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What is Leverage?

Leveraged ETFs aim to multiply the daily gains of the underlying index. For example, if an ETF increases by 1%, its 2x leveraged version increases by 2%. On the other hand, downwards movement is also amplified: If the underlying ETF falls 1%, the 2x leveraged version falls 2%.

The implementation of this is complex, but it can be thought of as investing borrowed money: You borrow $100 and invest it together with $100 of your own. The borrowed $100 are paid back at the end of the day, but you may keep the gains (or deal with the losses) of the $200 investment. One may naively believe those gains to be twice as big over the lifetime of the investment, but it is not quite that easy, as we will see.

Volatility Drag

Volatility drag is the phenomenon where fluctuating prices reduce your overall return, even if the average price stays the same. This happens because losses have a proportionally larger impact than gains.

Say you invested in an ETF that is worth 100 points. If this ETF falls 10%, it’s now worth 90 points and needs to grow by 11.11% to get back to 100 points.
This phenomenon is amplified by leverage: The 2x leveraged ETF would’ve fallen to 80 points and would then need to increase by 25%, which means that the unleveraged ETF needs to increase by 12.5%.

This means that leveraged ETFs need longer to recover from crashes.
And in times of a sideways trend, the leveraged ETF declines in value, even if the underlying ETF’s value stays the same.

This effect can be explained by the term “path dependence”: The value of a leveraged stock depends on the path of the unleveraged stock, not just its current value.

Compound Interest

Compound interest is an effect that counteracts volatility drag.
To illustrate, let’s imagine two scenarios with an asset that grows at a fixed annual rate of 8%:

• Scenario 1: No Leverage. After 5 years, a $100 investment would grow to approximately $147, a 47% increase.

• Scenario 2: 2x Leverage. If we could perfectly double the returns each year (ignoring volatility drag for a moment), our annual return would be 16%. After 5 years, the same $100 investment would rise to $210, a 110% increase.

As you can see, doubling the yearly returns more than doubles the total return over time.

Considering both volatility drag and compound interest, we see that leveraged stocks profit from low volatility and high returns, which begs the question: Is a lever < 1 possible, and if so, does it take advantage of high volatility and low returns, using some kind of “inverse volatility drag”?

Deleveraging

Deleveraging is indeed possible and can be practically accomplished in two different ways:

• Investing in the inverse ETF: Inverse ETFs short the original ETF to invert its price development. Investing in both an ETF and its inverse (with appropriate weights) implements deleveraging. For example, a lever of 0.5 is achieved by investing 75% in the original ETF and 25% in its inverse. Note that this is not recommended.

• Investing in risk-free assets: Investing a portion of your money in the ETF and the rest in a risk-free asset (such as your bank account) accomplishes the same effect as deleveraging: The expected returns shrink, but your investment is less prone to fluctuations in the capital market. This method requires regular rebalancing.

The second method is the reason why a lever < 1 can be interpreted as not investing all your available money into stocks. Taking this point of view, we all deleverage stocks!

Let’s compare the performance of a 0.5x and 2.0x lever to the unleveraged ETF:

In the above two heatmaps, the median values for different combinations of volatility and expected annual return are compared for a (de)leveraged stock and its unleveraged version. Blue indicates that the (de)leveraged stock performed better than the unleveraged one.

I have chosen to compare the medians instead of the averages because the volatility of the average of independent paths approaches zero as the number of simulations grows while the expected annual return stays the same. A portfolio of many independent stocks with the same volatility and annual return cannot be achieved in the real world.

Seeing that there are scenarios in which we should deleverage and others in which we should leverage, is there a formula for the theoretically optimal lever?

Determining the Optimal Lever

An in some sense optimal lever (maximizing log wealth) is given by the Kelly criterion. It assumes that one may invest some money into a volatile stock and the rest into a risk-free asset — similarly to how we interpreted deleveraging. In the world of the Kelly criterion, there are no third options.

Let r be the expected annual return of the stock, σ its volatility and f the (guaranteed) annual return of the risk-free asset. Then, according to the Kelly criterion,

is the optimal share to invest in the stock.

Let’s apply this by comparing the S&P 500 (estimated return 8%, volatility 15%) to the European Central Bank’s current interest rate of 3.25%. The formula gives a value of about 2.0. This means that we should invest 200% of our money into the stock, which is only possible if we borrow money — essentially leveraging the stock.
If the volatility was 30%, then a lever of 0.5 would be optimal, meaning that we should invest 50%, deleveraging the stock.

Real World Considerations

While there are good reasons to leverage stocks from a mathematical point of view, the above considerations fail to account for aspects that need to be accounted for in the real world:

• In reality, there is not just one stock to invest in. Reducing volatility by building a diverse portfolio is most often the sensible option.

• A leveraged ETF’s TER is usually higher than the non-leveraged ETF’s. For example, the TER for Amundi’s 2x Lev MSCI USA is currently 0.50% vs. 0.07% for Xtrackers’ MSCI USA.

• Volatility and drift need to be estimated, leading to additional uncertainty.

• This article focuses on an investment at a single point in time and therefore ignores effects such as the cost average effect.

• Furthermore, this article doesn’t account for taxes and regulations.

• The mathematical model assumed in the Kelly criterion (explained below) may be an oversimplification. For example, it assumes the drift and volatility do not change over time.

• The Kelly criterion maximizes expected long-term growth — this may not be optimal for you as an investor. For example, a near total loss right before retirement is a lot worse than doubling your fortune is good.

• The purely mathematical analysis ignores an investor’s irrationality.

• Leveraging is risky. Before you follow strategies that someone calls mathematically optimal, make sure you understand the implications.

Appendix: Mathematical Model

For my simulations, I modeled the price of an asset via a so-called geometric Brownian motion (GBM). A GBM “S” depends on:

• the starting price S(0), for example $100,

• the number of years t,

• the expected yearly return r, for example 8%,

• the volatility σ, for example 15%, and

• randomness, coming from a Brownian motion W, which can be thought of as compounding random fluctuations.

A formula for a GBM is given by

This formula tells us that:

• The price increases exponentially with time.

• The price is proportional to the starting price.

• There is uncertainty that depends on the volatility and that increases with time. The exact terms in the exponential are of technical nature.

Note that this is not the standard parameterization of a GBM; it is usually stated using “drift”, which I rewrote in terms of expected annual returns to make the formula more intuitive.

It can be shown that S(t) follows as log-normal distribution. S(1), that is, the price after one year, has mean (1+r) S(0) and standard deviation

The left part of this formula for the standard deviation is the expected price after one year and the square root-term is approximately equal to the volatility σ. Volatility is therefore roughly the standard deviation divided by the expected value.

Since S(t) follow a log-normal distribution, its median and the exponential of its expected logarithm are equal:

This means that our simulations involving the median path and the maximization of log-wealth in the Kelly criterion are equivalent.