Is there any interpretation to the imaginary component obtained when computing the geometric mean of a series of negative returns?

Question

When computing returns in finance geometric means are used because the return time series of a financial asset is a geometric series: $\mu_r = \sqrt[T]{\prod_{t=1}^T r_t}$ where the return is computed as $r_t = \log\left(\frac{p_{t+1}}{p_t}\right)$ and $p$ is the value of the asset. Negative returns are (sadly) a financial reality. But the geometric mean return obtained when there are negative returns does not lend itself to a straightforward interpretation because $r < 0 \implies \mu_r \notin \mathbb{R} $. Is there any financial interpretation to the imaginary component obtained when computing the geometric mean of a return time series (geometric series) including negative returns?

Answer 1

If you use the logarithmic values, then you will get the log of the geometric mean by calculating $$log(\mu_r)=\frac{1}{T} \cdot \sum_{t=1}^T r_t$$

The formula you have posted is only valid for returns, which have not been logarithmized.