To illustrate the mathematical impact of the S&P 500's movements versus a 3x leveraged ETF, let's break down each step of the performance using simple percentage calculations.

**Scenario:** S&P 500 and 3x Leveraged ETF

We assume the S&P 500 starts with a base value of 100, and we apply the following changes:

**1. 10% rise:** S&P 500 increases by 10%.

**2. 10% fall: **S&P 500 decreases by 10% from its new level.

**3. 10% rise:** S&P 500 increases again by 10% from the new level.

**4. 10% fall: **S&P 500 decreases by 10% from this new level.

The 3x leveraged ETF experiences these same percentage changes, but magnified by a factor of 3.

**Step-by-Step Calculations:**

**1. First 10% Rise:**

- S&P 500:

- Initial Value: 100

- After 10% rise: \( 100 \times 1.10 = 110 \)

- 3x Leveraged ETF:

- Initial Value: 100

- After 30% rise: \( 100 \times 1.30 = 130 \)

**2. First 10% Fall:**

- S&P 500:

- Starting Value: 110

- After 10% fall: \( 110 \times 0.90 = 99 \)

- 3x Leveraged ETF:

- Starting Value: 130

- After 30% fall: \( 130 \times 0.70 = 91 \)

**3. Second 10% Rise:**

- S&P 500:

- Starting Value: 99

- After 10% rise: \( 99 \times 1.10 = 108.9 \)

- 3x Leveraged ETF:

- Starting Value: 91

- After 30% rise: \( 91 \times 1.30 = 118.3 \)

**4. Second 10% Fall:**

-S&P 500:

- Starting Value: 108.9

- After 10% fall: \( 108.9 \times 0.90 = 98.01 \)

- 3x Leveraged ETF:

- Starting Value: 118.3

- After 30% fall: \( 118.3 \times 0.70 = 82.81 \)

**Summary of Final Values:**

**S&P 500: **Ends at 98.01, which is a slight overall decline from the original value of 100.

**3x Leveraged ETF: **Ends at 82.81, reflecting a significant overall decline due to the compounded effects of leverage.

**Key Takeaways:**

**Leverage Magnifies Losses: **The 3x leveraged ETF shows how gains are amplified by the leverage, but it also demonstrates how losses are exacerbated. After two cycles of 10% rises and falls, the S&P 500 is down about 2%, while the leveraged ETF has fallen by nearly 17.2%.

**Volatility Decay: **The leveraged ETF's performance illustrates the concept of "volatility decay," where frequent fluctuations lead to a lower end value than would be expected from a simple linear growth model.

This example can be used to explain how leveraged ETFs can lead to unexpectedly large losses in volatile markets, even when the underlying index ends close to its original value.

**Alpesh Patel OBE**

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**Disclaimer:** The content provided on this blog is for informational purposes only and does not constitute financial advice. The opinions expressed here are the author's own and do not reflect the views of any associated companies. Investing in financial markets involves risk, including the potential loss of your invested capital. Past performance is not indicative of future results.

You should not invest money that you cannot afford to lose. Mentions of specific securities, investment strategies, or financial products do not constitute an endorsement or recommendation. The author may hold positions in the securities discussed, but these should not be viewed as personalised investment advice.

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