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Why XIRR Requires Solving a Net Present Value Equation

Article Quant Q&A · Author: Joshua Fox
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Summary

The document frames the internal rate of return as the rate that makes a cash-flow stream’s net present value equal to zero. For regularly spaced cash flows, this means finding the rate that sets the sum of each cash flow discounted by its period-specific factor to zero.

The answer states that this equation has no closed-form solution and therefore requires iterative approximation. The explanation is brief: it gives the defining equation but does not show why arbitrary cash-flow amounts and timing prevent a general explicit formula, nor does it discuss numerical methods, convergence, or cases with special solvable cash-flow patterns. XIRR implementations handle irregular dates, so the displayed expression is a simplified representation rather than a full date-based formulation.

Key ideas

  • Internal rate of return is defined as the discount rate that makes net present value zero.
  • The cash-flow equation requires solving for the rate inside the discount factors.
  • The document says a general closed-form solution is unavailable, so numerical approximation is used.
  • Special cash-flow patterns and details of XIRR’s irregular date handling are not covered.

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# Why is there no closed-form equation for XIRR?


# Why is there no closed-form equation for XIRR?












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?

## Answer by chrisaycock (score 3)

https://quant.stackexchange.com/a/8577

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.