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?
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.