When using correlation coefficients to evaluate reliability, which of the following is undesirable?

9.2.1.2 Spearman correlation coefficient

The Spearman correlation coefficient is also called the rank correlation coefficient, and linear analysis is carried out with the help of the rank of the variables [31]. This coefficient does not require the analysis of original variables to meet specific requirements, and its scope of application is wider than that of Pearson, and it is a typical nonparametric statistical method [32]. Under normal circumstances, when either side of the variable to be measured is a nonnormally distributed variable, using the Spearman correlation coefficient will have a better effect than Pearson [33]. The Spearman correlation coefficient can be calculated by the following formula:

(9.2)S=∑i=1nx i−x¯⋅yi−y¯∑i=1nxi−x¯2 ⋅∑j=1nyj−y¯2

Linear and scatter correlation

Linear correlation coefficient. The calculation of correlation of the CTAs’ returns with other CTAs and benchmarks provide insight to diversification of returns and subsequently the reduction of risk. The linear correlation coefficient measures the degree of relationship between two series of returns, with the results being any value between – 1 and 1. The closer the value is to 1, the more related the two series of returns, and the more predictive one series is to the other. Similarly, the closer the value is to –1, the more the two series are diametrically opposite and would also be predictive of each other. A value around 0 means that the two series are unrelated and the returns of one series hold no predictive value in the returns of the other. In accordance with modern portfolio theory, significant reductions in volatility can be achieved by combining negatively correlated CTAs in a fund, significant reductions in volatility can be achieved.

While linear correlation coefficients are extremely useful in portfolio allocation and combining CTAs with the goal of minimizing volatility, there are some limitations to using them in evaluating CTAs. The main problem with the linear correlation coefficient is that in the search for negatively correlated investments, the important fact that negative correlation is not necessarily good and positive correlation is not necessarily bad was lost. For example, if the client has already invested with CTA A and is searching for a second CTA (CTA B), then a linear measurement does not differentiate between desirable negative correlation (e.g. CTA B is up when CTA A is down) and undesirable negative correlation (e.g. CTA B is down when CTA A is up). Furthermore, although the linear measurement does properly penalize undesirable positive correlation (e.g. CTA B is down when CTA A is down), it also wrongly penalizes desirable positive correlation (e.g. CTA B is up when CTA A is up).

Scatter correlation graphs. Scatter correlation graphs provide a terrific supplement to simple linear correlation coefficient values. Assuming that the client already has exposure to a benchmark index or is invested in a CTA (CTA A), and is looking to add a CTA B, the scatter correlation graph provides invaluable information. By graphing the distribution of returns for CTA A against those of CTA B, a clear picture of CTA B's suitability with CTA A develops. It is desirable if CTA B's returns are positive when CTA Ais down (negative correlation) as it is if CTA B's returns are positive when CTA A is up (positive correlation). Overall, a desirable scatter correlation graph would be in the general shape of an upward sloping parabola consisting of negative correlation on the left half of the graph and positive correlation on the right half of the graph.

Effect of Test Length on Criterion-Related Evidence of Validity

The correlation coefficient of a test with another variable is often called criterion-related evidence of validity for the test. For example if a company wants to use a test to select salespeople, the correlation of scores on the test with sales figure (i.e., the criterion) is criterion-related evidence of the validity. The formula in (7) shows the correlation of the longer test as a function of the number of parallel measurements, the correlation of each of these measurements with the criterion variable, and the reliability coefficient for each of the parallel measurements. Lengthening the test increases the correlation with the criterion. The impact of lengthening the test is larger when the reliability of the parallel measurements and the correlation of the parallel measurements with the criterion are smaller. The formula can be misleading if the error score variable for Y is correlated with the error score variables for one or more of X1–XK.

Pearson’s Product Moment Linear Correlation and Spearman’s Nonlinear Rank Correlation

Typically, when we use the term correlation, we usually mean a linear correlation. And, of course, correlations can take on any value between −1 and +1 inclusive, which means that the correlation coefficient has a sign (direction) and magnitude (strength). The problem arises when there is nonlinearity, and we use linear correlations. Figure 3.41 illustrates a few scatter charts with pairwise X and Y variables (e.g., hours of study and school grades). If we draw an imaginary best-fitting line in the scatter diagram, we can see the approximate correlation (we show a computation of correlation in a moment, but for now, let’s just visualize). Part A shows a relatively high positive correlation coefficient (R) of about 0.7 as an increase in X means an increase in Y, so there is a positive slope and, therefore, a positive correlation. Part B shows an even stronger negative correlation (negatively sloped; an increase of X means a decrease of Y and vice versa). It has slightly higher magnitude because the dots are closer to the line. In fact, when the dots are exactly on the line, as in Part D, the correlation is +1 (if positively sloped) or −1 (if negatively sloped), indicating a perfect correlation. Part C shows a situation where the curve is perfectly flat, or has zero correlation, where, regardless of the X value, Y remains unchanged, indicating that there is no relationship.

Problems arise when there are nonlinear relationships (typically the case in many real-life situations) as shown in Figure 3.42. Part E shows an exponential relationship between X and Y. If we use a nonlinear correlation, we get +0.9, but if we use a linear correlation, it is much lower at 0.6 (Part F), which means that there is information that is not picked up by the linear correlation. The situation gets a lot worse when we have a sinusoidal relationship, as in Parts G and H. The nonlinear correlation picks up the relationship very nicely with a 0.9 correlation coefficient; using a linear correlation, the best-fitting line is literally a flat horizontal line, indicating zero correlation. However, just looking at the picture would tell you that there is a relationship. So, we must therefore distinguish between linear and nonlinear correlations, because as we have seen in this exercise, correlation affects risk, and we are dealing with risk analysis!

The linear correlation coefficient is also known as the Pearson’s product moment correlation coefficient. It is computed by R=∑i=1n(Xi−X¯)(Yi−Y¯) ∑i=1n(Xi−X¯)2(Yi−Y¯) 2 and assumes that the underlying distribution is normal or near-normal, such as the t-distribution. Therefore, this is a parametric correlation. You can use Excel’s CORREL function to compute this effortlessly. The nonlinear correlation is the Spearman’s nonparametric rank-based correlation, which does not assume any underlying distribution, making it a nonparametric measure. The approach to Spearman’s nonlinear correlation is very simple. Using the original data, we first “linearize” the data, and then apply the Pearson’s correlation computation to get the Spearman’s correlation. Typically, whenever there is nonlinear data, we can linearize it by using either a LOG function (or, equivalently, an LN or natural log function) or a RANK function. The following table illustrates this effect. The original value is clearly nonlinear (it is 10x where x is from 0 to 5). However, if you apply a log function, the data becomes linear (1, 2, 3, 4, 5), or when you apply ranks, the rank (either high to low or low to high) is also linear. Once we have linearized the data, we can apply the linear Pearson’s correlation. To summarize, Spearman’s nonparametric nonlinear correlation coefficient is obtained by first ranking the data and then applying Pearson’s parametric linear correlation coefficient.

Table 3.1.

5.2.1 Linear correlation

The linear correlation coefficientρ, defined in (24), is a commonly misused measure of dependence. To illustrate the confusion involved in interpreting it, consider the following classic example. Let X∼N(μ,σ2) and let Y=X2. Then ρ( X,Y)=0, yet clearly X and Y are dependent. Unless we are willing to make certain assumptions about the multivariate distribution, linear correlation can therefore be a misleading measure of dependence. Since the copula of a multivariate distribution describes its dependence structure we would like to use measures of dependence which are copula-based. Linear correlation is not such a measure.

10.3.1 Formulation of an Adjusted Correlation Measure

The standard linear correlation coefficient is a measure of correlation between two time series Δxi and Δyi and is defined as follows:

(10.1)Q(xi,yi)≡ ∑i=1n(Δxi−〈Δx〉)(Δyi−〈Δy)〉∑i=1n (Δxi−〈Δx〉)2∑i=1n(Δyi−〈Δy〉)2

with the sample means,

(10.2)〈Δx〉≡∑i=1nΔxi nand〈Δy〉≡∑i=1n Δyin

The sample is of size T with n = T/Δt homogeneously spaced observations. Correlation values are unitless and may range from − 1 (completely anti-correlated) to 1 (completely correlated). A value of zero indicates two uncorrected series.

The two variables Δxi, and Δyi are usually returns of two financial assets. In risk assessment (but not in portfolio allocation), the deviation of returns from the zero level is often considered instead of the deviation from the sample means 〈Δx〉 and 〈Δy〉. In this special case, we can insert 〈Δx〉 = 〈Δy〉 = 0 in Equation 10.1.

An estimate of the local covolatility for each of these observations is defined by further dividing each time span (Δt) over which Δxi and Δyi, are calculated into m equal subintervals from which subreturn values, Δx˜j and Δy˜j, can be obtained. This redefined time series now consists of n˜=T/Δt˜ equally spaced return observations where Δt≡mΔt˜. The return definitions conform to Equation 3.7, based on logarithmic middle prices as in Equation 3.6. To obtain a homogeneous series, we need linear interpolation as introduced in Equation 3.2. The choice of linear interpolation method is essential.

For each of the previous coarse returns, Δxi, (as for Δyi), there exists a corresponding estimation of covolatility between the two homogeneous time series of returns

(10.3) ωi(Δx˜j;Δy˜j;Δt˜)≡∑j=1m(|Δx˜i·m−j−〈Δx˜i·m〉|·|Δy˜i·m−j−〈Δy˜i· m〉|)α

where

(10.4)〈Δx˜i·m〉=∑j=1mΔx˜ i·m−jmand〈Δy˜ i·m〉=∑j=1mΔy˜i·m−jm

The most obvious choice for α is 0.5, though this can be investigated as a way to magnify or demagnify the weight given to farther outlying return values. A value of 0.5 is used in all cases described in this discussion.

Equation 10.3 formulates covolatility around the mean rather than around zero and it therefore follows that ωi = 0 for the case of returns derived from two linearly interpolated prices existing outside of our region of interest, Δt. These covolatility estimates can be inserted as weights in all the sums computed to obtain the variances and covariance of the correlation calculation:

(10.5)Q˜(Δxi,Δyi,ωi)≡ ∑i=1T/Δt[(Δxi− 〈Δx〉)(Δyi−〈Δy〉)ωi]∑i=1T/Δt[(Δxi−〈Δx〉)2ωi] ∑i=1T/Δt[(Δyi−〈Δy〉)2ωi]

Note that Δxi and Δyi from Equation 10.5 are the same values as used in Equation 10.1, as they are logarithmic returns taken over the same time period, Δt. These coarse return values can then be defined as the sum of the fine return values

(10.6)Δxi≡∑j=1mΔx˜i·m−j

The sample means 〈Δx〉 and 〈Δy〉 have to be reconsidered in Equation 10.5. In the special case of risk assessment, we can still replace them by zero. Otherwise, we prefer that they are calculated again in a weighted fashion so that returns are considered only when observations over intervals of size Δt exist. Rather than keeping Equation 10.2, we define covolatility weighted mean values for both time series:

(10.7) 〈Δx〉≡∑i=1T/Δt(Δxi·ωi)∑i=1T/Δtωiand〈Δy〉≡ ∑i=1T/Δt(Δyi·ωi)∑i=1T/Δtωi

In this way, the means are calculated over the identically weighted data sample also used for the rest of the correlation calculation. The weights adjust for periods of lower or higher activity.

Equation 10.3 is formulated in such a way that ωi = 0 for the case of returns interpolated over a data gap—that is, a tick interval that fully contains the analyzed interval of size Δt. Data gaps have no influence on the means, and the sums of Equations 10.5 and 10.7 are not updated there. The covolatility adjusted measure of correlation described by Equation 10.5 also retains the desirable characteristics of the original, standard linear correlation coefficient; it is scale free, and completely different measurements are directly comparable. In addition, this alternative method is only slightly more complicated to implement than the standard linear correlation coefficient and can easily be implemented on a computer.

As will be applied later, this correlation measure easily fits into the frame-work of autocorrelation analysis. Given a time series of correlations Q˜t, it can be correlated with a copy of itself but with different time lags (τ) between the two, as shown in Equation 10.8:

(10.8)R(Q˜(Δxi,Δ yi,ωi),τ)=∑t=τ+1 n(Q˜t−〈Q˜1〉)(Q˜t−τ−〈Q˜2〉)[ ∑t=τ+1n(Q˜t−〈Q˜1〉)2∑t=τ+1n(Q˜t−τ−〈Q˜2〉)2 ]1/2

for τ < 0, where

(10.9)〈Q˜1〉=1n−τ∑t=τ+1nQ˜tand〈Q˜2〉=1n−τ∑t=τ+1nQ˜t−τ

For the discussions that follow, we measure correlation using the covolatility adjusted method described by Equation 10.5, unless otherwise stated, and always with m = 6 and α = 0.5 (see Equation 10.3). Any subsequent use of the commonly recognized linear correlation coefficient (Equation 10.1) will be referred to as the “standard” method.

5.2.3.2 Spearman rank-order correlation coefficient

The Spearman rank-order correlation coefficient, a nonparametric method, is used to measure the strength of association between two variables when one or both are measured on an ordinal/ranked (integer) scale, or when both variables are not normally distributed. If one of the variables is on an interval scale, then it needs to be transformed to a rank scale to analyze with the Spearman rank-order correlation coefficient, although this may result in a loss of information. Once two variables are ranked and sorted in an ascending order, the spearman correlation coefficient rs is calculated using the following equation.

(5.13)rs=1−6∑ i=1ndi2n(n2−1)

where di (d1,d2,….dn) are the differences in ranks of two variables xi (x1,x2,… .xn) and yi (y1,y2, ….yn).

The correlation coefficient takes on the values from −1.0 to +1.0. A value of −1.0 indicates a perfect negative linear relationship between the two variables, which means as one variable increases, the other decreases. Similarly, a value of +1.0 indicates a perfect positive linear relationship between the two variables, which means as one variable increases, the other increases too. If the value is 0.0, then it indicates no linear relationship. Any coefficient value between −1.0 and 0.0 or 0.0 and +1.0 indicate a negative or positive linear relationship but not an exact straight line. Hinkle et al. (2003) provided a rule of thumb for interpreting the correlation coefficient, as shown in Table 5.1.

Table 5.1. Interpreting of correlation coefficient (Hinkle et al., 2003).

15.2.3 FoHFs Return Drivers

In this section, we evaluate the dynamic relationships among FoHFs returns and some factors that may be considered as FoHFs return drivers. We study some hedge fund return drivers reported in the literature (e.g., Agarwal and Naik, 1999, 2004; Liang, 1999; Chan et al., 2005; Billio et al., 2006; Racicot and Théoret, 2007). The excess return drivers considered are: three factors of the Fama–French (1993) model (i.e., market risk premium factor (Rm – Rf), small minus big (SMB) factor, and high minus low (HML) factor); 1-month US Treasury bill (T-bill) rate; percentage change of monthly EUR/US$ exchange rate; percentage change of monthly crude oil per barrel price; percentage change of monthly volatility VIX index. Return driver data are collected for the same time span as FoHFs and hedge fund data.

Table 15.2 presents the dynamic correlation coefficients between each return driver and the excess return on each FoHFs index. Correlation coefficients are computed for the first lag, contemporaneous value, and first lead of each return driver. Bold numbers in Table 15.2 show that, for all FoHFs indices, significant contemporaneous correlation is evidenced for the market risk premium, percentage change of the VIX index, and percentage change of the EUR/US$ exchange rate.2 Furthermore, we find significant correlation between all FoHFs indices and the first lead of market risk premium, SMB factor, and percentage change of the VIX index. Finally, Table 15.2 shows that some FoHFs are correlated with the T-bill rate and the HML factor as well.

Table 15.2. Dynamic Correlation Coefficients of FoHFs and Hedge Fund Returns with Return Drivers

HFRI Fund of Funds Composite Index (FF); HFRI FOF Strategic Index (FF1); HFRI FOF Market Defensive Index (FF2); HFRI FOF Diversified Index (FF3); HFRI FOF Conservative Index (FF4). Asterisks denote test statistically significant at the ∗10%, ∗∗5% and ∗∗∗1% levels, respectively.

Data sources of return drivers: Bloomberg, Reuters, and DataStream.

3 Empirical Results

The graphs from time-varying correlation coefficients as computed from equation (5) between each stock market index and the crude oil prices are presented in Figures 4 and 5.

In 2003, there was a relatively lower dynamic correlation in the case of exporting countries (Dubai, Kuwait, and South Africa). This result is explained by the war in Iraq in March 2003 and the strike movement in Venezuela. We observe a breakdown in dynamic correlation for all exporting countries in 2006. We explain this decrease in interdependence between oil prices and the stock market index by a military attack in Nigeria, which caused the shutdown of more than 600,000 billion barrels a day.

Another period of interest is that spanning from 2006 until mid-2008, characterized by high oil prices due to rising demand, mainly from China. The level of correlation shows an increasing and positive pattern for all countries. This aggregate demand-related oil price shock had a positive impact on stock markets (both in oil-importing and oil-exporting countries) as it signaled an increase in world trade. These findings are in line with Hamilton (2009) and Kilian and Park (2009), who suggest that aggregate demand-related oil price shocks, originating from world economic growth, have a positive impact on stock prices.

From mid-2006 to early 2009, the correlation rises sharply and reaches a higher value (except for Venezuela). The main event during this period is the global financial crisis triggered by the export of US mortgages to the rest of the world as asset-backed securities, which can be regarded as an aggregate demand-related oil price shock (International Energy Agency 2009). The greater interaction between oil and stock market prices can be explained by the fact that the resulting crisis caused stock markets to enter bearish territory and oil prices to decline heavily, as also documented by Creti et al. (2013).

There are only three periods of noteworthy higher or lower correlation between oil prices and stock markets for exporting countries. These are early 2000 until 2001 (aggregate demand-oriented oil price shocks—higher correlation), 2003–2005 (aggregate demand-oriented oil price shocks—higher correlation), and 2007–2008 (aggregate demand-oriented oil price shock—positive correlation).

The years 2003–2005 (Figure 5) represent the sole period showing little difference between importing and exporting countries in terms of the correlation pattern of oil and stock market prices. The explanation for such findings may be due to the housing market boom in 2000, which created a positive environment for world markets and, at the same time, high demand for oil, driving the prices of both markets to higher levels. The 9/11 terrorist attack and the second war in Iraq also led to significant uncertainty in all economies, causing similar stock market movements and thus similar correlation with oil prices. In addition, Chinese growth and its impact on world trade caused euphoria in all stock markets regardless of the country of origin. Likewise, the recent global financial crisis impacted on all stock markets in a similar fashion and thus on their comovements.

Our analysis shows that aggregate demand-oriented oil price shocks (housing market boom, Chinese economic growth, and the recent global financial crisis) caused a significantly higher correlation between stock market prices and oil prices. Considerable precautionary demand-oriented oil price shocks (i.e., second war in Iraq, terrorist attacks) tended to cause higher correlation but are of less magnitude when compared with the aggregate demand-oriented oil price shocks. The origin of shock seems to be an important determinant of the magnitude of correlation between oil prices and stock markets, as when the oil shocks originate from major world turmoil events, such as wars or changes in global business cycles.

Overall, the findings of the previous literature are confirmed concerning the impact of oil shocks on oil-importing and oil-exporting country stock markets, whereas in the case of supply shocks, our findings show some aspects that have been neglected to date. In particular, we highlight the role of crisis periods in oil prices as drivers of the comovements between oil and stock markets.

2.3.1 Emerging vs Advanced Stock Market Excess Returns

Figure 1 reports the dynamics of the correlation coefficients between 28 emerging and 6 advanced stock market excess returns, and the excess returns of the US stock market. Figure 2 reports the dynamic unconditional correlation between the excess returns of 10 emerging stock market sector excess returns and the excess returns of the MSCI World. Dynamic unconditional correlations are obtained using a rolling window of 60 months.

Four empirical findings are worth noting: (i) correlation coefficients are increasing both across emerging and advanced stock markets; (ii) the correlation between the advanced stock market excess returns and the excess return of the US stock market is (on average) higher than the correlation between the emerging stock market excess returns and the excess return of the US stock market; (iii) differently from the 9/11 recession, during the 2007–2009 US subprime crisis financial contagion took place in all stock markets; and (iv) the correlation between emerging stock market sector excess returns and the world portfolio excess return is increasing over time.

While standard international asset pricing models predict that financial liberalization may reduce the cost of capital, an increasing integration process across financial and goods markets tends to reduce international portfolio diversification benefits, thus, forcing investors to look for alternative forms of investment.