Simple Compounding vs Continuous Compounding in return series

Question

I'm creating a log price series in MATLAB. This is fairly easy to do using standard functions. Given a price series prices:

r = diff(log(prices)) will give you the standard log return series calculated by

$r = \ln(P_t) - \ln(P_{t-1})$

However, I'd like to use the tick2ret function in the Financial Toolbox to keep my tooling consistent. Here's where the confusion starts:

I've always referred to r above as the "continually compounded return series". Perhaps this is incorrect. tick2ret calculates, by default, the "simple compounding return series". Curious, I made another return series:

r2 = tick2ret(prices)

and as it turns out r and r2 are virtually the same aside from rounding. I expected the "simple compounded returns" to be calculated using

$r = \frac{P_t - P_{t-1}}{P_{t-1}}$

However, according to my calculation they are using the log return calculation.

Am I misunderstanding simple and compound returns? Can the log return series be calculated using simple compounding? I think I am getting lost in terminology.

Thank you!

Answer 1

Let $\Delta P = P_t - P_{t-1}$ and expand the continuously compounded return in a Taylor series $$ r = \log\left(\frac{P_t}{P_{t-1}}\right) = \log\left(\frac{P_{t-1}+\Delta P}{P_{t-1}}\right) = \log\left(1+\frac{\Delta P}{P_{t-1}}\right) \approx \frac{\Delta P}{P_{t-1}} - \mathcal{O}\left(\left(\frac{\Delta P}{P_{t-1}}\right)^2\right) = \frac{P_t - P_{t-1}}{P_{t-1}} - \mathcal{O}\left(\left(\frac{\Delta P}{P_{t-1}}\right)^2\right) $$ If $\Delta P/P_{t-1}$ is small, which is typically the case for daily or shorter time frames, the higher order terms can be neglected and the continuously compounded return is approximately equal to the simple return.