Introduction

•Contrary to common beliefs, financial correlations display statistically significant  and expected properties.

•Correlation levels as well as correlation volatility are generally higher in economic  crises, which should be taken into consideration by traders and risk managers.

•Strong  mean  reversion  can  be  found  in  correlations  and  the  distribution  of  correlations is typically not normal or lognormal.

Equity Correlations in Different Economic States

THE STUDY –

•In their study, the authors observed daily closing prices of the 30 stocks in the Dow Jones  Industrial Average (Dow) from January 1972 to October 2012. This resulted in 10,303 daily  observations of the Dow stocks and hence 10,303 × 30 = 309,090 closing prices. Monthly  bins were built and 900 correlation values (30 × 30) were derived for each month, applying  the Pearson correlation approach. Since there were 490 months in the study, all together 490 ×  900 = 441,000 correlation values were derived. The unity correlation values on the diagonal  of each correlation matrix were eliminated and 441,000 — (30 × 490)  = 426,300  correlation values were obtained as inputs.

The composition of the Dow changes in time, with successful stocks being put into the Dow  and unsuccessful stocks being removed. This study included the Dow stocks that represent  the Dow at each particular point in time.

THE RESULT –

•The first figure shows the 490 monthly averaged correlation levels from 1972 to 2012 with  respect to the state of the economy. The second figure shows the volatility of the averaged  monthly correlations. The economy can be segregated into three states

1. an expansionary period with gross domes- tic product (GDP) growth rates of  3.5% or  higher,

2. a normal economic period with growth rates between 0% and 3.49%, and

3. a recession with two consecutive quarters of negative growth rates.

THE RESULT –

•From the previous figures, the erratic behaviour of Dow correlation levels and volatility is  observed. However, the following table reveals some expected results. It can be observed that  correlation levels are lowest in strong economic growth times. The reason may be that in strong  growth periods equity prices react primarily to idiosyncratic, not macroeconomic factors. In  recessions, correlation levels typically increase, as shown in this Table. It is also known that  correlation levels increased sharply in the Great Recession from 2007 to 2009. In a recession,  macroeconomic factors seem to dominate idiosyncratic factors, leading to a downturn of  multiple stocks.

Correlation Level and Correlation Volatility with Respect to the State of the Economy

	Correlation Level	Correlation Volatility
Expansionary period	27.46%	71.17%
Normal economic period	32.73%	83.40%
Recession	36.96%	80.48%

THE RESULT –

•A further expected result in the previous table is that correlation volatility is lowest in an  economic expansion and highest in worse economic states. Although a higher correlation  volatility could have been expected in a recession compared to a normal economic state, but it  seems that high correlation levels in a recession remain high without much additional volatility.  Generally, correlation volatility is high, as can also be observed from the somewhat erratic  correlation function in the previous figures. Altogether, the higher correlation risk in bad  economic times is displayed in the table, which traders and risk managers should consider in  their trading and risk management.

•Overall, a generally positive relationship between correlation level and correlation volatility is  observed.

Mean Reversion

•Mean reversion is the tendency of a variable to be pulled back to its long-term mean. In finance,  many variables, such as bonds, interest rates, volatilities, credit spreads, and more, are assumed to  exhibit mean reversion. Fixed coupon bonds, which do not default, exhibit strong mean  reversion: A bond is typically issued at par, for example at $100. If the bond does not default, at  maturity it will revert to exactly that price of  $100, which is typically close to its long-term  mean.

•Interest rates are also assumed to be mean reverting: In an economic expansion, typically  demand for capital is high and interest rates rise. These high interest rates will eventually lead to  a cooling off of the economy, possibly leading to a recession. In this process, capital demand  decreases and interest rates decrease from their high levels towards their long-term mean,  eventually falling below their long-term mean. Being in a recession, eventually economic activity  increases again, often supported by monetary and fiscal policy. In this reviving economy, demand  for capital increases, in turn increasing interest rates to their long-term means.

•Mean reversion is present if there is a negative relationship between the change of a variable, 𝑆_t — 𝑆_t–1, and the variable at 𝑡 — 1, 𝑆_t–1. Formally, mean reversion exists if

where

𝑆_t: price at time t

𝑆_t–1: price at the previous point in time 𝑡— 1

𝜕: partial derivative coefficient

The above equation tells us: If  𝑆_t–1  increases by a very small amount, 𝑆_t  — 𝑆_t–1, will decrease by  a certain amount, and vice versa. This is intuitive: If 𝑆_t–1 has decreased and is low at 𝑡 — 1  (compared to the mean of  S, 𝜇_S), then at the next point in time t, mean reversion will pull up 𝑆_t–1  to 𝜇_s and therefore increase 𝑆_t  — 𝑆_t–1. If 𝑆_t–1 has increased and is high in 𝑡  — 1 (compared to  the mean of S,𝜇_S), then at the next point in time t, mean reversion will pull down 𝑆_t–1, to 𝜇_s and therefore decrease 𝑆_t — 𝑆_t–1. The degree of the pull is the degree of the mean reversion, also called mean reversion rate, mean reversion speed, or gravity.

Mean Reversion Quantification

•The degree of mean reversion can be calculated by starting with the discrete Vasicek process,

where

𝑆_t  : price at time t

𝑆_t–1: price at the previous point in time 𝑡 — 1

𝑎: degree of mean reversion, also called mean reversion rate or gravity, 0 ≤ 𝑎 ≤ 1

𝜇_s: long-term mean of 𝑆

𝜎_s: volatility of 𝑆

𝜀: random drawing from a standardized normal distribution at time 𝑡, ε(𝑡): 𝑛 ~ (0,1)

To deal with mean reversion, the stochasticity part in the above equation (i.e. 𝜎_s𝜀  Δ𝑡) can be  ignored. If Δ𝑡 = 1, then a mean reversion parameter of a = 1 will pull 𝑆_t–1 to the long-term mean 𝜇_s  completely  at  every  time  step.  For  example,  if  is  80  and  𝜇_s  is  100,  then a × (𝜇_s — 𝑆_t–1) = 1 × (100 — 80) = 20, so the 𝑆_t–1 of 80 is mean reverted up to its long-  term mean of 100. Naturally, a mean reversion parameter a of 0.5 will lead to a mean reversion of  50% at each time step, and a mean reversion parameter a of 0 will result in no mean reversion.

•To quantify mean reversion. Setting Δ𝑡 to 1, and ignoring stochasticity, we get

To find the mean reversion rate 𝑎, a standard regression analysis can be done of the form

𝑌 = 𝛼 + 𝛽𝑋

Hence, 𝑆_t  — 𝑆_t–1 is being regressed with respect to 𝑆_t–1:

It can be observed that the regression coefficient 𝛽 is equal to the negative mean reversion  parameter 𝑎.

•After this the author runs a regression to find the empirical mean reversion of the earlier  obtained correlation data. Hence 𝑆 represents the 30 × 30 Dow stock monthly average  correlations from 1972 to 2012. The regression analysis is displayed in this figure.

•The regression function in this figure  displays a strong mean reversion of 77.51%.  This means that on average in every month,  a deviation from the long-term correlation  mean (34.83% in our study) is pulled back to  that long-term mean by 77.51%. This strong  mean reversion can also be observed by  looking at the first figure in this chapter. An  upward spike in correlation is typically  followed by a sharp decline in the next time  period, and vice versa.

Autocorrelation

•Autocorrelation is the degree to which a variable is correlated to its past values. Autocorrelation  can be quantified with the Nobel Prize-winning autoregressive conditional heteroscedasticity  (ARCH) model or its extension, generalized autoregressive conditional heteroscedasticity  (GARCH). However, we can also regress the time series of  a variable to its past time series  values to derive autocorrelation. This is the approach discussed here.

•In finance, positive autocorrelation is also termed persistence. In mutual fund or hedge fund  performance analysis, an investor typically wants to know if an above-market performance of a  fund has persisted for some time (i.e., is positively correlated to its past strong performance).

•The question whether autocorrelation exists is an easy one. Autocorrelation is the “reverse  property” to mean reversion: The stronger the mean reversion (i.e., the more strongly a variable  is pulled back to its mean), the lower the autocorrelation (i.e., the less it is correlated to its past  values), and vice versa.

•The autocorrelation can be derived as

where

𝐴𝐶: autocorrelation

𝜌_t: correlation values for time period 𝑡

𝜌_t–1: correlation values for time period 𝑡 — 1

𝐶𝑜𝑣: covariance;

This equation is algebraically identical with the Pearson correlation coefficient equation. The  autocorrelation just uses the correlation values of time period 𝑡 and time period 𝑡 — 1 as  inputs.

Equity Correlations and Autocorrelation

•It was found that the one-period lag autocorrelation of  the correlation values from 1972 to  2012 was 22.49%. As mentioned earlier, autocorrelation is the opposite property of mean  reversion. Therefore, not surprisingly, the autocorrelation of 22.49% and the mean reversion in  our study of 77.51% add up to 1. This Figure shows the autocorrelation with respect to  different time lags.

•It can be observed that time lag 2  autocorrelation is highest, so autocorrelation  with respect to two months prior produces  the highest autocorrelation. Altogether we  observe the expected decay in autocorrelation  with respect to time lags of earlier periods.

Distribution of Equity Correlations

•The input data of the distribution tests were daily correlation values between all 30 Dow stocks  from 1972 to 2012. This resulted in 426,300 correlation values. The distribution is shown in this  figure. It can be observed that correlations between the stocks in the Dow are mostly positive .  In fact, 77.23% of all 426,300 correlation values were positive.

•The distributions were tested for fitting the  histogram, applying three standard fitting tests:

(1)Kolmogorov-Smirnov,

(2)Anderson-Darling

(3)Chi-squared.

The versatile Johnson SB distribution with four  parameters, 𝛾 and 𝛿 for the  shape, μ for  location, and 𝜎 for scale, provided the best fit.

Standard distributions such as normal  distribution, lognormal distribution, or beta  distribution provided a poor fit.

Bond Correlations and Default Correlations

•Preliminary studies of 7,645 bond correlations and 4,655 default probability correlations display  properties similar to those of equity correlations. Correlation levels were 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 present in bond correlations (25.79%) and in default probability correlations  (29.97%). These levels were lower than the very high equity correlation mean reversion of  77.51%.

•The default probability correlation distribution is similar to the equity correlation distribution  and can be replicated best with the Johnson SB distribution. However, 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. The bond correlation and default  probabilities results are currently being verified with a larger sample database.

Equity, Bond and Default Correlations

•The following table summarizes the information:

Correlation Type	Average Correlation	Correlation Volatility	Reversion Rate	Best Fit Distribution
Equity	34.83%	79.73%	77.51%	Johnson SB
Bond	41.67%	63.74%	25.79%	Generalized Extreme Value
Default Probability	30.43%	87.74%	29.97%	Johnson SB