Correlation

The most obvious attempt at modeling possible diversification benefits consists of es- timating unconditional correlations. For the (raw) return series shown in Figure 3.9, the resulting linear correlation coefficient is shown in the first column of Table 3.1.

01/71 01/74 01/77 01/80 01/83 01/86 01/89 01/92 01/95 01/98 01/01 01/04 01/07 01/10 −20 −10 0 10 20 Stocks mm/yy R et u rn (% ) 01/71 01/74 01/77 01/80 01/83 01/86 01/89 01/92 01/95 01/98 01/01 01/04 01/07 01/10 −30 −20 −10 0 10 20 30 Commodities mm/yy R et u rn (% ) 01/71 01/74 01/77 01/80 01/83 01/86 01/89 01/92 01/95 01/98 01/01 01/04 01/07 01/10 −40 −20 0 20 40 Gold mm/yy R et u rn (% )

We find the lowest correlation—and thus the highest potential for diversification— between stocks and gold. However, the correlation between stocks and commodities is only slightly higher. The strongest link is found between commodities and gold, which is not surprising, since gold is one of the commodities included in the S&P GSCI index. The second and third column of Table 3.1 contains estimates for Kendall’s τ

and Spearman’sρS. Rank correlation estimates differ in values, but not in the relative

ordering with respect to the riskiness of asset class combinations.

Pair ρ τ ρS

Stocks and Commodities 0.1722 0.0946 0.1360

Stocks and Gold 0.1101 0.0553 0.0834

Commodities and Gold 0.2725 0.1562 0.2274

Table 3.1: Unconditional Correlation Estimates

However, correlations among asset classes may vary substantially, depending on the subsample chosen. As an illustration, Figure 3.10 shows linear correlations obtained from a one–year rolling window; that is, the first value (01/71) is calculated from re- turns between February 1970 and January 1971, the last one (10/10) is based on returns between November 2009 and October 2010. For all three asset class combinations, we observe large variations in correlation.

One finds periods in which correlation is clearly negative, as is the case for stocks and commodities in January 1976. At the same time, and more dangerously, there are periods which are characterized by large positive correlations. For example, between January 1980 and September 1981—thus, in the aftermath of the second oil crisis— all asset class combinations show a high positive correlation. Looking at Figure 3.9, we confirm that this period is one of high variability in returns. Similar effects are observed, for example, for the correlation between stocks and commodities after the bursting of the dotcom bubble in 2000, as well as during the recent financial crisis starting in the fourth quarter of 2008. In other words, we find evidence for correlation breakdown: During the turbulent times when diversification is most needed, it fails to protect against large portfolio losses. Using an unconditional correlation estimate as those of Table 3.1 may therefore be too optimistic.

Another hint at the appropriateness of a constant correlation capturing the entire de- pendency structured is given by the exceedance correlations. These are shown in Figure

01/71 01/74 01/77 01/80 01/83 01/86 01/89 01/92 01/95 01/98 01/01 01/04 01/07 01/10 −1

−0.5 0 0.5

1 Stocks and Commodities

mm/yy C or re lat ion 01/71 01/74 01/77 01/80 01/83 01/86 01/89 01/92 01/95 01/98 01/01 01/04 01/07 01/10 −1 −0.5 0 0.5

1 Stocks and Gold

mm/yy C or re lat ion 01/71 01/74 01/77 01/80 01/83 01/86 01/89 01/92 01/95 01/98 01/01 01/04 01/07 01/10 −1 −0.5 0 0.5

1 Commodities and Gold

mm/yy C or re lat ion

Figure 3.10: Linear Correlations, 12–Month Rolling Window.

3.11, where the empirical exceedance correlations are compared to those implied by the multivariate normal distribution. Here and in the following, we consider GARCH(1,1) filtered returns in order to circumvent effects of volatility clustering on the analyses of dependence structures. Therefore, the range of returns—and, thus, exceedance levelsθ

in Figure 3.11— is smaller than in the unfiltered case. As soon as there are less than ten joint observations below θ (or above it, for θ > 0), we stop decreasing (or increasing)

θin order to avoid exceedance correlation estimates to be driven by single observations.

Stocks and Commodities Stocks and Gold Commodities and Gold

−1 −0.5 0 0.5 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 θ ρ ( θ ) −1 −0.5 0 0.5 1 −0.2 0 0.2 0.4 0.6 θ ρ ( θ ) −1 −0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 θ ρ ( θ )

Figure 3.11: Empirical (Dashed) and Multivariate–Normal (Solid) Exceedance Cor-

For all three asset class combinations, we find asymmetric exceedance correlation struc- tures. That is, for negativeθ—adverse price movements—conditional correlations tend

to be higher than for positive returns. For example, given that the (GARCH–filtered) returns of stocks and commodities are both below 0.5%, their correlation is higher

than 0.4. This value, when compared to an unconditional correlation coefficient of

ρ= 0.1722 as reported in Table 3.1, illustrates once more the possible dangers of ne-

glecting asymmetric correlation structures. However, we do not find such clear–cut relations as earlier studies did. For example, Longin and Solnik (2001) find exceedance correlations to increase in bear, but not in bull market. For our data, exceedance cor- relations for θ < 0 increase only in two of three cases. Furthermore, we cannot find

clear directions of changes for bull markets (θ > 0). The pair coming closest to the

exceedance correlations observed by Longin and Solnik (2001) consists of stocks and commodities.

The solid lines in Figure 3.11 are the exceedance correlations implied by a multivari- ate normal distribution fitted to the data. The deficits of the multivariate normal become apparent in two aspects: Firstly, the multivariate normal is not able to cap- ture the asymmetries in correlation structures; secondly, and related to this, the level of exceedance correlation is underestimated by the multivariate normal. In fact, the implied exceedance correlations are at best as high as the empirical ones forθ >0, but

do not come close to those for negative returns.

We conclude that going from an unconditional to a conditional correlation cannot save the assumption of multivariate normality. For the multivariate normal, (theoretical) exceedance correlations are not constant, so that the mere fact of observing changes in correlation does not imply the rejection of this distribution; rather, the structure of these changes for returns is so different from that of the multivariate normal that we have to seriously doubt the appropriateness of the latter distribution.