Can logarithmic returns cumulatively sum to less than -1?

Question

I have a question regarding the time additive properties of logarithmic returns specifically regarding financial assets. My understanding of logarithmic returns is that they can be summed across time to return compound return rates. If this is true, I would guess that the minimum bound of the cumulative sum of log returns for financial assets would be -1 as one cannot lose more than 100% of their principle assuming they are long-only without leverage. This however does not appear to be the case! I have repeatedly encountered assets whose log returns sum cumulatively to less than -1. I am not sure if I simply misunderstand the properties of log returns or if I am in some way misapplying the calculation in my code.

Here is a minimal example in python demonstrating the phenomenon using Apple stock data from 2000 to present day:

import yfinance as yf

data = yf.download("AAPL", start="2000-01-01", interval="1d").get("Adj Close")
cum_log_returns = np.log1p(data.pct_change()).cumsum()

# Plot returns
cum_log_returns.plot()

Here was can clearly see that the cumulative sum of our log returns is below -1 between approx. 2002 to 2003. How can this be explained? Does the y-axis not represent percentage return of some sort? What am I missing here? I understand that we can convert these to arithmetic returns by taking the natural exponent, but how is this any different than just using the cumulative product of arithmetic returns?

Further resources on the mathematical properties of log returns would be great. I have read a number of articles on when to use them and how to calculate them, but not found any rigorous explanation of their properties.

Additionally, this is my first post here so please do let me know if I haven't provided enough information in some way.

Answer 1

First, for a price series $p_t$, the log-returns are defined like this:

$$r_t = \log{\left(\frac{Y_t}{Y_{t-1}}\right)}$$

If you sum these up over a period of time from $t_1$ to $t_2$, using the usual properties of logarithms, you get the telescoping sum:

$$\sum_{t=t_1}^{t_2}r_t = \log{\left(\frac{Y_{t_1}}{Y_{t_1-1}}\right)} + \log{\left(\frac{Y_{t_1+1}}{Y_{t_1}}\right)} + \cdots + \log{\left(\frac{Y_{t_2-1}}{Y_{t_2-2}}\right)}+\log{\left(\frac{Y_{t_2}}{Y_{t_2-1}}\right)}$$

Which is just:

$$\sum_{t=t_1}^{t_2}r_t = \log{\left(\frac{Y_{t_2}}{Y_{t_1-1}}\right)}$$

So, the "cumulative log-return" is just a normal log-return, over the whole period.

Second, since $Y_t$ is strictly positive but arbitrarily small, the log-return can be arbitrarily negative. For example, take a stock that was worth \$1000 at the start and is now worth \$0.01. That's a log-return of:

$$\log{\left(\frac{0.01}{1000}\right)} \approx -11.5 \ll -1$$

This is not the case for period returns, which are defined like this:

$$R_t = \left(\frac{Y_t}{Y_{t-1}}\right)-1$$

Period returns can't be smaller than -1 because that would require $Y_t$ to be negative. The two measures are different; they are only numerically similar when they are relatively small.

The fact that log-returns are supported on the whole real line has advantages with respect to the period returns: it's usually easier to build a model for $r_t$ than for $R_t$ because having to restrict the model's support is typically more difficult than not.