The computation and usage of Pearson’s correlation coefficient is quite ubiquitous in the quantitative analysis of financial markets. However, applied quantitative re- searchers are often unaware of the pitfalls involved in careless application and usage of correlations as a measure of risk. It is therefore appropriate to investigate this de- pendence concept in some detail and point out the shortcomings of this measure.
Financial Risk Modelling and Portfolio Optimization withR, Second Edition. Bernhard Pfaff.
© 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.
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k k Pearson’s correlation coefficient between two random variablesX1andX2with finite
variances is defined as
𝜌(X1,X2) = COV(X1,X2)
√VAR(X1)√
VAR(X2), (9.1) whereCOV(X1,X2) =E((X1−E(X1))(X2−E(X2)))denotes the covariance between the random variables andVAR(X1)andVAR(X2)are their variances, respectively.
As is evident from (9.1), the correlation coefficient is a scalar measure. In the case of a perfect linear dependence the value of this measure is𝜌(X1,X2) =|1|, where the perfect linear dependence is given byX2=𝛽+𝛼X1, with𝛼∈ℝ\0, 𝛽∈ℝ, oth- erwise the correlation coefficient can take values in the interval −1< 𝜌(X1,X2)<
1. Furthermore, the linear correlation coefficient is invariant with respect to linear affine transformations of the random variablesX1andX2:𝜌(𝛼1X1+𝛽1, 𝛼2X2+𝛽2) = signum(𝛼1𝛼2)𝜌{X1,X2}for𝛼1, 𝛼2∈ℝ\0, 𝛽1, 𝛽2∈ℝ.
Hence, Pearson’s correlation coefficient depicts the strength of a linear relation- ship between two random variables. The logical reasoning that𝜌(X1,X2) =0 implies independence betweenX1andX2is only valid in the case of multivariate elliptically distributed random variables. Put differently, lack of correlation is only a sufficient condition for independence and only in the case of elliptical distributions can the two concepts be used interchangeably. Elliptic distributions are characterized by the fact that the points of the density function which yield the same values represent an ellipse; that is, horizontal cuts through the probability mass are elliptically shaped.
The implication for multivariate risk modelling is that only in the case of jointly elliptically distributed risk factors can the dependence between these be captured adequately by the linear correlation coefficient. Given the stylized facts of financial market returns this assumption is barely met. It should be pointed out at this point that with respect to risk modelling one is usually more concerned with the dependence structure in the tail of a multivariate loss distribution than with an assessment of the overall dependence, but a correlation measure just depicts the latter, in principle.
Furthermore, the linear correlation coefficient is not invariant with respect to non- linear (e.g., log-transformed) random variables, or if nonlinear (e.g., quadratic or cubic) dependence between variables exists. This fact will be illustrated with the following two examples.
First, a standardized normal random variableX1∼N(0,1)with quadratic depen- denceX2=X12is investigated. Between these two variables a direct relationship is evident; for a given realizationx1ofX1the value forx2can be concluded. However, if one were to calculate the linear correlation coefficient between these two variables, the result would beCOV(X1,X2) =E(X1⋅(X12−1)) =E(X13) −E(X1) =0, because the skewness of normally distributed random variables is zero.
The purpose of the second example is to highlight the fact that the correlation co- efficient depends on the marginal distributions of the random variables in question and the possibility that the correlation coefficient cannot take all values in the inter- val−1≤𝜌≤1 if one views the multivariate distribution of these random variables.
k k Given two log-normal random variables X1 ∼log N(0,1) andX2∼log N(0, 𝜎2)
with𝜎 >0, the feasible range of values for the correlation coefficient has a lower bound of 𝜌min=𝜌(eZ,e−𝜎Z) and an upper bound of 𝜌max=𝜌(eZ,e𝜎Z), where Z∼ N(0,1). The sign of the exponent depends on whether the random variables move in the same direction or opposite directions. In the first case a positive dependence (co-monotonicity) results, and in the latter a negative one (counter-monotonicity). The lower bound of the correlation coefficient is therefore given by
𝜌min= e−𝜎−1
√(e−1)(e𝜎2−1)
(9.2) and the upper bound is determined by
𝜌max= e𝜎−1
√
(e−1)(e𝜎2−1)
. (9.3)
As is evident from (9.2) and (9.3), these bounds depend only on the variance of the marginal distribution forX2. Here, the lower bound is greater than−1 and the upper bound coincides with the case of perfect dependence only when𝜎=1, and is less than 1 in all other instances. Incidentally, with ever increasing variance the correlation coefficient approaches zero.
With respect to risk modelling the following conclusions can be drawn. For in- stance, a correlation coefficient as high as 0.2 would only indicate a weak relationship between two risk factors. Indeed, the opposite can be true and therefore a tentative diversification gain in the context of a portfolio is void when capital is allocated to the respective assets. By converse reasoning, the information of a correlation coeffi- cient as high as 0.7 between two log-normally distributed risk factors with expected values of zero and variances of 1 and 4, respectively, is void. If these two random vari- ables are jointly distributed, such a high value for the correlation coefficient cannot be achieved.
In summary, the dependence between financial instruments can only be depicted correctly with the linear correlation coefficient if these are jointly elliptically dis- tributed. It was also shown that the value of the correlation coefficient depends on the marginal distributions and that not all values in the range[−1,1]are attainable. In particular, a perfect positive dependence between two random variables is not inter- changeable with a correlation coefficient of one, but the value can be much lower and at the limit a value of zero is obtained. Conversely, a correlation coefficient of zero does not lead to a conclusion of independence. This reasoning is only valid in the case of jointly elliptical distributions. Last, but not least, it should be pointed out that the linear correlation coefficient is only defined for pairs of random variables with fi- nite variance. This assumption may be violated for risk factors with great probability masses located in the tails. As a consequence, the application of alternative concepts for measuring the dependence between risk factors becomes necessary. In addition to the rank correlation coefficients of Kendall and Spearman, the copula concept is a promising route to take. The methods will be introduced in the next section.
k k