Why cannot investors hold Leveraged Exchanged Traded Funds as a long-term investment instrument given the mathematical structure of LETFs?

Investments in Leveraged Exchange-Traded Funds have increased recently. Leverage is a key feature of these funds, and they're utilized mostly by short-term traders. They are also popular with individual investors who want to make leveraged or hedging investments in their portfolios. Critics, however, have charged that LETFs are unsuitable for longer-term investors, due to volatility decay. Volatility decay occurs when the returns of Leveraged Exchange Traded Funds diverge from the multiple returns of their underlying assets as the holding period lengthens. Volatility decay is demonstrated to be accelerated by increasing the leverage multiplier algebraically. In this paper, I will describe the mathematical structures of Leveraged Exchange-Traded Funds and Exchange-Traded Funds in terms of their performance in terms of arithmetic return, geometric return, rebalancing activity, and diffusion process. Using their mathematical structures, it is straightforward to demonstrate the long-term impact of volatility decay on these funds. In addition, I will use historical data to validate the influence of volatility decay, and I will examine the performance of LETFs on the assumption that the portfolio is not subject to volatility decay.