Uniqueness of IRR for Conventional Cash Flows

Why Uniqueness Matters

An internal rate of return (IRR) is an interest rate that makes a cash-flow stream’s net present value equal to zero. For a conventional investment—an initial outflow followed only by positive inflows—the NPV moves in one direction as the rate changes. That behavior is the key to showing that the stream cannot have two different IRRs in the admissible rate domain.

This result is useful when interpreting a calculated IRR. A numerical method may produce a candidate rate, but a proof of uniqueness establishes that no second admissible rate is being overlooked. The conclusion depends on the cash flows having the stated sign pattern; it is not a general guarantee for every stream.

Definition: A finite conventional cash-flow stream consists of an initial investment $-I$, where $I>0$, at time 0, followed by cash flows $C_k\geq 0$ at times $t_k>0$, with at least one $C_k>0$. Its NPV at effective annual rate $i>-1$ is

\operatorname{NPV}(i)=-I+\sum_{k=1}^{n} C_k(1+i)^{-t_k}.

An IRR is an admissible rate $i>-1$ satisfying $\operatorname{NPV}(i)=0$. If all cash-flow dates are integer years, the $t_k$ are positive integers. For other dated cash flows, $t_k$ is measured in years and $i$ is an annual effective rate.

The condition $i>-1$ makes each accumulation factor $1+i$ positive, so discounting to time 0 is defined. In some decisions, a rate must also be nonnegative to meet a particular economic requirement. That is an additional restriction: the unique mathematical IRR can be negative.

Why NPV Decreases as the Rate Increases

The initial outflow $-I$ does not depend on $i$. Each positive future inflow, by contrast, has a present value that falls when the rate rises. Differentiating the NPV function makes this precise.

Differentiate each discounted inflow.
For $i>-1$, the derivative of $C_k(1+i)^{-t_k}$ is $-t_kC_k(1+i)^{-t_k-1}$.

Add the contributions.
The initial outflow has derivative zero, so the derivative of the full NPV is the sum of the derivatives of the inflows.

Use the sign pattern.
Every term in the derivative is nonpositive, and at least one is strictly negative because some $C_k>0$ and $t_k>0$. Thus NPV is strictly decreasing on $i>-1$.

\frac{d}{di}\operatorname{NPV}(i) = -\sum_{k=1}^{n}t_kC_k(1+i)^{-t_k-1}<0.

This is a complete monotonicity argument: if $i_1<i_2$, then $\operatorname{NPV}(i_1)>\operatorname{NPV}(i_2)$. Consequently, two distinct rates in the domain cannot both make NPV zero. If they did, strict decrease would require the NPV at the lower rate to be greater than the NPV at the higher rate, even though both were zero.

Theorem: A finite conventional cash-flow stream with at least one positive future inflow has at most one IRR in the domain $i>-1$. In fact, it has exactly one IRR in that domain.

To establish existence as well as uniqueness, consider the ends of the rate domain. As $i$ approaches $-1$ from above, the present value of any positive future inflow grows without bound. As $i$ grows without bound, every future inflow’s present value approaches zero, so NPV approaches $-I$. NPV is continuous for $i>-1$; therefore it crosses zero between those limits. Combined with strict decrease, this proves there is exactly one root.

\lim_{i\to -1^+}\operatorname{NPV}(i)=+\infty, \qquad \lim_{i\to +\infty}\operatorname{NPV}(i)=-I<0.

The value at $i=0$ shows whether the unique root is positive, zero, or negative. At zero, no discounting is applied, so NPV is the sum of the future inflows less the initial investment.

\operatorname{NPV}(0)=-I+\sum_{k=1}^{n}C_k.

If $\sum C_k>I$, then $\operatorname{NPV}(0)>0$, and the unique IRR is positive.
If $\sum C_k=I$, then $\operatorname{NPV}(0)=0$, and the unique IRR is zero.
If $\sum C_k<I$, then $\operatorname{NPV}(0)<0$, and the unique IRR is negative but still greater than $-1$.

These conclusions follow from strict decrease: the NPV crosses zero only once. Comparing total inflows with the initial outflow determines which side of that crossing contains zero; it does not, by itself, calculate the IRR.

Applying the Result

Worked Example: Verify a Positive IRR

A project requires an outlay of \$100 at time 0. It pays \$60 at the end of year 1 and \$72 at the end of year 2. Find the IRR and establish that it is the only admissible IRR.

Method. Set the NPV equal to zero, solve for the rate, then use the theorem to establish uniqueness. A common mistake is to treat the sum of future cash flows, \$132, as the project’s present value at every rate.

The equation for the IRR is

-100+\frac{60}{1+i}+\frac{72}{(1+i)^2}=0.

Set $x=1+i$, where $x>0$. Multiplying the equation by $x^2$ gives

-100x^2+60x+72=0, \qquad 25x^2-15x-18=0.

The quadratic formula gives

x=\frac{15\pm\sqrt{(-15)^2-4(25)(-18)}}{50} =\frac{15\pm\sqrt{2025}}{50} =\frac{15\pm45}{50}.

The two algebraic values are $x=1.2$ and $x=-0.6$. Since $x=1+i>0$, the admissible solution is $x=1.2$, giving $i=0.2$, or 20% per year.

Check. At 20%, the present values are $60/1.2=50$ and $72/1.44=50$. Their sum is \$100, exactly offsetting the initial outflow. The cash flows are conventional, so the theorem guarantees that this 20% IRR is the only one in $i>-1$. Rejecting $x=-0.6$ is essential: it would imply $i=-1.6$, outside the admissible domain.

Worked Example: A Conventional Stream with a Negative IRR

An equipment purchase costs \$120 now and returns \$54 at the end of year 1 and \$48.60 at the end of year 2. Find its IRR. Is there an IRR at a nonnegative rate?

Method. Check NPV at zero to determine the root’s sign, then verify a rate by discounting each inflow. The common mistake is to assume that a conventional cash-flow stream must have a positive IRR.

At zero, total inflows are $54+48.60=\$102.60$. Therefore,

\operatorname{NPV}(0)=-120+102.60=-17.40.

Because NPV is negative at zero and strictly decreasing as $i$ increases, a root cannot be zero or positive. The theorem still guarantees a unique root somewhere in $i>-1$. Test $i=-0.10$. Then $1+i=0.90$, and the discounted inflows are

\frac{54}{0.90}=60, \qquad \frac{48.60}{(0.90)^2} =\frac{48.60}{0.81}=60.

Thus,

\operatorname{NPV}(-0.10)=-120+60+60=0.

The IRR is $-10\%$ per year. The calculation verifies the root, and strict decrease establishes that it is the unique admissible IRR. Since that root is negative, there is no IRR at a nonnegative rate.

Worked Example: Verify a Rate for Three Dated Inflows

An investment has an initial outflow of \$200, followed by inflows of \$55 at the end of year 1, \$121 at the end of year 2, and \$66.55 at the end of year 3. Find and verify the IRR, and determine its sign.

Method. First check the undiscounted cash flows to determine the sign of the root. Then test a candidate rate by calculating each present value. Do not add the inflows and call that total the NPV at a positive rate.

The total of the future inflows is

$$ 55+121+66.55=242.55. $$

Therefore, $\operatorname{NPV}(0)=-200+242.55=\$42.55>0$, so the unique IRR is positive. At $i=0.10$, the discount factors are $1/1.1$, $1/(1.1)^2$, and $1/(1.1)^3$. The corresponding present values are

\frac{55}{1.1}=50,\qquad \frac{121}{1.21}=100,\qquad \frac{66.55}{1.331}=50.

Their sum is \$200, so $\operatorname{NPV}(0.10)=-200+50+100+50=0$. The IRR is 10% per year. The stream has one initial outflow and only positive future inflows; therefore, strict decrease proves that no other admissible rate produces zero NPV.

Common Exam Trap: Monotonicity Has a Direction

A frequent error is to say that NPV increases with the interest rate because the rate appears in the denominator. The denominator does increase as the rate rises, but that makes each positive discounted inflow smaller, not larger. The derivative confirms the direction: it is strictly negative for a conventional stream.

For example, in the first worked example, at $i=0$ the NPV is $-100+60+72=\$32$. At $i=0.20$, it is \$0. A mistaken increasing-NPV argument would predict that increasing the rate from 0 to 20% raises NPV, contradicting the actual values. The correction is to track the present value of each inflow: discounting more heavily lowers it, so NPV falls.

A second trap is to infer that an IRR must be positive merely because there is one initial outflow followed by positive inflows. That sign pattern guarantees a unique root in $i>-1$, not a positive root. The sign of $\operatorname{NPV}(0)$, equivalently the comparison of total inflows with the initial outflow, determines whether the root is positive, zero, or negative.

Key takeaway: For a finite conventional cash-flow stream, NPV is continuous and strictly decreasing on $i>-1$, tends from positive infinity to $-I$, and therefore has exactly one admissible IRR. Its sign is determined by NPV at zero.

Check Your Understanding

For each stream, use the sign pattern and NPV at zero to determine what can be said about the IRR. All amounts are in dollars, and cash flows occur at the stated year-end unless otherwise specified.

A project pays \$500 now and receives \$300 at the end of year 1 and \$240 at the end of year 2. Is its unique IRR positive, zero, or negative? (Answer: Positive; total inflows are \$540, so $\operatorname{NPV}(0)=\$40$.)
An investment pays \$1,000 now and receives \$400, \$500, and \$300 at the ends of years 1, 2, and 3. How many admissible IRRs does it have, and what is the sign? (Answer: Exactly one, and it is positive; $\operatorname{NPV}(0)=\$200$.)
A cash-flow stream is $-\$100$ now, followed by \$40 at the end of year 1 and \$60 at the end of year 2. What is the sign of its unique IRR? (Answer: Exactly one IRR, equal to zero, because $\operatorname{NPV}(0)=0$.)
An investment costs \$200 and pays \$80 at the end of each of years 1 and 2. Is there an admissible IRR? Is there a nonnegative IRR? (Answer: Exactly one admissible IRR, and it is negative because total inflows are \$160; there is no nonnegative IRR.)
A project has an outflow of \$100 at time 0, an inflow of \$40 after half a year, and an inflow of \$90 after two years. Rates are annual effective. How many admissible IRRs are there, and what is their sign? (Answer: Exactly one, and it is positive; $\operatorname{NPV}(0)=-100+40+90=\$30$, and both future dates are positive.)

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