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The volatility tax is a
mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...
term, formalized by
hedge fund A hedge fund is a pooled investment fund that trades in relatively liquid assets and is able to make extensive use of more complex trading, portfolio-construction, and risk management techniques in an attempt to improve performance, such as ...
manager Mark Spitznagel, describing the effect of large investment losses (or volatility) on compound returns.Not all risk mitigation is created equal
''Pensions & Investments'', November 20, 2017
It has also been called volatility drag, volatility decay or variance drain. This is not literally a tax in the sense of a levy imposed by a government, but the mathematical difference between geometric averages compared to arithmetic averages. This difference resembles a tax due to the mathematics which impose a lower compound return when returns vary over time, compared to a simple sum of returns. This diminishment of returns is in increasing proportion to volatility, such that volatility itself appears to be the basis of a progressive tax. Conversely, fixed-return investments (which have no return volatility) appear to be "volatility tax free".


Overview

As Spitznagel wrote: Quantitatively, the volatility tax is the difference between the
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
and geometric average (or “
ensemble average In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents ...
” and “time average”) returns of an asset or portfolio. It thus represents the degree of “ non-ergodicity” of the geometric average. Standard quantitative finance assumes that a portfolio’s
net asset value Net asset value (NAV) is the value of an entity's assets minus the value of its liabilities, often in relation to open-end, mutual funds, hedge funds, and venture capital funds. Shares of such funds registered with the U.S. Securities and Exc ...
changes follow a
geometric Brownian motion A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It i ...
(and thus are log-normally distributed) with arithmetic average return (or “
drift Drift or Drifts may refer to: Geography * Drift or ford (crossing) of a river * Drift, Kentucky, unincorporated community in the United States * In Cornwall, England: ** Drift, Cornwall, village ** Drift Reservoir, associated with the village ...
”) \mu,
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
(or “volatility”) \sigma, and geometric average return :\mu-\sigma^2/2 So the geometric average return is the difference between the arithmetic average return and a function of volatility. This function of volatility :\sigma^2/2 represents the volatility tax. (Though this formula is under the assumption of log-normality, the volatility tax provides an accurate approximation for most return distributions. The precise formula is a function of the central moments of the return distribution.) The mathematics behind the volatility tax is such that a very large portfolio loss has a disproportionate impact on the volatility tax that it pays and, as Spitznagel wrote, this is why the most effective risk mitigation focuses on large losses: According to Spitznagel, the goal of risk mitigation strategies is to solve this “vexing non-ergodicity, volatility tax problem” and thus raise a portfolio’s geometric average return, or CAGR, by lowering its volatility tax (and “narrow the gap between our ensemble and time averages”). This is “the very name of the game in successful investing. It is the key to the kingdom, and explains in a nutshell
Warren Buffett Warren Edward Buffett ( ; born August 30, 1930) is an American business magnate, investor, and philanthropist. He is currently the chairman and CEO of Berkshire Hathaway. He is one of the most successful investors in the world and has a net ...
’s cardinal rule, ‘Don’t lose money.’”''The Volatility Tax''
Universa Investments, February 2018
Moreover, “the good news is the entire hedge fund industry basically exists to help with this—to help save on volatility taxes paid by portfolios. The bad news is they haven't done that, not at all.” As
Nassim Nicholas Taleb Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist whose work concerns problems of randomness ...
wrote in his 2018 book '' Skin in the Game'', “more than two decades ago, practitioners such as Mark Spitznagel and myself built our entire business careers around the effect of the difference between ensemble and time.”


See also

*
Annual growth % Annual growth rate (AGR) is the change in the value of a measurement over the period of a year. Economics Annual growth rate is a useful tool to identify trends in investments. According to a survey of nearly 200 senior marketing managers conduc ...
*
Arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
* Compound interest *
Ecological fallacy An ecological fallacy (also ecological ''inference'' fallacy or population fallacy) is a formal fallacy in the interpretation of statistical data that occurs when inferences about the nature of individuals are deduced from inferences about the g ...
(Averages do not predict individual performance) *
Exponential growth Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
*
Geometric Brownian motion A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It i ...
*
Geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
*
Log-normal distribution In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a norma ...
*
Mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...
*
Rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, cas ...


References

{{reflist Interest Mathematical finance Exponentials Risk management