The asymmetric impact of gains and losses

It is important to be aware of the fact that a positive and negative return of the same relative magnitude do not compensate each other in terms of terminal wealth. Mathematically, this can be seen from the identity \((1+x)*(1-x)=1-x^2\), which is less than one. A 50% loss is thus not compensated by a 50% gain. After a loss of 50%, what is the return needed to be at par again? Verify your answer in the R console.