Let us explore the ticker symbol GVP. We will test for mean reversion with the Hurst exponent and calculate the half life of mean reversion.

First, lets plot the daily closing prices:

library(ggplot2) ggplot(new.df, aes(x = Date, y = Close))+ geom_line()+ labs(title = "GVP Close Prices", subtitle = "19950727 to 20170608")+ theme(plot.title = element_text(hjust=0.5),plot.subtitle = element_text(hjust=0.5,size=9), plot.caption = element_text(size=7))

Lets run the Hurst exponent to test for mean reversion, we will do this over the entire history of GVP. For this test we will use a short term lag period of 2:20 days (Explanation Here).

# Hurst Exponent # Andrew Bannerman # 8.11.2017 require(lubridate) require(dplyr) require(magrittr) require(zoo) require(lattice) # Data path data.dir <- "D:/R Projects" output.dir <- "D:/R Projects" data.read.spx <- paste(data.dir,"GVP.csv",sep="/") # Read data read.spx <- read.csv(data.read.spx,header=TRUE, sep=",",skip=0,stringsAsFactors=FALSE) # Convert Values To Numeric cols <-c(3:8) read.spx[,cols] %<>% lapply(function(x) as.numeric(as.character(x))) # Convert Date Column [1] read.spx$Date <- ymd(read.spx$Date) # Make new data frame new.df <- data.frame(read.spx) # Subset Date Range #new.df <- subset(new.df, Date >= "2000-01-06" & Date <= "2017-08-06") #new.df <- subset(new.df, Date >= as.Date("2017-01-07") ) #Create lagged variables lags <- 2:20 # Function for finding differences in lags. Todays Close - 'n' lag period getLAG.DIFF <- function(lagdays) { function(new.df) { c(rep(NA, lagdays), diff(new.df$Close, lag = lagdays, differences = 1, arithmetic = TRUE, na.pad = TRUE)) } } # Create a matrix to put the lagged differences in lag.diff.matrix <- matrix(nrow=nrow(new.df), ncol=0) # Loop for filling it for (i in lags) { lag.diff.matrix <- cbind(lag.diff.matrix, getLAG.DIFF(i)(new.df)) } # Rename columns colnames(lag.diff.matrix) <- sapply(lags, function(n)paste("lagged.diff.n", n, sep="")) # Bind to existing dataframe new.df <- cbind(new.df, lag.diff.matrix) head(new.df) # Calculate Variances of 'n period' differences variance.vec <- apply(new.df[,9:ncol(new.df)], 2, function(x) var(x, na.rm=TRUE)) # Linear regression of log variances vs log lags log.linear <- lm(formula = log(variance.vec) ~ log(lags)) # Print general linear regression statistics summary(log.linear) # Plot log of variance 'n' lags vs log time xyplot(log(variance.vec) ~ log(lags), main="GVP log variance of price diff Vs log time lags", xlab = "Time", ylab = "Logged Variance 'n' lags", grid = TRUE, type = c("p","r"),col.line = "red", abline=(h = 0)) hurst.exponent = coef(log.linear)[2]/2 hurst.exponent

If we divide the log(logs) coefficient by 2 we obtain the Hurst exponent of 0.4598435.

Remember H value less than 0.5 = mean reversion.

0.5 = random walk

0.5 = momentum.

Great.

Lets apply a simple linear strategy to see how it performs over this series. We will setup a rolling z-score and we will buy when the zscore crosses below 0 and we will sell when it crosses back over 0. We use a arbitrarily chosen lookback of 10 days for this.

Here are the results:

The above plot is the compounded growth of $1 and since 1995 $1 has grown to over $800 or over 79,900 %.

Next lets calculate the half life of mean reversion. We do this with linear regression. For the independent variable we use the price difference between today’s close and yesterdays close. For the dependent variable we use the price differences between today’s and yesterdays close – the mean of the price difference between today’s close and yesterdays close.

Note we use the previous 100 days of data to produce this test:

# Calculate yt-1 and (yt-1-yt) y.lag <- c(random.data[2:length(random.data)], 0) # Set vector to lag -1 day y.lag <- y.lag[1:length(y.lag)-1] # As shifted vector by -1, remove anomalous element at end of vector random.data <- random.data[1:length(random.data)-1] # Shift data by -1 to make same length of vector y.diff <- random.data - y.lag # Subtract todays close - close from yesterday y.diff <- y.diff [1:length(y.diff)-1] # Adjust length of vector prev.y.mean <- y.lag - mean(y.lag) # Subtract yesterdays close from the mean of lagged differences prev.y.mean <- prev.y.mean [1:length(prev.y.mean )-1] # Adjust length of vector final <- merge(y.diff, prev.y.mean) # Merge final.df <- as.data.frame(final) # Create final data frame # Linear Regression With Intercept result <- lm(y.diff ~ prev.y.mean, data = final.df) half_life <- -log(2)/coef(result)[2] half_life

We obtain a half life of 4.503093 days.

Next lets see if we can set our linear strategy lookback period equal to the half life to see if it improves results. The original look back period was 10 days chosen arbitrarily. The result of a look back of 4.5 rounded to 5 days is below:

From 1995 to roughly present day the result did not improve significantly but looking at the plot we see a large uptick in the equity curve from 2013 onwards. Lets subset our data to only include data post 2013 and lets re-run the 10 day look back and also the 5 day look back to see if we can see the benefit of optimizing using the mean reversion half life.

First the result of the 10 day look back arbitrarily chosen:

We see that $1 has grown to $8 or 700% increase.

Next the look back of 4.5 rounded to 5 days derived from the mean reversion half life calculation:

We see that using a look back set to equal the mean reversion half life of 5 days rounded, we see $1 has grown to over $15 or a 1400% increase.

Lets run the Hurst exponent on both periods, the first from 1995 to 2013. The second from 2013 to roughly present day:

1st test: We see H = **0.4601632 **

2nd: We see H = **0.4230494**

Ok so we see the Hurst exponent become more mean reverting post 2013. If we test >= 2016 and >= 2017 we see:

H = 0.3890816 and 0.2759805 respectively.

Next lets choose a random time frame between 1995 and 2013.

From period 2000 to 2003, H = **0.5198083** which is more a random walk.

If we look at period 2003 to 2008 we have a H value of **0.4167166 ** which is more mean reverting, however, this H value of 0.41 is actually lower than the post 2013 H value of 0.4230494. So the H value in this case didnt say because H is this, then gains should be that.

This might be caused by other factors, frequency of trades, price range, fluctuations etc..

Note this post is largely theoretical no commissions are included in any of the trades. This demonstrates the combination of using statistical tools and performing a back test.