Log Returns vs Periodic Returns in Trading

The choice between log returns and periodic returns in trading can significantly impact your results. Let’s explore this in more detail.

Log Returns

Log returns are advantageous because compounding can be achieved through addition rather than multiplication. For periodic returns, the total return is calculated as: [ R_{total} = R_1 \cdot R_2 \cdot … \cdot R_n ]

This works because: [ R_{total} = \frac{P_t}{P_{t-1}} ] Therefore: [ R_{total} = \frac{P_n}{P_0} ]

For log returns, the total return is calculated as: [ r_{total} = r_1 + r_2 + … + r_n ]

This works because: [ r_t = \log\left(\frac{P_t}{P_{t-1}}\right) = \log(P_t) - \log(P_{t-1}) ] So: [ r_{total} = \log\left(\frac{P_{t+n}}{P_t}\right) ]

Important Takeaways from Approximation Error in Logs

1. The approximation error is small for values of x close to zero but becomes substantial for |x|>0.2.
2. The approximation error is asymmetric around zero for |x|>0.01, being larger for −x than x.
3. The error direction is also asymmetric, with negative values of x being amplified and positive values of x being dampened.

Key Observations

• During the technology bubble, P/D in periodic returns appears to grow exponentially, whereas in log returns, it looks almost linear (log(ex)=x explains this).
• During the financial crisis, the drop in log returns is much more pronounced than in periodic returns.
• This highlights a crucial point: a rising series appears more dramatic in periodic returns, while a falling series looks more severe in log returns.
• Additionally, the vast majority of stock returns fall between -0.05 and 0.05, where the approximation error is minimal.
• This discrepancy becomes significant during extreme events, where the approximation error causes noticeable differences between the two methods.