I am trying to calculate what the total return would be over a defined period where there is a natural attrition rate from the base. Input fields are therefore (i) Starting earnings (ii) Time period (years) and (iii) attrition/retention rate. For example $100,000 starting earnings over five years with a 90% retention (10% attrition) per annum.

I would have thought this would be straight forward however I have not found an obvious answer. Positive growth is easy: $100,000 x 5^(1+0.1).

Any guidance would be greatly appreciated.