I extensively explain the math of volatility decay here, but in a nutshell, this is where the myth that leveraged ETFs are ineffective for long-term investment started:

(1 — x)(1 + x) = 1 — x²

Let’s clarify this with an example. Imagine the market declines by x% one day and then increases by x% the next day. For instance, the market falls by 10% and subsequently rises by 10%. The net result is (1−0.1)×(1+0.1)=0.99, which equates to a decrease of 0.01, or 1%.

This can also be calculated with the leg 1-x² like this 1–0.1² = 0.99.

So whenever the market experiences volatility, the math says we lose money. The greater the value of x, the larger the impact of volatility drag. So for a leverage ETF that returns 2 times or 3 times of x, we will have multiples of the volatility drag.

But remember, even with a leverage of 1 (i.e., no leverage at all), we also have volatility drag. When a stock or ETF declines by 10%, it requires an 11% increase to return to the original level. If it drops 20%, it needs a 25% gain to recover. A 50% fall requires a 100% increase… and so on.

So yes, 2X leverage has drag… but so does 1X leverage!

The proponents of the myth have overlooked two critical factors that determine the returns of leveraged ETFs: benchmark returns and benchmark volatility (with the benchmark being, for example, the Nasdaq 100 or the S&P 500).

However, it is important to note that while the return is proportional to the leverage, the drag is also proportional to the square of the leverage. As a result, there is a threshold beyond which the volatility drag will outweigh the additional returns gained from leverage. Therefore, there’s a limit to the amount of leverage that can be effectively employed.

The paper from Tony Cooper ran the numbers and created a model that I have further applied to determine the current ideal leverage for any asset.

In a nutshell, it calculates the optimal leverage using the following formula:

optimal_leverage = mean_return / (volatility^2)

Optimal leverage achieves a balance by maximizing the difference between the amplified mean return and the volatility drag.

So after running the math, they produced a chart indicating that, over a 135-year period, the ideal leverage for the US stock market would have been 2X:

According to large data samples, the ideal leverage for different indexes would have been:

S&P 500 (60 years of data): 3X leverage
Dow Jones (80 years of data): 2X leverage
Nasdaq 100 (40 years of data): 2X leverage
Russell 2000 (22 years of data): 2X leverage

Here’s a (very short) list of Leveraged ETFs:

SSO — ProShares Ultra S&P 500 (2x the S&P 500).
UPRO — ProShares UltraPro S&P500 (3x the S&P 500).
QLD — ProShares Ultra QQQ (2x the Nasdaq 100).
DDM — ProShares Ultra Dow30 (2x the Dow 30).

You don’t need to leverage your S&P 500 by 3X. Perhaps adopting a conservative approach with 2X leverage might be preferable. Alternatively, you could leverage the Nasdaq 100 by 1.5X — instead of the 2X previously discussed — by combining QQQ and QLD.

The point is that having a moderate amount of leverage can improve your returns while still aligning with your risk tolerance.

Remember:

Leverage itself is not inherently bad; rather, the challenge lies in identifying the optimal leverage level.
Volatility drag affects both leveraged and non-leveraged investments, not exclusively leveraged ETFs.
Historical data shows that 2X leverage has been optimal for various indexes over extended periods.
The optimal leverage is directly proportional to the ratio of returns to the square of volatility.
My “Optimal Leverage Indicator” can help you determine the current optimal leverage for any given asset.