# Alpay Kaya, CFA

La vérité existe. On n'invente que le mensonge. —Braque (Truth exists. One only invents falsehood.)

# Are -1x Inverse Funds ‘Leveraged’?

It may be easy to say “no” since +1 and -1 are the same magnitude, meaning the percent change of the -1x fund is equal and opposite to that of the +1x fund. On the other hand, investors have noticed inverse funds exhibiting the value decay consistent with leveraged ETFs. Obviously, something is different and worthy of considered analysis.

EXAMPLE  Consider a 2-day example summarized below, where  $r_{\;\!\! i}^{}$  is continuously calculated (log) return and  $a_i^{}$  is percent change.

$\begin{array}{lcccc} & r_{\;\!\! 1}^{} & a_1^{} & r_{\;\!\! 2}^{} & a_2^{} \\[0.75ex] \mathtt{+1x\ ETF\ } & +4\% & +4.08\% & -4\% & -3.92\% \\ \mathtt{-1x\ ETF\ } & -4.17\% & -4.08\% & +3.85\% & +3.92\% \\ \mathtt{Leverage\ } & -1.04 & -1 & -0.96 & -1 \end{array}$

The ‘logical flow’ for each day is a clockwise circle:

$r_{\;\!\! i}^{} (\mathtt{index}) = r_{\;\!\! i}^{} (\mathtt{+1x}) \;\rightarrow\; a_i^{} (\mathtt{+1x}) \;\rightarrow\; a_i^{} (\mathtt{-1x}) \;\rightarrow\; r_{\;\!\! i}^{} (\mathtt{-1x})$

The final value of the index (and +1x ETF) equals its initial value since the log returns sum to zero.  The numbers in the chart are consistent with the general case.

1. When the index is up, the -1x ETF loses value with an effective log leverage magnitude greater than 1; and
2. When the index is down, the -1x ETF gains value with an effective log leverage magnitude less than 1.

In both cases, the arithmetic return leverage is a constant negative one. The resulting price series is shown here.

$\begin{array}{lccc} & \mathtt{P_0^{}} & \mathtt{P_1^{}} & \mathtt{P_2^{}} \\[0.75ex] \mathtt{+1x\ ETF\ } & 100 & 104.08 & 100 \\ \mathtt{-1x\ ETF\ } & 100 & 95.92 & 99.68 \end{array}$

But |-1| = |+1| so how is that possible? The behavior of asset prices is a few steps above (mathematically speaking) the basic arithmetic employed by some people (including some finance ‘professionals’).

These -1x inverse ETFs do share a characteristic with leveraged ETFs, the effective log leverage (which is the correct way of looking at such things) becomes warped. . . smaller-magnitude than nominal when the ETF gains value and larger-magnitude when it loses value. It can be shown analytically that -1x inverse ETFs have the same negative drift rate as +2x leveraged ETFs.

The difference between -1x and the tracked index (and +1x funds) becomes a matter of direction. . . why does growth imply a larger-magnitude percent change than decay? If a system evolves continuously as a function of its contemporaneous state of being, growth to infinity and decay to zero are symmetric concepts. This must be internalized before one can understand LETFs in particular or the math of finance in general.

So back to the title’s question: are -1x inverse funds ‘leveraged’? The best response is that the term leverage misses the point. The point is bias, and yes, inverse ETFs’ returns are biased. . . just as (long & short) leveraged ETFs’ returns are biased.