Correlation surprise
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Abstract
Soon after Harry Markowitz published his landmark 1952 article on portfolio selection, the correlation coefficient assumed vital significance as a measure of diversification and an input to portfolio construction. However, investors typically overlook the potential for correlation patterns to help predict subsequent return and risk. Kritzman and Li (2010) introduced what is perhaps the first measure to capture the degree of multivariate asset price ‘unusualness’ through time. Their financial turbulence score spikes when asset prices ‘behave in an uncharacteristic fashion, including extreme price moves, decoupling of correlated assets, and convergence of uncorrelated assets.’ We extend Kritzman and Li’s study by disentangling the volatility and correlation components of turbulence to derive a measure of correlation surprise. We show how correlation surprise is orthogonal to volatility and present empirical evidence that it contains incremental forwardlooking information. On average, after controlling for volatility, we find that periods characterized by correlation surprise lead to higher risk and lower returns to risk premia than periods characterized by typical correlations. This result holds across many markets including US equities, European equities and foreign exchange. Our results corroborate the predictive capacity of turbulence and suggest that its decomposition may also prove fruitful in forecasting investment performance.
Keywords
correlation turbulence volatility dislocation risk managementINTRODUCTION
Soon after Harry Markowitz published his landmark 1952 article on portfolio selection, the correlation coefficient assumed vital significance as a measure of diversification and an input to portfolio construction. More recently, investors have come to recognize the importance of correlation to a wide variety of investment activities. Analysts have used the parameter to detect regime shifts, describe markets as ‘riskon/riskoff’ and justify the underperformance of stock pickers. Of course, to monitor the tangled web of relationships between assets can be daunting. To cover a universe of 10 assets, one must track 45 pairwise correlations. Kritzman and Li (2010) introduced what is perhaps the first measure to capture the degree of correlation ‘unusualness’ across a set of assets through time. Their financial turbulence score spikes when asset prices ‘behave in an uncharacteristic fashion, including extreme price moves, decoupling of correlated assets, and convergence of uncorrelated assets’. For at least two reasons, this framework is well suited to the purpose of quantifying correlation surprises. First, it summarizes in a single measure the collective unusualness of correlations across any universe of assets. Second, rather than identify whether correlations are high or low, it measures the degree to which interactions depart from their historical norms, whatever those may be.
In this article, we extend Kritzman and Li’s study by disentangling the volatility and correlation components of turbulence to derive a measure of correlation surprise. We also show how correlation surprise is orthogonal to volatility and present empirical evidence that it contains incremental forwardlooking information. On average, after controlling for volatility, we find that periods characterized by correlation surprise lead to higher risk and lower returns to risk premia than periods characterized by typical correlations. This result holds across many markets including US equities, European equities and foreign exchange. Our results corroborate the predictive capacity of turbulence and suggest that its decomposition may also prove fruitful in forecasting investment performance.
This article is organized as follows. First, we review the methodology behind Kritzman and Li’s financial turbulence measure, also known as the Mahalanobis distance, and review its empirical features. Next, we show how to decompose turbulence to isolate the contribution of correlation surprises. We then present empirical evidence that correlation surprises contain incremental information about future risk and return at both daily and monthly frequencies.
THE MAHALANOBIS DISTANCE AS A MEASURE OF FINANCIAL TURBULENCE
 d _{ t }

turbulence for a particular time period t (scalar)
 y _{ t }

vector of asset returns for period t (1 × n vector)
 μ

sample average vector of historical returns (1 × n vector)
 Σ

sample covariance matrix of historical returns (n × n matrix)
 n

number of assets in universe
This statistic, which can be thought of as a multivariate zscore, measures the statistical unusualness of a contemporaneous crosssection of asset returns relative to its historical distribution. It captures the extent to which the riskadjusted magnitudes of the returns differ from their historical means as well as the extent to which their interaction is inconsistent with the historical correlation matrix. Turbulence is different from crosssectional volatility, which measures the dispersion around the crosssectional mean but ignores the time series means.^{1} It is also different from the rolling volatility of a portfolio because it describes the unusualness of a particular day rather than the dispersion in returns over a period of time.

It tends to spike during recognizable periods of market stress that are characterized by heightened volatility and correlation breakdowns.

It is linked to investment performance; on average, returns to a wide variety risk premia are significantly lower during turbulent periods.^{2}

It is persistent. Turbulent episodes tend to cluster in time and do not subside immediately after they arise.
Taken together, the persistence of turbulence and its link with returns suggest that investors could enhance performance by derisking when turbulence first strikes. For example, Kritzman and Li show how to improve the performance of a foreign exchange carry strategy by reducing exposure when currency turbulence rises. In this article, we put the persistence of turbulence – and its relationship with subsequent return and risk – under the microscope by disentangling correlation surprises from magnitude surprises.
ISOLATING CORRELATION SURPRISES
The turbulence methodology is ideally suited to detecting periods when the comovement between assets differs from what we would expect based on historical correlations. For a given day, the turbulence score captures both the average degree of unusualness in individual asset returns (magnitude surprise) and the degree of unusualness in the interaction between each pair of assets (correlation surprise). To disentangle these two components and isolate the degree of correlation surprise, we first compute the magnitude surprise. Magnitude surprise is equal to the turbulence score, given in equation (1), where all offdiagonal elements in the covariance matrix are set to zero.^{3} This ‘correlationblind’ turbulence measure captures magnitude surprises, but ignores whether comovement is typical or atypical. Next, we divide the standard turbulence score – which includes correlation effects – by the magnitude surprise. This ratio is the correlation surprise: the unusualness of interactions on a particular day relative to history.
 1
Magnitude surprise: a ‘correlationblind’ turbulence score in which all offdiagonals in the covariance matrix are set to zero.
 2
Turbulence score: the degree of statistical unusualness across assets on a given day, as given in equation (1).
 3
Correlation surprise: the ratio of turbulence to magnitude surprise, using the above quantities (2) and (1), respectively.
The dashed ellipse in Figure 1 is the isoturbulence ellipse. All observations that fall along this ellipse have the same turbulence score. Its slant reflects the positive correlation between the two assets: in any given period, it is more likely that A and B move in the same direction than in opposite directions. In other words, when A and B move in the same direction, we require a larger return magnitude to produce the same degree of turbulence. In this example, period 2 is more turbulent than period 1 despite the fact that the magnitudes of the two observations are identical (each has a magnitude surprise of 1.0). Period 2 has a higher correlation surprise than period 1 because it reflects an outcome where the two assets, which are expected to move together, diverge.
HOW DOES CORRELATION SURPRISE DIFFER FROM OTHER MEASURES THAT INCORPORATE CORRELATION?

Rolling correlation: Investors sometimes monitor the rolling correlation between two assets to help them identify regime shifts. Whereas rolling correlation is a pairwise measure, correlation surprise can capture in a single parameter the unusualness of comovement across a large universe of assets.^{5} Indeed, the number of pairwise correlations quickly becomes unmanageable as the asset universe expands. Again, to monitor a 10asset universe, we would be required to estimate 45 rolling correlations; for a 100asset universe, the number increases to 4950.^{6}

Crosssectional volatility: Crosssectional volatility is different from correlation surprise because it fails to account for the degree to which a particular correlation outcome is typical or atypical. Put differently, crosssectional volatility spikes when asset returns diverge, regardless of whether their typical correlation is 10 per cent or 90 per cent. All else equal, a divergence is far more unusual in the latter case than in the former.

Multivariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models: This class of models extends the univariate GARCH framework to derive volatility forecasts based on lagged covariance terms as well as lagged variance terms. For example, a multivariate GARCH model might forecast the volatility of a twostock portfolio based on lagged innovations in stock A, lagged innovations in stock B and lagged covariance terms between stocks A and B.^{7} Engle (2002) has proposed a Dynamic Conditional Correlation model, which is a simple class of multivariate GARCH models. However, to our knowledge, none of the myriad multivariate GARCH specifications account for interactions between variance and covariance terms.^{8} And, as we will demonstrate in the next section, volatile episodes characterized by atypical correlations tend to be more persistent and severe than volatile episodes characterized by typical correlations. The decomposition of the Mahalanobis distance enables us to analyze the intertemporal relationship between correlation and magnitude surprises in a way that multivariate GARCH models do not.
DATA AND RESULTS
Time series data
US equities  European equities  Currencies  

Lookback window  10 years of daily returns^{a}  10 years of daily returns^{a}  3 years of daily returns 
Index start date  26 November 1975  26 November 1975  24 November 1977 
Index end date  30 September 2010  30 September 2010  30 September 2010 
Data source  S&P US Sectors^{b}  MSCI Europe Sectors^{b}  WMR 4 pm London Fix Rates 
vs the US dollar  
Constituents used  Consumer discretionary  Consumer discretionary  Australian dollar 
to build index  Consumer staples  Consumer staples  British pound 
Energy  Energy  Canadian dollar  
Financials  Financials  Euro^{c}  
Healthcare  Healthcare  Japanese yen  
Industrials  Industrials  New Zealand dollar  
Information technology  Information technology  Norwegian krone  
Materials  Materials  Swedish krona  
Telecommunications  Telecommunications  Swiss franc  
Utilities  Utilities  — 
It is apparent from visual inspection of Figure 3 that the correlation and magnitude surprise scores capture different information (which is collectively captured in the turbulence score). In fact, on a contemporaneous basis the two metrics are negatively correlated. What is intriguing is that despite this contemporaneous negative correlation, we find strong empirical evidence that high correlation surprise tends to precede higher risk and also lower returns. We will discuss these findings in more detail shortly. Before we shift our focus to empirics, let us first consider a more theoretical question: why would we expect unusual correlations to precede heightened volatility and lower returns? There are several plausible reasons. Investors who build correlation assumptions into their models – either explicitly or through intuition – may underperform when correlations deviate from their historical norms, inducing them to derisk. In addition, financial markets are not perfectly efficient and it takes some measure of time for information to propagate from one segment to another. For example, consumer stocks may react immediately to a particular news item, but financial stocks may not react until investors have analyzed relationships between these sectors, many of which are obscure and opaque. In this scenario, a shock would register as a correlation event before it registers as a marketwide volatility event. It is also possible that there is a behavioral explanation. Perhaps investors tend to derisk when markets are ‘acting weird’ and are difficult to understand.
Whatever the reason, we find convincing empirical evidence that there is a leadlag relationship between correlation surprises, volatility and returns. For a vivid example of how correlation surprise manifests in practice, consider the month of September 2008, one of the most turbulent months in financial history. A correlation surprise score above one occurred on nine days that month.^{9} The average magnitude surprise score on the days following these correlation surprise events was 8.7, and the average daily S&P 500 return was −219 basis points (bps). In contrast, the days with correlation surprise less than one were followed by an average magnitude surprise of 4.0 and an average S&P 500 return of +86 bps. The worst oneday market loss during the month occurred on 29 September, when the S&P 500 lost 879 bps from the previous day’s close, accompanied by a very large magnitude surprise of 33. Correlation surprise on the day of this drawdown was low (0.4), but on the previous day it was very high (1.9), in part because of a large divergence in the daily return of the Materials sector (−252 bps) and the Financials Sector (+309 bps), which over the preceding 10 years had experienced a positive correlation of 57 per cent.^{10} A similar divergence occurred on Friday, 12 September, with correlation surprise of 1.7 reflecting a gain for Materials (+323 bps) and a loss for Financials (−106 bps). The market dropped 471 bps on Monday, following news of Lehman’s default. Monday had low correlation surprise, and the market rallied 175 bps on Tuesday. However, Tuesday’s correlation surprise was 2.0. The market dropped another 471 bps on Wednesday. This example is clearly anecdotal and in choosing it we are guilty of selection bias. However, we find that the same intuition holds on average across three different markets from the 1970s through 2010. We will now turn to a more robust empirical analysis.
 1
Identify the 20 per cent of days in the historical sample with the highest magnitude surprise scores.^{11}
 2
Partition the sample from step 1 into two smaller subsamples: days with high correlation surprise (greater than one) and days with low correlation surprise (less than one).
 3
Measure, for the full sample identified in step 1 and its two subsamples identified in step 2, the subsequent volatility and performance of relevant investments and strategies.
Conditional average magnitude surprise on the day of the reading
US equities  European equities  Currencies  

Day of top 20% MS with CS ⩽1  4.6  4.0  3.6 
Day of top 20% MS (all observations)  3.9  4.0  3.5 
Day of top 20% MS with CS>1  2.6  2.8  3.0 
Difference in means (high CS minus low CS sample)  −2.0  −1.2  −0.6 
Percentage increase (high CS vs low CS sample)  −43%  −29%  −17% 
Conditional average magnitude surprise on the day after the reading
US Equities  European Equities  Currencies  

Day following top 20% MS with CS⩽1  2.1  2.1  1.5 
Day following top 20% MS (all observations)  2.3  2.4  1.7 
Day following top 20% MS with CS>1  2.6  3.0  2.1 
Difference in means (high CS minus low CS sample)  0.5  0.9  0.6 
Percentage increase (high CS vs low CS sample)  21%  41%  38% 
tstatistic of difference in means test  1.54  3.12  3.02 
pvalue of difference in means test  0.06  0.00  0.00 
Investment performance on the day after the reading^{a}
Average(annualized)  Std. deviation(annualized) (%)  Hit rate(% days positive)  Number of days in sample  

US equities, S&P 500^{b}  Full sample  7.2%  17.1  51.0  9092 
Day following top 20% MS with CS⩽1  27.7%  23.8  57.3  1211  
Day following top 20% MS (all observations)  16.2%  24.5  54.3  1818  
Day following top 20% MS with CS>1  −6.7%  25.9  48.1  607  
Difference (high CS minus low CS sample)  −34.5%  2.2  −9.2  —  
tstatistic of difference in means test  −1.73  —  —  —  
pvalue of difference in means test  0.04  —  —  —  
European equities, MSCI Europe^{b}  Full sample  11.5%  16.1  55.0  9092 
Day following top 20% MS with CS⩽1  7.0%  21.2  54.0  1297  
Day following top 20% MS (all observations)  5.8%  22.9  53.3  1819  
Day following top 20% MS with CS>1  2.5%  26.7  51.5  522  
Difference (high CS minus low CS sample)  −4.5%  5.5  −2.5  —  
tstatistic of difference in means test  −0.22  —  —  —  
pvalue of difference in means test  0.41  —  —  —  
Currencies, G10 carry strategy^{b}  Full sample  3.4%  6.1  54.9  5024 
Day following top 20% MS with CS⩽1  −1.2%  6.9  54.6  784  
Day following top 20% MS (all observations)  −3.1%  7.8  53.7  1063  
Day following top 20% MS with CS>1  −8.3%  10.0  51.3  279  
Difference (high CS minus low CS sample)  −7.0%  3.1  −3.3  —  
tstatistic of difference in means test  −0.69  —  —  —  
pvalue of difference in means test  0.26  —  —  — 
For all three investable indices, Table 4 shows that the average return is lower following extreme magnitude days characterized by high correlation surprise than following extreme magnitude days characterized by low correlation surprise. These results are statistically significant at the 95 per cent level for US equities. Interestingly, magnitude surprise alone is not particularly effective for partitioning the next day’s returns. In fact, for US equities, the 20 per cent of days with the largest magnitude surprise scores actually foretold higher returns, on average, than the remaining 80 per cent of the sample. Table 4 also reveals that the hit rate (per cent of positive days) for all three indices is lowest following days characterized by both high correlation surprise and high magnitude surprise.^{13} Finally, Table 4 shows that the standard deviation of returns is higher following days where both magnitude surprise and correlation surprise are high.^{14} This result holds for all three asset classes.
Short straddle performance following conditioned and unconditioned spikes in magnitude surprise
Average(annualized)  Std. deviation(annualized) (%)  Hit rate(% days positive)  Number of days in sample  

US equities, Short S&P 500 straddle^{a}  Full sample  13.3%  11.2  63.5  5371 
Day following top 20% MS with CS⩽1  31.8%  16.5  64.0  726  
Day following top 20% MS (all observations)  24.9%  17.0  64.4  1241  
Day following top 20% MS with CS>1  15.1%  17.7  64.9  515  
Difference (high CS minus low CS sample)  −16.7%  1.2  0.8  —  
tstatistic of difference in means test  −1.06  —  —  —  
pvalue of difference in means test  0.14  —  —  —  
European equities, Short DAX 30 straddle^{a}  Full sample  4.9%  14.1  59.8  4,869 
Day following top 20% MS with CS⩽1  15.1%  18.0  59.1  804  
Day following top 20% MS (all observations)  13.2%  20.8  59.2  1186  
Day following top 20% MS with CS>1  9.1%  25.7  59.4  382  
Difference (high CS minus low CS sample)  −6.1%  7.7  0.3  —  
tstatistic of difference in means test  −0.26  —  —  —  
pvalue of difference in means test  0.40  —  —  —  
Currencies, Short a basket of G10 straddles vs USD^{a}  Full sample  3.4%  6.1  54.9  3543 
Day following top 20% MS with CS⩽1  10.2%  6.4  59.5  603  
Day following top 20% MS (all observations)  6.2%  6.4  57.6  807  
Day following top 20% MS with CS>1  −5.5%  6.6  52.0  204  
Difference (high CS minus low CS sample)  −15.6%  0.2  −7.5  —  
tstatistic of difference in means test  −1.86  —  —  —  
pvalue of difference in means test  0.03  —  —  — 
Table 5 reveals that the average annualized return for all three strategies was lowest following the subsample characterized by high magnitude surprise and high correlation surprise simultaneously. The volatility was highest following this subsample. The hit rate (per cent of days with a positive return) was lowest following the joint occurrence in the currency market but highest in the two equity markets, where the signal appears to suffer from some false positives. Nonetheless, overall, these results suggest that a short volatility investor would be well advised to hedge his or her exposure when magnitude surprise and correlation surprise spike simultaneously.
HOW QUICKLY DOES THE SIGNAL DECAY?

average magnitude surprise for periods following high correlation surprise minus average magnitude surprise for periods following low correlation surprise, and

average return during periods following high correlation surprise minus average return during periods following low correlation surprise.
Time decay^{a}
After event, statistics for the following …  …1 day  … days 2–5  …days 6–10  …days 11–20  

US equities  Average magnitude surprise  0.6  0.1  0.1  0.2 
(0.06)  (0.29)  (0.23)  (0.02)  
S&P 500 return (%, ann)  −34.5%  4.4%  −1.5%  −0.9%  
(0.04)  (0.3)  (0.42)  (0.43)  
S&P 500 short straddle return (%, ann)  −16.7%  −1.8%  −1.7%  −1.1%  
(0.14)  (0.39)  (0.38)  (0.38)  
European equities  Average magnitude surprise  0.9  0.4  0.4  0.3 
(0)  (0.02)  (0.01)  (0.02)  
MSCI Europe return (%, ann)  −4.5%  −2.1%  2.1%  −8.6%  
(0.41)  (0.41)  (0.4)  (0.05)  
DAX 30 short straddle return (%, ann)  −6.1%  −3.0%  −5.8%  0.8%  
(0.4)  (0.38)  (0.24)  (0.44)  
Currencies  Average magnitude surprise  0.6  0.4  0.2  0.1 
(0)  (0)  (0.01)  (0.03)  
G10 carry return (%, ann)  −7.0%  −0.2%  −5.3%  −4.3%  
(0.26)  (0.49)  (0.11)  (0.05)  
G10 short straddles return (%, ann)  −15.6%  −7.1%  0.1%  −0.8%  
(0.03)  (0.07)  (0.49)  (0.38) 
We report average differentials for direct exposure to the equity indices and G10 carry strategy as well as to the simulated shortvolatility strategies.
To better understand Table 6, consider the example of US equities. The average magnitude surprise one day after a joint signal (a topquintile magnitude surprise where correlation surprise was greater than one) was 0.6 higher than the average magnitude surprise one day after a magnitudeonly signal (a topquintile magnitude surprise where the correlation surprise was less than one). Over the next four days (days 2 through 5) this difference falls to 0.1. Table 6 also indicates that, based on our analysis spanning 1975 through 2010, the average return of US equities the day after a joint signal was 34.4 per cent lower than the day after a magnitudeonly signal.
To summarize, we find a relationship between the joint signal and nextday volatility (as measured by magnitude surprise) across all three markets. We also find a relationship between the joint signal and nextday returns. Of the three asset classes we studied, both of these relationships appear to decay most quickly in the US equity market. For the short volatility strategies, the results are most persistent in the currency market. In the next section, we explore a way to derive correlation surprise signals that apply to longer holding periods.
LONGER HORIZON SIGNALS
Investment performance the month following conditioned and unconditioned signals
Average(annualized)  Std. deviation(annualized)(%)  Hit rate(% months positive)  Number of observations  

US equities,  Full sample  10.3%  15.6  61.4  417 
S&P 500  Month following high MS with low CS  10.8%  23.7  65.3  49 
Month following high MS (all observations)  7.2%  22.3  59.8  87  
Month following high MS with high CS  2.5%  20.6  52.6  38  
Difference (high CS minus low CS sample)  −8.4%  −3.1  −12.7  —  
tstatistic of difference in means test  −0.51  —  —  —  
pvalue of difference in means test  0.31  —  —  —  
European equities,  Full sample  10.0%  16.5  63.5  417 
MSCI Europe  Month following high MS with low CS  16.4%  19.9  65.2  46 
Month following high MS (all observations)  8.2%  20.6  60.0  85  
Month following high MS with high CS  −1.5%  21.4  53.8  39  
Difference (high CS minus low CS sample)  −17.9%  1.4  −11.4  —  
tstatistic of difference in means test  −1.15  —  —  —  
pvalue of difference in means test  0.13  —  —  —  
Currencies,  Full sample  3.6%  5.8  65.2  227 
G10 carry strategy  Month following high MS with low CS  −2.5%  8.5  46.7  30 
Month following high MS (all observations)  −4.1%  8.0  45.7  46  
Month following high MS with high CS  −7.1%  7.3  43.8  16  
Difference (high CS minus low CS sample)  −4.5%  −1.2  −2.9  —  
tstatistic of difference in means test  −0.55  —  —  —  
pvalue of difference in means test  0.29  —  —  — 
Short straddle performance the month following conditioned and unconditioned signals
Average (annualized)  Std. deviation (annualized)(%)  Hit rate (% months positive)  Number of observations  

US equities,  Full sample  12.6%  8.7  73.6  246 
Short S&P 500 straddle  Month following high MS with low CS  13.1%  13.8  77.1  35 
Month following high MS (all observations)  11.3%  12.2  72.5  69  
Month following high MS with high CS  9.5%  10.5  67.6  34  
Difference (high CS minus low CS sample)  −3.6%  −3.3  −9.5  —  
tstatistic of difference in means test  −0.35  —  —  —  
pvalue of difference in means test  0.36  —  —  —  
European equities,  Full sample  5.9%  12.7  62.8  223 
Short DAX 30 straddle  Month following high MS with low CS  30.6%  17.1  90.3  31 
Month following high MS (all observations)  16.4%  15.5  72.3  65  
Month following high MS with high CS  3.4%  13.1  55.9  34  
Difference (high CS minus low CS sample)  −27.1%  −4.0  −34.4  —  
tstatistic of difference in means test  −2.06  —  —  —  
pvalue of difference in means test  0.02  —  —  —  
Currencies,  Full sample  4.0%  4.3  60.6  160 
Short a basket of G10  Month following high MS with low CS  8.0%  6.1  69.6  23 
straddles vs USD  Month following high MS (all observations)  7.4%  5.9  69.4  36 
Month following high MS with high CS  6.6%  5.6  69.2  13  
Difference (high CS minus low CS sample)  −1.4%  −0.5  −0.3  —  
tstatistic of difference in means test  −0.20  —  —  —  
pvalue of difference in means test  0.42  —  —  — 
SUMMARY
We describe how to decompose the Mahalanobis distance (also known as financial turbulence) to derive a measure of correlation surprise across a set of assets. We show, both conceptually and empirically, that our correlation surprise measure is distinct from and incremental to magnitude surprise. Whereas magnitude surprise describes the extent to which asset returns are extreme visàvis their historical distributions, correlation surprise describes the extent to which their interaction is unusual given the historical correlation matrix. Finally, we construct correlation surprise series for three different asset classes and present evidence that the joint occurrence of high magnitude surprise and high correlation surprise foretells higher volatility and lower return than high magnitude surprise in isolation.
These findings have several concrete investment implications. A superior understanding of correlation surprises could enhance trade execution algorithms, which rely on correlation estimates to manage opportunity costs. In addition, risk managers could improve their volatility forecasts by incorporating correlation surprises into their models. And, perhaps most interestingly, portfolio managers may be able to enhance their performance by derisking when they observe correlation surprises coupled with heightened volatility.
Footnotes
 1.
Imagine a period where all assets experience a 50 per cent loss. Turbulence would spike but crosssectional volatility would be zero.
 2.
Examples include the equity risk premium, small cap premium, growth minus value, the FX carry trade and hedge fund returns.
 3.
It is also equal to the average squared zscore across the assets in our universe.
 4.
A wide variety of conventional risk measures incorporate both volatility and correlation but differ from correlation surprise in obvious ways. For example, the rolling standard deviation of a portfolio does not necessarily capture correlation surprises among its components. Consider a portfolio that is allocated to two stocks with a historical correlation of 0.75. Imagine that on a particular day, one stock realizes a 20 per cent gain while the other experiences a 20 per cent loss. The returns offset one another perfectly, the portfolio’s return for the day is 0 per cent, and its rolling volatility declines. In this case, volatility fails to capture an extremely unusual correlation outcome. Volatilitybased metrics (such as value at risk) and volatility forecasting models (such as ARCH and GARCH) fail to capture correlation surprises in similar fashion.
 5.
Another difference between rolling correlation and correlation surprise is that the former is, as its name suggests, a rolling measure whereas the latter is a normalized measure of unusualness over a discrete period. The relationship between rolling correlation and correlation surprise is analogous to the relationship between rolling volatility and a zscore.
 6.
A correlation matrix of n assets contains ((n*n)−n)/2 distinct correlation coefficients.
 7.
Typically, innovation terms are equal to squared residuals and covariance terms are equal to the product of residuals.
 8.
Multivariate GARCH models suffer from the ‘curse of dimensionality’ in that the number of covariance terms increases nonlinearly with the number of assets. The addition of a variancecovariance interaction term would render them even more unwieldy. Indeed, much of the existing multivariate GARCH literature is dedicated to adaptations that restrict the number of parameters. For a nice discussion on this subject, see Fengler and Herwartz (2008).
 9.
The days with a correlation surprise score above one were 2, 3, 5, 10, 12, 16, 19, 24 and 26 September.
 10.
The worst daily return following a correlation surprise reading less than one was 9 September’s loss of 341 basis points.
 11.
We compute magnitude surprise series using the components and lookback windows listed in Table 3. We precondition the experiments on the 20 per cent of days with the largest magnitude surprise to isolate the correlation surprise signals that are likely to contain the most information; days characterized by high correlation surprise but miniscule returns are most likely noise.
 12.
To compute the carry returns, we assume equal sized long or short forward positions in each G10 currency (rebalanced monthly) depending on the currencies’ interest rate differentials versus the US dollar.
 13.
The hit rate also provides information about false positives. For example, Table 4 indicates that, in the US equity market, we observed positive returns after a joint signal 48.1 per cent of the time.
 14.
It should be noted that the subsample means are significantly different from one another; hence, if we were to compute these volatilities around the full sample means we would expect to see even larger differences.
 15.
These results do not depend on monthend rebalancing. Although not reported, we obtained very similar results based on overlapping monthly holding periods.
Notes
Acknowledgements
The authors thank Megan Czasonis, Mark Kritzman and Chirag Patel for helpful comments and assistance.
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