A near "geometric mean" that handles zeros and negative numbers

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.

• If you multiply all the numbers by the same constant $k$, the geometric mean is also multiplied by $k$. This means that, for example, it doesn't matter whether the input data are given in units of dollars, cents, or thousands of dollars, etc.; the geometric mean doesn't change. This is not true of your IHS mean.
– user856
Aug 26, 2016 at 4:33
• For example, $\text{GM}(\$0.01, \$1.00) = \$0.10 = 10¢ = \text{GM}(1¢, 100¢)$. But$\text{IHSM}(\$0.01, \$1.00) \approx \$0.46 \ne 11¢ \approx \text{IHSM}(1¢, 100¢)$...
– user856
Aug 26, 2016 at 4:56
• I don't think this question has much sense unless you explain what you mean when you say things like "IHS is more useful". Does "more useful" just mean "doesn't crash my code"? There are many things you could do which will not crash your code, for example the arithmetic mean of your data will also be defined when there are zeros and negative numbers. If you ask 10 people to come up with new a way of assigning a mean to a dataset with zeros and negatives, you'll probably get 10 different answers. What properties does your mean have that make it better than the other 9? Aug 26, 2016 at 5:15

1 Answer

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.

• What's your thoughts on this: worthwhile.typepad.com/worthwhile_canadian_initi/2011/07/… Aug 26, 2016 at 5:28
• Thank you for the insights. Aug 26, 2016 at 5:29
• Interesting. They cite two papers. The key claim is in the abstract of the 1988 article, that the Box-Cox and Inverse Hyperbolic Sine transformation are "reasonable on a priori grounds". Whether there are a priori arguments for the use of those transformations in economic settings, such as the scale-invariance that justifies the log transformation, is the whole question, but the articles are paywalled and I did not see the part that might explain what those reasonable grounds are. I would be interested to find out more about that if you have access to the articles.
– zyx
Aug 26, 2016 at 5:38
• Here is a different abstract I found ajae.oxfordjournals.org/content/83/5/1314.extract Aug 26, 2016 at 12:27
• I did some internet searching, and it seems that IHS is used as a data transformation in econometrics, but with no justifications other that it being approximately $x$ (therefore approximately linear) for small $x$, approximately logarithmic for large $x$, and having $f(-x) = -f(x)$ so that negative values are accomodated and symmetrically cancel positive ones.
– zyx
Aug 26, 2016 at 13:52