I know that the IRR can be computed using iterative methods, but why is this necessary? What makes it impossible to give an expression for IRR? How would you prove it to be impossible?
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From the Wikipedia article, solving for the IRR is the same as finding a solution to an arbitrary polynomial. In general, it is difficult to solve an arbitrary $n$-degree polynomial with $n>4$.
I'm sure, given specific cash flow values and NPV, you can come up with explicit solutions for $r$, but general polynomials require more complicated formula than just taking roots.
Most of what is known about the difficulty of solving polynomial equations is covered in the field of Galois theory.
I only wanted to comment on the answer yet as I don't have enough points thus no way to invoke the comment link
Solving roots of a polynomial entails a number of solutions that are same as the highest power of the given polynomial (albeit that none of these may be real).
In terms of an investment and finding an IRR - ( internal rate of return at which the investor breaks even on the investment ) takes the route of solving one of the many IRR equations ( may it be the net present value NPV=0, the net future value NFV=0, the benefit to cost ratio BCR=1, or the equivalent annual annuity EAA=0 ).
Now that there will be multiple solutions for any of these equations, but there will be only 1 or no real IRR value.
I say only one IRR of the many real roots of the equations will best represent the investor's return on investment - ROI.
Say now that you have multiple real IRR values of which some are negative and others are positive. Now deciding upon which of these IRR values is the actual IRR is a problem of its own to solve. That leads me to think that if we are only seeking a single IRR value then solving for the roots of the given IRR equation as a polynomial may not be the best approach. And we would have to look forward towards cornering an IRR formula that will take the cash flows and the timings of such cash flows and to either return 1 or 0 solutions (either a real or complex solution).
If there were to be such a formula to find a single IRR then finding the payback period using the same formula will be possible as well. This is so as the Payback period occurs when the sum of earlier cash flows of the investment is zero.