Engineering Economic Analysis

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240 RATEOF RETURN ANALYSIS

3., Find the future worth of the receipts with the investingrate.

FW = 3.5M(FjP, 15%,6)+2.5M(FjP, 15%,5) +1.5M(FjP,15%,4)+O.5M(FjP, 15%,3)

= 3.5M(2.313) + 2.5M(2.011) + 1.5M(1.749) + O.5M(1.521)=16.507M

4. Find the MIRR that makes the present and future worths equivalent.

0= (1+MIRRt .PW +FW

0= (1 +MIRR)7(-6.744M) + 16.507M

(1 + MIRR)7=16.507Mj6.744M=2.448

(1 + MIRR)=2.4481/7=1.1364

MIRR=13.64%

The MIRR does allow calculation of a rate of return foranyset of cash flows.However,

the result is only as realistic as the external rates that are used. The MIRR value can depend

as much on the external rates that are used, as it does on the cash flowsthat it is describing.

SUMMARY

In cash flows with more than one sign change, we find that solving the cash flow equation

can result in more than one positive rate of return. Typicalsituations include a new oil well

in an existing field, a project with a significant salvage cost, and staged construction.

In a sign change, successive nonzero values in the cash flow have different signs (that

is, they change from + to -, orvice versa).Zero sign changes indicates there is no rate of

return, as the cash flow is either all disbursements or all receipts.

One sign change is the usual situation, and a single positive rate of return generally

results. There will be a negative rate of return whenever loan repayments are less than the

loan or an investment fails to return benefits at least equal to the investment.

Multiple sign changes may result in multiplepositiveroots fori.When they occur, none

of the positive multiple roots are a suitable measure of the project's economic desirability.

If multiple roots are identified by graphing the present worth versus the interest rate, then

the modified internal rate of return can be used to evaluate the project.

Graphing the present worth versus the interest rate ensures that the analyst recognizes

that the cash flow has multiple sign changes. Otherwise a rate could be found and used that

is not in fact a meaningful descriptor of the project.

The modified internal rate of return (MIRR) relies on rates for investing and borrowing

that are external to the project. The number of sign changes are reduced to one, ensuring

that the MIRR can be found.

Problems

Unless the problem asks a different question or pro-

vides different data: (1) determine how many roots

are possible, and (2) graph the PW versus the inter-

est rate to see whether multiple roots occur. If the

root is a unique IRR, it is the project's rate of re-

turn. If there are multiple roots, then use anexternal

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