Engineering Economic Analysis

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Difficulties in Solving for an Interest Rate 231

Why Multiple Solutions Can Occur

, Example7A-l producedunexpectedresults.This couldhappenbecausethere were two

changes in the signs of the cash flows. Years 0 and 1 were positive, Years 2 and 3 were

negative, and Years 4 and 5 were positive. The cash flow series went from positive cash

flows to negative cash flows to positive cash flows.

Most cash flow series have only one shift in sign. Investments start with one or more

years of negative cash flows followed by many years of positive cash flows. Loans begin

with a positive cash flow that is repaid with later negative cash flows. These problems

have a unique rate of return because they have a single change in the sign of the cash

flows.

Having more than one sign change in the cash flow series can produce multiple points

or roots at which the PW equals O.To see how many roots are possible, we link solving an

economic analysis problem to solving a mathematical equation.

A project's cash flows are the values from CFo toCFn.

The equation to find the internal rate of return, where PW=0, is written as follows:

PW =0 -:-CFo + CFI(1 + i)-I + CFz(1 + i)-z +... +CFn(1+i)-n (7A-l)

If we letx= (1 +i)-I, then Equation 7A-I may be written

(7A-2)

Equation 7A-2 is an nth-order polynomial, andDescartes' ruledescribes the number

of positive roots forx.The rule is:

If a polynomial with real coefficients hasmsign changes, then the number of positive

roots will bem- 2k,wherekis an integer between 0 andm12.

A sign change exists when successive nonzero terms, written according to ascending

powers ofx,have different signs. Ifxis greater than zero, then the number of sign changes

in the cash flows equals the number in the equation. Descartes' rule means that the number

of positive roots (values ofx) of the polynomial cannot exceedm,the number of sign

changes. The number of positive roots forxmust either be equal tomor less by an even

integer.

III

Year Cash Flow

(^0) CFo
(^1) CFI
(^2) CFz
n CFn