Why Conventional Projects Have One IRR

Why the Shape of the NPV Profile Matters

An internal rate of return is a discount rate that makes a project’s net present value equal to zero. A project’s cash-flow pattern determines how its NPV changes as the discount rate changes. For a conventional project—one initial investment followed by nonnegative cash flows—the NPV profile has a particularly useful shape: it falls strictly as the rate rises, provided at least one later cash flow is positive. This rules out two distinct nonnegative rates both making NPV equal to zero.

The result concerns the number of IRRs, not whether an IRR must exist. A conventional project can have at most one nonnegative IRR; whether it has one depends on the NPV level at a zero rate. The distinction matters because a one-root guarantee is not a guarantee that a root exists.

Definition: A conventional project has an initial cash outflow of $I>0$ at time 0, followed by cash flows $C_t\geq 0$ at the ends of periods $t=1,\ldots,n$. Assume at least one $C_t>0$. Its NPV at a per-period discount rate $r\geq 0$ is $$ \operatorname{NPV}(r)=-I+\sum_{t=1}^{n}\frac{C_t}{(1+r)^t}. $$ An IRR is a rate $r\geq 0$ for which $\operatorname{NPV}(r)=0$.

The period in the rate must match the cash-flow period. For example, if the cash flows are annual, $r$ is an effective annual rate; if they are quarterly, $r$ is a per-quarter rate. The formula assumes each $C_t$ arrives at the end of period $t$, and that the same rate is used to discount all periods. A cash flow of zero is allowed in an intermediate period; it is neither a positive inflow nor an additional outflow.

The NPV Profile Is Strictly Decreasing

For $r\geq 0$, every positive future cash flow has a present value that falls as $r$ increases. Differentiating the finite NPV sum makes the result precise:

\frac{d}{dr}\operatorname{NPV}(r) = -\sum_{t=1}^{n}\frac{tC_t}{(1+r)^{t+1}}.

Each term in the sum is nonnegative, and every term associated with a positive $C_t$ is positive before the minus sign. Because at least one $C_t>0$, the derivative is strictly negative for every $r\geq 0$. The initial investment $-I$ does not affect the derivative because it is constant with respect to $r$.

Theorem: For a conventional project with at least one positive future cash flow, $\operatorname{NPV}(r)$ is strictly decreasing on $r\geq 0$. Therefore, it can equal zero at most once on that domain. This is a uniqueness result, not an existence result.

The derivative gives the slope, but the uniqueness conclusion can also be understood directly. If $r_2>r_1\geq 0$, then $(1+r_2)^{-t}<(1+r_1)^{-t}$ for every $t\geq 1$. Multiplying by $C_t\geq 0$ shows that each discounted inflow at $r_2$ is no greater than its value at $r_1$, and at least one is strictly smaller. The initial outflow stays unchanged. Thus $\operatorname{NPV}(r_2)<\operatorname{NPV}(r_1)$.

State the cash-flow pattern.
There is one negative cash flow at time 0, no later negative cash flows, and at least one positive future cash flow.

Write the NPV function.
Discount each end-of-period cash flow using the same per-period rate $r$.

Check the slope.
The derivative is strictly negative for every nonnegative rate.

Conclude uniqueness.
A strictly decreasing function cannot equal zero at two different nonnegative rates.

Existence and Uniqueness Are Different Questions

For a finite conventional project, the NPV function is continuous on $r\geq 0$, and as $r$ grows without bound, the discounted future cash flows approach zero. Consequently, NPV approaches $-I$, which is negative. If $\operatorname{NPV}(0)>0$, continuity means the profile must cross zero at some positive rate; strict decrease means it crosses only once. If $\operatorname{NPV}(0)=0$, zero itself is the only nonnegative root. If $\operatorname{NPV}(0)<0$, strict decrease means it cannot reach zero at a nonnegative rate. These conditions follow from the profile’s shape; they do not change the theorem that there is at most one root.

Worked Example: Solve for the Only Nonnegative IRR

A hypothetical equipment project costs $1,000 now and returns $600 at the end of year 1 and $550 at the end of year 2. Find its nonnegative IRR. The method is to set NPV to zero and solve using $x=1+r$; this also makes the rate domain explicit.

Step 1: Check the cash-flow pattern. The project has one initial outflow and two positive annual inflows, so it is conventional. Its NPV at a rate $r$ per year is $$ \operatorname{NPV}(r)=-1{,}000+\frac{600}{1+r}+\frac{550}{(1+r)^2}. $$

Step 2: Set NPV to zero. Let $x=1+r$. Multiplying by $x^2$, which is positive for $r\geq 0$, gives $$ -1{,}000x^2+600x+550=0, $$ or, after multiplying by $-1$, $$ 1{,}000x^2-600x-550=0. $$

Step 3: Solve the quadratic and check its roots. The discriminant is $ (-600)^2-4(1{,}000)(-550)=360{,}000+2{,}200{,}000=2{,}560{,}000$, whose square root is $1{,}600$. Thus $$ x=\frac{600\pm1{,}600}{2{,}000}, $$ so the roots are $x=1.1$ and $x=-0.5$. Since $x=1+r\geq 1$ for a nonnegative rate, only $x=1.1$ is admissible. Therefore $r=0.10$, or 10% per year.

Step 4: Check the result. At 10%, the present values are $600/1.1=\$545.4545\ldots$ and $550/1.21=\$454.5454\ldots$. They sum to $1,000, so NPV is zero. The NPV at a zero rate is $-\$1{,}000+\$600+\$550=\$150$, consistent with a crossing above zero. The uniqueness theorem confirms that no other nonnegative IRR exists. The common mistake to avoid is miscalculating the discriminant or treating an inadmissible quadratic root as an IRR.

Using the Profile to Bracket an IRR

A root does not always have a convenient exact form. The decreasing profile is still useful: calculate NPV at selected rates, and a change from positive to negative brackets a root. Strict decrease then establishes that the bracket contains the project’s only nonnegative IRR. Interpolation can estimate the rate within the bracket, but the interpolated value is an estimate rather than an exact solution.

Worked Example: Bracket an IRR and Estimate It

A hypothetical distribution project requires $1,000 immediately and produces annual inflows of $500, $400, and $300 at the ends of years 1, 2, and 3. Determine whether the IRR lies between 10% and 12%, estimate it by linear interpolation, and explain why it is the only nonnegative IRR. The method is to calculate the NPV at both rates and use the sign change.

Step 1: Evaluate NPV at 10%. $$ \operatorname{NPV}(0.10) =-1{,}000+\frac{500}{1.10}+\frac{400}{1.10^2}+\frac{300}{1.10^3}. $$ The three present values are $\$454.5455$, $\$330.5785$, and $\$225.3944$, respectively. Their sum is $\$1{,}010.5184$, so $\operatorname{NPV}(0.10)=\$10.5184$.

Step 2: Evaluate NPV at 12%. $$ \operatorname{NPV}(0.12) =-1{,}000+\frac{500}{1.12}+\frac{400}{1.12^2}+\frac{300}{1.12^3}. $$ The present values are $\$446.4286$, $\$318.8776$, and $\$213.5341$. Their sum is $\$978.8403$, so $\operatorname{NPV}(0.12)=-\$21.1597$. Since NPV is positive at 10% and negative at 12%, the root lies between those rates.

Step 3: Interpolate for an estimate. Linear interpolation gives $$ \widehat{r} = 0.10+\frac{10.5184}{10.5184+21.1597}(0.12-0.10) \approx 0.1066. $$ The estimated IRR is 10.66%. This estimate is based on a straight-line approximation between the two calculated NPV values; the NPV profile itself need not be a straight line.

Step 4: State the uniqueness conclusion. This project has one initial outflow and positive future inflows, so its NPV profile is strictly decreasing. The sign change shows that one root exists between 10% and 12%; strict decrease shows that this is its only nonnegative root. The common mistake to avoid is treating interpolation as proof of an exact IRR or assuming the profile is linear.

Zero Cash-Flow Periods Do Not Break the Result

A conventional project need not have a positive cash flow in every period. If one or more future cash flows are zero, their terms contribute nothing to either NPV or its derivative. The slope remains strictly negative as long as at least one future inflow is positive. This is why the relevant condition is “nonnegative future cash flows, with at least one positive,” not “positive cash flows in every year.”

Worked Example: A Project With a Delayed First Inflow

A hypothetical system upgrade costs $1,000 today, has no cash flow at the end of year 1, and yields $605 at the end of year 2 and $665.50 at the end of year 3. Show that 10% is an IRR and establish that it is the only nonnegative IRR. The method is to verify NPV directly and then check the slope, including the zero-cash-flow period.

Step 1: Write the NPV function. $$ \operatorname{NPV}(r)=-1{,}000+\frac{0}{1+r}+\frac{605}{(1+r)^2}+\frac{665.50}{(1+r)^3}. $$ The zero in year 1 does not change the value of the function.

Step 2: Check NPV at 10%. The year 2 present value is $605/1.21=\$500$, and the year 3 present value is $665.50/1.331=\$500$. Therefore, total present value is $\$1{,}000$, and $\operatorname{NPV}(0.10)=0$.

Step 3: Establish the slope and uniqueness. $$ \frac{d}{dr}\operatorname{NPV}(r) =-\frac{2(605)}{(1+r)^3}-\frac{3(665.50)}{(1+r)^4}<0 \qquad (r\geq 0). $$ Both future inflows are positive, so the derivative is strictly negative throughout the nonnegative-rate domain. The verified 10% root is therefore the only nonnegative IRR.

Step 4: Check the profile at another rate. At 20%, the year 2 present value is $605/1.44=\$420.1389$, and the year 3 present value is $665.50/1.728=\$385.1273$. Thus NPV is $-\$194.7338$, below its zero value at 10%, as the decreasing-profile result predicts. The common mistake to avoid is interpreting a zero intermediate cash flow as a sign change or as a reason the uniqueness result fails.

Common Mistake: Confusing a Slope With a Root Count

The derivative contains a factor $t$ from differentiating $(1+r)^{-t}$. Omitting that factor gives an incorrect slope. For the project in the preceding example, the correct derivative is $$ -\frac{2(605)}{(1+r)^3}-\frac{3(665.50)}{(1+r)^4}, $$ not $-605/(1+r)^3-665.50/(1+r)^4$. Both expressions happen to be negative, but only the first is the derivative of the stated NPV function. Show the derivative accurately when using it to justify the profile’s shape.

A second trap is to infer existence from uniqueness. The theorem says that a conventional project cannot have two nonnegative IRRs; it does not say that it must have one. A useful initial diagnostic is NPV at $r=0$, which is the initial investment plus all future cash flows without discounting. In the bracketing example, it is $-\$1{,}000+\$500+\$400+\$300=\$200$, so it starts above zero. In all cases, use the cash-flow pattern and the NPV values together: the pattern establishes the shape, while the values establish whether a crossing occurs.

Key takeaway: With one initial outflow, no later outflows, and at least one positive future inflow, each positive future cash flow loses present value as the nonnegative discount rate rises. NPV therefore falls strictly and can have at most one nonnegative IRR. A sign change in the NPV profile can establish that a root exists; strict decrease establishes that there cannot be another.

Check Your Understanding

For each problem, first identify whether the cash flows are conventional, then use the NPV profile or the stated calculations to reach a conclusion.

A project costs $500 today and returns $200 at each of the next three year-ends. Calculate NPV at 0%, and state how many nonnegative IRRs it can have. (Answer: NPV at 0% is $100; the project has exactly one nonnegative IRR.)
A project costs $1,000 and returns $450, $400, and $300 at the ends of years 1, 2, and 3. Calculate NPV at 10% to the nearest cent and use NPV at 0% to state where its unique nonnegative IRR lies. (Answer: NPV at 10% is $-\$34.94$; NPV at 0% is $150, so the unique IRR lies between 0% and 10%.)
A project costs $750 now, has zero cash flow in year 1, then returns $400 in year 2 and $500 in year 3. Write the derivative of its NPV and state its sign for $r\geq 0$. (Answer: $\frac{d}{dr}\operatorname{NPV}(r)=-800/(1+r)^3-1{,}500/(1+r)^4<0$ for $r\geq 0$.)
A project costs $900 now and returns $350 at each of the next three year-ends. Calculate NPV at 0% and 10% to the nearest cent, then state how many nonnegative IRRs it has and where the root lies. (Answer: NPV at 0% is $150; NPV at 10% is $-\$29.60$; it has exactly one nonnegative IRR, between 0% and 10%.)
A conventional project has NPV equal to zero at a nonnegative rate of 8%. A colleague claims that it may also have NPV equal to zero at 15%. Assess the claim using the project’s cash-flow pattern, without needing to calculate either NPV. (Answer: The claim is false; strict decrease means the NPV can equal zero at only one nonnegative rate, so 8% and 15% cannot both be IRRs.)

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