Crypto Returns: Logarithmic vs. Arithmetic

1. Defining and Calculating Returns

The return of an asset is the most fundamental measure of its performance. There are two primary ways to calculate it: arithmetic and logarithmic.

1.1. Arithmetic (Simple) Returns

The arithmetic, or simple, return is the one most people are familiar with. It represents the percentage change in price over a single period.

The formula is:
$R_t=\frac{P_t-P_{t-1}}{P_{t-1}}=\frac{P_t}{P_{t-1}}-1$
where $P_t$ is the price at time $t$ and $P_{t-1}$ is the price at the previous time step.

While simple returns are intuitive for a single period, they are not additive over time. To find the total return over $n$ periods, you must compound them:
$R_{total}=(1+R_1)(1+R_2)...(1+R_n)-1$

The following video provides a clear walkthrough of calculating simple and cumulative returns.

1.2. Logarithmic (Continuously Compounded) Returns

Logarithmic returns are based on the concept of continuous compounding, which should be familiar from your physics and math background. Instead of compounding at discrete intervals (daily, monthly), we assume it happens infinitely often.

The formula for the log return is:
$r_t=\ln \left(\frac{P_t}{P_{t-1}}\right)=\ln (P_t)-\ln (P_{t-1})$
where $\ln$ is the natural logarithm.

To understand the origin of this formula and its link to continuous compounding, the next video provides an excellent derivation.

The most important property of log returns is that they are time-additive. The total return over $n$ periods is simply the sum of the individual log returns:
$r_{total}={\sum }_{i=1}^n{r}_i$
To convert a total log return back to a simple return, you use the exponential function:
$R_{total}=e^{r_{total}}-1$

This time-additive property makes log returns incredibly convenient for statistical modeling and multi-period analysis.

2. Arithmetic vs. Log Returns: A Comparison

Choosing between simple and log returns involves a trade-off. The following article provides a concise summary of their pros and cons.

Here is a summary of the key points:

For small return values, arithmetic and log returns are nearly identical. The difference becomes significant with larger price changes, which are common in crypto markets.

The choice of chart scaling—arithmetic vs. logarithmic—is a direct visual application of this concept. A logarithmic scale chart plots price changes such that equal vertical distances correspond to equal percentage changes, which is effectively visualizing log returns.

3. Statistical Properties and the "Fat Tail" Reality

In quantitative finance, it is often assumed that log returns are normally distributed. This assumption is mathematically convenient, as the sum of normally distributed variables is also normally distributed, which simplifies modeling.

However, for most financial assets, and especially for cryptocurrencies, this assumption is dangerously inaccurate. Real-world returns exhibit properties that a normal distribution fails to capture.

The Problem of "Fat Tails" (Leptokurtosis)

Financial returns are leptokurtic, meaning their distribution has "fat tails." This signifies that extreme events (both large gains and large losses) occur far more frequently than a normal distribution would predict. Assuming normality leads to a systematic underestimation of risk and the probability of "black swan" events.

The following reading explains this limitation and demonstrates it with Python code.

As you saw in the article:

1. Both simple and log returns for real stock data fail the KS test for normality. The low p-value allows us to reject the null hypothesis that the data comes from a normal distribution.
2. A much better fit is achieved using the Student's t-distribution. This distribution has an additional parameter, degrees of freedom ($\nu$), which controls the "fatness" of the tails. A lower $\nu$ means fatter tails. As $\nu \to \infty$, the t-distribution converges to the normal distribution.
3. For financial modeling, especially in a volatile market like crypto, using a distribution that accounts for fat tails is crucial for realistic risk management and strategy evaluation.

4. Python Implementation

Calculating these returns in Python is straightforward using the pandas and numpy libraries.

• Simple Returns: The pct_change() method in pandas is designed for this.

  import pandas as pd
  
  # Assuming 'df' is a DataFrame with a 'Close' column
  df['simple_return'] = df['Close'].pct_change()
  

• Log Returns: You can use numpy.log() in combination with shift().

  import numpy as np
  
  # Assuming 'df' is a DataFrame with a 'Close' column
  df['log_return'] = np.log(df['Close'] / df['Close'].shift(1))
  

The Python implementation section of the first video provides a good visual demonstration of these calculations.

Conclusion

In this lesson, we have dissected the concept of returns, the fundamental input for any quantitative trading strategy.

Key Takeaways:

• Arithmetic (simple) returns are intuitive and asset-additive, making them useful for calculating portfolio performance at a single point in time.
• Logarithmic returns are time-additive, making them the standard for statistical modeling and multi-period analysis.
• For quantitative strategies like momentum, log returns are generally preferred due to their desirable mathematical and statistical properties.
• Crypto asset returns are not normally distributed. They exhibit fat tails, meaning extreme price moves are more common than a normal distribution suggests.
• The Student's t-distribution is a more realistic model for financial returns as it can account for fat tails. Acknowledging this is critical for robust risk management.

Now that we understand how to calculate returns from price data, the next logical step is to learn how to acquire that data efficiently and reliably.

In the next lesson, we will tackle the learning outcome: "Fetch historical OHLCV data for multiple crypto assets using REST APIs and handle exchange-specific rate limits and pagination." This will be a practical, hands-on lesson in building the data pipeline that will feed our future momentum calculations.