©2010 CFA Institute Skulls, Financial Turbulence, and Risk ... · ©2010 CFA Institute Skulls, Financial Turbulence, and Risk Management Mark Kritzman, CFA, and Yuanzhen Li Based

30 www.cfapubs.org ©2010 CFA Institute

Financial Analysts JournalVolume 66 • Number 5©2010 CFA Institute

Skulls, Financial Turbulence, and Risk Management

Mark Kritzman, CFA, and Yuanzhen Li

Based on a methodology introduced in 1927 to analyze human skulls and later applied to turbulencein financial markets, this study shows how to use a statistically derived measure of financialturbulence to measure and manage risk and to improve investment performance.

ost investors look to their domesticequity markets as the main engine ofgrowth for their portfolios, and theysearch for other assets to diversify this

exposure. The typical investor considers only aver-age correlations, however, when measuring anasset’s diversification benefits, and average corre-lations tend to be misleading. For example, whenboth U.S. and non-U.S. equities produce returnsgreater than one standard deviation above theirmeans, their correlation equals –17 percent; whenboth markets produce returns more than one stan-dard deviation below their means, their correlationrises to +76 percent.1 These differences explain whymany investors who believed their portfolios werewell diversified suffered catastrophic losses duringthe financial crisis of 2007–2008. Rather than rely onaverage measures of risk to structure portfolios andmanage risk, investors should use conditional mea-sures that take into account the behavior of assetsduring turbulent subperiods.

Chow, Jacquier, Lowrey, and Kritzman (1999)introduced a mathematical measure of financialturbulence, which originally was developed byMahalanobis (1927, 1936) to analyze human skulls.We extended this research by investigating theempirical features of financial turbulence and bydemonstrating how this methodology can be usedto stress-test portfolios, to construct turbulence-resistant portfolios, and to scale exposure to risk toimprove performance.

Measuring Financial TurbulenceWe define financial turbulence as a condition inwhich asset prices, given their historical patterns ofbehavior, behave in an uncharacteristic fashion,including extreme price moves, decoupling of cor-

related assets, and convergence of uncorrelatedassets. Financial turbulence often coincides withexcessive risk aversion, illiquidity, and devaluationof risky assets.

The method we used to measure turbulencefirst appeared as the “Mahalanobis distance.”Mahalanobis (1927) used 7–15 characteristics of thehuman skull to analyze distances and resemblancesbetween various castes and tribes in India. Theskull characteristics used by Mahalanobis includedhead length, head breadth, nasal length, nasalbreadth, cephalic index, nasal index, and stature.The characteristics differed by scale and variability.That is, Mahalanobis might have considered a half-inch difference in nasal length between two groupsof skulls a significant difference whereas he consid-ered the same difference in head length to be insig-nificant. Mahalanobis normalized differences ineach characteristic by the characteristic’s standarddeviation and then squared and summed the nor-malized differences, thus generating one compos-ite distance measure that was invariant to thevariability of each dimension. He later proposed amore generalized statistical measure of distance,the Mahalanobis distance, which takes into accountnot only the standard deviations of individualdimensions but also the correlations betweendimensions (see Mahalanobis 1936). In his system,with n characteristics for measurements, each skullcan be represented as an n-dimensional vector. TheMahalanobis distance of an individual vector yfrom a sample of vectors with average � and cova-riance matrix � is defined as

(1)

whered = Mahalanobis distance from sample aver-

age (scalar)y = vector of multivariate measurements (1 ×

n vector)Mark Kritzman, CFA, and Yuanzhen Li are at WindhamCapital Management, LLC, Cambridge, Massachusetts.

M

d = −( ) −( )′−y y� ��

1,

September/October 2010 www.cfapubs.org 31


� = sample average vector (1 × n vector)� = sample covariance matrix (n × n matrix)The Mahalanobis distance can be used to mea-

sure the similarity of a particular skull to a sampleof skulls belonging to a group of known anthropo-logical origin. Consider the following hypotheticalexample. Suppose we use two features, head lengthand head breadth, to measure skulls. Each skull canthen be represented as a point in a two-dimensionalspace. Two groups of skulls belonging to distinctanthropological origins form two clusters in thetwo-dimensional space, as shown in Figure 1.

Suppose we compare a skull of unknown ori-gin, represented by the square in Figure 1, with thetwo groups and categorize it. In terms of Euclideandistance, it lies closer to the center of Group 2 thanto the center of Group 1. The Mahalanobis distance,however, would consider this skull more similar toGroup 1 because its characteristics are less unusualin light of the more inclusive scatter plot of Group1’s characteristics.

This example reveals that the Mahalanobis dis-tance is scale independent, which is also apparentfrom Equation 1. The characteristic deviations arescaled by the covariance matrix.

Chow et al. (1999) independently derived anearly identical formula to detect turbulence infinancial markets.2 By substituting asset returns

for skull characteristics, Chow et al. determinedthe statistical unusualness of a cross section ofreturns on the basis of their historical multivariatedistributions. With n assets, the returns for a par-ticular period (such as a month or a day) constitutean n-dimensional vector. Suppose a historical sam-ple of such return vectors has an average vector �and covariance matrix �. The statistical measureof financial turbulence, which we term the “turbu-lence index,” is formally defined as

(2)

wheredt = turbulence for a particular time period t

(scalar)yt = vector of asset returns for period t (1 × n

vector)� = sample average vector of historical

returns (1 × n vector)� = sample covariance matrix of historical

returns (n × n matrix)Turbulence, as we have just defined it, can be

calculated for any group of n return series a usermay choose. Figure 2 illustrates this statistical mea-sure of turbulence for a simple example with tworeturn series—stocks and bonds. Each point repre-sents the returns of stocks and bonds for a particular

dt = −( ) −( )′y yt

1

t� ��−

,

Figure 1. Scatter Plot of Hypothetical Human Skull Characteristics

Note: Points on each ellipse have identical Mahalanobis distances from the corresponding group center.

Group 1

1

2

4

12

4

Group 2


Financial Analysts Journal

period. The center of the ellipse represents the aver-age of the joint returns of stocks and bonds. Theellipse itself represents a tolerance boundary thatencloses a certain percentage—for example, 75percent—of the bivariate Gaussian distribution ofstock and bond returns. All points on the ellipsehave equal Mahalanobis distances from the center.This boundary also embodies the threshold thatseparates “turbulent” from “quiet” observations.Points inside the ellipse represent return combina-tions associated with quiet periods because theobservations are within 75 percent of the distribu-tion and are thus not particularly unusual. Theobservations outside the ellipse are statisticallyunusual and are thus likely to characterize turbulentperiods. Notice that some returns just outside themiddle of the ellipse are closer to the ellipse’s centerthan some returns within the ellipse at either end.This pattern illustrates the notion that some periodsqualify as turbulent not because one or more of thereturns were unusually high or low but, instead,because the returns moved in the opposite direction

in that period despite the fact that the assets arepositively correlated, as evidenced by the positiveslope of the scatter plot.

Others have addressed the problem of financialturbulence differently. Correlations conditioned onupside or downside market moves (Ang and Chen2002); time-varying volatility models, such asGARCH (generalized autoregressive conditionalheteroscedasticity) models (Bollerslev 1986);Markov regime-switching models (Ang and Bekaert2002); mixture models, such as jump diffusion (Dasand Uppal 2004); and implied volatility (Mayhew1995) have all been proposed as measures of finan-cial duress. The statistical measure of turbulencedefined in this article has two particular advantagesover the (perhaps) most commonly used indicatorof financial stress—namely, implied volatility. Inour measure, turbulence can be estimated for anyset of assets rather than only for assets with liquidoption markets. And our measure captures interac-tions among combinations of assets in addition tothe magnitude of the assets’ returns.

Figure 2. Scatter Plot of Hypothetical Stock and Bond Returns

Stocks

Bonds



Analysts may be tempted to think that thevolatility of an index comprising the assets used tomeasure turbulence captures the same informationas our measure. After all, such a volatility estimateincorporates both the volatility of the individualassets and their correlations with each other. Bysummarizing the data in an index, however, onesacrifices the higher-dimensional information cap-tured by the turbulence index.

This distinction is illustrated in Figure 3. Theloosely clustered circles in the scatter plot are thereturns of two assets with relatively high volatili-ties and a negative correlation. The tightly clus-tered squares are the returns of two assets withrelatively low volatilities and a positive correlation.It turns out that an index comprising the circleassets has the same volatility as an index compris-ing the square assets, yet the turbulence estimatesof each index’s assets are substantially different.

Thus far, we have characterized returns asbelonging to distinct turbulent and nonturbulent

regimes. Depending on the data or particular appli-cation, this distinction may be somewhat arbi-trary.3 We can just as well characterize returnsalong a continuum ranging from quiescence to tur-bulence. If we follow the latter approach, we findthat our mathematical measure of turbulence coin-cides closely with well-known turbulent events, asevidenced by Figure 4.

Figure 4 shows a turbulence index calculatedaccording to Equation 2, for which we usedmonthly returns of six asset-class indices: U.S.stocks, non-U.S. stocks, U.S. bonds, non-U.S.bonds, commodities, and U.S. real estate. The aver-age vector � and covariance matrix � in Equation2 were calculated for the full sample from January1980 to January 2009. Spikes in this index canclearly be seen to coincide with events known tohave been associated with financial turbulence.Also interesting to note, but certainly not surpris-ing, is that the 2007–08 financial crisis is by far themost turbulent episode of recent history.

Figure 3. Scatter Plot of Asset Pairs from Indices with Equal Variances

Asset 1 Returns

Asset 2 Returns

Data Set 1 Data Set 2



Empirical Features of TurbulenceTwo empirical features of turbulence are particu-larly interesting. First, returns to risk are substan-tially lower during turbulent periods than duringnonturbulent periods, irrespective of the source ofturbulence. For example, the recent financial crisisbegan with a downturn in housing prices, which ledto a sharp devaluation of mortgage derivatives.What was somewhat surprising to many investorsis that this crisis in the mortgage derivatives marketcoincided with substantial losses in carry strategies.The carry strategy calls for long positions in cur-rency forward contracts that sell at a discount com-bined with short positions in currency forwardcontracts that sell at a premium. Why should a crisisin the mortgage market lead to losses in a currencystrategy? As mortgage derivatives fell in value,many investors, especially hedge funds, wererequired to raise capital. But the mortgage deriva-tives market is relatively illiquid. To meet margincalls from their prime brokers and redemptions fromtheir investors, these hedge funds thus turned to themost liquid components of their portfolios—theircurrency positions and investments in publiclytraded securities. Figure 5 provides evidence thatreturns to risk are substantially lower during epi-sodes of financial turbulence, even though thecauses of turbulence may differ over time.

These differences in return suggest that know-ing whether the period ahead will be turbulentwould be helpful, which leads to the second empir-ical feature of turbulence. Financial turbulence ishighly persistent. It is similar to the turbulenceencountered in air travel. Weather turbulence mayarrive unexpectedly, but once it begins, we knowthat it will take time for the airplane to pass throughthe weather system or for the pilot to find asmoother altitude. The same process is true offinancial turbulence. Although we may not be ableto anticipate the initial onset of financial turbu-lence, once it begins, it usually continues for aperiod of weeks as the markets digest it and reactto the events causing the turbulence. Table 1 pro-vides evidence of the persistence of turbulence.

Table 1 shows the level of average daily turbu-lence following the initial arrival of a day that ismore turbulent than 10 percent of the most turbu-lent days in the sample for the next 5 days, 10 days,and 20 days, not including the initial turbulent day.The turbulence values are multivariate distancescalculated according to Equation 2 for full-sampleaverages and covariance matrices. These turbulencescores were normalized to 1 to facilitate compari-sons of the different sets of assets. The rightmostcolumn shows the normalized turbulence scoresseparating the 10 percent most turbulent days from

Figure 4. Historical Turbulence Index Calculated from Monthly Returns of Six Global Indices, 1980–2009

Note: Measured as of January each year.

Index

Stagflation

Black Monday

Gulf War

Russian Default

Technology Bubble

9/11

Global Financial Crisis

70

40

20

30

10

60

50

0

Quiet Turbulent

80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09



the rest of the sample. The 10 percent threshold foreach market was identified from the full sample ofreturns. The percentile rankings of the average dailyturbulence scores shown next to each score clearlyshow that markets tend to remain turbulent for upto a month or longer once turbulence begins.

ApplicationsThis statistical measure of finance turbulence hasseveral useful applications. Analysts can use it tostress-test portfolios more reliably than whenusing conventional methods. Analysts can use it tostructure portfolios that are relatively resilient to

Figure 5. Returns to Risk during Turbulent and Nonturbulent Periods

Notes: Turbulent periods were identified from USD-denominated daily values of the turbulence indexconstructed for global asset allocation (World Equities), U.S. sectors based on the small-capitalizationpremium (Small – Large) and the growth premium (Growth – Value), and developed country currencies(Carry Trade) for 4 January 1993–31 December 2008. Monthly turbulence index values for global assetallocation over the period January 1993–December 2008 were used for Hedge Funds. Raw turbulencevalues are multivariate distances based on a full-sample covariance matrix. The market returns are dailyreturns of the MSCI World Index (for World Equities), the Russell 2000 Index minus the S&P 500 Index(Small – Large), the Russell 1000 Growth Index minus the Russell 1000 Value Index (Growth – Value),and a naive carry strategy over the same time period. The monthly Hedge Fund returns are from theHFRI Fund of Funds Composite Index.

Source: State Street Associates.

Annualized Daily Return (%)

10

0

−10

−20

−30

Nonturbulent Turbulent

WorldEquities

Small −Large

Growth −Value

CarryTrade

HedgeFunds

Table 1. Persistence of TurbulenceNext 5 Days Next 10 Days Next 20 Days 10th

Percentile ThresholdMarket Level

Percentile Rank Level

Percentile Rank Level

Percentile Rank

Global assets 2.31 7 2.22 8 2.13 8 1.93U.S. assets 2.98 5 2.9 5 2.79 6 1.95U.S. sectors 3.12 5 3.04 6 2.87 6 2.03Currencies 2.08 8 1.93 9 1.8 11 1.83U.S. fixed income 4.05 4 3.85 5 3.6 5 2.12U.S. Treasury notes 3.19 5 3.13 6 2.96 6 2.00U.S. credit 4.17 4 4.09 4 3.69 4 1.61

Note: Time periods used for the calculations are as follows: January 1993–April 2009 for “Global assets,”August 1989–April 2009 for “U.S. assets,” January 1973–April 2009 for “U.S. sectors,” January 1975–April 2009 for “Currencies,” December 2000–April 2009 for “U.S. fixed income,” September 1998–April2009 for “U.S. Treasury notes,” and August 1998–April 2009 for “U.S. credit.”



turbulent episodes without significantly compro-mising their long-term growth prospects. Andfinally, analysts can use turbulence as a signal forscaling a strategy’s exposure to risk.

Stress-Testing Portfolios. Investors oftenuse value at risk (VaR) to measure a portfolio’sexposure to loss. VaR provides the largest loss aportfolio might experience at a certain level of con-fidence. The conventional approach for measuringVaR uses the full-sample covariance matrix to com-pute the portfolio’s standard deviation and consid-ers the probability distribution only at the end ofthe investment horizon. We can measure exposureto loss more reliably by estimating covariancesfrom the turbulent subperiods, when losses aremore likely to occur, and by accounting for interimlosses as well as losses that occur only at the con-clusion of the investment horizon.4

Table 2 shows three portfolios—conservativeto aggressive—together with assumptions for theirexpected returns and two estimates of standarddeviation. One estimate of standard deviation,“Full-sample risk,” is based on the full-samplecovariance matrix of monthly returns beginning inJanuary 1977 and ending in December 2006. Theother estimate of standard deviation, labeled “Tur-bulent risk,” is based on the covariance matrix fromthe turbulent subsample. Turbulence was calcu-lated according to Equation 2, in which each returnvector consisted of returns of the five asset-levelindices for a particular month, and average vector� and covariance matrix � were calculated frommonthly returns during the entire 30-year history.The threshold for identifying turbulent periodswas set at 75 percent, which means that roughly 25percent of the months fell within turbulent subpe-riods.5 Note how risk increases for each portfoliowhen turbulence risk is used.

Table 3 shows the VaR, given a 1 percent con-fidence level, for each portfolio as of December2006. The first column shows VaR estimated as ofthe end of a five-year horizon on the basis of stan-

dard deviations estimated from the full-samplecovariance matrix. The second column reports VaRwhen we used the standard deviations estimatedfrom the covariances that prevailed during the tur-bulent subsample and we took into account lossesthat could occur throughout the investment hori-zon. The third column shows the maximum lossesactually experienced by each of these portfoliosfrom inception during the financial crisis. Thefourth column reports the maximum drawdownsrealized by each of these portfolios.

If we consider the 2007–08 financial crisis as aonce-in-a-century event, Table 3 shows that theconventional approach to measuring exposure toloss badly underestimated the riskiness of theseportfolios. The turbulence-based approach, in con-trast, anticipated the exposure to loss of these port-folios much more accurately. To be clear, we pointout that the turbulence-based approach does notoffer a more reliable estimate of when an extremeevent will occur; rather, it gives a more reliableestimate of the consequences of such an event. Alsonote that turbulence is a relative measure. If theworld becomes more turbulent, for example, thethreshold for separating turbulent periods fromnonturbulent periods will rise.

Table 2. Efficient Portfolios, Expected Returns, and Two Estimates of Risk

Asset ClassConservative

PortfolioModeratePortfolio

AggressivePortfolio

U.S. stocks 22.86% 35.23% 48.15%Non-U.S. stocks 16.59 24.22 32.19U.S. bonds 49.95 32.81 14.89Real estate 3.85 2.59 1.28Commodities 6.75 5.16 3.49

Expected return 7.60% 8.37% 9.17%Full-sample risk 7.77 10.12 12.86Turbulent risk 10.68 13.68 17.33

Note: “Full-sample risk” was estimated from the full-samplecovariance matrix; “Turbulent risk” was estimated according tothe covariance matrix of the turbulent subsample.

Table 3. VaR and Realized Returns, End of 2006

Portfolio

VaR for Full Sample,

End of Horizon

VaR forTurbulent Sample,

within Horizon

Maximum Loss from Inception

(Jan/07–Sep/09)

MaximumDrawdown

(Jan/07–Sep/09)

Conservative 2.10% 26.20% 19.60% 25.80%Moderate 9.90 35.10 29.42 35.50Aggressive 18.70 45.00 38.96 45.30

Note: The horizon is five years.

Source: Windham Capital Management, LLC.



Building Turbulence-Resistant Portfolios.We have shown how analysts—by focusing on thebehavior of assets during periods of turbulence,especially how they interact with each other—canconstruct portfolios that are conditioned to betterwithstand turbulent events and, at the same time,perform relatively well in all market conditions.Analysts can also modify two methods of optimiza-tion, mean–variance optimization and full-scale opti-mization, to derive turbulence-resistant portfolios.

We modified mean–variance optimization byblending the differences between the realized tur-bulent returns and full-sample returns with equi-librium returns to estimate expected returns. Wealso blended the turbulent subsample covarianceswith the full-sample covariances in proportion totheir sample sizes.

We applied a modified version of full-scaleoptimization. Full-scale optimization uses a searchalgorithm to maximize expected utility for a givensample of returns.6 We modified this optimizationmethod by increasing the representation of the tur-bulent subsample returns beyond their empiricalfrequency. The details of these modified optimiza-tion methods are described in Appendix A.

To evaluate the two methods, we performed1,000 random trials of training and out-of-sampletesting. For each trial, we drew a random half fromthe historical sample to use as training data. Theother half was used as testing data. From the train-ing data, we identified a turbulent subsample bycalculating the turbulence index according to Equa-tion 2 and, subsequently, selecting the periods withthe highest turbulence index values (the highestquartile). Using the full training sample, we built anunconditioned optimal portfolio that did not taketurbulence into account. Using the turbulent sub-sample combined with some information from thefull training sample, we built a conditioned optimalportfolio that was expected to be more resistant toturbulence than an unconditioned portfolio. Wethen used the testing data to test the unconditionedand the conditioned optimal portfolios out of sam-ple. We performed two types of testing: one on thefull testing sample and the other on a turbulentsubsample within the testing sample.

Figure 6 compares the performance of the con-ditioned, turbulence-resistant, optimal portfoliowith that of the unconditioned optimal portfolio.The conditioned portfolios substantially outper-formed the unconditioned portfolios in the out-of-sample turbulent periods and only marginallyunderperformed the unconditioned portfolios, onaverage, in all market conditions.

The conditioned portfolios also outperformedthe unconditioned portfolios much more fre-quently in the out-of-sample turbulent periodsand outperformed almost as often in all marketconditions, on average, as shown in Table 4.

This evidence strongly suggests that byunderstanding the conditional behavior of assets,portfolio managers can construct turbulence-resistant portfolios without substantially compro-mising average performance.

Figure 6. Differences between Median Annualized Returns of Conditioned and Unconditioned Portfolios

Notes: The median was taken over 1,000 out-of-sample tests.“Subsample Gain” stands for outperformance (gain in medianreturn) of the conditioned portfolio during turbulent periods.“Full-Sample Cost” stands for underperformance (loss inmedian return) of the conditioned portfolio across quiet andturbulent conditions.

Table 4. Frequency of Conditioned Portfolio Outperforming Unconditioned Portfolio

Out-of-Sample Period

ModifiedMean–VarianceOptimization

ModifiedFull-Scale

Optimization

Turbulent periods 85% 86%

All market conditions 45 41

Median Annualized Return (%)

5

0

1

2

3

4

−1

Subsample Gain Full-Sample Cost

Modified Mean−Variance Optimization

Modified Full-ScaleOptimization



Scaling Exposure to Risk. The differentialperformance of risky strategies during turbulentand nonturbulent periods, together with the persis-tence of turbulence, raises the tantalizing prospectthat portfolio managers might be able to improveperformance by conditioning exposure to risk onthe degree of turbulence. Indeed, we show here thata simple scaling rule applied to the carry strategydoes significantly improve performance.7 We mea-sured the 30-day moving average of turbulenceeach day from the returns of the G–10 currenciesand recorded whether the level of turbulence thatday fell into the first, second, third, fourth, or fifthquintile of turbulence on the basis of a trailingthree-year window.8 We then weighted exposure tothe carry strategy in inverse proportion to turbu-lence as shown in Table 5. We assumed a one-daylag for implementation.

We applied the same scaling rule in usingother signals of market stress, including VIX(index of volatility on the S&P 500), swap spreads,and yield spreads. Table 6 shows the performanceof the unfiltered carry strategy as well as its filteredperformance. The evidence shows that reducingexposure to the carry strategy in proportion toturbulence substantially improves performanceand by a wider margin than using any other signalof market stress. Moreover, filtering on turbulenceeliminates most of the negative skewness of thecarry strategy.

ConclusionWe showed that a mathematically derived measureof financial turbulence, which measures the statis-tical unusualness of a set of returns given theirhistorical pattern of behavior, coincides with well-known episodes of market turbulence.

We also showed that this measure of financialturbulence is mathematically equivalent to theMahalanobis distance, first motivated by the studyof similarity in human skulls. The followinganalogy provided intuition of this equivalence.The Mahalanobis distance captures several dimen-sions of human skulls that set them apart from aperfect sphere. If one constructed a scatter plot ofthree return series emanating from a single multi-variate normal distribution, it would form a per-fect ellipsoid. This measure of financial turbulencecaptures deviations from this idealized three-dimensional scatter plot.

Financial turbulence has two intriguing fea-tures. First, returns to risk are substantially lowerduring turbulent periods than during nonturbulentperiods, and second, turbulence is persistent. Itmay arrive unexpectedly, but it does not immedi-ately subside. It typically continues for weeks asmarket participants digest it and react to its cause.

This measure of financial turbulence can beapplied in a variety of useful ways. Portfolio man-agers can use it to stress-test portfolios by estimat-ing VaR from the covariances that prevailed duringthe turbulent subsample. They can also constructportfolios that are relatively resistant to turbulenceby conditioning inputs to the portfolio constructionon the performance of assets during periods ofturbulence. Finally, they can enhance the perfor-mance of certain risky strategies by using turbu-lence as a filter for scaling exposure to risk.

We thank Erin Abouzaid, Timothy Adler, JordanAlexiev, Robin Greenwood, Sébastien Page, and DavidTurkington for helpful comments and assistance.

This article qualifies for 1 CE credit.

Table 5. Exposure to Carry Strategy Weighted in Inverse Proportion to Turbulence

TurbulenceQuintile Exposure

1st 20%2nd 403rd 604th 805th 100

Table 6. Filtered Carry Trade Performance

MeasureUnfiltered

CarryTurbulence

Index VIXFive-Year

Swap SpreadTED

SpreadYield

Spread

Return (%) 2.81 3.02 1.98 2.67 2.21 2.10Standard deviation (%) 5.55 3.29 3.95 4.04 3.51 3.07Information ratio 0.51 0.92 0.50 0.66 0.63 0.68Skewness –0.48 –0.15 –0.26 –0.25 –0.31 –0.56

Notes: The swap spread is based on U.S. dollars and is measured against the comparable U.S. Treasurysecurity. The bond yield spread represents the difference between 10-year and 2-year Treasury yields.“TED Spread” stands for Treasury–Eurodollar spread.



Appendix A. Two Methods for Building Turbulence-Resistant PortfoliosThe investment universe we used for building tur-bulence-resistant portfolios is as follows: U.S. equi-ties, non-U.S. equities (the MSCI EAFE Index), U.S.T-bonds, U.S. corporate bonds, U.S. Treasury Infla-tion-Protected Securities (TIPS), commodities,REITs, and cash. Our historical sample includesdata between 1973 and 2009. (We back-filled TIPSdata for the period between 1973 and 1996 on thebasis of U.S. Consumer Price Index returns andreturns of bond indices with similar durations.)

The conditioned and unconditioned portfolioswere constructed 1,000 times from a randomlyselected half of the training data and tested out ofsample on the remaining unused data. From thetraining sample, we identified a turbulent subsam-ple by comparing each period’s turbulence indexwith a threshold. The turbulence index was calcu-lated according to Equation 2, and the thresholdwas the 70th percentile of all the training-sampleturbulence index values.9

Using the full training sample, we derived anunconditioned optimal portfolio that was expectedto be optimal in all market conditions, quiet orturbulent. Using the turbulent subsample com-bined with information from the full training sam-ple, we derived a conditioned optimal portfolio forwithstanding market turbulence.

To evaluate the performance of the uncondi-tioned and the conditioned optimal portfolios, weused both the full testing sample and a turbulentsubsample, which was identified by placing athreshold on the testing-period turbulence index.The threshold was set equal to the one used fordissecting the training sample.

We performed this process 1,000 times. Figure6 and Table 4 in the body of this article demonstratethat the conditioned optimal portfolios derived byincorporating the turbulence index are indeed moreresistant to market turbulence yet perform reason-ably well, on average, in all market conditions.

Modified Mean–Variance Optimization1. We estimated the unconditioned expected

returns as equilibrium returns on the basis ofthe full training sample. To estimate equilib-rium returns, we used a portfolio consisting of60 percent U.S. equities, 30 percent T-bonds,and 10 percent U.S. corporate bonds as themarket portfolio.

2. We estimated the conditioned expectedreturns as follows:• We computed the average returns of the

turbulent subsample and compressedthem toward the return of the minimum-variance portfolio. The subsample, condi-tioned on its turbulence index being high,was often a relatively small sample. There-fore, expected returns estimated by takingthe subsample average were subject topossibly severe estimation errors, leadingto inferior out-of-sample performance.The small-sample problem can be miti-gated by compression, also known asshrinkage, where the sample average isblended with another estimator thatenforces more structure. Any commonreturn constant can function as a shrink-age target and improve out-of-samplerisk-adjusted performance of the resultingportfolio. In our experiments, we chose touse the return of the minimum-varianceportfolio, proposed by Jorion (1986), as theshrinkage target:

(A1)

where e = [1, 1, . . . , 1]� is the unit vector,� is the sample covariance matrix, and is the sample mean.The shrinkage strength (i.e., the weightapplied to the shrinkage target for linearlyblending it with the sample mean) wasdetermined by the following formula:

(A2)

where n is the number of dimensions (i.e.,number of assets in the portfolio) and T isthe sample size (i.e., number of observa-tions in the sample). The return estimateafter shrinkage was found by

(A3)

• We computed the average returns of thefull training sample and compressedthem toward the return of the minimum-variance portfolio. The shrinkage targetand shrinkage strength were calculated inthe same fashion as in the previous stepwith the use of Equations A1 and A2, butthe calculations were performed on thefull training sample instead of the turbu-lent subsample.

��

�

�

�m

e

e e=

−

−

1

1

,

�̂

wn

n T=

+

+ + − −( ) ( )−

2

2 � � � � �� m me e

1,

� � �s m= + −( )w w1 .



• We used ratios proportional to samplesizes to blend the differences between thecompressed subsample and full-samplereturns with equilibrium returns.

3. We used the full training sample to estimatethe unconditioned covariances by shrinkingthe sample covariance matrix toward a one-factor market model covariance matrix (Ledoitand Wolf 2003).

4. Using ratios proportional to sample sizes, weestimated the conditioned covariances byblending the turbulent subsample covariancematrix (which was also shrunk toward a one-factor market model covariance matrix) withthe full-sample covariance matrix.

5. We identified the unconditioned and condi-tioned optimal portfolios in a mean–varianceframework by using expected returns andcovariances estimated in Steps 1–4. We choserisk aversions in such a way that the uncondi-tioned and the conditioned optimal portfolioswould have similar risk profiles in the train-ing data, both with standard deviationsaround 10 percent.

6. We computed performances of unconditionedand conditioned optimal portfolios for the fulltesting sample and the turbulent subsample.The results are provided in the first portions ofTable 4 and Figure 6.

Modified Full-Scale Optimization1. We estimated the unconditioned expected

returns as equilibrium returns on the basis ofthe full training sample. To estimate equilib-rium returns, we used a portfolio consisting of60 percent U.S. equities, 30 percent T-bonds,and 10 percent U.S. corporate bonds as themarket portfolio.

2. We scaled the training sample by the uncondi-tioned expected returns estimated in Step 1 sothat the average asset returns would be equalto equilibrium returns.

3. We modified the training data by increasingthe representation of the turbulent subsample.The increased representation was achieved viaboot strapping: The subsample was sampled atthree times the frequency at which the fulltraining sample was sampled.

4. We identified the unconditioned optimal port-folio in a mean–variance framework by usingunconditioned expected returns estimated asequilibrium returns (Step 1) and uncondi-tioned covariances estimated for the unmodi-fied full training sample.

5. We used the modified training data to identifythe conditioned full-scale optimal portfolio.Full-scale optimization uses a search algorithmto identify a set of asset weights that maximizeexpected utility given a sample of returns anda utility function (see Cremers et al. 2005 for adetailed description). We used a kinked utilityfunction in which utility changes abruptly at athreshold return level. Utility was defined as alog-wealth function above the threshold returnand a linear function with a steep slope belowthe threshold return:

(A4)

where x stands for return, θ stands for thethreshold, and γ stands for the slope of utilitybelow the threshold. In our experiments, xwas annually compounded portfolio return, θwas –15 percent, and γ was 2.25. For a partic-ular set of weights, we calculated a portfolio’sexpected utility as the sum of its utility foreach period. We considered many sets of can-didate portfolio weights and identified the setthat yielded the highest expected utility as theset of optimal weights.

6. We computed performances of unconditionedand conditioned optimal portfolios on the fulltesting sample and its turbulent subsample.The results are provided in the right-hand por-tions of Table 4 and Figure 6.

Notes1. These correlations are based on monthly returns of the S&P

500 Index and MSCI World ex US Index from the periodstarting January 1970 and ending February 2008.

2. The only difference between the two measures is that theMahalanobis distance is the square root of the turbulencemeasure as defined in Equation 2. In many of the applica-tions in Chow et al. (1999), the authors used the ordinalrelationships of the turbulence measure, which are notaltered by squaring, to identify turbulent periods.

3. One approach for separating turbulent from nonturbulentregimes is to compute the absolute differences in correla-tions, cell by cell, between the 5 percent, 10 percent, 15percent, 20 percent, and so forth, most turbulent regimesand the nonturbulent remaining observations and then plotthe average of these differences as a function of sample sizeand look for obvious correlation shifts. Alternatively, onecan determine the threshold as a function of returns to risk.

Ux for x

x for x=

+( ) ≥−( ) + +( ) <

⎧⎨⎪

⎩⎪

ln ,

ln ,,

1

1

θγ θ θ θ



4. VaR that takes into account within-horizon losses is called“continuous VaR.” It is estimated numerically from firstpassage probabilities. See Kritzman and Rich (2002) for athorough description of continuous VaR.

5. The results are not particularly sensitive to the 75th per-centile threshold.

6. See Cremers, Kritzman, and Page (2005) for a detaileddescription of full-scale optimization.

7. We implemented the carry strategy by first estimating eachcurrency forward contract’s expected return as its annual-

ized monthly discount or premium. Then, we constructedan equally weighted portfolio of long and short exposures(long discount currencies and short premium currencies).We rebalanced the portfolio monthly.

8. The G–10 countries are Belgium, Canada, France, Germany,Italy, Japan, the Netherlands, Sweden, Switzerland, theUnited Kingdom, and the United States.

9. The results are not particularly sensitive to the 70th per-centile threshold.

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