# 1.37 General MIRR Formula (Derivation)

The general formula for MIRR conforms to our basic TVM formula:

PV (1+R)= FV or [ (FVĂˇPV)^{1/n}] â 1= R

While this works well for a case involving just one outflow and one inflow, we will usually have multiple inflows. Thus, a more general formula, which you may use for the MIRR is:

(Future Value Ăˇ Present Value of Cost) ^{1/n} â 1 = MIRR

Where the Future Value (âFVâ) is:

ÎŁ [CFs (1 + REIN) ^{n}] = FV

The projectâs Terminal Value (âTVâ) is the future value (âFVâ) of the reinvested cash flows the project is expected to produce, assuming an âexogenousâ (i.e., external) reinvestment rate, to the horizon.

Where âREINâ is the compound reinvestment rate for the cash inflows, and the âterminal valueâ is thereby determined by compounding the interim cash flows at the reinvestment rate to the horizon.

Again, the CFs are the nominal future cash flows, each of which is bumped up respectively to its terminal value and then aggregated. In calculating the terminal values, be careful about the manner in which you are employing exponents when dealing especially with uneven cash flows; if the cash inflows are an annuity, the process is much simpler. Remember you must figure how many periods are left to the horizon for each cash flow; âthe arrows are going to the right.â

The âpresent value of the costâ (PV) is the original cost of the investment, or its initial outlay.

Or more simply: [(FV Ăˇ PV) ^{1/n}] â 1 = MIRR

We know this formula from the â$1 to $2â discussion. It is the same formula as used there!

**Problem****:** Suppose someone had $3,050 nine years ago and today it has grown to $3,950. What is his compound annual rate of return?

**Solution:** (3,950 Ăˇ 3,050) ^{1/9} â 1 = 0.0291

Note that, in this problem, the IRR = MIRR because there is no cash flow to reinvest.