Long Term Behaviour of Leveraged ETF Calculator

Formula

The formula to calculate the compound daily growth rate of a leveraged ETF is:

$R = k\mu - \frac{1}{2} \frac{k^2 \sigma^2}{1 + k\mu}$

Where:

$R$ is the compound daily growth rate of the ETF
$k$ is the ETF leverage
$\mu$ is the mean daily return of the benchmark
$\sigma$ is the daily volatility of the benchmark

What is the Rate of Convergence?

The rate of convergence refers to the speed at which a sequence of numbers approaches a certain value or limit. In the context of leveraged ETFs, it helps determine how many iterations (days) are needed to achieve a certain level of accuracy in returns.

Example Calculation

Let's assume the following values:

Leverage ( $k$ ): 2
Mean Daily Return ( $\mu$ ): 0.05%
Daily Volatility ( $\sigma$ ): 1%

Step 1: Convert the mean daily return and daily volatility to decimals:

$\mu = 0.0005 \quad \text{and} \quad \sigma = 0.01$

Step 2: Calculate the compound daily growth rate:

$R = 2 \times 0.0005 - \frac{1}{2} \frac{2^2 \times 0.01^2}{1 + 2 \times 0.0005} = 0.001 - \frac{0.0002}{1.001} \approx 0.001 - 0.0001998 = 0.0008002$

Step 3: Convert to annualized percentage:

$\text{Annualized } R = 0.0008002 \times 252 \times 100 \approx 20.16\%$

Therefore, the annualized compound daily growth rate is approximately 20.16%.

Resouce:https://www.ddnum.com/articles/leveragedETFs.php