# Average return per period for a total loss

Short version: Is there a meaningful notion of "average return per period" for an investment whose value falls to zero over time?

Long version: Call the gross return on an investment the ratio $\frac{Pt}{Pt-k}$, where $P_t$ is the asset price at time $t$ and $P_{t - k}$ is the asset price $k$ periods before $t$. Then $$\frac{P_t}{P_{t-k}} = \frac{P_{t-k+1}}{P_{t-k}}...\frac{P_{t-1}}{P_{t-2}}\frac{Pt}{P_{t-k}}$$ the product of the single-period gross returns from $t-k$ to $t$. For nonzero gross returns over $k$ periods, we can compute a meaningful average return via the geometric mean, i.e. the $k^{th}$ root of the $\frac{P_t}{P_{t-k}}$. Choosing all of the single-period returns to be this value would result in the desired $k$-period return.

In the case of a total loss, where the final asset price is zero, the geometric mean would be zero. This makes sense in that, if all single-period returns were zero, the $k$-period return would also be zero. But it fails to account for the number of periods it takes for the asset price to fall to zero. It seems sensible that the gross daily return for an asset whose value falls from \$100 to \$0 in a single day is 0. But what if the asset price falls to linearly to zero over the course of 100 days? It seems that the gross daily return should be higher in this example than in the first, since the value of the investment declines less rapidly. Under the geometric mean approach, though, both of these scenarios result in the same gross daily return: 0.

Is there a measure of average return that accounts for the time difference between these two scenarios?