MIRR, eh?

I got e-mail asking about MIRR, the Modified Internal Rate of Return. As usual, I ain't never heard of it. So I googled and found something like this: You borrow $A to start some company (or some project). The Financing rate is F%. (F% is the interest charged on the loan.) Profits from the company are Invested at a rate I%. Expected profits are $B after 1 year and $C after 2 years. (These you intend to invest at that annual rate of I%.) Then comes the BIG question: Is $A too much to pay for the company? The profits will be worth (at the end of n years): P = B (1+I)n-1 + C (1+I)n-2. Suppose the company is worth $K, at the end of n years. The total value of profits + company will then be their sum, P + K. So what "return" are you getting on your enterprise? There's some magic rate of return (which we'll call MIRR). Our A would be worth A (1+MIRR)n (after n years at this magic return). Setting that equal to P + K we'd get the equation: A (1+MIRR)n = B (1+I)n-1 + C (1+I)n-2 + K. Now all we have to do is solve for MIRR, eh? To buy that pizzeria, suppose we get MIRR = 8%. For the burger stand we get MIRR = 7%. Then we buy the pizzeria. ------------------------------ Note: If A is the only loan, then MIRR ain't got nothin' to do with the Finance rate F. However, if there are other loans, we gotta calculate their present value (at the Finance rate F%) and add all these present values to A. If that give A', then we'd solve for MIRR from: A' (1+MIRR)n = B (1+I)n-1 + C (1+I)n-2 + K. 'course, there may be other profits as well as B and C, so ... uh ... I reckon y'all can take it from here. If an initial loan of A0 is followed by subsequent loans (after 1, 2, ... years) of A1, A2 ... and the Financing rate is F ... and annual profits (after 1, 2, ... years) are B1, B2 ... which are invested at an Investment rate I ... then define: PV = A0 + A1/(1+F)1 + A2/(1+F)2 + ... That's a Present Value FV = B1(1+I)n-1 + B2(1+I)n-2 + ... That's a Future Value then MIRR is defined so that: PV (1+MIRR)n = FV So the PV gives FV after n years with annual return MIRR. Solving: MIRR = [ FV / PV ] 1/n - 1 I forgot to mention that one (often) takes the PV as a series of negative cash flows, so - PV is (often) used in the formula for MIRR ... just so you get a positive ratio: -FV/PV. Further, if I = F = IRR, then MIRR ain't no different than IRR. I might also mention that there's FMRR. In the above analysis, all cash flows are discounted to Present Value at the same rate. With FMRR, there are a couple of rates for this ritual ... and the discounting is not (necessarily) to the "present" value.