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I'm trying to learn how to calculate the Extended Internal (XIRR). One website says the following: "The XIRR is not an empirical formula though. It is a solution for a particular state of the the XNPV (eXtended Net Present Value) calculation."

That leaves me with 2 questions:

  • What is an "empirical formula"?
  • What is a "solution for a particular state" of something else?
Userq49874
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    Empirical usually means a model derives from observed data rather than any underlying principles. I have no idea what the particular state phrase means. – copper.hat Sep 13 '19 at 17:43
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    It looks like they are using lofty language and intimidating acronymns to sound smart. From what I see they have an equation and the left-hand-side they are calling XNPV and the variable $r$ they are calling XIRR. They could have just continued to call it $r$. They are just saying they do not have a direct expression for $r$ in terms of XNPV, rather, they only have an expression for XNPV in terms of $r$. – Michael Sep 13 '19 at 17:45
  • I think that there's a better discussion of the XIRR here – saulspatz Sep 13 '19 at 17:50

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They have an equation $$XNPV = \sum_{i=1}^N \frac{P_i}{(1-r)^{(d_i-d_1)/365}}$$ They are calling the $r$ parameter the XIRR value.

If we know the values $P_i, d_i$ for $i \in \{1, …, 365\}$ and we also know $r$, then we can calculate $XNPV$ in terms of the above closed-form formula. In their language I believe they use "empirical formula" to mean "closed form formula."

On the other hand, if we know $P_i, d_i$ for $i \in \{1, …, 365\}$ and we also know the particular value of XNPV (they are calling it the particular state of XNPV) then we do not have a closed form expression for $r$. Mathematically, we do not know how to "invert" that formula. You can only get $r$ by a numerical search. For example, you can guess a value of $r$ (call it $r_{guess}$) then calculate $\sum_{i=1}^N \frac{P_i}{(1-r_{guess})^{(d_i-d_1)/365}}$, see if this is larger or smaller than the target XNPV value, then increase or decrease $r_{guess}$ accordingly.

Suppose $P_i, d_i$ are fixed for all $i \in \{1, ..., 365\}$. If we assume $d_i\geq d_1$ for all $i \in \{1, …, 365\}$ and $0\leq r < 1$, then the expression $\sum_{i=1}^N \frac{P_i}{(1-r)^{(d_i-d_1)/365}}$ is increasing in $r$ and: $$ \sum_{i=1}^N P_i \leq XNPV < \infty$$ So for a given XNPV value that satisfies the above inequality, there is one unique value of $r$ that works. It can be found by formalizing the above guess method according to a standard bisection search.

Michael
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