# Mathematical solution to return decay of daily leveraged products (leveraged ETFs)

Daily leveraged ETFs have an inherent path dependence. An index performing (5%, -5%, 5%) on 3 days has an overall performance of 2.9%. A -1x leveraged ETF would perform -3.1%. At a higher volatility, say (10%, -10%, 10%), the index would make 8.9% yet the -1x product would lose -10.9%. The decay increased.

We can Monte Carlo simulate the return of leveraged products, see my R script at the end of this post.

My question is if there is a mathematical solution based on just the standard deviation to calculate the expected performance (given the standard deviation and an underlying index performance).

In the prospectus of the Proshares -1x Short S&P500 ("SH") we find the following table (Link page 5).

How did Proshares calculate these returns? My Monte Carlo simulation results in different values at high volatility.

Here is my result (using the script below). At high volatility it's completely off.

library(tidyverse)
library(scales)

mu <- c(-6:6/10)
sigma <- c(0.1, 0.25, 0.5, 0.75, 1)
N <- 252
n <- 100

simulation <- tibble()
for (m in mu) {
for (s in sigma) {
print(paste0("mu: ", m))
print(paste0("sigma: ", s))
for (i in 1:n) {
out <- tibble(t = 1:N,
r1 = rnorm(n = N, mean = (1+m)^(1/252)-1, sd = s/sqrt(252)),
r-1 = -1*r1) %>%
mutate(mu = m,
sigma = s,
i = i)

simulation <- bind_rows(out, simulation)
}
}
}

simulation %>%
group_by(i, mu, sigma) %>%
summarise(r-1 = prod(1+r-1)-1) %>%
group_by(mu, sigma) %>%
summarise(r-1 = mean(r-1)) %>%
pivot_wider(names_from = sigma, values_from = r-1) %>%
mutate_all(percent, accuracy = 0.01)

• See this answer. Commented Sep 19, 2021 at 10:04

Is this what you are looking for?

The Dynamics of Leveraged and Inverse Exchange-Traded Funds, Minder Cheng and Ananth Madhavan, Journal Of Investment Management (JOIM), Fourth Quarter 2009

and here p 31

Dynamics of Leveraged and Inverse ETFs, Minder Cheng and Ananth Madhavan

And here where it comes from

Structural Slippage of Leveraged ETFs, Marco Avellaneda, Doris Dobi

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2127738

Hope it helps

$$r_{x} = (1+\mu)^x \times \text{exp}\Bigg(\frac{(x-x^2)\sigma^2t_N}{2}\Bigg)$$
With $$x$$ being the leverage factor, $$r_x$$ the expected return of the leveraged product, $$\mu$$ the daily expected return (annual divided by number of days), $$\sigma$$ the daily expected standard deviation (annual divided by square-root of days), and $$t_N$$ the number of days (e.g. 252).