 15 Day = 15-day moving average of absorption ratio AR 1 Year = one-year moving average of absorption ratio = standard deviation of one-year absorption ratio

As Exhibit 7 shows, all of the 1% worst monthlydrawdowns were preceded by a one-standard-deviationspike in the absorption ratio, and a very high percentageof other significant drawdowns occurred after theabsorption ratio spiked.

We should not conclude from Exhibit 7 that aspike in the absorption ratio reliably leads to a signifi-cant drawdown in stock prices. In many instances, stocksperformed well following a spike in the absorption ratio.We would be correct to conclude, though, that a spike

in the absorption ratio is a near necessary condition fora signif icant drawdown, just not a sufficient condition.Again, a high absorption ratio is merely an indicationof market fragility.

Even though a spike in the absorption ratio doesnot always lead to a major drawdown in stock pr ices, onaverage, stocks perform much worse following spikes inthe absorption ratio than they do in the wake of a sharpdrop in the absorption ratio. Exhibit 8 shows the averageannualized one-day, one-week, and one-month returnsfollowing a one-standard-deviation increase or decreasein the 15-day absorption ratio relative to the one-yearabsorption ratio.

Exhibit 8 offers compelling evidence that signif i-cant increases in the absorption ratio are followed bysignificant stock market losses, on average, while sig-nif icant decreases in the absorption ration are followedby significant gains. This differential performance sug-gests that it might be profitable to reduce stock exposuresubsequent to an increase in the absorption ratio and toraise exposure to stocks after the absorption ratio falls,which is what we next test.

Exhibit 9 shows the performance of a dynamic

trading strategy in which the stock exposure of an other-wise equal-weighted portfolio of stocks and governmentbonds is raised to 100% following a one-standard-devia-tion decrease in the 15-day absorption ratio relative to theone-year absorption ratio and reduced to 0% followinga one-standard-deviation increase. These rules are sum-

marized as follows:

These rules are applied daily with a one-daylag following the signal for the period January 1,1998, through January 31, 2010, using the MSCIUSA stock index and Treasury bonds. 9 Exhibit 9shows that these rules triggered only 1.72 tradesper year on average, which should not be sur-prising given that one-standard-deviation eventsoccur infrequently. Nonetheless, these infrequent

E X H I B I T 7Absorption Ratio and Drawdowns

Notes: One standard deviation, 15 days/1 year; 1/1/1998 through 5/10/2010.


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shifts improved return by more than 4.5% annuallywhile increasing risk by only 0.61%, thereby raising thereturnrisk ratio from 0.47 to 0.83.

Although most investors might be reluctant toshift entirely in or out of stocks given a single signal, thisexperiment does offer persuasive evidence of the potentialvalue of the absorption ratio as a market-timing signal.Exhibit 10 reveals that the absorption ratio would havekept investors out of stocks during much of the dot-commeltdown as well as the global financial crisis. It alsoreveals that the absorption ratio produced some false posi-tives, but not enough to offset its net beneficial effect.

Although daily MSCI industry data extend backonly to 1995, Datastream offers less granular industrydata which allows us to include the 1987 stock marketcrash in our analysis. Exhibit 11 shows how well theabsorption ratio served as an early warning signal of the

great stock market crashes of the past three decades.If investors withdrew from stocks within one dayof a one-standard-deviation spike in the standard-ized shift of the absorption ratio, they would haveavoided all of the losses associated with the 1987

crash and the global financial crisis, and nearly70% of the losses associated with the dot-commeltdown.

This trading rule appears to improve perfor-mance in other stock markets as well. Exhibits 12and 13 show the absorption ratio estimated fromstock returns in Canada, Germany, Japan, and theU.K. and that applied as a timing signal in thosemarkets yielded similar improvement in total returnas well as risk-adjusted return. 10

The Absorption Ratio and the HousingBubble

Former Federal Reserve Chairman AlanGreenspan stated that although regional housingmarkets often showed signs of unsustainable specu-lation resulting in local housing bubbles, he didnot expect a national U.S. housing bubble. 11 Hadthe Fed examined the absorption ratio of the U.S.housing market, it might have learned that regionalhousing markets were becoming more and moreunified as early as 1998, setting the stage for anational housing bubble.

Exhibit 14 shows the absorption ratio esti-mated from 14 metropolitan housing markets in the

United States, along with an index of the CaseShiller10-City National Composite Index. 12

Exhibit 14 reveals that the housing market absorp-tion ratio experienced a signif icant step up from 47.60%in October 1996 to 67.04% in March 1998, just as thenational housing bubble got underway. It reacheda historic peak of 72.76% in September 1998 and thenanother peak in July 2004 at 85.63% as the housingbubble continued to inf late. It again reached a historicpeak of 89.07% in December 2006 within a few monthsof the housing bubble peak. Then as the housing bubbleburst, the absorption ratio climbed sharply, reaching anall-time high of 94.22% in March 2008. As housingprices stabilized and recovered slightly in 2009, theabsorption ratio began to retreat modestly.

E X H I B I T 8Absorption Ratio and Subsequent Returns

E X H I B I T 9Absorption Ratio as a Market-Timing Signal

Notes: One standard deviation, 15 days/1 year; 1/1/1998 through 5/10/2010.

Notes: 1/1/1998 through 5/10/2010.


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It is quite clear from Exhibit 14 that systemicrisk in the national housing market increased signifi-cantly leading up to the beginning stages of the housingbubble and as the bubble inf lated, and that it increasedeven further after the bubble burst and housing pricestumbled.

The Absorption Ratio andFinancial Turbulence

Next, we turn to the relationship between systemicrisk and financial turbulence. We define financial tur-bulence as a condition in which asset prices behave in anuncharacteristic fashion given their historical pattern ofbehavior, including extreme price moves, decoupling ofcorrelated assets, and convergence of uncorrelated assets.We measure financial turbulence as

t

1(3)

where,

d t = turbulence for a particular t ime period t yt = vector of asset returns for period t

= sample average vector of historical returns = sample covariance matrix of historical returns

We interpret this formula as follows. By subtractingthe historical average from each assets return, we cap-ture the extent to which one or more of the returns wasunusually high or low. In multiplying these differencesby the inverse of the covariance matrix of returns, we

E X H I B I T 1 0Absorption Ratio Stock Exposure

E X H I B I T 1 1The Absorption Ratio as an Early Warning Signal ofCrashes


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both divide by variance, which makes the measure scale

independent, and we capture the interaction of the assets.By post-multiplying by the transpose of the differencesbetween the asset returns and their averages, we convertthis measure from a vector to a single number. Previousresearch has shown that this statistical characterization offinancial turbulence is highly coincident with events infinancial history widely regarded as turbulent, as shownin Exhibit 15. 13

In order to measure the connection between sys-temic risk and financial turbulence, we first identify the10% most turbulent 30-day periods, based on averagedaily turbulence, of the MSCI USA stock index cov-ering the period from January 1, 1997, through January

10, 2010. We then synchronize all of these turbulentevents and observe changes in the 15-day absorptionratio relative to the one-year absorption ratio estimatedfrom industry returns as described earlier, leading up toand following the turbulent events. Exhibit 16 showsthe results of this event study.

Prior to turbulent events in the stock market,the median of the standardized shift in the absorptionratio increased beginning about 40 days in advance ofthe event, and continued to rise throughout the tur-bulent periods. It then fell following the conclusion ofthe turbulent episodes. This evidence suggests that theabsorption ratio is an effective precursor of both theinception and conclusion of turbulent episodes, whichcould prove to be quite valuable. In addition to per-sistence, another feature of turbulence is that returnsto risk are much lower during turbulent periods thannonturbulent periods. 14

The Absorption Ratio and GlobalFinancial Crises

The previous subsections examined the perfor-

mance of the absorption ratio in the domestic economy.We now analyze its implications in the global economy.To calculate the global absorption ratio, we collecteddaily stock market returns for 42 countries (and someregional indices) from February 1995 to December2009.

Exhibit 17 shows that the global absorption ratioshifts within a range of 65% to 85%. Also, it shows thatthe global absorption ratio increased in October 1997(Hong Kongs speculative attack after the Asian f inancialcrises) and August 1998 (the Russian and LTCM col-lapses), which were two of the most significant emergingmarket crises in the last 20 years (the other being theTequila Crisis in 1994, which is outside our sample).

During the recovery in emerging markets (1999 2001), the global absorption ratio decreased. It increasedfollowing the boom that came after the severe declines ininterest rates that took place after September 11, 2001; theaccounting standard scandals; and the dot-com collapsein 2001. Finally, the last significant increase took place

E X H I B I T 1 2Global Performance of Absorption Ratio

E X H I B I T 1 3Global Performance of Absorption Ratio


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E X H I B I T 1 5Financial Turbulence Index

Note: The index is calculated using returns of global stocks, bonds, real estate, and commodities.

E X H I B I T 1 4Absorption Ratio and the National Housing Bubble


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absorption ratio of the U.S. housing market pro-vided early signs of the emergence of a nationalhousing bubble, long before the Fed recognizedthis fact. We have presented evidence showing thatincreases in the absorption ratio anticipate by more

than a month subsequent episodes of financial tur-bulence. Finally, we have shown that variation in theabsorption ratio coincided with many global finan-cial crises and tracked other more complex measuresof financial contagion. In short, the absorption ratioappears to serve as an extremely effective measureof systemic risk in f inancial markets.

A P P E N D I X

THE HERFINDAHL INDEX AS A

MEASURE OF IMPLIED SYSTEMIC RISKAs alternative to our methodology of estimating

the absorption ratio as the fraction of total varianceexplained by a subset of eigenvectors, we consideredusing the Herfindahl index to infer systemic risk. Specifi-cally, we squared the fraction of total var iance explained byeach eigenvector and took the square root of the sum of thesesquared values,

AR i j

N = )2 2 (A-1)

where,AR H = absorption ratio estimated from the Herfindahl

indexN = number of assets

Ei 2 = variance of the i th eigenvector, sometimes called

eigenportfolio

Aj 2 = variance of the j th asset

Exhibit A1 clearly shows that our approach leads tobetter results than using a Herfindahl index to estimate sys-temic risk. Our conjecture is that the less important eigen-vectors, which are included in the Herfindahl index, arerelatively unstable and introduce noise to the estimate ofsystemic risk.

ENDNOTES

We thank Timothy Adler, Robin Greenwood, RichardRoll, and participants of seminars at the European Quantita-tive Forum, the International Monetary Fund, JOIM , MIT

Sloan School, PIMCO, QWAFAFEW, and State Street GlobalMarkets in Cambridge, MA, New York, and London forhelpful comments.

1This is a euphemism for accounting practices such asLehman Brothers use of swaps to conceal $50 bil lion of debtshortly before its demise.

2We could instead calculate the number of eigen-vectors required to explain a fixed percentage of variance,but for no particular reason we chose to fix the number of

eigenvectors.3See, for example, Ang and Bekaert [2002], Kritzman,

Lowry, and Van Royen [2001], and Baele [2003] on regimeshifts; Van Royen [2002b] and Hyde, Bredin, and Nguyen[2007] on financial contagion; and Longin and Solnik [2001],Butler and Joaquin [2002], Campbell, Koedijk, and Kofman[2002], Cappiello, Engle, and Sheppard [2006], and Hyde,Bredin, and Nugyen [2007] on correlation asymmetr ies.

4There are a variety of techniques to identify eigen-vectors. We can use matrix a lgebra to identify eigenvectorsgiven a small set of observations. For larger datasets, it is moreeff icient to resort to numerical procedures.

5The number of eigenvectors never exceeds the numberof assets; however, total variation could be explained by fewereigenvectors than assets to the extent assets are redundant. Tobe precise, the total number of eigenvectors equals the rankof the covariance matrix.

6Pukthuanthong and Roll [2009] provided a formalanalysis of the distinction between average correlation andmeasures of integration based on principal components.

E X H I B I T A 1Herfindahl Index vs. Sum of Subset


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7In principle, we should condition the number of eigen-vectors on the rank of the covariance. Because the covariancematrices in our analysis are nearly ful l rank, we are effectivelydoing this.

8As an alternative to measuring market unity by thefraction of tota l variance explained by a subset of eigenvec-tors, we could construct a Herfindahl index, by which wesquare the fraction of variance explained by each of the eigen-vectors and sum the squared values. Our experiments showthat this latter approach is significantly less informative thanour method. We present this evidence in the appendix.

9We require two years to estimate the covariance matrixand eigenvectors and another year to estimate the standarddeviation of the one-year moving average; hence, our databegin January 1, 1995.

10As with the U.S. stock market, we set the number ofeigenvectors at about 1/5th the number of industries.

11He has since rejected this view.12The returns are computed from the CaseShiller Sea-

sonally Adjusted Indices that go back to 1987. The metropolitanareas include Los Angeles, San Diego, San Francisco, Denver,Washington, DC, Miami, Tampa, Chicago, Boston, Char-lotte, Las Vegas, New York, Cleveland, and Portland. In thiscase, the covariance matrix is based on five years of monthlyreturns beginning January 1987 and ending March 2010, andthe absorption ratio is based on the first three eigenvectors,roughly 1/5th of the number of assets. The half-l ife for expo-nentially weighted variances is set to two-and-a-half years.

13For more about this measure of f inancial turbulence,including its derivation, empirical properties, and usefulness,see Kritzman and Li [2010].

14See Kritzman and Li [2010].15Because both series are quite noisy, we smooth them

by taking a 60-day moving average and then compute the cor-relation of the percentage changes of the smoothed series.

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