### Empirical Properties of Correlation: Behaviors of Correlation in the Real World

Statistically significant and expected properties are displayed by financial correlations as we will show in this chapter. In economic crises, correlation volatility and correlation levels are higher. Risk managers and traders should take this factor into consideration.

# The Behavior Equity Correlations in a Recession, Normal Economic Period or Strong Expansion

Assuming the daily closing prices of the stocks at each particular point in time, in the Dow Jones Industrial Average, has been under observation for the past thirty plus years. The observation made is that the Dow’s composition changes with time due to successful stocks being added while unsuccessful ones get removed. Idiosyncratic factors are dominated by macroeconomic factors during a recession, hence causing a downturn in multiple stocks. Compared to normal economic state, correlation volatility is higher in a recession and remains so without additional volatility. In an economic expansion, correlation volatility is lowest and highest in worse economic states.

# Do Equity Correlations Exhibit Mean Reversion?

Variables like bonds, interest rates, volatilities, credit spreads, etc. tend to be pulled back to their long-term mean. This tendency is referred to as mean reversion. For interest rates, due to high demand for capital, interest rates rises causing the economy to cool off and consequently a recession.

A decrease in demand for capital leads to a decline in the interest rates to their long-term mean and probably below. Again as a result of better fiscal policies economic activities increases leading to an eventual increase in interest rates back to their long-term mean.

## Quantifying mean Reversion

A negative relationship between the changes of a variable, $${ S }_{ t }-{ S }_{ t-1 }$$ and the variable at $$t – 1$$, $${ S }_{ t-1 }$$.

Therefore:

$$\frac { \partial \left( { S }_{ t }-{ S }_{ t-1 } \right) }{ \partial { S }_{ t-1 } } <0$$

Where:

$${ S }_{ t }\Rightarrow Price\quad at\quad time\quad t$$,

$${ S }_{ t-1 }\Rightarrow Price\quad at\quad a\quad previous\quad point\quad in\quad time\quad t–1$$.

A decrease in $${ S }_{ t-1 }$$ and is low at $$t – 1$$ implies that a mean reversion will pull up $${ S }_{ t-1 }$$ to $${ \mu }_{ s }$$ hence increasing $${ S }_{ t }-{ S }_{ t-1 }$$.

To the contrary, an increase in $${ S }_{ t-1 }$$ with high $$t – 1$$ implies that mean reversion will pull down $${ S }_{ t-1 }$$ at the next point in time t to $${ \mu }_{ s }$$ thus decreasing $${ S }_{ t }-{ S }_{ t-1 }$$. The degree of the pull is the mean reversion rate or speed or gravity.

By the discrete Vasicek process, quantifying the degree of mean reversion:

$${ S }_{ t }-{ S }_{ t-1 }=a\left( { \mu }_{ s }-{ S }_{ t-1 } \right) \Delta t+{ \sigma }_{ S }\varepsilon \sqrt { \Delta t }$$

Where:

$${ S }_{ t }\Rightarrow$$Price at time $$t$$,

$${ S }_{ t-1 }\Rightarrow$$price at a previous point in time $$t–1$$,

$$a$$ is the degree of mean reversion and $$o\le a\le$$,

$${ \sigma }_{ S }$$ volatility of $$S$$,

$${ \mu }_{ s }$$ is the long term mean of $$S$$ and

$$\varepsilon$$ is a random drawing from a standardized normal distribution at time $$t$$,$$\varepsilon \left( t \right) :n\sim \left( 0,1 \right)$$.

Setting $$\Delta t=1$$ then:

$${ S }_{ t }-{ S }_{ t-1 }=a{ \mu }_{ S }-a{ S }_{ t-1 }\quad \quad \quad \quad \quad \left( a \right)$$

Running a standard regression analysis of the form $$Y=\alpha +\beta X$$, will enable the determination of mean reversion rate.

Thus regressing $${ S }_{ t }-{ S }_{ t-1 }$$with respect to $${ S }_{ t-1 }$$ gives:

$${ S }_{ t }-{ S }_{ t-1 }=a{ \mu }_{ S }-a{ S }_{ t-1 }$$

Where:

$${ S }_{ t }-{ S }_{ t-1 }=Y$$,

$$a{ \mu }_{ S }=\alpha$$ and

$$a{ S }_{ t-1 }=\beta X$$.

# Do Equity Correlations Exhibit Autocorrelation?

A variable is usually correlated to its past values up to a certain degree. This degree of correlation is referred to as autocorrelation. To derive autocorrelation, the time series of the variable is regressed to its past time series values.

Autocorrelation is the reverse property to mean reversion. The following equation is a derivation of autocorrelation ($$AC$$) for a 1-period time lag.

$$AC\left( { { \rho }_{ t },{ \rho }_{ t-1 } } \right) =\frac { Cov\left( { \rho }_{ t },{ \rho }_{ t-1 } \right) }{ \sigma \left( { \rho }_{ t } \right) \sigma \left( { \rho }_{ t-1 } \right) }$$

Where $${ \rho }_{ t }$$ and $${ \rho }_{ t-1 }$$ are correlation values for time period $$t$$ and $$t – 1$$.

Assume the above histogram was generated from the correlations of the Dow stocks for thirty plus years with a continuous line of best fit showing the Johnson SB distribution with parameters $$\gamma$$ and $$\delta$$ for shape, $$\mu$$ for location and $$\sigma$$ for scale.

From this figure, positive correlations are the most observed between the Dow stocks. A number of distributions are tested using Kolmogorov-Smirnov, Anderson Darling,and Chi-Squared tests to fit the histogram.

# Is Equity Correlation an Indicator for Future Recessions?

The following observations in recessions were made for the more than thirty years study of the data,i.e., 1972 to 2012:

1. Between 1973 and 1974 the first shock of oil prices caused a severe recession.
2. There was a short recession in 1980.
3. Due to the second shock in oil prices, there was another severe recession between 1981 and 1982.
4. Between 1980 and 1981, a mild recession was experienced.
5. The internet bubble bust of 2001 led to another mild recession.
6. Finally, the global financial crisis that lasted from 2007 to 2009 led to another recession which is assumed to be the greatest of all times.

It has been observed that correlation volatility experiences a downturn prior to every recession. Furthermore, an expansionary period comes before a recession with low correlation volatility.

# Properties of Bond Correlations and Default Probability Correlations

Similar properties are observed between bond correlations and equity correlations when studies were conducted. Compared to equity correlation levels, in bonds correlation levels were higher and in default probabilities slightly lower.

Again, in comparison to volatilities in equity correlation, correlation volatility in default probability was slightly higher and lower for bonds.

For mean reversion, equity correlation had quite high amounts which were lower in bond correlations and default probability correlations.

By use of Johnson SB distribution, replicating default correlation distribution which is similar to equity correlation distribution is possible.

# Summary

In summary, a terrible economic state implies higher correlations of equities. There is a positive relation between equity correlation levels and equity correlation volatility with equity correlations having strong mean reversion. This calls for the need to add mean reversion when modeling correlation. A low autocorrelation displayed by equity correlation results from strong mean reversion.

The Johnson SB distribution replicates equity correlation distribution. There is a similarity between features of bond correlations and those of equity correlations. Similar observations were made when features of default correlations were compared to those of equity correlations.

# Practice Questions

1) A correlation data has a long-term mean of 43.6%. The averaged correlation was again observed as 41.32% in June 2014 for the 30 x 30 Dow correlation matrices. Given that the average mean reversion was 79.1% from the regression function for 40 years, determine the expected correlation one month later. What is the implication?

1. 42.69%
2. 43.12%
3. 39.51%
4. 42.98%

Using equation:

$${ S }_{ t }-{ S }_{ t-1 }=a{ \mu }_{ s }-a{ S }_{ t-1 }$$

Then:

$${ S }_{ t }=a\left( { \mu }_{ s }-{ S }_{ t-1 } \right) +{ S }_{ t-1 }$$

Where

$$a$$ = 79.1%,

$${ \mu }_{ s }$$ = 43.6%,

$${ S }_{ t-1 }$$ = 41.32%

Therefore,

$${ S }_{ t }=0.791\left( 0.436–0.4132 \right) +0.4132 = 0.4312 = 43.12\%$$

The mean reversion rate of 79.1% increases correlation of 41.32% in June 2014 to an expected correlation of 43.12% in July 2014.

2) Given that the change in time $$\Delta t$$ is 1.2, use the Vasicek 1977 processtp quantify the degree of mean reversion assuming that the price of a stock at a previous point in time is $15.98 and the long-term mean is$21.55 with a mean reversion rate of 2.4.

1. $18.71 2.$7.80
3. $19.93 4.$17.99

From the Vasicek process, we have that:

$${ S }_{ t }-{ S }_{ t-1 }=a\left( { \mu }_{ s }-{ S }_{ t-1 } \right) \Delta t+{ \sigma }_{ s }\varepsilon \sqrt { \Delta t }$$

Where

$$a$$ = 1.4,

$${ S }_{ t-1 }$$=15.98,

$${ \mu }_{ s }$$ = 21.55.

Since we are only interested in the mean reversion, we ignore the $${ \sigma }_{ s }\varepsilon \sqrt { \Delta t }$$, part of the equation.

Therefore:

$${ S }_{ t }-{ S }_{ t-1 }=a\left( { \mu }_{ s }-{ S }_{ t-1 } \right) \Delta t$$

$$\Rightarrow { S }_{ t }-{ S }_{ t-1 }=2.4\left( 21.55–15.98 \right) \times 1.4=18.71$$

3) In February 2013, a set of data is given such that the Dow correlation matrices have an averaged correlation of 0.3352 along with a term mean of 0.1857. Compute the expected correlation for March 2013 if the average mean reversion is 0.4599.

1. 18.57%
2. 26.64%
3. 22.23%
4. 20.85%

Using equation:

$${ S }_{ t }-{ S }_{ t-1 }=a{ \mu }_{ s }-a{ S }_{ t-1 }$$

Then:

$${ S }_{ t }=a\left( { \mu }_{ s }-{ S }_{ t-1 } \right) +{ S }_{ t-1 }$$

Where

a =0.4599,

$${ \mu }_{ s }$$= 0.1857,

$${ S }_{ t-1 }$$ = 0.3352

Therefore,

$${ S }_{ t }=0.4599\left( 0.1857–0.3352 \right) +0.3352 = 0.2664 = 26.64\%$$