# LO 7.2: Calculate a mean reversion rate using standard regression and calculate the

LO 7.2: Calculate a mean reversion rate using standard regression and calculate the corresponding autocorrelation.
Mean reversion implies that over time, variables or returns regress back to the mean or average return. Empirical studies reveal evidence that bond values, interest rates, credit spreads, stock returns, volatility, and other variables are mean reverting. For example, during a recession, demand for capital is low. Therefore, interest rates are lowered to encourage investment in the economy. Then, as the economy picks up, demand for capital increases and, at some point, interest rates will rise. If interest rates are too high, demand for capital decreases and interest rates decrease and approach the long-run average. The level of interest rates is also a function of monetary and fiscal policy and not just supply and demand levels of capital.
Mean reversion is statistically defined as a negative relationship between the change in a variable over time, S St l , and the variable in the previous period, St p
d(St – S t_!)
a s t_!
In this equation, S is the value of the variable at time period r, S j is the value of the variable in the previous period, and d is a partial derivative coefficient. Mean reversion exists when S j increases (decreases) by a small amount causing S S j to decrease (increase) by a small amount. For example, if S j increases and is high at time period t 1, then mean reversion causes the next value at S to reverse and decrease toward the long-run average or mean value. The mean reversion rate is the degree of the attraction back to the mean and is also referred to as the speed or gravity of mean reversion. The mean reversion rate, a, is expressed as follows:
St St_j a(p St_ i) At + a S VAt
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If we are only concerned with measuring mean reversion, we can ignore the last term, crgeVAt , which is the stochastic part of the equation requiring random samples from a distribution over time. By ignoring the last term and assuming A t = 1, the mean reversion rate equation simplifies to:
Example: Calculating mean reversion
Suppose mean reversion exists for a variable with a value of 50 at time period t 1. The long-run mean value, /i, is 80. What are the expected changes in value of the variable over the next period, S S 15 if the mean reversion rate, a, is 0, 0.5, or 1.0?
If the mean reversion rate is 0, there is no mean reversion and there is no expected change. If the mean reversion rate is 0.5, there is a 50% mean reversion and the expected change is 15 [i.e., 0.5 x (80 – 50)]. If the mean reversion rate is 1.0, there is 100% mean reversion and the expected change is 30 [i.e., 1.0 x (80 50)]. Thus, a stronger or faster mean reversion is expected with a higher mean reversion rate.
Standard regression analysis is one method used to estimate the mean reversion rate, a. We can think of the mean reversion rate equation in terms of a standard regression equation (i.e., Y = a + (3X) by applying the distributive property to reformulate the right side of the equation:
^t-l ”
a^t-l
Thinking of this equation in terms of a standard regression implies the following terms in the regression equation:
St – St l = Y; ap, = a ; and – aSt l = (3X
A regression is run where S St l (i.e., the Yvariable) is regressed with respect to St l (i.e., the X variable). Thus, the (3 coefficient of the regression is equal to the negative of the mean reversion rate, a.
>From the 1972 to 2012 study, the data resulted in the following regression equation:
Y = 0.27 – 0.78X
The beta coefficient of 0.78 implies a mean reversion rate of 78%. This is a relatively high mean reversion rate. Thus, if there is a large decrease (increase) from the mean correlation
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for one month, the following month is expected to have a large increase (decrease) in correlation.
Example: Calculating expected correlation
Suppose that in October 2012, the average monthly correlation for all Dow stocks was 30% and the long-run correlation mean of Dow stocks was 33%. A risk manager runs a regression, and the regression output estimates the following regression relationship: Y = 0.273 0.78X. What is the expected correlation for November 2012 given the mean reversion rate estimated in the regression analysis? (Solve for S in the mean reversion rate equation.)
There is a 3% difference from the October 2012 and long-run mean correlation (35% 30% = 5%). The (3 coefficient in the regression relationship implies a mean reversion rate of 78%. The November 2012 correlation is expected to revert 78% of the difference back toward the mean. Thus, the expected correlation for November 2012 is 33.9%:
St = a(M– St_i) + St_i
S = 0.78(35% – 30%) + 0.3 = 0.339
Autocorrelation measures the degree that a current variable value is correlated to past values. Autocorrelation is often calculated using an autoregressive conditional heteroskedasticity (ARCH) model or a generalized autoregressive conditional heteroskedasticity (GARCH) model. An alternative approach to measuring autocorrelation is running a regression equation. In fact, autocorrelation has the exact opposite properties of mean reversion.
Mean reversion measures the tendency to pull away from the current value back to the long-run mean. Autocorrelation instead measures the persistence to pull toward more recent historical values. The mean reversion rate in the previous example was 78% for Dow stocks. Thus, the autocorrelation for a one-period lag is 22% for the same sample. The sum of the mean reversion rate and the one-period autocorrelation rate will always equal one (i.e., 78% + 22% = 100%).
Autocorrelation for a one-period lag is statistically defined as:
AC(pt,pt_ I) =
cov(pt,ptt) a(pt) x cr(pt_ i)
The term AC(pt, pt l) represents the autocorrelation of the correlation from time period t and the correlation from time period t – 1. For this example, the pt term can represent the correlation matrix for Dow stocks on day r, and the pt l term can represent the correlation
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matrix for Dow stocks on day t 1. The covariance between the correlation measures, cov(pt, pt l), is calculated the same way covariance is calculated for equity returns.
This autocorrelation equation was used to calculate the one-period lag autocorrelation of Dow stocks for the 1972 to 2012 time period, and the result was 22%, which is identical to subtracting the mean reversion rate from one. The study also used this equation to test autocorrelations for 1- to 10-day lag periods for Dow stocks. The highest autocorrelation of 26% was found using a two-day lag, which compares the time period t correlation with the t 2 correlation (two months prior). The autocorrelation for longer lags decreased gradually to approximately 10% using a 10-day lag. It is common for autocorrelations to decay with longer time period lags.
Professor Note: The autocorrelation equation is exactly the same as the correlation coefficient. Correlation values for time period t and t 1 are used to determine the autocorrelation between the two correlations.
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