Understanding the Impact of Compounding on Leveraged ETFs Over Time

Introduction: Compounding is a concept that plays a crucial role in the world of finance and investments. It can have a significant impact, especially when it comes to leveraged exchange-traded funds (ETFs). In this article, we’ll explore how compounding daily leveraged returns differs from delivering a leveraged return over an arbitrary period. We’ll use a real example involving ETFs X and Y, and we’ll also provide a comprehensive explanation of this concept.

The Difference Between Daily Leveraged Returns and Arbitrary Period Returns

Compounding daily leveraged returns means delivering returns that are a multiple of the daily return of an underlying asset. In our example, ETF Y is designed to deliver twice the daily return of ETF X. Let’s illustrate this with a two-day scenario:

• Day 1: Both ETFs start at $100. ETF X rallies by 10% to $110, while ETF Y rallies by 20% to $120, delivering twice the daily return of X.
• Day 2: ETF X drops back to $100, a decrease of 9.09%. ETF Y, designed to deliver twice the daily return of X, drops by 18.18%, bringing its price to $98.18.

At the end of the second day, ETF X remains unchanged, while ETF Y has lost 1.82%. This demonstrates the impact of daily compounding on returns and how it can differ from what you might expect.

Delivering a Leveraged Return Over an Arbitrary Period

When you evaluate the performance of these ETFs over a more extended period, the results deviate from what you might intuitively expect due to the compounding effect. Suppose you evaluate these ETFs over a 10-day period, starting at $100:

• ETF X would be back at $100 after 10 days, having experienced daily fluctuations.
• ETF Y, due to the compounding of its daily returns, would not be at $100 but instead lower, showcasing the impact of compounding.

To demonstrate this more clearly, we can calculate the final prices for both ETFs over a 10-day period:

# Initial values
initial_price = 100
daily_returns_x = [0.10, -0.0909]  # 10% gain and then 9.09% loss for ETF X
daily_returns_y = [0.20, -0.1818]  # 20% gain and then 18.18% loss for ETF Y

# Create lists to store the values
values_x = [initial_price]
values_y = [initial_price]

# Calculate the values for each day
for day in range(1, 11):
    value_x = values_x[-1] * (1 + daily_returns_x[day % len(daily_returns_x)])
    value_y = values_y[-1] * (1 + daily_returns_y[day % len(daily_returns_y)])
    values_x.append(value_x)
    values_y.append(value_y)

df=pd.DataFrame([values_x,values_y])

You’ll find that the final price of ETF Y is less than $100 due to the compounding of daily returns, while ETF X remains at $100. This highlights how compounding can lead to a deviation from the expected leveraged return over arbitrary periods.

Visualizing the Impact

To visualize the impact of compounding, we can create a graph showing the price evolution of ETFs X and Y over a more extended period. In our example, we’ll consider a 100-day period:

import matplotlib.pyplot as plt

# Initial values
initial_price = 100
daily_returns_x = [0.10, -0.0909]  # 10% gain and then 9.09% loss for ETF X
daily_returns_y = [0.20, -0.1818]  # 20% gain and then 18.18% loss for ETF Y

# Ensure both ETFs have the same number of daily returns
assert len(daily_returns_x) == len(daily_returns_y)

# Create lists to store cumulative prices
prices_x = [initial_price]
prices_y = [initial_price]

# Number of days
num_days = 100

for _ in range(num_days):
    r_x, r_y = daily_returns_x[_ % len(daily_returns_x)], daily_returns_y[_ % len(daily_returns_y)]
    prices_x.append(prices_x[-1] * (1 + r_x))
    prices_y.append(prices_y[-1] * (1 + r_y))

# Create the time axis
days = list(range(num_days + 1))

# Plot the prices
plt.figure(figsize=[15,6])
plt.plot(days, prices_x, label="ETF X")
plt.plot(days, prices_y, label="ETF Y")
plt.xlabel("Days")
plt.ylabel("Price")
plt.title("Price Evolution of ETF X and ETF Y over 100 Days")
plt.legend()
plt.grid(True)

# Show the plot
plt.show()

Conclusion

Compounding daily leveraged returns can have a significant impact on the performance of leveraged ETFs over time. It’s essential to understand that delivering a leveraged return over an arbitrary period is not the same as compounding daily returns. The compounding effect can lead to deviations from the expected performance, making it crucial for investors to be aware of these dynamics when considering leveraged ETFs in their portfolios. This phenomenon is a mathematical fact and underscores the importance of understanding how compounding works in the world of finance.