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:
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.
Let's assume the following values:
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
