~-.. -. -. - ....
.~.- ~._~.238 RATEOF RETURN ANALYSIS
In this case, there is 1 positive root of 21.69%. The value can be used as an IRR, and the project
i~ very attractive.
~ -.When there are two or more sign changes in the cash flow, we know that there are
several possibilities concerning the number of positive values ofi.Probably the greatest
danger in this situation is to fail to recognize the multiple possibilities and to solve for a
value ofi.The approach of constructing the PW plot both establishes whether there are
multiple roots and what their values are. This may be tedious to do by hand but is very easy
with a spreadsheet (or a graphing calculator).
If there is a single positive value ofi, we have no problem. On the other hand, if
there is no positive value ofi,or if there are multiple positive values, the situation may be
attractive,unattractive, or confusing. Where there are multiple positive values ofi,none of
them should be considered a suitable measure of the rate of return or attractiveness of the
cash flow.Modified Internal Rate of Return (MIRR)
Two external rates of return can be used to ensure that the resulting equation is solvable
for a unique rate of return-the MIRR. The MIRR is a measure of the attractiveness of the
cash flows, but it is also a function of the two external rates of return.
The rates that areexternalto the project's cash flows are (1) the rate at which the
organization normally invests and (2) the rate at which it normally borrows. These are
external rates for investing, einv,and for financing, efin.Because profitable firms invest at
higher rates than they borrow at, the rate for investing is generally higher than the rate for
financing.Sometimes a singleexternalrate is used for both, but this requires the questionable
assumption that investing and financinghappen at the same rat~.
The approach is:- Combine cash flows in each period into a single net receipt,R"or net expense,Et.
- Find the present worth of the expenses with the financingrate.
- Find the future worth of the receipts with the investingrate.
- Find the MIRR which makes the present and future worths equivalent.
The result is Equation 7A-3. This equation will have a unique root, since it has a single
negative present worth and a single positive future worth. There is only one sign change in
the resulting series.(F/P,MIRR,n)L Et(P /F, efin,t)= LRtCF/P,einv,n- t)
t t(7A-3)There are other external rates of return, but the MIRR has historically been the most
clearly defined. All of the external rates of return are affectedby the assumed values for the
investing and financing rates, so none are atruerate of return on the project's cash flow.
The MIRR also has an Excel function, so it now can very easily be used. Example 7A-7
illustrates the calculation, which is also summarized in Figure 7A-5.r1
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