Sizing Inverse and Leveraged ETFs for a Target Index Exposure
Summary
The document derives allocation weights for two ETFs tracking the same index with different daily leverage: one returns negative one times the index move, and the other returns 1.5 times the move. It assumes the full portfolio is invested across the two funds and seeks a specified combined return as a multiple of the index return.
The method sets up two linear relationships: the weights sum to one, and their leverage-weighted sum equals the target exposure. Solving these equations gives a direct formula for each allocation. A zero-exposure example uses 40% in the leveraged long ETF and 60% in the inverse ETF, which offset for a single-day index move. The equations can also be applied to the requested positive and negative exposure targets. The analysis is a single-period calculation based on stated daily leverage; it does not address how leveraged ETF returns compound over multiple days, fees, tracking differences, or changing exposures.
Key ideas
- The portfolio weights must sum to one when all capital is allocated.
- The net index exposure is the sum of each ETF weight multiplied by its leverage.
- Solving these two conditions gives the allocations for a chosen target exposure.
- The zero-exposure example balances the two funds at 40% and 60%.
- Daily leveraged returns do not by themselves specify multi-day portfolio performance.
Tags
Full text
# calculating net exposure to an index
# calculating net exposure to an index
I would like to invest in a portfolio of two inversely correlated ETFs, with funds allocated at a ratio. The two funds track the performance of the same underlying index: the share price of NVDA (called NVDAX). The first fund, NVDD, tracks the underlying index NVDAX at a multiple of -1X the index's daily return. The other fund, NVDL, tracks NVDAX at a multiple of 1.5X the index's daily return.
Given this, can you tell me the ratio of funds I should allocated to each ETF in order to achieve the following net exposure levels to the underlying index: -25%, -20%, -15%, -10%, 0%, 10%, 15%?
By way of example, assuming a portfolio consists of USD 60,000 worth of NVDD and USD 40,000 worth of NVDL, and NVDAX goes down by 1% on a given day. The portfolio would not change in value, as the net exposure to NVDAX is 0%: the 60,000 of NVDD would increase in value by 1% or 600, while the 40,000 of NVDL would decrease by 1.5%, also 600.
My goal now is to understand the ratio of allocations necessary the achieve other exposure levels. Thus, if NVDAX goes down by 1%, with a 25% exposure level to NVDAX, for example, the portfolio would go down by .25%.
I hope I formed this question correctly. Thank you for any help.
## Answer by Attack68 (score 1)
https://quant.stackexchange.com/a/79191
So the return on one fund is: $1.5\alpha$ And the return on the other is: $-1.0\alpha$ Where $\alpha$ is the return of your index.
And you want to target a specific return as a multiple of your index, say $\lambda \alpha$ where all your capital is allocated, i.e. $x_1 + x_2 = 1.0$, if the x's are fractions of your capital. Then you have a system of two equations:
$$ 1.5x_1 -1.0x_2 = \lambda \\ x_1 + x_2 = 1.0 $$
Which implies
$$x_1 = \frac{\lambda + 1}{2.5}\\ x_2 = \frac{1.5 - \lambda}{2.5} $$
In your example you target a net exposure of zero. I.e. $\lambda=0.0$, and you obtain the weights: 40% and 60%.Shown 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.