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The Math of the Fixed Rate Equivalent (FREQ)

If we multiply both sides of Equation (1) by and then move the constant coefficient to the right hand side, we obtain

To make this equation look simpler, let us introduce the following new notation:

= , the number of cash flows except for the last
= , the number of periods between the th and the last cash flow
= , the negative of the last cash flow

Equation (2) now becomes

Finally, we successively pull the highest possible power of out of the leftmost two summands in Equation (3):

If one wanted to evaluate the left hand side of Equation (4) for a particular value of , one would do it from the inside out, successively evaluating the following expressions:

This way of evaluating a polynomial is known as the Horner scheme, and it is in fact used in mathematical software as the most efficient way of evaluating a polynomial. In the case of the internal rate of return, it reveals the alternate interpretation of the IRR that we were looking for: if we interpret the first cash flow as an initial deposit, then the left hand side of Equation (4) is the ending balance of an account with a fixed interest rate of , subjected to the given deposits and withdrawals except for the last one. If we also interpret the last cash flow as the withdrawal that empties the account (or, equivalently, as the negative of the ending account balance), then solving the equation for means finding the fixed interest rate that leads to the same ending balance as the actual investment. In other words, the internal rate of return aims to be the fixed interest rate that replicates the actual investment. Equations (5) above demonstrate this nicely: the expressions

are the account balances right after the th cash flow, and is the ending balance.