# Testing leveraged ETF decay: Are leveraged ETFs really a bad long-term investment? [2021]

In this article we will look at leveraged ETFs and how they act long-term (known as leveraged ETF decay). The conventional wisdom is that you should never hold a leveraged ETF for longer than the rebalance period (often one day), since there is significant decay in value over that time.

But is this really true? Can leveraged ETFs perform well long term? Let’s find out.

## What is leveraged ETF decay?

Let’s take a look at a simple example. Say we have an underlying index that starts at $100/share. Over the next 4 days it goes up or down with equal probability. Our returns look something like this:

Instrument | Day 1 Value | Day 2 Value | Day 3 Value | Day 4 Value | Day 5 Value |
---|---|---|---|---|---|

Index | $100 | $90 | $100 | $90 | $100 |

Now, let’s take a look at what a daily rebalanced leveraged ETF would do over the same period, assuming it also starts at $100/share:

Instrument | Day 1 Value | Day 2 Value | Day 3 Value | Day 4 Value | Day 5 Value |
---|---|---|---|---|---|

Index | $100 | $90 | $100 | $90 | $100 |

Percent change | -10% | +11% | -10% | +11% | |

2x Percent change | -20% | +22% | -20% | +22% | |

2x leveraged ETF value | $100 | $80.00 | $97.60 | $78.08 | $95.26 |

As you can see, the value of the 2x leveraged ETF decays over time, even if the underlying index remains flat. The larger the up and down movement of the index, the larger the decay.

## Suggested reading

- Path-Dependence Properties of Leveraged Exchange-Traded Funds: Compounding, Volatility and Option Pricing
- Do Stocks Exhibit Momentum? A reality check [2021]

## Relating a leveraged ETF to its underlying ETF

In the linked article above, Zhang derives a relationship between percentage change in the underlying ETF and percentage change in the leveraged ETF. That relationship is a bit complicated, but we can approximate it by this:

%∆L ≈ (%∆S)^{β} * exp(.5(β-β^{2})σ^{2}∆t)

where β is the leverage ratio (for example 3 for a 3x leveraged ETF) and σ^{2} is the variance of the underlying ETF.

You can see that the exp(…) term will be negative for betas > 1, meaning that the change in the leveraged ETF is decaying based on the volatility.

Let’s see if this formula is accurate in practice. From August 12, 2020 to August 12, 2021, the QQQ gained 40.0%, and TQQQ (the 3x ETF) gained 213.8%, so this equation should hold

2.138 ≈ (1.400)^{3} * exp(.5(-6)σ^{2}(252))

where the variance of QQQ over that time is .0135^{2}. It turns out that the equation is roughly correct, although it diverges over longer time periods.

What’s clear from all of this is that there is significant decay in relation to the underlying ETF. But does that mean leveraged ETFs are bad long-term investments?

We leave that up to you to decide.

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