Why Daily Leverage Changes Tracking Difference Over Time
Summary
The document asks whether leverage should multiply an index or ETF’s tracking difference by the leverage factor. The response explains that this equality need not hold when leverage is applied each day or over another subperiod, because compounded returns depend on the path of gains and losses. Comparing geometric annual returns over multiple periods with a one-period hypothetical can therefore produce different results.
An example uses consecutive daily returns of -20% and +10%. Compounding gives the unleveraged fund a two-day loss of 12%, while doubling each daily return produces a 28% loss, exceeding twice the unleveraged loss. The difference illustrates volatility drag: a decline followed by an equal-sized percentage gain does not restore the starting value. The discussion focuses on daily reset leverage; actual tracking differences also depend on the measurement period and how the products implement leverage.
Key ideas
- A leverage factor does not necessarily scale a multi-period tracking difference by the same factor.
- Daily leverage changes each period’s return before the returns are compounded.
- The example shows that a loss followed by a smaller percentage gain compounds asymmetrically.
- Volatility drag can make a leveraged fund’s longer-period return differ from a simple multiple of the underlying return.
Tags
Full text
# Leverage and Tracking difference
# Leverage and Tracking difference
I am a little confused. I have calculated the tracking difference of an Index and an ETF using the return getting 0,4% tracking difference per year. I have then leveraged both, the Index and the ETF to a lever of 2 getting 0.63% tracking difference per year. I have then done some testing with hypothetical value and got 10% unleveraged and 20% levaraged with an lever of 2. So, $$ \text{leveraged tracking difference} = \text{lever}\times(\text{unleveraged tracking difference}) $$ in the hypothetical case. In the real case, I am not getting an equality.
However the real case used a 5-year period and I calculated the annual return using the geometric mean. In the hypothetical case, I only simulated one year of return.
My question is. Should the tracking difference always be equal to my formula when leverage? if that is the case, I might have done some mistake in the real case. Else, it differs when a 5-period is used along with the geometric mean.
## Answer by D Stanley (score 1)
https://quant.stackexchange.com/a/85728
> Should the tracking difference always be equal to my formula when leverage?
Not if the leverage is applied daily (or applied to any sub-period).
Take an extreme example:
Fund ABC has daily returns of -20% and +10%. the two-day total return is then
`0.80 * 1.1 = 0.88 or -12%`
2X Leveraged fund DABC multiplies the daily returns by 2, and has a two-day return of
`0.60 * 1.2 = 0.72 or -28%`
which is more than twice the overall loss of the unlevered fund.
Leverage amplifies the fact that relative returns are asymmetric, meaning negative relative returns cannot be reversed by an equivalent positive return, and is called volatility drag. It could be better if there are no large negative days, but over time drops have a much bigger impact than increases, which tends to drag down the return over time.
It's the main reason that levered funds have tons of disclaimers that they are "not suitable for long-term investing" but only seek to multiply the returns over a short period (typically daily).
## Answer by Vitomir (score 0)
https://quant.stackexchange.com/a/46382
It is probably coming from AM-GM inequality, i.e. from the fact that geometric mean is a concave functionShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.