Compare an index and an ideal daily-reset leveraged fund after an equal percentage rise and fall on successive days. The calculation shows the effect of compounding over that specific two-day path.

Enter the daily move magnitude and signed daily leverage; −1 represents an inverse daily objective.

Two-day ideal daily-reset model: the index first rises by m%, then falls by the same m%. Fees, financing costs, tracking error, distributions and taxes are excluded.

Examples: 2 for twice the daily return, −1 for inverse daily return.

Starting value (blank = 100)

Blank normalizes both paths to 100.

Two-day compounding

Let m be the daily move as a decimal and L the signed daily leverage. The index factor is (1 + m)(1 − m) = 1 − m². The fund factor is (1 + Lm)(1 − Lm) = 1 − L²m².

Thus the modeled loss magnitudes are 100m²% for the index and 100L²m²% for the fund. For a nonzero move, the loss ratio is L². A negative L represents an inverse daily objective.

Example

With a 2% move and 3× daily leverage, a starting value of 100 becomes 102 then 99.96 for the index. The fund becomes 106 then 99.64. The index loses 0.04% and the ideal fund loses 0.36%, a ninefold loss ratio.

Frequently asked questions

Is an equal rise and fall a flat index return?

No. A fall is applied to the larger value after the rise. For example, +2% followed by −2% multiplies the starting value by 0.9996.

What happens at a 100% daily modeled fund loss?

The path wipes out modeled fund value. An implied daily loss greater than 100% is rejected because this simplified nonnegative fund-value model cannot represent it.

Does the result forecast long-term ETF decay?

No. It is one two-day identity. Actual funds depend on the complete return path, fees, financing, tracking, distributions and taxes. The SEC explains that daily objectives need not equal a leverage multiple of longer-period performance.

References