To find the "rate of return", I assume you want the average rate of return (as market fluctuations continuously change the instantaneous rate of return). Assume it is a constant interest rate. Assume it is compounded monthly. Now, this is a problem we can solve. Set it up like you are calculating compound interest. Let's say the client invests for months. Let's say is the amount the client invests in month . So, is the principle investment: £55,186. The total value (after compounding interest) would be:
As some have told you, this would involve solving a complicated polynomial (the variable I am using is , which is "annual interest", or annual rate of return). However, there are techniques for solving complicated polynomials that take relatively very little time (especially when you know the approximate range for zeros of the polynomial).
Alternately, if you use the average monthly contribution, then you can simplify the problem slightly to:
The LHS simplifies to this:
This is a simpler polynomial, but will still require a computer algebra system to solve.