By definition of a geomean, a geometric mean of a set of numbers containing zero is 0. (See Geometric mean of a dataset containing zeros) However in financial data that is commonplace and sometimes finance data has negative numbers.
The inverse hyperbolic sine mean doesn't suffer from these shortcomings.
Let's define that as: sinh(mean(asinh(money)))
It handles negatives, positives and zeros with ease.
Why won't people use the inverse hyperbolic sine mean (IHS) mean? Why don't they teach the IHS mean? Is there a better term for this? Surely someone else has thought of it.
This isn't a question about R as the code is only given to be informative.
set.seed(12)
(money=sort(round(rlnorm(100,6,4)))+1)
sinh(mean(asinh(money))) #gives 393.7934
exp(mean(log(money))) #gives 391.2577
prod(money)^(1/length(money)) #gives 391.2577 but can overflow
median(money) #261.5
Notice with positive numbers, the IHS mean is nearly identical.
set.seed(12)
(money=sort(round(rlnorm(100,6,4))))
sinh(mean(asinh(money))) #gives 369.4342
exp(mean(log(money))) #gives 0
prod(money)^(1/length(money)) #gives 0
median(money) #260.5
Notice with positive numbers and zeros, the IHS mean is more useful.
set.seed(12)
(money=sort(round(rlnorm(100,6,4)))-10000)
sinh(mean(asinh(money))) #gives -201.5435
exp(mean(log(money))) #gives NaN
prod(money)^(1/length(money)) #gives Inf so WRONG due to overflow
median(money) #-9739.5
Notice with mixed negative/positive numbers and zeros, the IHS mean is more useful.
set.seed(12)
(money=sort(round(rlnorm(100,6,4)))*-1)
sinh(mean(asinh(money))) #gives -369.4342
exp(mean(log(money))) #gives NaN
prod(money)^(1/length(money)) #gives 0
median(money) #-260.5
Notice with purely negative, the IHS mean is still useful.
Because only the arithmetic mean and geometric mean have a meaningful financial interpretation.
Financial processes are scale-invariant for "normal" sizes and durations of the process. A $k$ times larger quantity of stock is thought of as having $k$ times the value, although this is violated for very small or very large $k$, or on very short time scales.
Therefore, proportional changes in value of a portfolio are often more relevant than absolute changes. Tracking the proportional changes is equivalent to tracking log(PortfolioValue). Taking the average (arithmetic mean) in the log scale is the same as taking geometric mean of the values without the logarithms.
This explains the use of the geometric mean. Other changes of the scale such as the inverse hyperbolic sine do not mean anything financially which explains why they are not used.
The appearance of negative numbers does not invalidate this approach. In finance, quantities that can change sign almost always appear due to subtractions of positive quantities (income minus expenses, assets minus liabilities) and the positive pieces can still be modeled using scale-invariant proportional changes.