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).

SH prospectus

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.

My simulation result


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)

1 Answer 1


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


Hope it helps

Edit (adding the formula):

From Cheng and Madhavan (2009), first linked paper in this answer, page 13.

$$r_{x} = (1+\mu)^x \times \text{exp}\Bigg(\frac{(x-x^2)\sigma^2t_N}{2}\Bigg)$$

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).

This gives exact same values as the ones reported by Proshares in their prospectus.

  • $\begingroup$ Thanks. I know some will tell me to do my own homework, but putting the right formula in the answer would be great and might also help others. Unfortunately I cannot access the first paper (not available for download) $\endgroup$
    – Martin
    Commented Sep 18, 2021 at 20:01
  • 3
    $\begingroup$ This seems to be a working link: semanticscholar.org/paper/… $\endgroup$
    – Bob Jansen
    Commented Sep 18, 2021 at 20:04
  • 1
    $\begingroup$ Please try the paper that Bob Jansen provided, it's in chapter 4 and 4.1 in particular the last equation on page 13. I'm traveling this weekend. If you can't access the paper I will post the equation in an edit to my answer. $\endgroup$
    – T123
    Commented Sep 18, 2021 at 20:12

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