Under some circumstances, altruists might prefer to leverage their investments. The easiest way to get leverage is to buy leveraged ETFs. But leveraged ETFs charge high fees and incur other hidden costs. These costs vary substantially across different funds and across time, but on average, leveraged ETFs have historically had annual excess costs of about 2%, or around 1.5% on top of the expense ratio.

Given reasonable expectations for future returns, leveraged ETFs most likely have substantially higher arithmetic mean returns than their un-leveraged benchmarks. They also appear to have higher geometric mean returns than their benchmarks, but only by a small margin. Slightly more pessimistic estimates would find that adding leverage decreases geometric return.

Note: Many investors can get leverage more cheaply via other methods, such as margin loans or futures. Even if leveraged ETFs appear better than un-leveraged investments, other forms of leverage might be better still.

Disclaimer: This should not be taken as investment advice. Any given portfolio results are hypothetical and do not represent returns achieved by an actual investor.

## Measuring the cost of leveraged ETFs

To calculate costs, I created a list of leveraged ETFs and looked up their historical returns from the start of 2016 to the end of 2020. I paired each leveraged ETF with an equivalent un-leveraged benchmark ETF. I calculated what return you would have earned if you had taken that ETF and applied the appropriate amount of leverage (either 2x or 3x, depending on how much the corresponding leveraged ETF uses), paying only the risk-free rate.1 Then I subtracted the actual return of the leveraged ETF to find the excess cost.

My analysis included these ETFs:

ETF Leverage Index Index ETF
SPXL 3 S&P 500 SPY
UPRO 3 S&P 500 SPY
UMDD 3 S&P 400 (mid cap) IJH
URTY 3 Russell 2000 (small cap) IWM
EFO 2 EAFE EFA
EURL 3 Europe VGK
EZJ 2 Japan EWJ
EET 2 emerging markets EEM
EDC 3 emerging markets EEM

The following table shows the total excess cost and after-fee cost for various leveraged ETFs. Excess cost is shown per 100% leverage (so the excess on a 3x ETF is divided by two). After-fee cost tells us the excess cost minus the expense ratio.2 r gives the correlation between the leveraged ETF and the simulated leveraged benchmark ETF.

ETF Excess Cost After Fee r
SPXL 2.23% 1.72% 0.995
UPRO 2.12% 1.64% 0.995
UMDD 1.31% 0.84% 0.995
URTY 2.08% 1.61% 0.993
EFO 2.47% 1.14% 0.978
EURL 1.97% 1.44% 0.984
EZJ 2.79% 1.84% 0.981
EET 2.18% 1.25% 0.994
EDC 1.67% 1.20% 0.994

The leveraged ETFs generally had an excess cost of about 2%, or around 1.5% after the ETF expense ratio. All fund pairings had high correlations, which tells us that the leveraged ETFs do a good job of tracking their benchmarks.

Excess costs were not stable—they varied substantially from year to year. The next table shows excess cost for each individual year:

ETF 2016 2017 2018 2019 2020 Min Max
SPXL 1.39 2.02 1.22 3.25 3.01 1.22 3.25
EET 2.73 2.63 1.55 2.71 1.58 1.55 2.73
UPRO 1.03 1.90 1.27 3.51 2.76 1.03 3.51
UMDD 1.58 0.85 1.04 2.03 0.06 0.06 2.03
URTY 1.05 1.59 0.87 2.48 3.88 0.87 3.88
EURL 1.51 3.08 1.15 4.21 1.34 1.15 4.21
EZJ 2.05 2.07 2.11 4.71 1.87 1.87 4.71
EDC 1.45 2.55 0.73 2.89 1.30 0.73 2.89
EFO 0.83 2.73 2.51 4.02 1.32 0.83 4.02
Min 0.83 0.85 0.73 2.03 0.06 0.06 2.03
Average 1.51 2.16 1.38 3.31 1.90 1.03 3.47
Max 2.73 3.08 2.51 4.71 3.88 1.87 4.71

For individual years, excess costs ranged from as low as 0.06% to as high as 4.71%, with an average low of 1% and an average high of nearly 4%.

Some factors that might contribute to this year-to-year variance:

1. The ETFs’ counterparties charge different rates based on perceived risk.
2. Market liquidity varies over time.
3. The ETFs might experience some tracking error.

Excess cost does not include the risk-free rate, so the year-to-year variance has nothing to do with changes in the risk-free rate.

Some people observe that 3x leveraged funds usually provide less than a 3x return due to volatility drag. To be clear, the excess cost I found is not related to volatility drag. Volatility drag equals the difference between the arithmetic mean and geometric mean of an investment. If two investments A and B have the same arithmetic return but B has higher volatility, then B will have lower geometric return.

Specifically, the geometric mean of an investment equals $\alpha - \sigma^2/2$, where $\alpha$ is the arithmetic mean and $\sigma$ is the standard deviation.

The geometric mean of a 3x leveraged ETF is less than 3x the arithmetic mean. For example, if the benchmark has a geometric mean of 5% and an a standard deviation of 16%, then we calculate the geometric mean of the leveraged ETF as:

1. Arithmetic mean of the benchmark is $5\% + 16\%^2/2 = 6.28\%$
2. Arithmetic mean of the leveraged ETF is $3 \times 6.28\% = 18.84\%$
3. Standard deviation of the leveraged ETF is $3 \times 16\% = 48\%$
4. Geometric mean of the leveraged ETF is $18.84\% - 48\%^2/2 = 7.32\%$

In this example, adding 3x leverage only increases the expected geometric return from 5% to 7.32%. If we assume a 3x leveraged ETF has a 2.5% excess cost, that reduces its expected geometric return to 4.82%, which is lower than the return of an un-leveraged ETF.

## Are leveraged ETFs worth it?

Mathematically speaking, an agent with a logarithmic utility function wants to maximize geometric mean, while a risk-neutral agent wants to maximize arithmetic mean. A logarithmic utility function is probably still too aggressive for self-interested investors, but it might be appropriate for altruists. Under some circumstances, altruists might even be close to risk-neutral.

For risk-neutral investors, leveraged ETFs pretty clearly have higher expected utility than normal ETFs. (Although they might prefer to get leverage some other way if they can, since leveraged ETFs are one of the most expensive forms of leverage.)

Are leveraged ETFs worth it for investors with logarithmic utility? That is, can we expect leveraged ETFs to have a higher geometric return than ordinary ETFs?

That depends on three things: the expected return of the benchmark, volatility, and excess cost.

Let’s assume an excess cost of 2% and a risk-free rate of 0%. Long-run market volatility generally varies from 15% (for more stable markets like the S&P 500) to 22% (for more volatile markets like US small-caps or emerging markets). Then we can ask, what expected geometric return does an index need to have for a leveraged ETF to be worth it? Let’s assume we can use the optimal amount of leverage, not just 2x or 3x. (We can combine a 3x ETF with an un-leveraged ETF to get an intermediate amount of leverage.)

At a 15% standard deviation, an investment needs an expected (geometric) return higher than 3.125% for leverage to be worth it. At a standard deviation of 22%, the investment must return more than 4.42%.3

If we assume a total cost of 2.5% (e.g., an excess cost of 2% plus a risk-free rate of 0.5%, or an excess cost of 2.5% when the risk-free rate is zero), then an investment with 15% standard deviation must return 3.625%, and one with 22% standard deviation must return 4.92%.

What returns can we expect from various market indexes? We can’t know with high confidence, but we can try to estimate it. The following table gives return estimates for various equity regions according to Research Affiliates (RA) and AQR, and standard deviation estimates from Research Affiliates.4

Index Return (RA) Return (AQR) Stdev (RA)
US large 2.0% 5.9% 15.4%
US small 4.3% N/A 20.7%
EAFE 6.3% 6.5% 17.3%
Europe 6.4% 6.3% 19.2%
Japan 5.8% 6.3% 17.6%
emerging 7.6% 7.0% 21.7%

The next table gives the expected (geometric) return of an optimally-leveraged portfolio5 minus the expected return of the index, constructed using a leveraged ETF with a 2% cost.6

Index Improvement (RA) Improvement (AQR)
US large <0% 1.6%
US small 0.0% N/A
EAFE 1.3% 1.5%
Europe 0.9% 0.8%
Japan 0.8% 1.2%
emerging 1.1% 0.7%

According to the Research Affiliates estimate, we shouldn’t apply leverage to a US large-cap index, but in every other case, applying leverage at least marginally increases expected geometric return. However, leverage provides only a modest improvement, with the best possible improvement being 1.6 percentage points.

Vanguard’s return expectations produce similar numbers (not shown).

Of course, these results heavily depend on what inputs we use for expected return, volatility, and costs, and the true numbers might differ significantly. If costs turn out to be closer to 3%, or if true (ex-ante) expected returns are one percentage point lower, that would eliminate the benefit of using leverage on most equity regions.

Or suppose you are trying to decide between two ways of investing:

2. Buy un-leveraged ETFs in a taxable account, and use margin to get leverage.

Say your DAF has to pay a 0.3% fund fee plus a 1% fee for an investment manager, and margin costs 0.5% over the risk-free rate. That means the DAF has to pay an extra 2.8% cost (including DAF fees plus the leveraged ETF cost) above what the taxable account pays, whereas the taxable account must pay taxes. 2.8% is a lot. It’s not obvious whether it outweighs the tax benefits of a DAF, but I personally would lean toward keeping my money in a taxable account.

# Notes

1. I used the 3-month T-bill yield as the risk-free rate.

2. I looked at the ETFs’ current expense ratios (as of 2021-02-11) and assumed that they were the same over the whole five years.

3. In general, a leveraged ETF has the same geometric return $\mu$ as the index when $\mu = \sigma^2 / 2 + R + c$ for standard deviation $\sigma$, risk-free rate $R$, and excess cost $c$.

Another way to put it: when $\mu = \sigma^2 / 2 + R + c$, optimal leverage is exactly 1x. For any higher value of $\mu$, adding leverage increases geometric return.

4. Research Affiliates estimates are as of 2021-01-31; AQR estimates are as of January 2021.

5. If geometric mean maximizing leverage is less than 3x (which it usually is), then we can construct an optimally-leveraged portfolio by holding both a 3x leveraged ETF and an un-leveraged ETF and rebalancing their weights daily.

6. Improvements can be calculated using the following Python code:

def return_improvement(mu, sigma, cost):
leverage = (mu + sigma**2/2 - cost) / (sigma**2)
return (
leverage * (mu + sigma**2/2) - (leverage * sigma)**2/2
- (leverage - 1) * cost
- mu
)