Why Conventional Cash Flows Have a Unique IRR

Why the Cash-Flow Sign Pattern Matters

An internal rate of return is a rate at which a project’s net present value is zero. The earlier material on solving IRR equations shows how to test or solve for such a rate. A separate question is whether the same cash-flow stream could have two different IRRs. For a conventional investment, the answer is no: its NPV falls strictly as the effective rate rises, so it can cross zero at most once.

This conclusion depends on the cash-flow pattern. A conventional project has one initial investment outflow, followed only by nonnegative future cash flows. A later outflow would break the sign condition used in the argument. The result is therefore a precise statement about a particular class of cash flows, not a general guarantee about every project.

Definition: A finite cash-flow stream is conventional if its time-0 cash flow is $-I$, where $I>0$, and its cash flow $C_t$ at each later time $t=1,\ldots,n$ is nonnegative, with at least one $C_t>0$. Its NPV at effective rate $i$ is $$ \operatorname{NPV}(i)=-I+\sum_{t=1}^{n}\frac{C_t}{(1+i)^t}. $$ For annual periods, the admissible effective-rate domain is $i>-1$, so each discount factor is defined and positive.

The dates in this definition are integer numbers of equal periods, and the rate $i$ is effective per period. If the periods are years, $i$ is an effective annual rate. Cash flows and rate periods must match: for example, annual cash flows require an annual effective rate in this formula. The time-0 outflow is not discounted; each later cash flow is discounted from its own time.

Deriving the Direction of NPV Change

For a fixed future cash flow $C_t$, its present value is $C_t(1+i)^{-t}$. Differentiating this term with respect to $i$ gives $$ \frac{d}{di}\left(\frac{C_t}{(1+i)^t}\right) =-\frac{tC_t}{(1+i)^{t+1}}. $$ The initial amount $-I$ is constant with respect to the rate, so its derivative is zero. Adding the derivatives of all future present values gives

\operatorname{NPV}'(i) =-\sum_{t=1}^{n}\frac{tC_t}{(1+i)^{t+1}}.

For every $i>-1$, the denominator $(1+i)^{t+1}$ is positive. Each $t$ is positive, and every $C_t$ is nonnegative. Thus every term in the sum is nonnegative before the minus sign. At least one $C_t$ is positive, so at least one term is strictly positive before that sign. It follows that

\operatorname{NPV}'(i)<0\qquad\text{for every }i>-1.

A differentiable function with a negative derivative throughout an interval is strictly decreasing there. Consequently, if $i_1<i_2$, then $\operatorname{NPV}(i_1)>\operatorname{NPV}(i_2)$. The NPV function cannot equal zero at both rates.

Theorem: For a conventional finite cash-flow stream, there is at most one IRR in the domain $i>-1$. The proof is that NPV is strictly decreasing throughout this domain, so it cannot take the value zero twice.

The phrase “at most one” is important. Strict decrease proves uniqueness if a root exists; by itself, it does not prove that a root exists in every restricted rate range. The distinction is especially useful when a question asks for a nonnegative IRR rather than any IRR greater than $-1$.

Worked Applications

Worked Example: Confirming a Candidate IRR and Its Uniqueness

A project requires an initial outlay of \$1,000. It returns \$420 at the end of year 1, \$470.40 at the end of year 2, and \$351.232 at the end of year 3. Verify that 12% is an IRR, then determine whether any other IRR is possible for $i>-1$.

Method. First evaluate NPV at the proposed annual effective rate. Then apply the strict-decrease result to the cash-flow pattern.

At $i=0.12$, the year-1 present value is $$ \frac{420}{1.12}=375. $$ The year-2 present value is $$ \frac{470.40}{(1.12)^2} =\frac{470.40}{1.2544}=375. $$ The year-3 present value is $$ \frac{351.232}{(1.12)^3} =\frac{351.232}{1.404928}=250. $$ Therefore, $$ \operatorname{NPV}(0.12)=-1{,}000+375+375+250=0. $$ So 12% is an IRR.

The initial flow is negative and each later flow is positive, so this is a conventional project. At least one future inflow is positive; hence its NPV is strictly decreasing for all $i>-1$. Since it is zero at 12%, it cannot be zero at any other admissible rate. Check: the three discounted inflows sum to exactly \$1,000, equal to the initial outlay.

Common mistake avoided: verifying a rate makes NPV zero identifies an IRR, but does not by itself establish uniqueness. Uniqueness follows here from the conventional sign pattern and the strictly negative NPV derivative.

Worked Example: A Unique IRR That Is Not Nonnegative

A project has cash flows $-\$100$ at time 0, $+\$40$ at the end of year 1, and $+\$32$ at the end of year 2. Verify its IRR and decide whether it has a nonnegative IRR.

Method. Test a rate in the full admissible domain, then use monotonicity to answer the question about nonnegative rates.

Try $i=-0.20$, or $-20\%$ per year. This rate is admissible because $-0.20>-1$. The year-1 and year-2 present values are $$ \frac{40}{1-0.20}=\frac{40}{0.8}=50 \qquad\text{and}\qquad \frac{32}{(0.8)^2}=\frac{32}{0.64}=50. $$ Thus, $$ \operatorname{NPV}(-0.20)=-100+50+50=0. $$ The cash flows are conventional, so strict decrease proves that $-20\%$ is the only IRR for $i>-1$.

At a zero rate, $$ \operatorname{NPV}(0)=-100+40+32=-28. $$ For any $i>0$, strict decrease means $\operatorname{NPV}(i)<\operatorname{NPV}(0)=-28$. NPV therefore cannot be zero at a nonnegative rate. The project has one admissible IRR, but that IRR is negative. Check: the future receipts total only \$72; the negative rate increases their present value to \$100.

Common mistake avoided: “at most one IRR” does not mean “one nonnegative IRR.” The rate domain must be stated when answering an existence question.

Worked Example: Zero-Cash-Flow Periods Do Not Break Uniqueness

An investment pays $-\$500$ at time 0, receives \$220 at the end of year 1, \$181.50 at the end of year 2, nothing at the end of year 3, and \$219.615 at the end of year 4. Verify the IRR at 10% and explain why the zero in year 3 does not affect the uniqueness conclusion.

Method. Discount each nonzero receipt to time 0, retain the zero period in the timeline, and check the signs of all future cash flows.

At $i=0.10$, the year-1 present value is $$ \frac{220}{1.10}=200. $$ The year-2 present value is $$ \frac{181.50}{(1.10)^2} =\frac{181.50}{1.21}=150. $$ The year-3 present value is $0/(1.10)^3=0$. The year-4 present value is $$ \frac{219.615}{(1.10)^4} =\frac{219.615}{1.4641}=150. $$ It follows that $$ \operatorname{NPV}(0.10)=-500+200+150+0+150=0. $$ The future flows are all nonnegative, and there are positive future receipts. A zero flow simply contributes zero to the NPV and derivative sums; it does not introduce a negative term. Therefore NPV is strictly decreasing, and 10% is the only IRR for $i>-1$. Check: the discounted receipts total \$500, matching the initial outlay.

Common mistake avoided: a conventional stream does not need a positive payment in every period. Zero-cash-flow periods are allowed; what matters is that no later flow is negative and at least one later flow is positive.

When Does an IRR Exist?

For a conventional project with at least one positive future receipt, the same model also gives a useful existence result over the full domain $i>-1$. As $i$ approaches $-1$ from above, the present value of any positive receipt at time $t\geq1$ grows without bound. Thus NPV tends to positive infinity. As $i$ tends to positive infinity, every future present value tends to zero, so NPV tends to $-I$. NPV is continuous for $i>-1$. It must therefore pass through zero somewhere in this domain, and strict decrease means it does so exactly once.

That conclusion changes if the question restricts the search to $i\geq0$. At zero, NPV is the sum of the undiscounted future receipts less $I$. If this value is negative, then NPV is even smaller at every positive rate, so there is no nonnegative IRR. If NPV at zero is positive, continuity and the limit $-I$ as $i$ grows imply one positive IRR. If NPV at zero is exactly zero, the unique nonnegative IRR is zero.

These existence statements rely on a finite stream, at least one positive future receipt, and the full admissible domain $i>-1$ when using the limit near $-1$. They do not change the central uniqueness argument: within any interval in the admissible domain, strict decrease permits no more than one root.

Common Exam Traps

Reversing the derivative sign. The derivative of $(1+i)^{-t}$ is $-t(1+i)^{-t-1}$, not a positive quantity. Discounted future receipts fall as the rate increases.
Confusing uniqueness with existence. Strict decrease prevents two roots; it does not automatically establish a root in a restricted range such as $i\geq0$.
Ignoring the rate domain. With annual periods and an effective annual rate, $i>-1$. The formula is not defined at $i=-1$, and the full-domain existence argument uses rates below zero as well as above zero.
Applying the theorem to the wrong sign pattern. A later negative cash flow violates the conventional-project assumption. The derivative terms can then have opposing signs, so this proof does not establish strict decrease.
Misdating a flow. The exponent in $(1+i)^t$ must match the number of periods from time 0 to the payment. A flow at the end of year 4 is discounted by $(1+i)^4$, even if an earlier year has no payment.

Key takeaway: For an initial outflow followed only by nonnegative future inflows, with at least one positive inflow, every future present value decreases with the effective rate. Their sum therefore decreases strictly, and NPV has at most one zero in the admissible domain. Whether that zero lies in a restricted range, such as nonnegative rates, is a separate existence question.

Check Your Understanding

For each problem, identify the relevant sign pattern and rate range before drawing a conclusion.

A project pays $-\$200$ at time 0, then receives \$88 at the end of year 1 and \$145.20 at the end of year 2. Verify whether 10% is an IRR and state how many IRRs are possible for $i>-1$. (Answer: At 10%, NPV $=-200+88/1.10+145.20/(1.10)^2=-200+80+120=0$; it is the unique IRR for $i>-1$.)
A conventional project has an initial outflow of \$150 and future receipts of \$60 at year 1 and \$80 at year 2. Determine whether it can have a nonnegative IRR. (Answer: NPV at zero is $-150+60+80=-\$10$; since NPV decreases strictly, there is no nonnegative IRR.)
For cash flows $-\$300$ at time 0, $+\$100$ at year 1, and $+\$200$ at year 3, write the derivative of NPV at $i=0.05$ and state its sign. (Answer: $\operatorname{NPV}'(0.05)=-100/(1.05)^2-600/(1.05)^4<0$.)
A project pays $-\$100$ at time 0, $+\$44$ at year 1, and $+\$72.60$ at year 2. Verify its IRR at 10%, then state how many IRRs it has over $i>-1$ and over $i\geq0$. (Answer: NPV at 10% is $-100+44/1.10+72.60/(1.10)^2=-100+40+60=0$; it has exactly one IRR in each range, at 10%.)
A project has cash flows $-\$100$ at time 0, $+\$30$ at year 1, $-\$20$ at year 2, and $+\$120$ at year 3. Can the conventional-cash-flow uniqueness result be used to establish at most one IRR? Explain. (Answer: No. The year-2 flow is negative, so the stream is not conventional and the strict-decrease proof does not apply.)

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