*This article assumes the reader understands how inverse and leveraged ETFs work*
Zhang’s 2010 paper, Path-Dependence Properties of Leveraged Exchange-Traded Funds: Compounding, Volatility and Option Pricing, gives a closed-form formula for estimating the decay of leveraged & inverse etfs.
t=time in years (t=1 means one year)
b=is the leverage factor (typically -1,-2,-3,2,3)
λ=cost to borrow (the annual fee paid to short sell an inverse etf)
f=expense ratio (typically 1 (100 basis points))
r=3-month LIBOR, which is around 1.25 (if b is positive, the 3-month libor rate is used; if negative, the federal funds rate is used.)
σ=annualized realized/actual volatility 
An additional variable can also be added e^(k) where k is a dividend, a drift, etc.
Because λ and σ are not functions of t, the integrals are trivial to evaluate and we have λ*t and σ*t
L_t/L_o is how much the leveraged or inverse ETF gains or falls, which I will denote as ∇
S_t/S_o is how much the underlying rises or falls, which I will denote as Δ
However, there is a problem. Although many of these variables are obvious (such as interest rates, expense ratio, and the leverage factor), others (such as actual volatility, the borrow fee, and k) are much harder to know. Does this mean we’re stuck? No. We can simply use existing leveraged and inverse ETF prices to reverse engineer the variables, by solving a system of equations.
S_t/S_o can be put in exponential form, and combined with the dividend/drift factor k, we have:
Example 1: Treasury Bond ETF
Consider TBT, the inverse 2x 30-year leveraged treasury bond etf; b=-2
…and TMF, the 3x 30-year leveraged treasury bond etf; b=3
…and TMV, the inverse 3x 30-year leveraged treasury bond etf; b=-3
…and TBF, the inverse 1x 30-year leveraged treasury bond etf; b=-1
All inverse and levered etfs have an expense ratio of about 1% (f=.01)
We know that the non-leveraged 30 year treasury bond ETF TLT pays out dividends, but the inverse and leveraged ones do not. The dividends are either added (for non-inverse funds) or subtracted (inverse funds) from the nav (net asset value) of the fund. Treasury bond funds are also affected by the underlying 30-year rate as well as the coupons converging to par. This is a lot of variables and it can be difficult disentangling them. Formula [1.1] is really helpful here because we can just treat e^(lnΔ+kt) as a single variable, w, for our system of equations. W is the total return, which includes dividends, price appreciation, etc. and may be a function of t. However, the current yield on a 30-year treasury bond is 2.8%, which means that the un-leveraged version will gain that much a year in terms of both dividends and drift towards par.
For the part (b^2-b)t*σ^2/2, I’m going to modify this slightly by multiply it by 5/7 to account for weekends, so we have (b^2-b)*5*t*σ^2/14 instead
The actual volatility, borrow fee, and Δ and k are unknown and this information is not found anywhere online or difficult to determine.
The 9-month period between 12/2/2016 to 9/2/2017 will be used for t , which means t=.75
The 3-month libor is used for r (r=.0125) and f=.01 for the expense ratio. According to the Ishares website, TLT, an un-leveraged 30 year bond fund, gained 8% in the 9-month period, which includes dividends. That means w=1.08.
Doing some algebra, the lhs is:
It is easier to work with positive b, because λ=0.
For TMF, the 3x 30-year treasury fund (b=3), we know w,r, and f. ∇=22.03/18.20 We need to find σ.
Using wolfram alpha, we obtain σ=.093
To test this, we can try to predict UBT, the 2x 30-year treasury ETF (b=2).
Using σ=.093, the predicted price of UBT 9 month later is 83.5; the actual closing price for 9/1/2017 is 83.3, a very small difference.
To find λ, TBF will be used. Solving gives λ=.009.
To test this, we can try to predict TBT, the 2x inverse 30-year treasury ETF (b=-2), and we obtain 34.58, which differs from the actual by 4 cents.
Example 2: UPRO, SSO, and dividend reinvestment
Is it commonly but erroneously assumed that all leveraged ETFS decay. Some do, but others such as SSO (the 2x S&P 500) and UPRO (the 3x version) obviously have not, far surpassing the performance of the S&P 500. Instead of decay, there is acceleration, meaning that a portfolio that is 2/3 cash and 1/3 UPRO beats a 100% fully-invested SPY portfolio. The reasons for the lack of decay are two-fold: SPY’s generous 2% per year dividend is automatically and continuously added to the NAV of SSO (+4% a year) and UPRO (+6% a year); second, the actual volatility of the S&P 500 is very low. This also makes SSO and UPRO ideal for tax purposes because the dividends are not counted as a capital gain when added to the NAV, and they are automatically ‘reinvested’ by being added to the NAV. This is the first time anywhere on the internet in the all the 10 years that these ETFs have existed has anyone realized this, but me.
So let’s prove that the SSO and UPRO dividend reinvestment is real.
Like before, the time period is from 12/2/2016 to 9/2/2017, so t=.75.
Because b is positive, the 3-month Libor is used for r, which is 1.25%, and λ=0.
The ∇’s are obtained just from inspecting the performance of SSO and UPRO, but we need to find w, which is the total return of the S&P 500 including dividends. And also σ. From inspection, we also know the performance of the S&P 500 from this period (excluding dividends), which is 13% (Δ=1.13).
Solving the pair of equations, we have w=1.147 (14.7% total return) and σ=9.3%.
Solving e^(lnΔ+kt)=1.147, we obtain k=.02, which is very close to the exact dividend yield of the S&P 500.
Example 3: VXX and UVXY
As shown in the post Why to Always Short UVXY Instead of VXX, UVXY and VXX decay significantly, but UVXY seems to decay much more so than predicted by the fact that it is 2x version of VXX. VXX keeps a rolling 30-day basket of VIX futures. UVXY is a 2x version of this, meaning b=2.
For this example, I’m ignoring r, λ, and f because the are negligible. As always, t=.75
How much volatility does VXX have? Once we know this, we can predict how much UVXY decays relative to VXX. Regarding UVXY, ∇=29/229. And w=46.2/116 (based off VXX).
As expected, the volatility is huge σ=.66 or 66%.
This means if VXX is flat, for example, over a 120-day period, UVXY will have decayed around 11%.
Various trading strategies can be created to take advantage of this decay, which I will discuss in a forthcoming post.
 This is not the expected volatility. You cannot plug the current value of the Vix for the formula and expect to get an accurate reading. You have to use the actual volatility.