Harry. M Kat

English

Research Papers in Economics

 
 
 
 
       
 
 
 
THE BUSINESS SCHOOL 
FOR FINANCIAL MARKETS 
The University of Reading 
 
 
 
 
 
The Dangers of Using Correlation to Measure Dependence 
 
 
ISMA Centre Discussion Papers in Finance 2002-23 
This version: 9 October 2002 
 
 
 
 
Harry M. Kat 
Associate Professor in Finance, 
ISMA Centre, University of Reading, UK 
Whiteknights Park 
Reading RG6 6BA 
United Kingdom 
Tel. +44-118-9316428 
E-mail: h.kat@ismacentre.rdg.ac.uk 
From 2003: harry@airc.info or harry@harrykat.com  
 
 
Comments by Mark Salmon are gratefully acknowledged. As always, all errors are mine. 
 
Copyright ©  2002 Harry M. Kat. All rights reserved. 
 
The University of Reading • ISMA Centre • Whiteknights • PO Box 242 • Reading RG6 6BA  • UK 
Tel: +44 (0)118 931 8239  • Fax: +44 (0)118 931 4741 
Email: research@ismacentre.rdg.ac.uk  • Web: www.ismacentre.rdg.ac.uk 
 
Director: Professor Brian Scott-Quinn, ISMA Chair in Investment Banking 
The ISMA Centre is supported by the International Securities Market Association   
   
   
  ISMA Centre Discussion Papers in Finance 2002-23 
 
Copyright 2002 Harry M. Kat 
 
1 
 
I. INTRODUCTION 
Correlation is without doubt the single most important parameter in modern portfolio 
theory, where it is used to measure the dependence between the returns on different 
assets or asset classes. The rule is simple: low correlation makes for good 
diversification and highly correlated assets or asset classes are to be avoided. Fifty 
years after Markowitz this way of thinking has become so common that nowadays 
most people use the terms ‘correlation’ and ‘dependence’ interchangeably. When 
dealing with the normal distributions that modern portfolio theory is based on there is 
nothing wrong with this. Unfortunately, however, the returns on most assets and asset 
classes are not exactly normally distributed and tend to exhibit a relatively high 
probability of a large loss (known formally as ‘negative skewness’) and/or a relatively 
high probability of extreme outcomes (known as ‘excess kurtosis’). In cases like this 
correlation is not a good measure of dependence and may actually be seriously 
misleading. Another problem is that even in cases where correlation is a valid 
measure of dependence, people do not seem to fully appreciate its exact nature. 
Although it appears quite surprising at first sight, at least part of the finding that the 
correlation between hedge fund returns and stock market returns is higher in down 
than in up markets for example can be attributed purely to technicalities. Even a 
normal distribution with a constant correlation coefficient will exhibit this sort of 
behaviour. In this brief note I will discuss these matters in some more detail and 
provide some examples.
1  
 
II. CORRELATION AND NON-ELLIPTICAL DISTRIBUTIONS 
When it comes to correlation and dependence the big question is whether the 
correlation coefficient is sufficient to describe the complete dependence structure 
between two variables.  Although beyond the scope of this note, it can be shown that 
this is only the case when the joint (or bivariate) probability distribution of both 
variables is elliptical. Elliptical simply means that when the joint distribution is 
viewed from above, the contour lines of the distribution are ellipses. The best-known 
member of the family of elliptical distributions is of course the normal distribution. 
From statistics 101 we all know that each and every bivariate normal distribution can ISMA Centre Discussion Papers in Finance 2002-23 
 
Copyright 2002 Harry M. Kat 
 
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be fully described by just two expectations, two variances and one correlation 
coefficient. Figure 1 shows a plot of a number of drawings from a standard normal 
bivariate distribution with correlation coefficient 0.5. From the plot we clearly see the 
elliptical contour shape of the distribution. 
 
Most real-life distributions exhibit positive or negative skewness and/or some degree 
of excess kurtosis. Elliptical distributions are therefore nothing more than an ideal 
type that is rarely encountered in practice. Elliptical distributions, however, are also 
the easiest distributions to work with mathematically. As a result, the assumption of 
normality has become the single most important assumption in econometrics, which 
has left us in the awkward situation where 95% of the econometric tools we have at 
our disposal assume a distribution that is hardly ever observed in reality.
2 As said, if 
the joint distribution is not elliptical, the correlation coefficient is not a good measure 
for the dependence structure between the two variables involved. An example is 
provided in figure 2, which shows plots of a large number of drawings from a normal 
(on the left) and a non-elliptical (on the right) bivariate distribution. Both plots look 
very different, implying a completely different dependence structure. The so-called 
‘tail dependence’ in the non-elliptical distribution is quite pronounced as it shows a 
clear tendency to generate extreme values for both variables simultaneously. 
Surprisingly, however, both these distributions have the same correlation coefficient, 
which immediately shows how dangerous it can be to rely on just the correlation 
coefficient to measure dependence. Another example can be found in figure 3. Again, 
despite the fact that in the distribution on the right there is a much stronger tendency 
for extreme values to go together, both these distributions have the same correlation 
coefficient. 
 
In the above context two other points are important as well. First, if two variables are 
both normally distributed this does not automatically mean that their joint distribution 
is normal as well. This is only the case if we assume that the joint distribution is 
elliptical. If not, there are an infinite number of bivariate distributions that fit this 
description. Second, we all know that because the correlation coefficient equals the 
normalized covariance, it will always lie between +1 and –1. However, whether it is 
actually possible for the correlation coefficient to take on these extreme values is ISMA Centre Discussion Papers in Finance 2002-23 
 
Copyright 2002 Harry M. Kat 
 
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another matter. For non-elliptical distributions the actually attainable interval might 
well be smaller. For some distributions the attainable interval can be very small, say 
between –0.1 and +0.2 for example. If this was indeed the case, finding a correlation 
coefficient of 0.2 and concluding that there was only very weak dependence between 
both variables involved would be a terrible mistake as both variables in question are 
actually perfectly dependent. 
 
The above emphasizes how limited the available econometric toolbox still is and how 
preconditioned we all have become on the assumption of normality. Fortunately, this 
is changing. More and more econometric research is turning towards non-normal 
distributions. Also, empirical research no longer aims to show that real-life 
distributions can be assumed to be normal after all. Partly driven by the fast 
development of the risk management profession, we seem to be taking things much 
more the way they really are, i.e. not normally distributed. 
 
III. CONDITIONAL CORRELATIONS 
One way to deal with the problem mentioned above is to calculate what are known as 
‘conditional correlations‘. This means splitting up the available data sample based on 
the size or the volatility of one or both of the variables involved and subsequently 
calculate correlation coefficients for these sub-samples separately. This technique has 
been used to study the correlation between different assets and asset classes in up and 
down markets for example. The conclusion is always the same: during major market 
events correlations go up dramatically. Based on this many investors have come to 
believe that during times of large moves in financial markets the benefits of 
(international) diversification are dramatically reduced. 
 
In a recent book, Lhabitant (2002, p. 171) uses the same technique to study the 
dependence between hedge funds and the stock market. Using data over the period 
January 1994 to August 2001 he finds that the correlation between most hedge fund 
indices and US and European equity is much higher in down markets than in up 
markets. Overall (as measured by the CSFB/Tremont index), the correlation between 
hedge funds and US equity is 0.18 in up markets but a whopping 0.53 in down ISMA Centre Discussion Papers in Finance 2002-23 
 
Copyright 2002 Harry M. Kat 
 
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markets. The biggest differences between up and down market correlations are 
observed in convertible arbitrage, emerging markets, and event driven strategies. 
Similar results can also be found in Schneeweis and Spurgin (2000) and Jaeger (2002, 
p. 124).    
 
At first sight, findings like the above seem quite worrisome. Fortunately, however, 
this need not necessarily be the case. It is not difficult to show that conditional 
correlations should actually display some of the observed behaviour for purely 
technical reasons. Suppose we had two random variables X and Y with a bivariate 
standard normal distribution and a correlation coefficient of 0.5, just like the one 
shown in figure 1. Now suppose we split the range of possible outcomes of X into 10 
different segments such that X had exactly 10% chance ending up in either one of 
these segments. Subsequently, we made a large number of drawings from the 
bivariate distribution and then calculated the conditional correlation for every segment 
separately. This would produce the results in table 1. From the table we clearly see 
that the conditional correlation in segment 1 and 10 is several times higher than in 
segment 5 or 6, suggesting that large drawings are much more correlated than smaller 
ones. This is not an empirical observation though, but a purely technical matter as the 
actual correlation coefficient is fixed.
3  
 
It turns out that what is important is the ratio of the conditional variance of X, i.e. the 
variance within the chosen segment, and the overall variance of X, i.e. the variance 
calculated over all segments. In our example, the former is given in the last column of 
table 1 while the latter equals 1 by assumption. The higher the conditional variance 
relative to the overall variance, the higher the conditional correlation we will find. 
This means that when we move from the normal distribution to distributions with 
significant skewness and/or excess kurtosis, the effect may be much stronger. When a 
distribution is skewed it means that compared to the normal distribution it has a long 
tail to the left or right. This long tail will raise the conditional variance relative to the 
overall variance and therefore produce a higher conditional correlation. The same 
happens with excess kurtosis, i.e. when the distribution in question has ‘fatter’ tails 
than the normal distribution. When a distribution is sufficiently skewed or leptokurtic 
this may raise the conditional variance so much that, unlike in the above example, the ISMA Centre Discussion Papers in Finance 2002-23 
 
Copyright 2002 Harry M. Kat 
 
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conditional correlation becomes higher than the unconditional correlation. Since this 
is a purely technical matter, however, one would be wrong to conclude from this that 
more extreme movements are more correlated than overall movements. 
Given the above, it is interesting to take another look at Lhabitant’s conditional 
correlations mentioned before. In a recent paper Brooks and Kat (2001) studied a 
large number of hedge fund indices, including those studied by Lhabitant. From their 
study it is clear that the indices that exhibit the biggest difference between up and 
down market correlation in Lhabitant’s study also happen to be the indices that 
exhibit the highest levels of negative skewness and excess kurtosis. This strongly 
suggests that part of Lhabitant’s findings is attributable to technicalities. How much is 
real is very hard to determine, however, as to do so we would have to know each 
index’s exact distribution so we could calculate what conditional correlations to 
expect on purely technical grounds.
4  
 
Another interesting example of the dangers of conditional correlations can be found in 
research that aims to find out whether correlation is higher in more volatile times. 
This case is quite straightforward because the conditioning is done on the conditional 
variance itself. We therefore know in advance that the higher we set the threshold of 
what constitutes high volatility, the more likely we are to find a high conditional 
correlation. Let’s return to the bivariate standard normal distribution with a fixed 
correlation of 0.5. Suppose we made a large number of drawings from this 
distribution, sorted them on whether the absolute value of X was higher or lower than 
0.674 and subsequently calculated the conditional correlations of the two resulting 
sub-samples. Doing so, we would find a conditional correlation of 0.21 for the small 
and 0.62 for the high values sample. Many people would be inclined to conclude from 
this that correlation differs dramatically between volatile and quiet periods. However, 
this would clearly be incorrect as the correlation is constant by construction. If we 
sorted the data directly on the variance of X we would get the results displayed in 
table 2. From the table we clearly see how the conditional correlation rises with the 
variance threshold, again suggesting that correlation is higher in more volatile markets 
while by construction it is not.  
 ISMA Centre Discussion Papers in Finance 2002-23 
 
Copyright 2002 Harry M. Kat 
 
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IV. CONCLUSION 
The above are only simple examples but they make it painfully clear how misleading 
(conditional) correlation can be. Low (high) correlations do not necessarily imply low 
(high) dependence. We therefore need other methods to investigate whether for 
example more extreme movements in financial markets are indeed more highly 
correlated than overall movements. Given the complexity of real-life distributions, 
however, such methods are likely to be a lot more complicated than our old friend the 
correlation coefficient.  
 ISMA Centre Discussion Papers in Finance 2002-23 
 
Copyright 2002 Harry M. Kat 
 
7 
 
Footnotes 
1. I do not claim any originality with respect to the ideas or even the examples 
discussed in this note. More details can be found in Boyer et al. (1999), 
Loretan and English (2000), Embrechts et al. (1999, 2002), or Malevergne 
and Sornette (2002) and the references therein.   
2. Note that this is somewhat of an exaggeration as strictly speaking most 
econometric tools only rely on asymptotic convergence to normality and not 
on finite sample normality.  
3. The basic problem here is that the correlation coefficient is dependent on the 
marginal distributions of the variables in question. Conditioning changes the 
marginal distributions and thereby changes the correlation coefficient.  
4. Amin and Kat (2002) find that when forming portfolios of hedge funds, 
stocks and bonds the negative skewness of the portfolio return distribution 
increases substantially when the hedge fund allocation is increased. This 
suggests that a significant part of the observed effect is indeed real.   
 
 
   
 
 ISMA Centre Discussion Papers in Finance 2002-23 
 
Copyright 2002 Harry M. Kat 
 
8 
 
REFERENCES 
Amin, G. and H. Kat (2002), Stocks, Bonds and Hedge Funds: Not a Free Lunch!, 
Working Paper, ISMA Centre, University of Reading. 
 
Boyer, B., M. Gibson and M. Loretan (1999), Pitfalls in Tests for Changes in 
Correlations, International Finance Discussion Paper Board of Governors of the 
Federal Reserve System.  
 
Brooks, C. and H. Kat (2001), The Statistical Properties of Hedge Fund Index Returns 
and Their Implications for Investors, forthcoming Journal of Alternative Investments. 
 
Embrechts, P., A. McNeil and D. Straumann (1999), Correlation: Pitffalls and 
Alternatives, RISK Magazine, May, pp. 69-71. 
 
Embrechts, P., A. McNeil and D. Straumann (2002), Correlation and Dependence in 
Risk Management: Properties and Pitfalls, in M. Dempster (ed.), Risk Management: 
Value at Risk and Beyond, Cambridge University Press,  pp. 176-223. 
 
Jaeger, L. (2002), Managing Risk in Alternative Investment Strategies, FT Prentice 
Hall.  
 
Lhabitant, F. (2002), Hedge Funds: Myths and Limits, Wiley. 
 
Loretan, M. and W. English (2000), Evaluating Correlation Breakdowns During 
Periods of Market Volatility, International Finance Discussion Paper Board of 
Governors of the Federal Reserve System. 
 
Malevergne, Y. and D. Sornette (2002), Investigating Extreme Dependences: 
Concepts and Tools, Working Paper Laboratoire de Physique de la Matiere 
Condensee, University of Nice – Sophia Antipolis.  
 
Schneeweis, T. and R. Spurgin (2000), Hedge Funds: Portfolio Risk Diversifiers, 
Return Enhancers or Both, Working Paper CISDM. 
 
 ISMA Centre Discussion Papers in Finance 2002-23 
 
Copyright 2002 Harry M. Kat 
 
9 
 
 
 
Table 1: Correlations bivariate standard normal distribution  
conditioned on outcome range. 
 
Segment  Range  Cond. Correlation  Cond. Variance 
1  < -1.282  0.2310  0.1690 
2  -1.282 / -0.842   0.0725  0.0159 
3  -0.842 / -0.524  0.0526  0.0083 
4  -0.524 / -0.253  0.0451  0.0061 
5  -0.253 / 0.000  0.0421  0.0053 
6  0.000 / 0.253  0.0421  0.0053 
7  0.253 / 0.524  0.0451  0.0061 
8  0.524 / 0.842  0.0526  0.0083 
9  0.842 / 1.282  0.0725  0.0159 
10  > 1.282  0.2310  0.1690 
 
 
 
 
 
 
 
Table 2: Correlations bivariate standard normal distribution  
conditioned on sub-sample volatility. 
 
Variance greater than   Cond. Correlation 
1.8  0.615 
1.7  0.610 
1.6  0.603 
1.5  0.595 
1.4  0.585 
1.3  0.575 
1.2  0.564 
1.1  0.553 
1.0  0.541 
 
 
 
 
 ISMA Centre Discussion Papers in Finance 2002-23 
 
Copyright 2002 Harry M. Kat 
 
10 
 
 
 
 
 
Figure 1: Drawings from a bivariate standard normal distribution with correlation 
coefficient of 0.5. Taken from Loretan and English (2000). 
 
 
 
 
 
 
 
 
 ISMA Centre Discussion Papers in Finance 2002-23 
 
Copyright 2002 Harry M. Kat 
 
11 
 
 
 
 
 
 
Figure 2: Drawings from a normal (left) and non-elliptical (right) bivariate 
distribution, both with the same correlation coefficient. Taken from Embrechts et al. 
(1999). 
 
 
 
 
 
 
 
 
 
 
 ISMA Centre Discussion Papers in Finance 2002-23 
 
Copyright 2002 Harry M. Kat 
 
12 
 
 
 
 
 
 
 
Figure 3: Drawings from two bivariate distributions with a different dependence 
structure but the same correlation coefficient. Taken from Embrechts et al. (2002). 
 
 
 

Correlation and Dependence in Risk Management: Properties and Pitfalls,

Correlation: Pitffalls and Alternatives,

Evaluating Correlation Breakdowns During Periods of Market Volatility, International Finance Discussion Paper Board of Governors of the Federal Reserve System.

Hedge Funds: Myths and Limits,

Hedge Funds: Portfolio Risk Diversifiers, Return Enhancers or Both, Working Paper CISDM.

Investigating Extreme Dependences: Concepts and Tools, Working Paper Laboratoire de Physique de la Matiere Condensee,

Managing Risk in Alternative Investment Strategies,

Stocks, Bonds and Hedge Funds: Not a Free Lunch!, Working Paper,

The Statistical Properties of Hedge Fund Index Returns and Their Implications for Investors, forthcoming Journal of Alternative Investments.

The Dangers of Using Correlation to Measure Dependence

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The Dangers of Using Correlation to Measure Dependence

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Research Papers in Economics