Empirical Properties of Correlation

LOS 1. Describe how equity correlations and correlation volatilities behave throughout various economic state.

Gunter Meissner et al analyzed the behavior of correlations estimated from daily closing prices of 30 stocks in Dow Jones index (for period Jan 1972 –Oct 2012). The composition of the index varies with time so the values derived are representative of the stocks that represent the index at each particular time. We can gather the following from the study.

1. Correlation levels are lowest in strong economic growth times wherein equity prices react primarily to idiosyncratic, not macroeconomic factors.
2. In recessions, correlation levels typically increase, as macroeconomic factors seem to dominate idiosyncratic factors, leading to a (simultaneous) downturn of multiple stocks.
3. Correlation volatility is lowest in an economic expansion and higher in normal periods and recessions. It seems that high correlation levels in a recession remain high without much additional volatility (correlation volatility is lower in recession than in normal times although correlation levels are higher).

The authors observe a generally positive relationship between correlation level and it’s volatility.

LOS 2. Calculate a mean reversion rate using standard regression and calculate the corresponding autocorrelation.

Mean reversion is present if there is a negative relationship between the change of a variable, i.e $\rho_{t+1} – \rho _t$, and the variable at the current time $\rho_t$. The degree of the pull towards mean is also called mean reversion rate. We model mean reversion behavior via the Vasicek / Ornstein-Uhlenbeck (OU) process: $$\rho_{t+1} – \rho_t = k \left(\mu_\rho – \rho_t\right) \Delta t + \sigma _\rho \varepsilon \sqrt{\Delta t}$$ We use $\rho$ to denote (monthly average) correlation, we set $\Delta t$ to 1 (in reality, a, month but for ease of explanation a single period) and we ignore stochastic term $(\sigma_{\rho} \varepsilon \sqrt{\Delta t})$. We have: $$\rho_{t+1} -\rho_t = k \mu_{\rho}- k\rho_t$$ To find the mean reversion rate $k$, we can run a standard regression analysis regressing $\rho_{t+1}-\rho_t$ with respect to $\rho_t$. $k$ is the negative of the slope parameter of regression. We find a strong mean reversion of $k=77.51\%$ which implies that an upward spike in correlation is typically followed by a sharp decline in the next time period. The expected correlation level can be obtained by $$ E(\rho _{t+1}) = k\mu_{\rho}+(1-k) \rho_t$$ Autocorrelation is the “reverse property” to mean reversion – the stronger the mean reversion, the lower the autocorrelation i.e., the less it is correlated to its past values, and vice versa. Positive autocorrelation is also termed persistence. Autocorrelation can be derived by regressing the time series of a variable to its past time series values. An autocorrelation with lag (obtained by regressing $(\rho_{t-L})$ and $\rho_t$) will be: $$AC (\rho_t, \rho_{t-L}) = \frac{cov(\rho_t, \rho_{t-L})}{\sigma(\rho_t )\sigma(\rho_{t-L})}$$ The 1-lag autocorrelation of 22.49% and the mean reversion of 77.51% add up to 1. Autocorrelation with lag 2 is highest and we observe the expected decay in autocorrelation with respect to time lags of earlier periods.

LOS 3. Identify the best-fit distribution for equity, bond, and default correlations.

The authors observed mostly positive correlations between the stocks in the Dow (77.23% values were positive). The versatile Johnson SB distribution with four parameters, $Y$ and $\delta$ for the shape, $\mu$ for location, and $\sigma$ for scale, provided the best fit.

The authors notice that except for the mild recession in 1990–1991, before every recession a downturn in correlation volatility occurred. Correlation volatility is low in an expansionary period, which often precedes a recession. The relationship between a decline in volatility and the severity of the recession is seen to be statistically non significant.

Correlation levels were found to be higher for bonds (41.67%) and slightly lower for default probabilities (30.43%) compared to equity correlation levels (34.83%). Correlation volatility was lower for bonds (63.74%) and slightly higher for default probabilities (87.74%) compared to equity correlation volatility (79.73%).

Mean reversion was also present in bond correlations $(k=25.79\%)$ and in default probability correlations $(k=29.97\%)$. These levels were lower than the very high equity correlation mean reversion of $k=77.51\%$.

The default probability correlation distribution can be replicated best with the Johnson SB distribution. The bond correlation distribution shows a more normal shape and can be best fitted with the generalized extreme value distribution and quite well with the normal distribution.