# Modified Internal Rate of Return (MIRR)

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As I promised in the post titled “How to Calculate IRR After Financing”, I want to dedicate a post showing how and why Modified Internal Rate of Return (MIRR) is calculated. But before we delve into Modified Internal Rate of Return (MIRR), let’s briefly restate what Internal Rate of Return (IRR) is and is not.

The Internal Rate of Return (IRR) is the rate which when used to discount future cash flows will yield a zero Net Present Value (NPV) and for which there is no equation; it is found by iteration (please see “How to Calculate the Internal Rate of Return (IRR)” for review). The IRR is not intentionally misleading, but there are some assumptions implicit in its calculation that one needs to be aware of.

The Internal Rate of Return (IRR) has a “Reinvestment Assumption”, but its existence is not some machination of crafty financiers or brokers, but  is rather a consequence of its mathematical definition: discounting cash flows in the future back to the present at a single rate is just the opposite of compounding an amount today at a single rate out into the future. Before delving further into the reinvestment assumption, let’s satisfy ourselves that discounting is the mathematical opposite of compounding: As you can see, compounding 100 by 10% yields 110 in year one, 121 in year two and so on. In mathematical terms we are just multiplying the present value of the cash flow by 1+(discount rate). Discounting is just the reverse process; by dividing the future value by (1+discount rate) we find the present value of the year preceding next year’s cash flow. So the Present Value of year five’s cash flow of 161.05 is 146.41, or 161.05/(1+10%)^1. Here, we raise the denominator by the power of one (^1) because we are just discounting for one year prior to the future value, in this case from year five to year four. If we wanted to discount all the way back to year zero, we would want to divide 161.05 by (1+10%)^5:

The Reinvestment Assumption implicit in IRR calculations stems from this math; the NOI from a property in every year needs to be reinvested and compounded at the IRR for every subsequent year if the IRR is to equal the Return on Investment (ROI), which will be calculated at the end of the investment by the simple equation:

(Cash Flows from Investment – Investment)/Investment

or

Cash Flows from Investment/Investment – 1

If you invested your free cash flows at something less than the IRR – say if you put them in your bank account at a 2.5% annual interest whereas the IRR is 9.15% – your ROI will be less than IRR. Modified Internal Rate of Return is used to account for the fact that free cash flows are reinvested and compounded at a rate different than the discount rate/IRR, a reality for most investors. Note however that by using unique Reinvestment Rates we are introducing subjectivity into the measurement of returns because your Reinvestment Rate is likely to be different from mine.

Let’s take a look at the formula for MIRR (from Wikipedia), which is in the same format as the simplified ROI formula: where n is the number of equal periods at the end of which the cash flows occur (not the number of cash flows), PV is present value (at the beginning of the first period), and FV is future value (at the end of the last period).

Therefore, you can see that the Total Cash Flow from Investment in the numerator is compounded at the Reinvestment Rate (say a CD at a bank or savings account), and the Investment (negative amount) in the denominator is discounted at the discount rate (aka Cost of Capital). Let’s see an example of this: The MIRR of the above stream of cash flows is 8.63%, whether calculated with the equation  (cell D14) or found through Excel’s MIRR formula (cell D16). This is less than the IRR of 9.15% but is the Return on the Investment (ROI) if cash flows from this investment are put in a bank account compounding at 2.5%.

MIRR calculations are also helpful if you have multiple instances in which cash flows go from negative to positive and back again. For example, if you run IRR on a series of numbers that have more than one change in the signs, say -100, 50, 150, -100, 275, you will have two unique Internal Rates of Return. The rule is that you will have as many unique IRRs as there are changes in signs. MIRR solves this problem as negative and positive numbers are treated separately.

To cap off, MIRR is a more realistic but more subjective measure of performance. For a given investment in real estate most people don’t have a chance to reinvest their annual cash flows in that particular investment or in alternate investments at or above the IRR of the investment in question (realistic). But the rate at which one person does actually reinvest their annual cash flows is different from another person (subjective).

For example, a real estate investment has an IRR of 20%. You have only a 2% savings account in which to reinvest your proceeds. But I could reinvest those same cash flows into hard money loans at 12% – our reinvestment rates are different. Therefore, our MIRR are different.

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