Evaluated at a project's own internal rate of return, its present worth, annual worth and future worth are all
Answer: A
The three measures differ only by positive conversion factors, so a rate that drives one to zero drives all three there. That is why the rate-of-return and worth criteria never disagree about a single conventional project.
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A project is acceptable by the rate of return criterion if its internal rate of return is
Answer: B
Correct answer: at least the minimum attractive rate of return
The minimum attractive rate is the return the firm could get on its next best use of the money, so a project earning at least that much is worth doing and one earning less is not. Comparing the internal rate with zero or with inflation ignores the opportunity cost of the capital entirely.
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For a single project, the rate of return and present worth criteria
Answer: C
Correct answer: always agree on acceptance or rejection
For a conventional cash flow the present worth falls steadily as the discount rate rises and crosses zero exactly once, at the internal rate of return. So a positive present worth at the required rate and an internal rate above that required rate are two statements of the same fact. They can disagree only when choosing between alternatives, not when accepting one.
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The internal rate of return is found in practice by
Answer: D
The equation is a polynomial of degree equal to the number of periods and has no general closed-form solution, so two trial rates bracketing the root are found and the answer interpolated. Spreadsheets do the same thing iteratively.
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Linear interpolation between two trial rates gives an approximate answer because the present worth function is
Answer: A
Present worth falls with the rate along a convex curve, so a straight line between two points cuts across it and the interpolated root is slightly off. Narrowing the bracket reduces the error to whatever precision is wanted.
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The internal rate of return method implicitly assumes intermediate receipts are reinvested at
Answer: B
That is what makes a very high computed rate suspect, since such returns are rarely available for reinvestment. The present worth method assumes reinvestment at the minimum attractive rate, which is the more defensible claim.
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A cash flow whose sign changes more than once may have
Answer: C
Correct answer: more than one internal rate of return
The present worth equation is a polynomial in the discount factor, and each sign change in the cash flow admits another possible positive root. A project requiring a large restoration outlay at the end - a mine or a nuclear plant - has two sign changes and may well have two rates at which the present worth is zero, neither of which means anything on its own.
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By Descartes' rule of signs, the number of real positive roots cannot exceed the number of
Answer: D
Correct answer: sign changes in the cash flow sequence
Descartes’ rule bounds the number of positive real roots of a polynomial by the number of sign changes in its coefficients. Applied to the present worth equation, the coefficients are the cash flows, so the number of possible internal rates cannot exceed the number of times the cash flow changes sign. It is an upper bound, not a count.
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A simple or conventional investment is one whose cash flow has
Answer: A
Correct answer: exactly one sign change
A conventional investment spends first and receives afterwards: one or more negative flows, then only positive ones. That single sign change guarantees a unique internal rate of return, which is why the method can be used without qualification on such projects and needs care on any others.
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The external or modified rate of return resolves multiple roots by
Answer: B
Correct answer: reinvesting intermediate flows at an assumed external rate
The multiple-root problem arises because the internal rate implicitly assumes intermediate receipts are reinvested at itself, which may be absurd. The modified rate states the reinvestment rate explicitly, compounds the positive flows forward at it to a single future sum, and solves for the one rate that equates that sum to the present outlays. A single sign change remains, so a single root.
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