Leveraged ETF Decay: Toy Model 1.0

To get a better feel for the decay rate of these leveraged ETFs, I was considering the following toy model with very simple assumptions.

1. Benchmark daily movements can only take: x% up, y% down
2. For trading days N, there are k up days, N-k down days
3. Leverage Factor: F
4. Composition of p% F Leveraged and (1 – p)% inverse or “-“F leverage
5. No favorable transaction costs to shorts: management fees
6. No unfavorable transaction costs to shorts: SLB interest, bid-ask spreads, etc.

Return of Benchmark Index: R_index = (1 + x)^k (1 – y)^(N-k)
Return on F-Leveraged: R_lev = (1 + F*x)^k (1 – F*y)^(N-k)
Return on “-“F-leveraged: R_inv = (1 – F*x)^k (1 + F*y)^(N-k)
Return of pair: R_combo = p*R_lev + (1 – p)*R_inv

Delta(index-leveraged) = R_index – R_lev
Delta(index-inverse) = R_index – R_inv
Delta(index-combo) = R_index – R_combo

Free parameters:
x, y, N, k, F, p

Much more work to be done incorporating the following, as well as with real data, but this was a start. May want to consider the following for starters, at the risk of much greater complexity:

1. Distribution of percentage movements: d(x, t), d(y, t)
2. Multiple leverage factors F_i
3. Time dependence of leveraged and inverse: p(t), 1-p(t)
4. Representative classes of leveraged & inverse leveraged
3x SPX: SSO, UPRO, SPXL
-3x: SPX: SPXS, SPXU

I wrote a simple Python script that generated some data: