An Application of the Geometric and Arithmetic Means<\/h4>\n

Over the past several years there has been a proliferation of ETFs for every imaginable purpose.<\/p>\n

One particularly interesting type of ETF is the leveraged ETF. These ETFs are designed to provide 2X or 3X the daily return of a target index such as the S&P500, and they do a reasonable job of achieving this goal.<\/p>\n

For example, if the S&P500 is up 1% for the day, the 2X S&P500 fund will be up 2% and the 3X fund will be up 3%. If the S&P500 is down 0.5% for the day, then the 2X fund will be down 1% and the 3X fund will be down 1.5%. (Note that for simplicity I\u2019m not considering the fees and implicit borrowing costs associated with leveraged funds in this post).<\/p>\n

Based on this description, you might think that if the S&P500 is up 10% for the year, then the 2X S&P500 fund would be up 20% and the 3X S&P500 fund will be up 30%. However, this is not the case. In fact, under some conditions, the 2X and 3X leveraged funds may be up even less than the S&P500 index. How can this be? If the daily leveraged returns are 2X or 3X the returns of the target fund, then why won\u2019t the annual returns be 2X or 3X the annual returns?<\/p>\n

To understand why there is discrepancy between daily returns and long-term returns, we need to review the difference between the arithmetic and geometric means.<\/p>\n

Arithmetic Mean<\/h5>\n

The arithmetic mean is what most of us think of when we hear the term \u201caverage\u201d, and it is defined by this equation.<\/p>\n

<\/p>\n

Geometric Mean<\/h5>\n

The geometric mean is the nth root of the product of n numbers, and it is defined by this equation:<\/p>\n

<\/p>\n

The geometric mean can also be calculated using logarithms:<\/p>\n

<\/p>\n

The geometric mean only applies for positive numbers. So, when using investment returns, the geometric mean is calculated using the decimal multiplier equivalent values. For example, a 6% return is 1.06, and a -4% return is 0.96.<\/p>\n

Relationship Between the Arithmetic and Geometric Means<\/h5>\n

The geometric mean is always less than the arithmetic mean unless every number in the series is identical. The AM-GM inequality states that for a series of non-negative numbers:<\/p>\n

<\/p>\n

For a large series of returns that is normally distributed, the geometric mean is approximately equal to the arithmetic mean minus half of the variance.<\/p>\n

<\/p>\n

Intuition Behind the Difference in the Means<\/h5>\n

The intuition behind this difference can be seen the following example. Consider two investments A and B which are held for two periods. Investment A earns 10% in the first period and -10% in the second period. Investment B earns 30% in the first period and -30% in the second period. For both of these investments, the arithmetic mean return\u00a0is zero percent. However, if an investor invested $100 in each investment, then the final value would be:<\/p>\n

\n<\/p>\n

So, even though the arithmetic mean return for the two investments is the same, the realized final value depends on the volatility.<\/p>\n

Which Mean Should We Use for Investment Returns?<\/strong><\/h5>\n

When thinking about investment returns, both the arithmetic mean and the geometric mean are useful.\u00a0 The arithmetic return is the most familiar measure of the mean, and it is also the most commonly used metric in many statistical models such as the CAPM.<\/p>\n

However, it is not correct to compound the arithmetic mean to calculate the return realized over multiple periods.\u00a0 Whenever we are compounding returns, the geometric mean must be used.<\/p>\n

For example, if, over a ten year period, our arithmetic mean for the annual return was 10%, and the geometric mean over the same period was 8%, then the correct calculation of the current account value for an investor who invested $1000 at the beginning of the 10 year period would be:<\/p>\n

<\/p>\n

It would be incorrect, and would substantially overstate the final value, if we used the arithmetic mean of 10% in this calculation.<\/p>\n

Application to Leveraged ETFs<\/strong><\/h5>\n

With leveraged ETFs, an investor earns 2X or 3X the daily returns of the target index (less fees and borrowing costs), so the daily arithmetic mean of the 2X ETF is 2X the daily arithmetic mean of the target index, and the daily arithmetic mean of the 3X ETF is 3X the daily arithmetic mean of the target index.<\/p>\n

However, since the daily variance is also increased, the daily geometric means do not go up by the same 2X or 3X factor.\u00a0\u00a0 If we use our approximation equation from above, we can estimate the geometric return for the target index and for the 2X and 3X leveraged ETFs.<\/p>\n

Geometric Mean Return of Target Index:<\/strong><\/p>\n

<\/p>\n

Geometric Mean Return of 2X Leveraged Fund:<\/strong><\/p>\n

<\/p>\n

Geometric Mean Return of 3X Leveraged Fund:<\/strong><\/p>\n

<\/p>\n

Empirical Data vs. Theoretical Estimates<\/strong><\/h5>\n

Since our equation for estimating the geometric mean from the arithmetic mean and standard deviation is only an approximation (and our daily return data is a “fat tailed” distribution), it is a good idea to check the approximation equation against the empirical data.<\/p>\n

In the plot below, I\u2019ve used the actual daily returns of the U.S. market (based on the CRSP daily market return data) from 1964 through 2010.\u00a0 For each year in the sample, I\u2019ve calculated the daily arithmetic and geometric means, and I\u2019ve also calculated the geometric and arithmetic means after doubling and tripling the daily returns.<\/p>\n

The individual points on the plot show the difference between the geometric and arithmetic means vs. the daily standard deviation over each year.\u00a0 The lines in the plot are based on the approximation equation.\u00a0 Note that the curves match up very well with the empirical data and validate our approximation equation.<\/p>\n

<\/a><\/a><\/strong><\/p>\n

Interestingly, the empirical data points that show a large error relative to our approximation equation are based on the return averages for 1987\u2026which, of course, had one huge down day (October 19th or Black Monday) which was many standard deviations from the mean daily return.<\/p>\n

Since the\u00a0difference between the actual\u00a0arithmetic and geometric mean\u00a0returns match up very\u00a0well with the approximation equation, we can use this equation to generate a reasonable prediction of\u00a0what will happen under various volatility scenarios.<\/p>\n

The arithmetic mean of the daily return over the full range of our data (1964-2010) is about 0.04% per day.\u00a0 The plot below shows how the geometric means vary with standard deviation for this “typical” level of daily return.<\/p>\n

<\/a><\/p>\n

This plot shows that although the leveraged ETFs can outperform under some scenarios, they will\u00a0do relatively poorly when volatility is high.\u00a0 The crossover points will occur at lower volatilities in years when the daily returns have an arithmetic mean of less than 0.04%, and they will occur at higher volatilities in years when the arithmetic mean is higher.\u00a0\u00a0Of course, the leveraged funds\u00a0will always underperform for years with negative arithmetic mean.\u00a0 Also, if we consider the borrowing costs, the leveraged ETFs will do even worse.<\/p>\n

Conclusion<\/h5>\n

As the final graph shows, leveraged ETFs have some undesirable characteristics which investors should take the time to understand before investing in these funds. \u00a0It is a common misconception that these funds will return 2X or 3X the index returns over the long term, and, as the analysis shows, they\u00a0are certain to\u00a0fall short of this level of return.\u00a0\u00a0In fact, there is a\u00a0strong possibility that they will return even less than the the underlying target index.<\/p>\n

If you feel you are capable of predicting short term market moves (good luck!), these funds may be an appropriate tool. \u00a0However,\u00a0they are probably not a good choice for the long term investor.<\/p>\n

Additional Info:<\/h5>\n

Data:<\/strong><\/p>\n

Daily CRSP market return data can be found at the Kenneth French data library<\/a>.<\/p>\n

Code:<\/strong><\/p>\n

This R code was used to generate the plot comparing the historical data to the approximation equation:<\/p>\n

\r\n# Download Daily Return Data\r\nff_daily <- read.table("F-F_Research_Data_Factors_daily.txt")\r\n\r\n# Add daily excess return to daily t-bill return to get total return\r\ndret <- ff_daily[,2] + ff_daily[,5]\r\n\r\n# Compute stats for full sample\r\ndarith <- mean(dret)\r\ndstd <- sd(dret)\r\n\r\n# Convert "date" column to year only for easy sorting\r\nyears <- trunc(ff_daily[,1]\/10000)\r\n\r\n# bind years variable to daily return variable\r\ndata <- cbind(years,dret)\r\n\r\n# Init variables\r\nd1x_a <- NULL\r\nd1x_sd <- NULL\r\nd1x_g <- NULL\r\nd2x_a <- NULL\r\nd2x_sd <- NULL\r\nd2x_g <- NULL\r\nd3x_a <- NULL\r\nd3x_sd <- NULL\r\nd3x_g <- NULL\r\n\r\n# Calculate arithmetic and geometric returns for every year from 1964-2010\r\nfor(i in 1:47) {\r\n  returns <- subset(data,years==i+1963)\r\n  d1x_a[i] <- mean(returns[,2])\r\n  d1x_sd[i] <- sd(returns[,2])\r\n  d1x_g[i] <- 100*(exp(mean(log(returns[,2]\/100+1)))-1)\r\n\r\n  d2x_a[i] <- mean(2*returns[,2])\r\n  d2x_sd[i] <- sd(2*returns[,2])\r\n  d2x_g[i] <- 100*(exp(mean(log(2*returns[,2]\/100+1)))-1)\r\n\r\n  d3x_a[i] <- mean(3*returns[,2])\r\n  d3x_sd[i] <- sd(3*returns[,2])\r\n  d3x_g[i] <- 100*(exp(mean(log(3*returns[,2]\/100+1)))-1)\r\n}\r\n\r\n# Calculate actual mean differences (geometric mean - arithmetic mean)\r\nrdiff_1x <- d1x_g - d1x_a\r\nrdiff_2x <- d2x_g - d2x_a\r\nrdiff_3x <- d3x_g - d3x_a\r\n\r\n# Calculate Theoretical Returns using approximation\r\nrth_1 <- -50*((sort(d1x_sd)\/100)^2)\r\nrth_2 <- -50*((sort(d2x_sd)\/100)^2)\r\nrth_3 <- -50*((sort(d3x_sd)\/100)^2)\r\n\r\n# Plot Actual and Theoretical Data\r\ndev.new(width=12,height=8)\r\nrds <- cbind(rdiff_1x,rdiff_2x,rdiff_3x)\r\nmatplot(d1x_sd,rds,xlab="",ylab="",type="p",pch=c(1,2,3),lwd=c(2,2,2),cex.axis=1.3)\r\nlegend(0.35,-0.1,c("S&P 1X Actual","S&P 2X Actual","S&P 3X Actual", "S&P 1X Theoretical","S&P 2X Theoretical", "S&P 3X Theoretical"),col=c(1,2,3,1,2,3),pch=c(1,2,3,NA,NA,NA),lty=c(0,0,0,1,1,1),lwd=c(0,0,0,3,3,3),cex=1.75)\r\ntitle("Return Difference vs. Standard Deviation",xlab="Daily Standard Deviation (%)",ylab="Geometric Mean - Arithmetic Mean (%)",cex.main="2.25",cex.lab="1.75")\r\nlines(sort(d1x_sd),rth_1,lwd=2,col=1)\r\nlines(sort(d1x_sd),rth_2,lwd=2,col=2)\r\nlines(sort(d1x_sd),rth_3,lwd=2,col=3)\r\n<\/pre>\n
This R code was used to generate the second plot based on the approximation equation and the long term arithmetic average:<\/p>\n
\r\n\r\n# Application Leveraged ETFs\r\n\r\n# Set daily arithmetic mean to 0.04%\r\ndailymean <- 0.0004\r\n\r\n# Create a vector of standard deviations\r\nstds <- 0:10 * 0.0025\r\n\r\n# Calculate Geomean estimate using arithmean and standard deviations\r\nsp1x <- dailymean - 0.5*stds^2\r\nsp2x <- 2*dailymean - 0.5*(2*stds)^2\r\nsp3x <- 3*dailymean - 0.5*(3*stds)^2\r\n\r\n# Bind series and plot\r\nmeans <- cbind(sp1x,sp2x,sp3x)\r\ndev.new(width=12,height=8)\r\nmatplot(100*stds,100*means,type="l",xlab="",ylab="",lwd=c(3,3,3),cex.axis=1.3)\r\nlegend(1.75,0.12,c("S&P 1X","S&P 2X","S&P 3X"),lwd=c(3,3,3),col=c(1,2,3),cex=1.75)\r\ntitle("Geometric Return vs. Standard Deviation",xlab="Daily Standard Deviation (%)",ylab="Daily Geometric Return (%)",cex.main="2.25",cex.lab="1.75")\r\n<\/pre>\n","protected":false},"excerpt":{"rendered":"
An Application of the Geometric and Arithmetic Means Over the past several years there has been a proliferation of ETFs for every imaginable purpose. One particularly interesting type of ETF is the leveraged ETF. These ETFs are designed to provide 2X or 3X the daily return of a target index such as the S&P500, and […]<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/posts\/2647"}],"collection":[{"href":"https:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/comments?post=2647"}],"version-history":[{"count":116,"href":"https:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/posts\/2647\/revisions"}],"predecessor-version":[{"id":2777,"href":"https:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/posts\/2647\/revisions\/2777"}],"wp:attachment":[{"href":"https:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/media?parent=2647"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/categories?post=2647"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.calculatinginvestor.com\/wp-json\/wp\/v2\/tags?post=2647"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}