mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field.
In general, there exist two separate branches of finance that req ...
term first published by Rick Ashburn, CFA in a 2003 column, and formalized by
hedge fund
A hedge fund is a Pooling (resource management), pooled investment fund that holds Market liquidity, liquid assets and that makes use of complex trader (finance), trading and risk management techniques to aim to improve investment performance and ...
manager
Mark Spitznagel
Mark Spitznagel (; born March 5, 1971) is an American investor and hedge fund, hedge fund manager. He is the founder, owner, and chief investment officer of Universa Investments, a hedge fund management firm based in Miami, Florida.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 branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.
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and geometric average (or “ ensemble average” 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 Liability (financial accounting), liabilities, often in relation to open-end fund, open-end, mutual fund, mutual funds, Hedge fund, hedge funds, and Venture capital, v ...
standard deviation
In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
(or “volatility”) , and geometric average return
:
So the geometric average return is the difference between the arithmetic average return and a function of volatility. This function of volatility
:
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 investor and philanthropist who currently serves as the chairman and CEO of the conglomerate holding company Berkshire Hathaway. As a result of his investment success, Buffett is ...
’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 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.”
Arithmetic mean
In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...
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Compound interest
Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.
Compo ...
Exponential growth
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast ...
Geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
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Log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
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Mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field.
In general, there exist two separate branches of finance that req ...
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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 over a specified time period, such as i ...