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 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 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
  • an expansionary period with gross domes- tic product (GDP) growth rates of 3.5% or higher,
  • a normal economic period with growth rates between 0% and 3.49%, and
  • a recession with two consecutive quarters of negative growth rates.
    
• 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.
  
• 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.
• 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_s ε√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 S_t-1 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

  S_t – S_t-1=a(μ_s– S_t-1)
  or
  S_t – S_t-1= aμ_s – a S_t-1

  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.
• The long-term mean of the correlation data is 34.83%. In February 2012, the averaged correlation of the 30×30 Dow correlation matrices was 26.15%. From the regression function from 1972 to 2012, we find that the average mean reversion is 77.51%. What is the expected correlation for March 2012 ?
• 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.
• 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