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6
The FREQ Vs. the IRR (Internal Rate of Return)

The ending balance comes out the same as that of the brokerage account, namely, $672.08. That demonstrates that 10.42% is indeed the FREQ for the brokerage account: the 10.42% fixed rate account replicates the performance of the brokerage account. We see that the replicating fixed rate account encounters negative balances despite the fact that the original brokerage account did not.

The chart on the previous page also shows the account balances over time for accounts with a 12% interest rate and a 9% interest rate, respectively. That should provide a bit of an intuition for why the FREQ is unique. There is no fixed interest rate other than the 10.42% FREQ that results in the $672.08 ending balance: if you increase the interest rate, the ending balance goes up, and if you decrease it, the ending balance goes down.

Now let us see how the IRR fares on this example. If you enter the numbers of Table 1 above into your favorite software for calculating the IRR, then, depending on which software you use and what initial guess you enter (if any), the answer could be 5.5% or 8.66%. The chart below demonstrates how under the IRR's model, where positive and negative balances are treated equally as far as rates are concerned, both rates replicate the brokerage account's performance. We are thus looking at an example of the internal rate of return (IRR) having multiple solutions.