Leveraged ETFs and return stacking: path dependence, costs and research

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Leveraged ETFs deliver a multiple of the index return on each trading day, and inverse ETFs the opposite. Over weeks and years, their result therefore depends on the price path: in strong trends they beat the simple multiple, in a zigzag market they lose. This article adds to leverage in a portfolio with scenarios, costs, the effect on markets, studies of investors and return stacking, in which leverage does not raise the equity allocation but makes room for a second building block.

How a leveraged ETF works

A leveraged ETF usually builds up its exposure through a total return swap or futures amounting to L times its assets. At the end of each trading day it adjusts that exposure: after a winning day it adds, after a losing day it reduces. This is the daily leverage reset. An inverse ETF does the same with negative leverage, such as −1 or −2. Besides ETFs, there are leveraged products in the form of ETNs or factor certificates; these are debt securities, so you also bear the risk of the issuer (issuer risk).

Trending versus sideways market

Without interest and costs, the following holds approximately, according to Avellaneda and Zhang (2010): fund value = (index level)^L × exp(−L × (L − 1) / 2 × σ² × T), with σ as the volatility and T in years (derivation under leverage in a portfolio). The first factor rewards trends, the second penalizes fluctuation. The table shows both for one year; the simple multiple of the index return is given in brackets.

In an uptrend, the 2x ETF ends above the double if the trend is strong enough for the fluctuation: at 15% volatility from an index return of about +18%, at 25% from about +33%, and at 30% only from about +42%. In a downtrend, it loses less than double, because compounding works in its favor in trends. In a sideways market it loses, and the more so the more the index fluctuates. This loss is called volatility decay, an intensified form of volatility drag. It grows with L × (L − 1): for 2x and inverse the factor is 2, for 3x it is 6.

In the long run: the mean rises, the median falls

If the index follows a geometric Brownian motion, the daily leveraged fund is also lognormally distributed. Two very different metrics follow from this: the expected value, which rises with leverage, and the median, i.e. the typical outcome, which falls beyond a certain level of leverage.

The growth-optimal leverage L* = (μ − r_fin) / σ² is the Kelly criterion. With the assumptions above, 2x and 3x are both above it. Over the years, leveraged ETFs therefore do well when the return is high, volatility is low and the market moves in trends; they do badly when volatility is high and there is no direction. The model calculation knows neither price jumps nor phases of high volatility. Both do additional damage to daily leveraged funds.

Costs: financing, swaps, ongoing charges

• Financing: the borrowed part costs (L − 1) × (money market rate + spread). At an interest rate of 3%, a 2x ETF therefore costs about 3% a year and a 3x ETF about 6%. Rising interest rates make leveraged products more expensive.
• Ongoing charges and swaps: on top of the management fee come swap spreads, which are not always included in the ongoing charges. They become visible in the tracking difference relative to the multiple.
• Trading: the daily re-leveraging costs spreads. Charupat and Miu (2011) found that deviations from the multiple are small for holding periods of up to a week and grow with longer holding periods.
• Premiums and discounts: the market price can deviate from the net asset value. On average the deviations are small, but large ones do occur (Charupat and Miu 2011).

How strongly the price path and costs act together was shown by FINRA, the US self-regulatory organization, with an example from the crisis winter of 2008/09: from December 2008 to April 2009, an index of US oil and gas stocks rose by 2%. An ETF offering twice the daily result fell by 6%, and the ETF offering twice the inverse daily result fell by 26% (FINRA 2009). This led to a warning to distributors:

End-of-day rebalancing and the markets

For the leverage to be right again the next morning, the fund has to trade shortly before the close. The amount required follows from a simple calculation (derivation below):

Does this amplify the move? Cheng and Madhavan (2009) pointed out early on that re-leveraging can increase volatility before the close. The findings since then are mixed. Shum et al. (2016) found, for 2006 to 2011, a positive, statistically significant relationship between end-of-day volatility and potential re-leveraging relative to trading volume; it was not economically significant everywhere, and was largest on the most turbulent days. Bai et al. (2015) show for real estate stocks that six leveraged and inverse ETFs moved their prices at the end of the day, increased volatility and amplified momentum. The results of Charupat and Miu (2011) are also consistent with higher volatility at the close.

Ivanov and Lenkey (2018) argue the opposite: investor inflows and outflows offset a large part of the re-leveraging, even in periods of stress. Once this is taken into account, the effect on end-of-day returns and volatility is economically insignificant. Our assessment: in large, liquid markets the re-leveraging hardly matters; in small segments and on days with large moves it matters more.

What the research shows about investors

According to Charupat and Miu (2011), leveraged ETFs are traded mainly by retail investors with very short holding periods, i.e. by the people the product is built for. Problems arise with long holding periods. Avellaneda and Zhang (2010) found that in 2008 most leveraged ETFs lagged behind the corresponding static strategies; their conclusion: as they are constructed, leveraged ETFs may not be suitable for buy-and-hold investors. If you want a fixed multiple over a longer period, you have to rebalance yourself; in their data, about one adjustment per week was enough.

Return stacking: leverage for a second building block

Return stacking uses leverage differently: not for more equities, but to hold two sources of return with the same capital. Examples are a 90/60 portfolio (90% equities, 60% bonds) or 100% equities plus 100% managed futures as an overlay via futures. The financing is embedded in the price of the futures (cost of carry) and corresponds roughly to the money market rate.

The idea is old. With portable alpha, investors separate the market return (beta) from the selection skill (alpha) and transfer the alpha to another portfolio, for example via futures on the target market (Kung and Pohlman 2004). Why leverage on low-volatility building blocks can pay off is explained by leverage aversion: because many investors do not want to or are not allowed to use leverage, safe assets offer more return per unit of risk (Asness et al. 2012; Frazzini and Pedersen 2014).

When stacking equities and trend following, correlation determines the risk. With 16% volatility for equities and 15% for the trend overlay, the total volatility is 19.62% at a correlation of −0.2, 21.93% at 0 and 25.00% at +0.3. That is considerably more than for equities alone. Whether an overlay improves the Sharpe ratio depends on its excess return after costs; more on this in the article CTAs and managed futures.

Correlations are not stable, however. According to Brixton et al. (2023), the correlation between equities and bonds was negative in the first two decades of the 21st century and often positive in the century before; greater inflation uncertainty can turn it positive again. And in bear markets, correlations between equity markets rise markedly (Longin and Solnik 2001). If you leverage diversifiers, you should be prepared for exactly these phases.

Advanced: two derivations

Trading requirement: the fund starts with assets V and exposure L × V. If the index rises by r, the exposure grows to L × V × (1 + r) and the assets to V × (1 + L × r). The new target is L × V × (1 + L × r). The difference is L × V × (L × r − r) = L × (L − 1) × r × V. Because L × (L − 1) is positive for L above 1 and for negative L, both types of fund trade with the market; only for L between 0 and 1 would it be the other way round.

Trend or zigzag: over two days with returns a and b, the fund delivers (1 + L × a) × (1 + L × b) − 1, whereas the simple multiple would be L × ((1 + a) × (1 + b) − 1). The difference is L × (L − 1) × a × b. If both days have the same sign, the fund gains relative to the multiple; otherwise it loses. With L = 2 and +10% on each day, that is +2 percentage points (+44% instead of +42%); with +10% and −10%, it is −2 percentage points. Positive autocorrelation of daily returns therefore helps, and negative autocorrelation hurts. This is the path-dependent option described by Cheng and Madhavan (2009).

Limitations and common mistakes

• The multiple only applies to one day: over months and years, a leveraged ETF is a bet on a trend and low volatility.
• Total loss is possible: if the index falls by 1 / L or more in a single day (by 33.3% for 3x), a daily leveraged fund is mathematically worthless.
• Inverse ETFs as a hedge: they hedge for short periods. Over longer periods, they lose in sideways phases just like a 2x ETF.
• Monthly data are misleading: a backtest with monthly reset leverage shows a different product from an ETF reset daily.
• Stacking needs financing and collateral: rising interest rates make leverage more expensive, margin calls on futures require liquidity, and the additional building block can itself lose for years.

This article is not investment advice; leveraged products can lead to a total loss.

How Portfolio Lab calculates

In the Backtest, each security can be given its own leverage between 0.1 and 5 if you wish. The app then calculates as for a daily leveraged fund: in each period, R = L × r + (1 − L) × g − max(L − 1, 0) × s, with g as the money market rate from the settings and s as the spread on the financed part (default 1% p.a., adjustable from 0 to 10%). With daily prices, the leverage is reset daily; with asset classes, monthly. If the return for a period reaches −100% or less, the security is worthless. Inverse funds with negative leverage cannot be modeled in this way; the ongoing charges and swap spreads of a real product can only be roughly accounted for via the spread.

Return stacking can be approximated because the backtest normalizes a portfolio's weights to 100%: a 60/40 portfolio with leverage of 1.5 on both building blocks corresponds to 90/60, financed at the money market rate plus the spread. Half equities and half the asset class “Trend following (managed futures)” (@TRENDFOLGE), both with leverage of 2, gives 100% equities plus 100% trend following minus one unit of money market rate and spread, i.e. an overlay. With a spread of 0, the financing comes close to that of futures. Between the portfolio's rebalancing dates, the weights drift. Formulas and examples are in the backtest methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.