2. Volatility drag: 2x exposure is not 2x return

Risk 4: Leverage

The previous module worked with a single period: the asset rises 10%, with a multiple of 3 you make 30%. That arithmetic is correct and it only holds for one period. As soon as you string several together, a phenomenon appears that surprises even experienced investors: twice the exposure does not hand back twice the accumulated return, and the difference is not some detail about commissions.

The round trip that does not come back

An index closes at 100. On Monday it rises 10% and stands at 110. On Tuesday it falls 10% and stands at 99. The unleveraged position has lost 1%. Nothing new so far: the asymmetry of percentages is familiar.

Now a product that replicates twice the daily move. On Monday it rises 20% and goes from 100 to 120. On Tuesday it falls 20% and stands at 96. It has lost 4%, not 2%. The daily triple loses 9%: it rises to 130 and falls to 91.

Look at the series: 1, 4, 9. The loss on the round trip grows with the multiple squared, not in proportion to it. Doubling the exposure quadruples what a swing costs you.

With the index exactly flat, you lose

The previous case leaves the index in negative territory, so let us take that excuse away. The index rises 10% to 110 and the next day falls 9.09%, so that it comes back to exactly 100. The index's return over the two days: zero.

The double daily exposure product rises 20% to 120 and then falls 18.18%, leaving it at 98.18. It has lost 1.8% with the index flat. The triple rises to 130, falls 27.27% and ends at 94.55: a 5.45% loss on a market that has not moved.

This is not a fault in the product or a hidden fee. It is the inevitable result of multiplying daily moves and then compounding them.

Repetition turns the drip into a haemorrhage

A single cycle looks anecdotal. Repeat it. Ten cycles of a 10% rise and a 10% fall on the same asset, that is, twenty sawtooth sessions:

• The index: 0.99 to the power of 10, a loss of 9.6%.
• The daily double: 0.96 to the power of 10, a loss of 33.5%.
• The daily triple: 0.91 to the power of 10, a loss of 61.1%.

The index loses less than ten points and the triple almost two thirds, without anybody having got the direction wrong: all it takes is for the market to go and come back.

Where the number comes from: volatility squared

The size of the drag is well estimated by subtracting from the multiplied return a term that depends on the square of volatility: for a multiple *k* and an annualized volatility *s*, it comes to around (*k* squared minus *k*) divided by two, multiplied by *s* squared.

With figures, to make it concrete:

• Double exposure, volatility of 15%: the drag comes to around 2.25% a year. Annoying and bearable.
• Double exposure, volatility of 30%: around 9% a year. That already eats an enormous part of the result.
• Triple exposure, volatility of 30%: around 27% a year. A market that rises 6% over the year leaves the triple clearly in the red despite having got the direction right.

Two readings. The first: the drag grows with the square of volatility, so going from a quiet market to a turbulent one does not double it, it quadruples it. The second: it builds up over time, so the gap against the index's multiple widens the longer the holding period.

The daily reset, and when it works in your favor

A product promising twice the *daily* move has to start each session with exactly twice the exposure on the previous close's assets. If it rose yesterday, today it has more assets and buys more; if it fell yesterday, today it has less and sells. It buys strength and sells weakness, every day, mechanically.

In a sawtooth that is the worst thing possible: it buys dear before the fall and sells cheap before the rise. But the opposite is worth stating too: in a sustained trend with no pullbacks, the same mechanism works in your favor and the product beats the index's multiple. Twenty consecutive sessions at 0.8% leave the index at around 17.3% accumulated and the daily double at close to 37.4%, considerably more than double. The reset is not a defect: it is a bias that pays in a trend and charges dearly in a sideways market. Since markets spend far more time going sideways than in a clean trend, the sums usually come out badly.

And one consequence that gets ignored: a constant multiple never takes the product to zero through daily compounding, because the exposure is cut back after every loss. In exchange, neither does it recover: an 80% fall demands 400% to come back.

Case study: daily leveraged products under the microscope

This is not a recent theory. In August 2009, after the previous year's storm, the American securities authorities and the industry regulator issued a joint investor alert warning that these products pursue a daily objective and that holding them for long periods can produce results very different from the stated multiple. The alert came after episodes were documented in which a leveraged bull product and its bear equivalent on the same sector both lost money over the same period: impossible with fixed exposure, perfectly possible with a daily reset.

The first half of 2020 repeated it on a grand scale. With falls and rebounds of several points a day, leveraged products on broad indices ended well below the index's multiple over the year as a whole, and several on commodities were closed or restructured by their issuers.

The conclusion is that a daily multiple and a horizon multiple are different things, and that the distance between them is set by volatility and time, not by the quality of the issuer.