Reading IRR from a Net Present Value Profile

What an NPV Profile Shows

An internal rate of return can be viewed as a rate that makes net present value equal to zero. An NPV profile displays how that present value changes as the trial discount rate changes. It turns an IRR equation into a picture: the horizontal axis is the rate, the vertical axis is NPV, and each intersection with the horizontal axis is a rate at which NPV is zero.

A profile is useful for seeing whether a zero is bracketed by two trial rates, estimating where it lies, and noticing that the graph may have more than one zero. It is not enough to look at the cash flows and assume the profile must slope downward: the shape depends on the timing and signs of all cash flows.

Definition: For cash flows $C_0,C_1,\ldots,C_n$ occurring at times $0,1,\ldots,n$, with each $C_t$ recorded as positive for an inflow and negative for an outflow, the NPV profile is the function

\operatorname{NPV}(i)=\sum_{t=0}^{n} C_t(1+i)^{-t}, \qquad i>-1.

At a zero of this function, $\operatorname{NPV}(i)=0$, so $i$ is an IRR for that cash-flow stream. This formula assumes the cash flows occur at the stated integer periods and $i$ is the effective rate per period. If periods are not equally spaced, discount each cash flow using its actual time.

To construct the profile, choose several trial rates, calculate NPV at each rate using the same timeline, and organize the results in a table or graph. A positive NPV is plotted above the horizontal axis; a negative NPV is plotted below it. If one value is positive and a later value is negative, a continuous NPV function has at least one zero between those rates. Interpolation can estimate its location, but it does not generally give an exact IRR.

Worked Example: Bracketing a Conventional Project’s IRR

A hypothetical project requires an outflow of $100 at time 0 and pays $60 at the end of each of the next two years. Construct a small NPV profile at 0%, 10%, 15%, and 20%, then use it to locate and estimate the IRR. Rates are effective annual rates, and cash flows occur annually.

Method. Use the same two-year timeline at every trial rate:

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

At 0%, each payment is undiscounted:

\operatorname{NPV}(0)=-100+60+60=20.00.

At 10%:

\operatorname{NPV}(0.10) =-100+\frac{60}{1.10}+\frac{60}{1.10^2} =-100+54.545455+49.586777 =4.132232.

At 15%:

\operatorname{NPV}(0.15) =-100+\frac{60}{1.15}+\frac{60}{1.15^2} =-100+52.173913+45.368620 =-2.457467.

At 20%:

\operatorname{NPV}(0.20) =-100+\frac{60}{1.20}+\frac{60}{1.20^2} =-100+50.000000+41.666667 =-8.333333.

The profile values are therefore $20.00, $4.13, -$2.46, and -$8.33, respectively, rounded to the nearest cent. NPV is positive at 10% and negative at 15%, so there is a zero between those rates. Linear interpolation gives

10\%+\frac{4.132232}{4.132232+2.457467}(15\%-10\%) \approx 13.14\%.

This is an estimate from the two profile values, not an exact solution. As a check, solving the NPV equation with $v=1/(1+i)$ gives $3v^2+3v-5=0$, whose positive solution is $v=(-3+\sqrt{69})/6$. Thus $i=1/v-1\approx 13.07\%$, which lies in the bracket. The method avoids the common mistake of treating linear interpolation as an exact IRR solver.

Why the Profile Often Slopes Down for Conventional Cash Flows

A conventional investment has one initial outflow followed by nonnegative inflows: $C_0<0$ and $C_t\geq 0$ for $t\geq1$, with at least one later inflow strictly positive. For this pattern, increasing the discount rate reduces the present value of each future inflow, so NPV decreases. The initial outflow at time 0 does not change with the rate.

The slope of the profile can be justified by differentiating each discounted cash flow. For $i>-1$,

\frac{d}{di}\operatorname{NPV}(i) =\frac{d}{di}\left(\sum_{t=0}^{n}C_t(1+i)^{-t}\right) =-\sum_{t=1}^{n}tC_t(1+i)^{-t-1}.

For a conventional investment, every term in the final sum is nonnegative before the minus sign, and at least one is positive. Hence the derivative is strictly negative. The profile is strictly decreasing and can have at most one zero. This conclusion depends on the conventional cash-flow pattern; it should not be carried over to cash flows that change sign more than once.

Interpretation: A profile that moves from positive NPV to negative NPV as the rate rises identifies a bracket containing at least one IRR. For a conventional investment, the profile’s strict decrease means there can be only one. With other sign patterns, profile shape must be examined rather than assumed.

Zero Crossings, Touches, and What the Graph Can Establish

A graph may pass through the horizontal axis, or it may just touch it and turn back. Both events give NPV equal to zero, so both can indicate an IRR. Consequently, counting only sign changes in a table can miss a root that occurs exactly at a trial rate or a root where the profile touches zero without changing sign.

A finite table also has limits. Opposite signs at two rates guarantee at least one zero between them, but the table alone does not show how many zeros lie in that interval. Likewise, values with the same sign at the endpoints do not prove that the profile stays on that side of zero between them.

Worked Example: Reading Two Zero Crossings from a Profile

A project has cash flows of $C_0=-200$, $C_1=500$, and $C_2=-300$, in dollars. The cash flows occur at annual intervals, and $i$ is an effective annual rate. Construct the NPV function, evaluate it at selected trial rates, and identify the nonnegative rates at which it is zero.

Method. Discount each dated amount to time 0:

\operatorname{NPV}(i)=-200+\frac{500}{1+i}-\frac{300}{(1+i)^2}.

Let $v=1/(1+i)$. Then

\operatorname{NPV}(i)=-200+500v-300v^2 =-300\left(v-1\right)\left(v-\frac{2}{3}\right).

The factorization shows that NPV is zero when $v=1$ or $v=2/3$. Since $v=1/(1+i)$, these correspond to $i=0\%$ and $i=50\%$. Profile values help show the pattern around these zeroes:

At 0%, $v=1$, so $\operatorname{NPV}(0)=-200+500-300=0$.
At 25%, $v=0.8$, so $\operatorname{NPV}(0.25)=-200+400-192=8.00$.
At 50%, $v=2/3$, so $\operatorname{NPV}(0.50)=-200+333.333333-133.333333=0$.
At 75%, $v=4/7$, so $\operatorname{NPV}(0.75)=-200+285.714286-97.959184=-12.244898$.

The profile is positive between the two zeroes and negative above 50%. Thus the two nonnegative IRRs are 0% and 50%. This example illustrates why a profile for cash flows with changing signs need not behave like the strictly decreasing profile of a conventional investment. The mistake to avoid is assuming that one initial outflow followed by one inflow pattern applies here: the final outflow changes the shape.

Worked Example: A Zero That Does Not Change Sign

Consider annual cash flows of $-$100$ at time 0, $+$240$ at time 1, and $-$144$ at time 2. Construct the NPV expression, locate its zero, and decide whether the profile crosses the horizontal axis there. Rates are effective annual rates.

Method. Discount the two future cash flows and write $v=1/(1+i)$:

\operatorname{NPV}(i)=-100+240v-144v^2 =-144\left(v-\frac{5}{6}\right)^2.

NPV is zero when $v=5/6$. Therefore $1/(1+i)=5/6$, so $1+i=6/5$ and $i=20\%$. Because a square is nonnegative, the negative coefficient gives $\operatorname{NPV}(i)\leq0$ for every $i>-1$, with equality only at 20%. The profile touches the horizontal axis at 20% and is negative on either side; it does not cross it.

For a numerical check, at 10% the NPV is $-100+240/1.10-144/1.10^2\approx-$0.8264$. At 30% it is $-100+240/1.30-144/1.30^2\approx-$0.5917$. Both are negative, consistent with the factorization and the zero at 20%. A sign-change-only search would miss this IRR if it tested rates on either side but never evaluated the zero itself. The correction is to recognize that an IRR is any rate at which NPV equals zero, whether or not the profile changes sign.

A Practical Profile-Reading Process

Set the timeline and rate unit.
Write every cash flow with its time. Use a trial rate per period consistent with the spacing; do not discount an annual cash flow using a monthly rate without converting units.

Calculate NPV at selected rates.
Keep the cash-flow signs and exponents tied to the timeline. A time-0 amount has exponent zero and is not discounted.

Mark the signs and zeroes.
Positive and negative NPV values lie on opposite sides of the horizontal axis. Opposite signs bracket at least one zero; an NPV of exactly zero is already a root.

Interpret the shape cautiously.
Use the conventional cash-flow pattern to justify a decreasing profile when it applies. Otherwise, do not infer the number of zeroes from a single pair of trial rates.

Report what the profile establishes.
Distinguish an exact zero from a bracket or an interpolation estimate. State whether the available information proves one root, at least one root, or merely suggests a location.

Common Profile-Reading Mistakes

Using inconsistent periods. If cash flows are annual, the trial rate and exponents must correspond to annual periods unless the cash flows are first converted to another consistent frequency.
Reversing the signs. Record the project’s outflow as negative and its inflows as positive throughout. Changing signs changes the NPV profile.
Discounting time 0. The initial cash flow is multiplied by $(1+i)^0=1$; it remains unchanged across the profile.
Calling an interpolation estimate exact. Interpolation estimates a zero from selected NPV values. The true profile may be curved between those points.
Counting only visible sign changes. A profile can have a zero at a sampled rate or touch zero without changing sign.
Assuming every profile slopes downward. Strict decrease follows for a conventional investment, not for every possible sequence of signed cash flows.

Key takeaway: An NPV profile is a map of NPV against the trial discount rate. Its zeroes are IRRs; opposite NPV signs bracket at least one zero, while a zero can also occur without a sign change. Use the cash-flow pattern to justify conclusions about the profile’s shape.

Check Your Understanding

For each problem, use the stated cash-flow timing and rate convention. Distinguish exact zeroes from estimates or brackets.

A project has cash flows of $-$500$ at time 0 and $+$300$ at the end of each of years 1 and 2. Find NPV at 0% and 20%, and state what the signs establish about an IRR. (Answer: NPV at 0% is $100.00; NPV at 20% is $-$41.67$; at least one IRR lies between 0% and 20%.)
A project pays $-$200$ now and $+$120$ at each of the next two year-ends. Calculate NPV at 10% and 20% and use the profile to bracket an IRR. (Answer: NPV at 10% is $8.26; NPV at 20% is $-$16.67$; at least one IRR lies between 10% and 20%.)
For annual cash flows $-$100$, $+$230$, and $-$132$ at times 0, 1, and 2, factor the NPV expression using $v=1/(1+i)$, then identify its nonnegative IRRs. (Answer: $\operatorname{NPV}(i)=-132(v-10/11)(v-5/6)$; the nonnegative IRRs are 10% and 20%.)
For annual cash flows $-$100$, $+$240$, and $-$144$ at times 0, 1, and 2, determine the IRR and whether the NPV profile changes sign there. (Answer: 20%; the profile touches zero and does not change sign.)
An NPV profile has values $40 at 5%, $12 at 10%, $-$9$ at 15%, $-$18$ at 20%, $-$3$ at 25%, and $8 at 30%. What can these values establish about zeroes? (Answer: At least one zero lies between 10% and 15%, and at least one lies between 25% and 30%; the table alone does not establish the total number of zeroes.)

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