Cash Flows with No Positive IRR

When a Cash-Flow Pattern Has No Positive IRR

An internal rate of return (IRR) is a rate that makes the net present value of a sequence of cash flows equal to zero. Some cash-flow sequences have no such rate greater than zero. This can happen because the cash flows, even after discounting at every positive rate, remain entirely on one side of zero. With cash flows that change sign more than once, however, the net present value (NPV) may rise and fall, so checking only one or two trial rates is not enough to establish that no positive IRR exists.

The task is therefore not just to fail to find a root with a calculator or solver. It is to show that no admissible root exists. A useful approach is to express NPV in terms of a discount factor, inspect the whole range corresponding to positive rates, and use bounds or extrema to verify its sign.

Definition: Suppose cash flows $C_0,C_1,\ldots,C_n$ occur at times $0,1,\ldots,n$, respectively, with positive amounts received and negative amounts paid from one consistent perspective. At effective annual rate $r$, their net present value is $$ \operatorname{NPV}(r)=\sum_{t=0}^{n} C_t(1+r)^{-t}. $$ An IRR is a rate $r$ for which $\operatorname{NPV}(r)=0$. A positive IRR is such a root with $r>0$. Unless stated otherwise, this tutorial assumes discrete annual periods and considers positive rates only.

Map Positive Rates to Discount Factors

For a positive rate, define $v=1/(1+r)$. Then $0<v<1$, and NPV can be written as a polynomial in $v$:

\operatorname{NPV}(r)=C_0+C_1v+C_2v^2+\cdots+C_nv^n, \qquad 0<v<1.

This substitution gives a convenient way to analyze all positive rates at once. The limit $r\to 0^+$ corresponds to $v\to 1^-$, so NPV approaches the sum of the cash flows. The limit $r\to\infty$ corresponds to $v\to 0^+$, so NPV approaches $C_0$. For finite positive rates, the relevant part of the profile is the open interval $0<v<1$; the endpoint values are limits, not necessarily values attained at a positive rate.

The transformation follows directly from the discounting rule: a payment at time $t$ has present value $C_t/(1+r)^t=C_tv^t$. Adding those present values produces the polynomial. In particular, if that polynomial is strictly positive or strictly negative throughout $0<v<1$, it cannot equal zero for any positive $r$.

Bounds That Rule Out a Positive Root

A bound is often the simplest proof of absence. For example, suppose the initial cash flow is negative and all later cash flows are nonnegative. At a positive rate, each later cash flow is discounted by a factor no greater than 1. Therefore, if the sum of all cash flows at rate zero is already negative, discounting the later receipts cannot raise NPV to zero. This conclusion depends on the stated sign pattern; it does not hold for every sequence with mixed signs.

Worked Example: Negative NPV Under Every Positive Rate

A project requires an outlay of $100, followed by receipts of $60 at the end of year 1 and $30 at the end of year 2. Determine whether it has a positive IRR. Use the investor's perspective, so the initial outlay is negative and receipts are positive.

Method: bound the discounted receipts. For $r>0$, let $v=1/(1+r)$, so $0<v<1$. The NPV is

\operatorname{NPV}(r)=-100+60v+30v^2.

Since $v<1$, we have $60v<60$ and $30v^2<30$. Thus

\operatorname{NPV}(r)<-100+60+30=-10.

The endpoint check agrees: at a zero rate the cash flows sum to $-100+60+30=-10$, and as $r\to\infty$, NPV approaches the initial outlay, $-100$. The bound is stronger than an endpoint check because it covers every positive rate. NPV is strictly negative throughout that rate range, so the project has no positive IRR.

Check and common mistake: The undiscounted receipts total $90, less than the $100 outlay, so even before discounting the project falls short by $10. A common mistake is to treat a solver's failure to return a rate as proof. Here the inequality proves the absence of a root directly.

A related argument applies when the initial cash flow is positive and every later cash flow is nonpositive: if the initial amount exceeds the total absolute value of all later payments, NPV remains positive at every positive rate. More generally, look for a bound that holds over the entire interval $0<v<1$, not just at selected rates.

Why Endpoint Signs Are Not Enough

The endpoint limits help describe a profile but do not, by themselves, rule out roots. When cash flows change sign, the profile can turn between the endpoints. It might cross zero and return, even if its limiting values at both ends have the same sign. To prove absence in that situation, examine the interior: find critical points, establish monotonicity, or derive a bound for the whole interval.

Worked Example: Checking an Interior Maximum

A three-period cash-flow sequence is $(-10,30,-25)$, with amounts at times 0, 1, and 2. Determine whether any positive rate makes NPV zero.

Method: locate the global maximum of the NPV profile. With $v=1/(1+r)$, where $0<v<1$,

\operatorname{NPV}(v)=-10+30v-25v^2.

This is a concave quadratic because its $v^2$ coefficient is negative. Its derivative is

\frac{d\,\operatorname{NPV}}{dv}=30-50v.

Setting the derivative equal to zero gives $v=30/50=0.6$. This point is inside the admissible interval and is the quadratic's maximum. Its NPV is

\operatorname{NPV}(0.6)=-10+30(0.6)-25(0.6)^2 =-10+18-9=-1.

Equivalently, this maximum occurs at $r=1/0.6-1=2/3$. Since the maximum NPV is $-1$, the NPV is negative everywhere on $0<v<1$; no positive IRR exists. As an endpoint check, the limiting values are $-10$ at $v\to0^+$ and $-5$ at $v\to1^-$, both also negative.

Check and common mistake: The profile never reaches zero because even its highest value is negative. It would be incorrect to infer absence merely because both endpoint limits are negative: a profile could rise above zero between them. Here, the calculated interior maximum verifies that it does not.

For comparison, the polynomial $-1+5v-5v^2$ has endpoint values $-1$ at $v=0$ and $v=1$, but its interior maximum occurs at $v=0.5$ and equals $0.25$. It therefore crosses zero within the interval despite having negative values at both endpoints. This illustrates why an endpoint-sign argument requires additional conditions, such as a proven monotonicity or a valid global bound.

Using Monotonicity to Establish a Sign

A derivative can also prove absence by showing that the profile is monotone and comparing its limiting values. The derivative is taken with respect to $v$, and its sign must be interpreted on $0<v<1$. For example, if the derivative is strictly negative throughout that interval, NPV decreases as $v$ increases. Its smallest limiting value is then at $v\to1^-$; if that limit is positive, NPV must remain positive throughout the interval.

Worked Example: A Positive Profile with Future Obligations

A hypothetical contract produces an immediate receipt of $15, followed by payments of $4, $3, and $2 at the ends of years 1, 2, and 3. From the holder's perspective, determine whether the sequence has a positive IRR.

Method: establish monotonicity and evaluate the limiting minimum. The cash flows are $(15,-4,-3,-2)$. Let $v=1/(1+r)$, so

\operatorname{NPV}(v)=15-4v-3v^2-2v^3.

Differentiate with respect to $v$:

\frac{d\,\operatorname{NPV}}{dv}=-4-6v-6v^2.

For every $0<v<1$, this derivative is negative. Thus NPV decreases as $v$ increases, and its lowest limiting value on the interval is the limit at $v\to1^-$:

\lim_{v\to1^-}\operatorname{NPV}(v)=15-4-3-2=6.

Because NPV is decreasing in $v$ and its limiting value at the upper endpoint is 6, it is greater than 6 for $0<v<1$. It is therefore strictly positive for every positive rate, and no positive IRR exists. The other endpoint limit is $15$, consistent with the profile remaining positive.

Check and common mistake: The future payments reduce present value by less than their undiscounted total of $9 at a positive rate, leaving a positive NPV. Do not assume that a sequence beginning with a receipt must have no IRR; the conclusion here follows from the derivative and the endpoint bound, not from the first cash flow alone.

A Practical Verification Process

When a cash-flow sequence appears to have no positive IRR, use a proof that matches its pattern. The following sequence of checks helps prevent an unsupported conclusion from a few trial rates.

Set the perspective and timeline.
Record each cash flow with its correct sign and period. A sign reversal caused by switching perspectives changes the NPV equation.

Write the NPV equation.
Use $\operatorname{NPV}(r)=\sum C_t(1+r)^{-t}$, assuming the stated cash-flow timing and an effective rate per period.

Restrict the domain correctly.
Set $v=1/(1+r)$. A positive rate corresponds to $0<v<1$, not to every real value of $v$.

Prove a sign over the whole interval.
Use a bound, monotonicity, or the extrema of the polynomial. A finite list of trial-rate calculations cannot establish that no root exists between the trials.

Distinguish endpoint roots from positive roots.
A root at $r=0$ is not a positive IRR. Check whether the sign result applies to the open interval $0<v<1$.

Common Exam Traps

Assuming equal endpoint signs rule out roots. A profile can cross zero twice between endpoints. Check its interior extrema or establish a global sign bound.
Confusing a solver result with a proof. A numerical method that fails to find a root may have used an unsuitable starting value or missed another root. Analyze the NPV profile itself.
Including $r=0$ among positive rates. A zero-rate root is not a positive IRR. For example, a profile may equal zero at $v=1$ but be strictly negative for every $v<1$.
Using a bound without checking signs. Discounting reduces positive future receipts, but it also reduces the absolute present value of future payments. State the sign pattern before applying an inequality.
Ignoring timing. The term $C_t(1+r)^{-t}$ uses the cash flow's period $t$. Shifting a payment by one period changes the profile and can change whether a root exists.

Key takeaway: To show that a cash-flow sequence has no positive IRR, prove that its NPV never equals zero for $0<v<1$, where $v=1/(1+r)$. Endpoint behavior is useful, but a complete argument may also require a bound, monotonicity, or an interior-extremum check.

Check Your Understanding

For each sequence, use the stated perspective and assume annual cash flows at times 0, 1, and so on. Decide whether the sequence has a positive IRR, and use a bound or profile argument rather than relying only on a solver.

A project has cash flows $(-120,50,55)$. Use the undiscounted future receipts to determine whether NPV can reach zero at a positive rate. (Answer: No positive IRR; NPV is less than $-15$ for every positive rate.)
A contract has cash flows $(8,-3,-4)$. Write its profile in $v$ and determine its sign on $0<v<1$. (Answer: No positive IRR; $\operatorname{NPV}(v)=8-3v-4v^2>1$ for $0<v<1$.)
An investment has cash flows $(-6,17,-12)$. Find the roots in $v$, then convert them to positive rates. (Answer: Two positive IRRs, $r=1/3$ and $r=1/2$.)
A project has cash flows $(-40,20,15,10)$. Use endpoint behavior and continuity to decide whether at least one positive IRR must exist; an exact rate is not required. (Answer: At least one positive IRR; NPV approaches $5$ as $r\to0^+$ and $-40$ as $r\to\infty$.)
A sequence has cash flows $(-20,10,10)$. Evaluate NPV at $r=0$, then determine whether the sequence has a positive IRR. (Answer: No positive IRR; NPV is zero at $r=0$, but $-20+10v+10v^2<0$ for every $0<v<1$.)

What You'll Learn