Leverage Time Machine

from to leverage, reset daily costs a year (fee and borrowing above the bill rate)

the index

the leveraged fund

what "L times" sounds like

1 + L × the index's return

the closed form

end point, variance and bills only

the variance tax

(L² − L) σ² / 2, a year

this window's best leverage

(μ − r) / σ²

One dollar, on a log scale · the index · the leveraged fund · ◆ the closed form's end point

Growth against leverage in this window · ★ the best · ● yours

Each day the fund holds L times the index and borrows the rest at the 3-month Treasury bill rate; the costs slider takes a further amount off every year. The closed form is V = (S_T / S_0)^L · exp(−(L² − L)σ²T/2 − (L − 1)∫r − costs·T): only the end point and the variance of the path matter. A day that takes the fund below zero wipes it out.