The conventional way to show volatility drag for a daily leveraged ETF goes something like this. Suppose the market goes up 25% one day and down 20% the next. An investor with a leverage of one ends up back where they started:

1.25 * 0.8 = 1

While an investor with a leverage of two ends up down ten percent:

1.5 * 0.6 = 0.9

There are two things at work here that make the leveraged ETF look extremely bad. One is that the market returned zero percent over the time frame, the other is the extreme volatility. A 25% gain followed by a 20% loss corresponds to a VIX level just above 350. Historically, the market has returned more than zero percent and has had a VIX level below 350.

What if a leverage of 0.5 were used instead of 1 or 2? The investor would actually make money.

1.125 * 0.9 = 1.0125

Why is a leverage of less than one never discussed? Isn't this just as much a result of volatility drag as in the two times leverage case? Maybe volatility drag only exists when leverage is greater than one? Then what exists when leverage is less than one? Volatility boost? What about leverage less than 0?

In order to look further into this I'll use the adjusted closing value of SPY, as provided by Yahoo from January 29th, 1993 to September 24th, 2014, to examine varying values of leverage. I'll define x as the adjusted close of a day divided by the adjusted close of the previous day. Then, to calculate the theoretical leveraged return (I'll denote this as y), the following equation can be used:

y = (x - 1) * leverage + 1

When leverage equals one this can be seen to simplify to:

y = x

When leverage equals two this simplifies to:

y = 2 * x - 1

and when leverage equals 0.5 this simplifies to:

y = 0.5 * x + 0.5

Now I realize that many assumptions have been made, but it is the purpose of this article to examine the effects of leverage on volatility drag. Because of all of the simplifying assumptions, the absolute results won't be exact (for example, the performance of a leverage 2 ETF), but relative results (for example, the performance of leverage 2 compared to leverage 2.1) will still be insightful.

The leveraged equation can be applied to all 5,453 days contained in the date range noted above, and a CAGR can be calculated as the product of all the days raised to the (252/5453), then subtracting one.

For a leverage of 0 this gives a CAGR of 0.00%.

For a leverage of 0.5 this gives a CAGR of 5.00%.

For a leverage of 1.0 this gives a CAGR of 9.26%.

For a leverage of 2.0 this gives a CAGR of 15.10%

This is then calculated for all leverages from 0 to 5. The CAGR can then be plotted against leverage for SPY. The results are shown below.

This shows that a leverage just shy of three, rebalanced daily, would have returned the largest return over the time period in question.

Now if we assume that a leverage of 0 has no volatility drag, and a leverage of 1 has no volatility drag, we can draw a line passing between those two points.

Now volatility drag can be thought of as the difference between the orange line and the blue line. It looks to be about 3.5% for a leverage of two and 11% for a leverage of three (yikes). But also notice that the blue line is above the orange line between a leverage of 0 and 1. To see this easier, we can plot the blue line minus the orange line and zoom in a bit.

So while it is easy to see that volatility drag is harmful above a leverage of 1, it is also helpful at a leverage between 0 and 1.

This same plot can be generated for any stock or ETF (the values will be different but the shape will be the same) so we can draw some generalities. What this effectively means is that a leverage of 0.5 has a higher Sharpe ratio than a leverage of 1 assuming a risk free rate of 0%. Or that you can increase your return by holding 50% of two uncorrelated assets (diversification) that have the same expected return and rebalancing daily. Even the leverage of 0.5 can be seen as a diversification between equities and cash (which happens to be perfectly uncorrelated but cash generally has a lower expected return). We started with leveraged ETFs and have now shown that diversification and rebalancing are good things. That's certainly a good sanity check.

My only gripe with this is that volatility drag hurts performance with leverage greater than 1, but helps with leverage between 0 and 1. It seems inconsistent for volatility "drag" to be causing a "boost". What if, instead, we draw the straight line still passing through 0 at 0%, but now tangent to the expected return parabola. (This requires calculus and I'd rather not discuss the specifics unless someone asks).

And once again, subtracting the orange line from the blue line.

Now we have volatility drag everywhere. Yes, it is worse at 2 than 1, and yes it is much worse at 3 than 2, but it still exists everywhere. In my opinion it doesn't matter which way you calculate drag. In the first method, a leverage of 0.5 is better than a leverage of 1.0, while in the second method a leverage of 0.5 isn't as bad as a leverage of 1.0. It's like saying a penny saved is a penny earned. I personally prefer the second method - but this method also suggests that volatility drag exists at leverage 1. This would mean that even SPY suffers volatility drag.

Now I've already pointed out that holding two positions at 50% each results in less drag than one position at 100%. But you might also be able to see that four positions at 25% each have even less drag, and 100 positions at 1% each have even less thanks to the parabolic nature of the curve.

Now the point of this article wasn't to show that SPY suffers from volatility drag. It really just depends on how you define volatility drag. The purpose was to show that volatility drag doesn't exist in 2x and 3x leveraged ETFs alone. Volatility drag is a continuously decreasing function of leverage. Sure, drag is worse at 2x and 3x leverage, but the returns are also better (in this example - not always the case). There are plenty of times where holding 1x leverage outperforms 2x leverage, but there are also plenty of times when 0.5x leverage outperforms 1x leverage. It's all relative.

Perhaps you decide that a leverage of 1.1x SPY looks just right for you. You can accomplish this with a weighted average of 95% SPY and 5% UPRO (3x SPY) and hold indefinitely. To exactly match the theory presented here, you would need to rebalance daily (not strictly necessary) but I wont get into the results here if you choose not to rebalance daily. What is easy to see is that in an increasing market, UPRO will increase faster than SPY, increasing its weighting and your leverage. SPY and UPRO also become less correlated over time, (increasing the benefits of diversification). This also results in a very reasonable weighted expense ratio.

One final point of interest. In option terms, a leveraged ETF has a constant delta and as a result, 0 gamma. Gamma could allow a leveraged ETF to avoid volatility drag. If a 2x leveraged ETF also had a gamma of 2 and speed of 0 (a 3x leveraged ETF with a delta of 3 requires a gamma of 6 and speed of 6 - these values can be derived by taking the derivatives of x^2 and x^3), then it would not suffer from volatility drag. But you get this gamma and speed with an option by paying a premium. This premium is "paid" by a leveraged ETF with volatility drag. It's not a coincidence that volatility drag and option premiums both increase with volatility.

**Disclosure:** The author is long UPRO.

The author wrote this article themselves, and it expresses their own opinions. The author is not receiving compensation for it. The author has no business relationship with any company whose stock is mentioned in this article.