Term for reciprocal percentage

Question

I find myself struggling to succinctly express the concept of gain or loss percentages that are equivalent in the sense that they are reciprocal. For instance, take the following sentence:

An investor with log-utility will experience the same magnitude of utility change for a gain or loss of a certain percentage of wealth.

What is intended is that the investor would lose as much utility from a 20% decrease in wealth as they would gain from a 25% increase (because (1+25%)=1/(1+-20%)). Obviously, I can say "change in log" but that makes the above sentence feel redundant. And I could also say "for percentage gains or losses that are reciprocal when added to unity", but that's pretty verbose. Is there a nice succinct term/way to express this idea of "same" reciprocal percentages?

Answer 1

The log-utility function is $U(c)=\log(c)$ where $c$ is the investor's wealth. We know that $$ \frac{dU(c)}{dc}=\frac{1}{c}\,. $$ In other words, the absolute change $dU$ of the utility under an absolute change $dc$ of wealth is $$ dU=\frac{dc}{c}\,. $$ The right hand side of this is obviously the relative change of wealth.

In words: the absolute change of log-utility equals the relative change of wealth, regardless if the latter is a gain or a loss.

Regarding your numerical example. The relation $$ 1+25\%=\frac{1}{1-20\%}\,. $$ says nothing else than a $25\%$ increase of wealth followed by a $20\%$ loss brings us back to the initial wealth $c_0$: $$ c_1=c_0(1+25\%)\,,\quad\quad c_2=c_1(1-20\%)=c_0(1+25\%)(1-20\%)=c_0\,. $$ I'd say that is is the succint way of expressing "same reciprocal percentage".

Closely related to this is the following asymmetry of the log-utility: The corresponding log-utility does not got back to the initial utility: $$ U_1=U_0+25\%\,,\quad\quad U_2=U_1-20\%=U_0+5\%\,. $$ This happens only when an absolute gain is followed by an equal absolute loss. The log-utility function has an asymmetry which is used to model risk aversion.