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The '''volatility tax''' is a [[mathematical finance]] term,<ref>https://www.tastytrade.com/tt/shows/from-theory-to-practice/episodes/the-volatility-tax-08-26-2016</ref> devised originally by [[hedge fund]] manager [[Mark Spitznagel]] in a [http://www.pionline.com/article/20171120/ONLINE/171119874/commentary-not-all-risk-mitigation-is-created-equal| 2017 ''Pensions & Investments'' article], describing the effect of large investment losses (or [[Volatility (finance)|volatility]]) on [[Compound interest|compound returns]].<ref name="VolTax1">[http://www.pionline.com/article/20171120/ONLINE/171119874/commentary-not-all-risk-mitigation-is-created-equal Not all risk mitigation is created equal], ''Pensions & Investments'', November 20, 2017</ref> It is also known as "volatility drag"<ref>https://blogs.cfainstitute.org/investor/2015/03/23/the-myth-of-volatility-drag-part-1/</ref>
The '''volatility tax''' is a [[mathematical finance]] term, formalized by [[hedge fund]] manager [[Mark Spitznagel]], describing the effect of large investment losses (or [[Volatility (finance)|volatility]]) on [[Compound interest|compound returns]].<ref name="VolTax1">[http://www.pionline.com/article/20171120/ONLINE/171119874/commentary-not-all-risk-mitigation-is-created-equal Not all risk mitigation is created equal], ''Pensions & Investments'', November 20, 2017</ref> It has also been called “volatility drag”.<ref>https://blogs.cfainstitute.org/investor/2015/03/23/the-myth-of-volatility-drag-part-1/</ref>


==Overview==
==Overview==
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{{quote|It is well known that steep portfolio losses crush long-run [[Compound annual growth rate|compound annual growth rates (CAGRs)]]. It just takes too long to recover from a much lower starting point: lose 50% and you need to make 100% to get back to even. I call this cost that transforms, in this case, a portfolio’s +25% average arithmetic return into a zero CAGR (and hence leaves the portfolio with zero profit) the “volatility tax”: it is a hidden, deceptive fee levied on investors by the negative compounding of the markets’ swings.<ref name="VolTax1"/>}}
{{quote|It is well known that steep portfolio losses crush long-run [[Compound annual growth rate|compound annual growth rates (CAGRs)]]. It just takes too long to recover from a much lower starting point: lose 50% and you need to make 100% to get back to even. I call this cost that transforms, in this case, a portfolio’s +25% average arithmetic return into a zero CAGR (and hence leaves the portfolio with zero profit) the “volatility tax”: it is a hidden, deceptive fee levied on investors by the negative compounding of the markets’ swings.<ref name="VolTax1"/>}}


Quantitatively, the volatility tax is the difference between the [[Arithmetic mean|arithmetic]] and [[Geometric mean|geometric average]] returns of an asset or portfolio.
Quantitatively, the volatility tax is the difference between the [[Arithmetic mean|arithmetic]] and [[Geometric mean|geometric average]] (or “[[ensemble average]]” and “time average”) returns of an asset or portfolio. It thus represents the degree of “[[Ergodic process|non-ergodicity]]” of the geometric average.


Standard quantitative finance assumes that a portfolio’s [[net asset value]] changes follow a [[geometric Brownian motion]] (and thus are [[Log-normal distribution|log-normally distributed]]) with arithmetic average return (or “[[Stochastic drift|drift]]”) <math>\mu</math>, [[standard deviation]] (or “volatility”) <math>\sigma</math>, and geometric average return
Standard quantitative finance assumes that a portfolio’s [[net asset value]] changes follow a [[geometric Brownian motion]] (and thus are [[Log-normal distribution|log-normally distributed]]) with arithmetic average return (or “[[Stochastic drift|drift]]”) <math>\mu</math>, [[standard deviation]] (or “volatility”) <math>\sigma</math>, and geometric average return
Line 17: Line 17:
:<math>\sigma^2/2</math>
:<math>\sigma^2/2</math>


represents the volatility tax (under the assumption of log-normality).
represents the volatility tax. (Though this is under the assumption of log-normality, the volatility tax is in fact independent of the assumed or actual 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:
The volatility tax equation is a [[quadratic function]] of <math>\sigma</math> (or “[[variance]]”). Extreme returns (typically extreme losses, or [[Market crash|market “crashes”]]) thus have an [[Exponential function|exponentially magnified]] impact on the volatility tax—their [[Square (algebra)|squared]] deviations (in the mathematical calculation of standard deviation) magnify their impact on <math>\sigma</math>, which is then itself squared to further magnify the impact on the volatility tax <math>\sigma^2/2</math>.
{{quote|We can see how this works by considering that the compound (or geometric) average return is mathematically just the average of the [[logarithms]] of the arithmetic price changes. Because the logarithm is a [[concave function]] (it curves down), it increasingly penalizes negative arithmetic returns the more negative they are, and thus the more negative they are, the more they lower the compound average relative to the arithmetic average—and raise the volatility tax.<ref name="VolTax3">[http://www.pionline.com/article/20180309/ONLINE/180309846/commentary-thanks-to-volatility-you-cant-always-get-what-you-want-in-investing Thanks to volatility, you can’t always get what you want in investing], ''Pensions & Investments'', March 9, 2018</ref>}}


According to Spitznagel, the goal of risk mitigation strategies is to raise a portfolio’s geometric average return, or CAGR, by lowering its <math>\sigma^2/2</math> more than they lower its <math>\mu</math>. 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]]’s cardinal rule, ‘Don’t lose money.’” (Very few, if any, risk mitigation strategies or hedge funds are successful at accomplishing this goal—as they tend instead to lower a portfolio’s <math>\mu</math> more than they lower its <math>\sigma^2/2</math>.)<ref name="VolTax2">[https://www.scribd.com/document/370967187/Universa-Mark-Spitznagel-Volatility-Tax ''The Volatility Tax''], Universa Investments, February 2018</ref>
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”).<ref name="VolTax3"/> 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]]’s cardinal rule, ‘Don’t lose money.’”<ref name="VolTax2">[https://www.scribd.com/document/370967187/Universa-Mark-Spitznagel-Volatility-Tax ''The Volatility Tax''], Universa Investments, February 2018</ref>

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.”<ref name="VolTax3"/>


== See also ==
== See also ==

Revision as of 15:56, 28 March 2018

The volatility tax is a mathematical finance term, formalized by hedge fund manager Mark Spitznagel, describing the effect of large investment losses (or volatility) on compound returns.[1] It has also been called “volatility drag”.[2]

Overview

As Spitznagel wrote:

It is well known that steep portfolio losses crush long-run compound annual growth rates (CAGRs). It just takes too long to recover from a much lower starting point: lose 50% and you need to make 100% to get back to even. I call this cost that transforms, in this case, a portfolio’s +25% average arithmetic return into a zero CAGR (and hence leaves the portfolio with zero profit) the “volatility tax”: it is a hidden, deceptive fee levied on investors by the negative compounding of the markets’ swings.[1]

Quantitatively, the volatility tax is the difference between the arithmetic 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 changes follow a geometric Brownian motion (and thus are log-normally distributed) with arithmetic average return (or “drift”) , standard deviation (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 is under the assumption of log-normality, the volatility tax is in fact independent of the assumed or actual 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:

We can see how this works by considering that the compound (or geometric) average return is mathematically just the average of the logarithms of the arithmetic price changes. Because the logarithm is a concave function (it curves down), it increasingly penalizes negative arithmetic returns the more negative they are, and thus the more negative they are, the more they lower the compound average relative to the arithmetic average—and raise the volatility tax.[3]

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”).[3] 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’s cardinal rule, ‘Don’t lose money.’”[4]

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.”[3]

See also

References

  1. ^ a b Not all risk mitigation is created equal, Pensions & Investments, November 20, 2017
  2. ^ https://blogs.cfainstitute.org/investor/2015/03/23/the-myth-of-volatility-drag-part-1/
  3. ^ a b c Thanks to volatility, you can’t always get what you want in investing, Pensions & Investments, March 9, 2018
  4. ^ The Volatility Tax, Universa Investments, February 2018
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