7

The Math of the Fixed Rate Equivalent (FREQ)
Here is a chart for the data of Table 2:

Chart 1

Imagine that you start with the 5.5% graph, then increase the interest rate so as to morph it into the 8.66% graph. The positive account values during the first three years will increase faster, while the account values during the subsequent five years will decrease faster. That’s how it is possible for the ending balance to be the same for the two rates. The intuition behind Theorem 1 is that this kind of “opposite movement” cannot happen unless the account balance dips into the negative.

So far we have seen:

- The internal rate of return, when interpreted as the interest rate paid by the fixed rate investment that replicates the actual investment, uses a model that applies the same compounding rate regardless of whether the balance is positive or negative. In the world of individual investing, this is unlikely to be deemed an appropriate model.
- This model of applying interest is the root cause of the multiple solutions issue of the IRR.

How can this situation be amended? There is really only one acceptable way of modifying the model of the fixed rate investment: do what is done in real life, that is, charge the cost of borrowing that was in effect at the respective time period whenever the balance of the modeling fixed rate investment becomes negative. The following theorem says that this model leads to a rate of return that no longer suffers from the multiple solutions issue. For simplicity, the theorem will assume that the first cash flow is positive (initially positive account balance) and the last cash flow is negative (ending account balance positive). This will exclude some odd corner cases such as the one where the investor is always in the red. The mathematically precise version of the theorem that is given in the extended version of this paper uses more refined assumptions.