Bracketing an IRR with Net Present Value Signs

Why Bracket an IRR?

For an uneven conventional cash-flow stream, the IRR is generally found by solving an equation rather than by rearranging a simple two-payment relationship. Before using a numerical method, it is useful to identify an interval that contains the root. NPV evaluations at trial rates provide a direct way to do this: if NPV is positive at one rate and negative at a higher rate, the rate at which NPV is zero lies between them, under the conventional-cash-flow assumptions below.

A bracket is more than a pair of guesses. The signs of the NPVs provide evidence about the location of a root. A close pair of trial rates with opposite signs gives a narrower interval than a widely spaced pair. This tutorial focuses on establishing and checking that interval, not on the numerical procedures used to refine an estimate.

Definition: For cash flows $C_0,C_1,\ldots,C_n$ occurring at times $0,1,\ldots,n$, the net present value at effective rate $i$ per period is $$ \operatorname{NPV}(i)=\sum_{t=0}^{n} C_t(1+i)^{-t}, \qquad i>-1. $$ An IRR is a rate $i$ for which $\operatorname{NPV}(i)=0$. A conventional investment cash-flow stream has an initial outflow $C_0<0$ and nonnegative later cash flows, with at least one later cash flow strictly positive.

The period in the formula must match the spacing of the cash flows. If the cash flows are annual, $i$ is an effective annual rate. If they are quarterly, $i$ is a rate per quarter. For cash flows at dates $0,1,\ldots,n$, a rate quoted per year cannot be substituted as though it were a rate per period unless it is first converted consistently. The earlier material on consistent periods and cash-flow timing applies to every NPV evaluation here.

The bracket claim uses two standard facts. NPV is continuous for $i>-1$, because each term $C_t(1+i)^{-t}$ is continuous there. For a conventional stream, NPV decreases as $i$ increases: higher rates reduce the present value of each positive future receipt, while the initial outflow at time zero is unchanged. In particular, opposite signs at two trial rates establish a zero between them by the intermediate value theorem. For conventional cash flows, the decreasing NPV also means that this zero is unique.

How the Sign Test Works

Choose trial rates $i_L<i_H$, both greater than $-1$, and calculate NPV at each using the same cash flows and time units. If $\operatorname{NPV}(i_L)>0$ and $\operatorname{NPV}(i_H)<0$, then the IRR is between $i_L$ and $i_H$. Reversing the signs also establishes a root between the rates, though for a conventional stream the lower rate normally has the higher NPV.

\operatorname{NPV}(i_L)\operatorname{NPV}(i_H)<0 \quad\Longrightarrow\quad \text{there is an IRR in }(i_L,i_H)

The sign test is about the computed NPV values, not about how large the values are. An NPV close to zero suggests that its trial rate may be near the IRR, but only opposite signs at the two endpoints certify a bracket this way. If both NPVs are positive, the chosen pair has not bracketed a root; for a conventional stream, a higher trial rate may be needed. Likewise, two negative NPVs call for trying a lower rate.

Important condition: The sign-change conclusion requires an NPV function continuous between the trial rates and a valid rate domain throughout that interval. In the standard annual model, this means using trial rates greater than $-1$. The conventional-cash-flow assumption gives the additional result that there is only one IRR. With cash flows that change sign more than once, opposite signs still guarantee at least one root under continuity, but do not establish uniqueness.

Worked Brackets

In the examples, cash flows occur at the ends of successive years, and all rates are effective annual rates. Calculations retain full precision until the stated final rounding. Present values are displayed to six decimal places, and NPV is reported to the nearest cent.

Worked Example: Bracketing a Two-Year Investment

A project requires an outlay of $1,000 now and pays $600 at the end of each of the next two years. Evaluate NPV at 10% and 20%, then state what the signs establish. Round NPVs to cents.

Method. Discount each year’s receipt by the trial rate raised to its year number; include the initial outlay at time zero without discounting.

At 10%:

\operatorname{NPV}(0.10) =-1000+\frac{600}{1.10}+\frac{600}{1.10^2} =-1000+545.454545+495.867769 =41.322314

Thus, the NPV at 10% is $41.32, positive.

At 20%:

\operatorname{NPV}(0.20) =-1000+\frac{600}{1.20}+\frac{600}{1.20^2} =-1000+500.000000+416.666667 =-83.333333

Thus, the NPV at 20% is $-$83.33$, negative. The cash flows are conventional, and both trial rates are in the valid domain. The sign change and continuity establish that the unique IRR lies between 10% and 20%.

Check and common mistake avoided. The calculation uses 1.10² and 1.20² for the year-two receipt. Discounting both receipts by only one period would misdate the second cash flow and could give an incorrect sign.

Worked Example: An Uneven Conventional Stream

A hypothetical equipment upgrade costs $2,500 at time zero and is expected to produce receipts of $700, $900, and $1,200 at the ends of years 1, 2, and 3. Test trial rates of 5% and 15%. Determine whether these rates bracket the IRR. Round NPVs to cents.

Method. Evaluate the dated cash flows separately at each annual trial rate, then inspect the signs of the two totals.

At 5%, the discounted receipts are:

\frac{700}{1.05}=666.666667,\qquad \frac{900}{1.05^2}=\frac{900}{1.1025}=816.326531,\qquad \frac{1200}{1.05^3}=\frac{1200}{1.157625}=1036.605118.

Therefore,

\operatorname{NPV}(0.05) =-2500+666.666667+816.326531+1036.605118 =19.598316.

The NPV at 5% is $19.60, positive.

At 15%, the discounted receipts are:

\frac{700}{1.15}=608.695652,\qquad \frac{900}{1.15^2}=\frac{900}{1.3225}=680.529301,\qquad \frac{1200}{1.15^3}=\frac{1200}{1.520875}=789.019477.

Therefore,

\operatorname{NPV}(0.15) =-2500+608.695652+680.529301+789.019477 =-421.755570.

The NPV at 15% is $-$421.76$, negative. Since the stream is conventional, the IRR is unique; since NPV is continuous and changes sign between 5% and 15%, that IRR lies within this interval.

Check and common mistake avoided. The receipts are discounted for one, two, and three years respectively. Applying the three-year discount factor to every receipt would not represent the stated timing.

Choosing a Useful Initial Interval

A very wide bracket is valid but may be inconvenient for subsequent calculation. A practical approach is to test plausible rates, use the sign and size of each NPV to decide which direction to explore, and then test a rate between the endpoints. For conventional cash flows, a positive NPV means the trial rate is below the IRR; a negative NPV means it is above the IRR. This direction follows from NPV’s decreasing behavior.

The size of an NPV is not itself a rate estimate. For example, an NPV of $60 does not mean the IRR is 60%, nor does the ratio of NPV to the initial investment give a generally valid rate adjustment. Use the sign to identify which side of the root the trial rate is on, and evaluate another rate to check whether an interval is actually bracketed.

Worked Example: Narrowing the Trial Interval

A proposed project has cash flows of $-$5,000$ now, followed by receipts of $2,000, $2,500, and $1,800 at the ends of years 1, 2, and 3. Check trial rates of 8%, 12%, and 13%. Use the results to select a narrower bracket. Round NPVs to cents.

Method. Evaluate the dated cash flows at each rate. If two tested rates have opposite NPV signs, use those two rates as the bracket.

At 8%:

\operatorname{NPV}(0.08) =-5000+\frac{2000}{1.08}+\frac{2500}{1.08^2}+\frac{1800}{1.08^3}

$$ =-5000+1851.851852+2143.347737+1428.898032 =424.097621. $$

The NPV at 8% is $424.10, positive.

At 12%:

\operatorname{NPV}(0.12) =-5000+\frac{2000}{1.12}+\frac{2500}{1.12^2}+\frac{1800}{1.12^3}

$$ =-5000+1785.714286+1992.984694+1281.204452 =59.903432. $$

The NPV at 12% is $59.90, still positive. The rates 8% and 12% do not bracket the IRR.

At 13%:

\operatorname{NPV}(0.13) =-5000+\frac{2000}{1.13}+\frac{2500}{1.13^2}+\frac{1800}{1.13^3}

$$ =-5000+1769.911504+1957.866865+1247.490287 =-24.731344. $$

The NPV at 13% is $-$24.73$, negative. The useful bracket is therefore 12% to 13%, not 8% to 12%. The positive value at 12% and negative value at 13% establish that the unique IRR lies between those rates.

Check and common mistake avoided. The 8% and 12% trial NPVs are both positive, so that pair cannot certify a bracket. A smaller positive NPV at the higher rate suggests that the IRR is nearer, but a sign change is still needed to confirm the interval.

Derivation: Why Opposite Signs Certify a Root

The bracket test follows from continuity, not from an approximation formula. For a finite sequence of cash flows, each discounted term is a constant multiplied by $(1+i)^{-t}$. That expression is continuous whenever $i>-1$, so their finite sum, NPV, is also continuous. Suppose two valid trial rates satisfy $i_L<i_H$, with positive NPV at $i_L$ and negative NPV at $i_H$. A continuous function that takes values of opposite signs must equal zero somewhere between those rates. That zero is an IRR.

For a conventional stream, the later positive receipts lose present value as the rate increases. More formally, differentiating NPV gives:

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

The strict inequality holds because $C_t\geq 0$ for future cash flows and at least one $C_t>0$. The time-zero term contributes no derivative. Thus NPV is strictly decreasing, so it cannot cross zero more than once. This justifies calling the bracketed root the unique IRR for a conventional stream.

Common Exam Traps

Using the same discount exponent for every receipt. A payment at the end of year $t$ is discounted by $(1+i)^t$, not by a common factor unrelated to its date.
Mixing rate and time units. Annual cash flows require a rate per year. Convert rates before applying the NPV formula if the stated rate uses another period.
Assuming any two rates form a bracket. A bracket requires opposite NPV signs, or an endpoint where NPV is exactly zero. Two NPVs with the same sign do not establish a root between those rates.
Ignoring the rate domain. In the standard discrete-period model, the effective rate must be greater than $-1$, so each accumulation factor $1+i$ is positive.
Claiming uniqueness without the right cash-flow pattern. The opposite-sign test establishes a root by continuity, but uniqueness follows here from the conventional pattern. It should not be assumed for cash flows with multiple sign changes.

A reliable bracket is established by three checks: the cash-flow dates and rate period agree, both trial rates are in the valid domain, and the calculated NPVs have opposite signs. When all three hold, the bracket gives a sound starting interval for further numerical work.

Check Your Understanding

For each problem, use the specified effective annual rates and year-end cash-flow timing. Round NPVs to cents when calculating.

A project has cash flows of $-$800$ at time zero and $500 at the end of each of the next two years. Evaluate NPV at 10% and 20%. Do these rates bracket the IRR? (Answer: NPV at 10% is $67.77 and NPV at 20% is $-$36.11$; yes, the IRR lies between 10% and 20%.)
An investment costs $1,200 now and returns $400, $500, and $500 at the ends of years 1, 2, and 3. What do trial rates of 5% and 10% establish? (Answer: NPV at 5% is $66.39 and NPV at 10% is $-$47.48$; the IRR lies between 5% and 10%.)
A project has cash flows of $-$2,000$ now and $900 at each of the next three year-ends. At 8% and 12%, do the NPVs establish a bracket? If not, what does a further test at 20% show? (Answer: NPV at 8% is $319.39 and at 12% is $161.65, so those rates do not bracket the IRR; NPV at 20% is $-$104.17$, so 12% and 20% do bracket it.)
A project requires $1,500 now and pays $300, $500, and $900 at the ends of years 1, 2, and 3. Evaluate at 5% and 10% and state the bracket. (Answer: NPV at 5% is $16.68 and at 10% is $-$137.87$; the IRR lies between 5% and 10%.)
For a conventional project, NPV is positive at 9% and positive at 11%. What can be concluded about a bracket, and which direction should the next trial rate move? (Answer: These rates do not bracket the IRR; test a higher rate, since NPV is decreasing and the root, if not already passed, is above 11%.)

What You'll Learn