# 1.35 The Reinvestment Rate Assumption of the Internal Rate of Return

**Problem:**

The IRR calculation assumes that the future cash flows will each be reinvested, to the horizon, at the project’s calculated IRR rate. This reinvestment rate assumption is most likely unrealistic because, in fact, the firm may choose to invest interim cash flows in some other project or investment. Should this be true, the project’s true rate of return will be more or less than the IRR, depending on whether the reinvestment rate is greater or less than the IRR.

To understand the nature of this problem, let’s recalculate the IRR formula using the IRR as the reinvestment rate, which will thereby enable us, first, to figure the project’s future – or “terminal” – value. (If you wish, it may be helpful to draw a timeline.) Then, we will compare the terminal value to the project’s initial cost (or investment value), which using TVM, will imply a discounted rate of return. If this discounted rate of return is the same as the IRR, it will mean that the IRR calculation embeds the redundancy implicit in the assumption that the IRR reinvests the interim cash flows at the IRR.

Let’s show the mathematics of this problem (so as to be sure that it is correct as stated), using **“Example 3”** above:

If the project’s reinvestment rate = 28% (i.e., the IRR), we have the following:

Project’s Future Value (FV) | = CF × FVAF |

= $1,000 × 8.6999 | |

Project’s Cost | = $2,532 |

Implied Rate of Return | = $8,699 ÷ $2,532 |

= 3.44× |

This means that for every dollar ($1) invested, the project will produce, at the horizon or terminal point, $3.44 – assuming a reinvestment rate equal to the IRR, which in this instance is 28%. We can solve for the annual compounded rate or return as follows:

(3.44) ^{1/5}– 1 = .28

Alternatively, using the interest rate tables, one would look for the 3.44 multiplier along the five-period row in the simple FV chart; this cell will be (approximately) located under the 28% column.

In any event, this implies a rate of return of 28%, i.e., the same as the originally derived IRR! In other words, the problem with the IRR is that it assumes reinvestment at the IRR, an unlikely possibility!!

The general IRR formula is now **R = [(FV / PV)] ^{1/(n x p) }-1**.