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Everything I have read about XIRR (e.g., as calculated in Excel) says that there is no closed-form equation and it must be calculated by iterated approximation.

Could someone give a brief mathematical explanation of why there is no closed-form solution?

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The internal rate of return is simply the rate at which the net present value is zero. So solve for $r$ in

$$ \sum{\frac{C_n}{(1+r)^n}} = 0 $$

There is no closed-form solution to this.

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