The Internal Rate of Return Equation

From a Cash-Flow Timeline to an IRR Equation

Net present value measures the value, at a selected discount rate, of a project’s future cash flows after accounting for its initial cost. The internal rate of return, or IRR, asks a related question: what annual effective rate makes the project’s net present value exactly zero? To answer it correctly, first place every cash flow at its actual date and discount it for the corresponding amount of time.

This tutorial focuses on defining that rate and writing its equation. It does not assume that an equation has only one solution or that every set of cash flows has an IRR. Those are separate questions. Here, the essential task is to build the present-value equation from a dated timeline and check a proposed rate by substitution.

Definition: Let $C_t$ denote a project's net cash flow at time $t$, measured in years from the valuation date. Cash inflows are positive and cash outflows are negative. Under annual effective discounting, an internal rate of return $i^*$ is a rate greater than $-1$ for which the project's net present value is zero: $$ \operatorname{NPV}(i^*)=\sum_t \frac{C_t}{(1+i^*)^t}=0. $$ The sum includes each dated cash flow. A cash flow at time zero is not discounted.

The equation uses a consistent time scale. If the stated annual effective rate is $i$, then the accumulation factor over $t$ years is $(1+i)^t$, and the corresponding present-value factor is $1/(1+i)^t$. Thus, a cash flow at six months has $t=0.5$, not $t=1$. In a timeline with no cash flow in some year, later cash flows retain their actual dates: an inflow at year 3 is discounted for three years even if year 2 has no flow.

The cash flows in a capital-budgeting equation are the project cash flows being valued, not accounting earnings simply because those earnings appear on a financial statement. As in NPV analysis, the amounts, signs, valuation date, and timing must be specified before writing the equation. A project may have an initial outflow, subsequent inflows, and additional outflows; the definition does not require every later cash flow to be positive.

Why the Equation Sets NPV to Zero

The equation follows directly from the meaning of present value. At a candidate annual effective rate $i$, a dollar received at time $t$ is worth $1/(1+i)^t$ dollars at time zero. Therefore, a cash flow $C_t$ at that date contributes $C_t/(1+i)^t$ to the value at time zero. Adding the contributions from all dates gives the project's NPV at rate $i$.

An IRR is a rate at which the present value of the project’s inflows and outflows balances. Because outflows have negative signs, that balance is represented by a sum equal to zero—not by a sum of inflows alone equal to zero. This gives the defining equation:

\operatorname{NPV}(i)=\frac{C_0}{(1+i)^0}+\frac{C_{t_1}}{(1+i)^{t_1}}+\frac{C_{t_2}}{(1+i)^{t_2}}+\cdots, \qquad \operatorname{IRR}=i^* \ \text{such that}\ \operatorname{NPV}(i^*)=0.

Since $(1+i)^0=1$, the time-zero cash flow appears unchanged. The notation $t_1,t_2,\ldots$ represents the actual dates in years, not merely the order in which nonzero cash flows occur. This distinction matters whenever a timeline includes a gap or a fraction of a year.

Worked Example: Checking a Candidate Rate for Annual Cash Flows

A hypothetical equipment project requires an outflow of $1,000 today. It is expected to generate inflows of $550 at the end of year 1 and $605 at the end of year 2. Write the IRR equation and determine whether 10% is an IRR.

Method. Assign time zero to today and record each cash flow with its sign and date: $C_0=-1{,}000$, $C_1=550$, and $C_2=605$. The candidate rate is annual effective, so the year 1 and year 2 discount factors are $1/1.10$ and $1/(1.10)^2$, respectively.

The project’s NPV at a general annual effective rate $i$ is:

\operatorname{NPV}(i)=-1{,}000+\frac{550}{1+i}+\frac{605}{(1+i)^2}.

Substitute the candidate $i=0.10$. First, $1.10^2=1.21$, so the two present values are $550/1.10=500$ and $605/1.21=500$. Therefore:

\operatorname{NPV}(0.10)=-1{,}000+500+500=0.

Conclusion and check. The project’s NPV at 10% is exactly zero, so 10% satisfies the IRR equation. The discounted inflows total $1,000, exactly offsetting the $1,000 time-zero outflow. This method avoids the common mistake of discounting the year 2 receipt for only one year.

Fractional Dates and Annual Effective Rates

A timeline need not contain only year-end cash flows. When the rate is annual effective, a cash flow at a fraction of a year is discounted with that fractional exponent. For example, six months is $0.5$ years, so its discount factor is $1/(1+i)^{0.5}$. Equivalently, for a positive annual accumulation factor, it is $1/\sqrt{1+i}$. This is not the same as dividing by $1+i$, which discounts for a full year.

Worked Example: A Receipt at Six Months

A hypothetical project costs $1,200 today, pays $630 after six months, and pays $660 after one year. Under annual effective discounting, determine whether a candidate rate of 10% makes NPV zero.

Method. Express each date in years: the initial outflow occurs at $t=0$, the first receipt at $t=0.5$, and the second receipt at $t=1$. The general equation is:

\operatorname{NPV}(i)=-1{,}200+\frac{630}{(1+i)^{0.5}}+\frac{660}{1+i}.

At 10%, the six-month discount factor is $1/(1.10)^{0.5}=1/\sqrt{1.10}$. Since $\sqrt{1.10}\approx1.048808848$, the present value of the six-month receipt is $630/1.048808848\approx600.68$. The one-year receipt has present value $660/1.10=600.00$.

Thus the NPV, rounded to the nearest cent, is:

\operatorname{NPV}(0.10)\approx -1{,}200+600.68+600.00=\$0.68.

Conclusion and check. The NPV is approximately positive $0.68, not zero. Therefore, 10% does not satisfy this project’s IRR equation. The present-value total is $1,200.68, which exceeds the initial $1,200 outflow by $0.68. The calculation rounds only the final NPV to cents.

Keep Actual Dates When a Period Has No Cash Flow

A blank period in a timeline is not a reason to renumber later cash flows. If a project has no cash flow in year 2 but does have one in year 3, that final flow is discounted by $(1+i)^3$. One may omit an explicitly stated zero term from the sum, but not shorten the exponent of a later nonzero term.

Worked Example: A Zero-Cash-Flow Year

A hypothetical project has a $1,800 outflow today, a $770 inflow at the end of year 1, no cash flow at the end of year 2, and a $1,464.10 inflow at the end of year 3. Check whether 10% satisfies the IRR equation.

Method. Write the dates before substituting a rate. The general equation is:

\operatorname{NPV}(i)=-1{,}800+\frac{770}{1+i}+\frac{0}{(1+i)^2}+\frac{1{,}464.10}{(1+i)^3}.

At $i=0.10$, the year 1 present value is $770/1.10=700$. The zero year 2 flow contributes $0/(1.10)^2=0$. For year 3, $(1.10)^3=1.331$, and $1{,}464.10/1.331=1{,}100$. Consequently:

\operatorname{NPV}(0.10)=-1{,}800+700+0+1{,}100=0.

Conclusion and check. The 10% rate satisfies the equation: the present values of the two nonzero inflows total $1,800, matching the initial outflow. The zero term may be left out, but the year 3 receipt remains discounted for three years.

A Reliable Translation Process

A disciplined timeline-to-equation process helps prevent sign, date, and rate-unit errors. It also makes the result easier to check before drawing any conclusion.

Set the valuation date.
Use time zero for the date at which the project is being evaluated. Record each cash flow as an inflow or outflow with its sign.

Convert dates into years.
Use the actual elapsed time: for example, six months is 0.5 years. Do not count only the number of nonzero cash flows.

Write the present-value terms.
For an annual effective candidate rate $i$, divide each $C_t$ by $(1+i)^t$. A time-zero flow is unchanged.

Set the sum equal to zero.
The IRR equation is the NPV equation with its value set to zero. Preserve zero-flow dates when they clarify the timeline.

Check any proposed rate.
Substitute it into every dated term, calculate the NPV, and verify that the result is zero, allowing for stated rounding where appropriate.

Common Exam Trap: Treating Dates as Period Counts

A frequent error is to use one full year of discounting for a six-month receipt. For the second worked example, that incorrect calculation would be:

-1{,}200+\frac{630}{1.10}+\frac{660}{1.10} =-1{,}200+572.73+600.00 =-\$27.27.

That is not the correct NPV at 10%, because $630$ arrives after half a year, not one year. The correct term is $630/(1.10)^{0.5}$, giving an NPV of approximately positive $0.68. A different timing error is to treat the third-year payment in the zero-flow example as a year 2 payment just because there are only two nonzero inflows. That would use the wrong exponent: the flow at year 3 must remain divided by $(1+i)^3$.

These mistakes change the equation itself, not just the arithmetic. Before calculating, check that every exponent matches the cash flow’s elapsed time and that every rate is interpreted using the stated compounding convention. The resulting equation is the starting point for later methods of solving and interpreting IRR.

Key takeaway: An IRR is a rate that makes the sum of all dated cash flows, each discounted to the valuation date, equal zero. Write the timeline first, use actual year fractions as exponents, and check a candidate rate by calculating its NPV.

Check Your Understanding

For each problem, use the stated annual effective rate convention and keep each cash flow at its actual date.

A hypothetical project has a $900 outflow today and a $990 inflow at the end of year 1. Does 10% satisfy its IRR equation? (Answer: Yes; $-900+990/1.10=0$.)
A project has a $2,000 outflow today, a $630 receipt after six months, and a $1,452 receipt after two years. Calculate its NPV at 10%, rounded to cents, and state whether 10% is an IRR. (Answer: $-2000+630/\sqrt{1.10}+1452/(1.10)^2\approx-\$199.32$; no.)
A project has a $1,600 outflow today, a $600 receipt at year 1, no year 2 cash flow, and a $1,331 receipt at year 3. Calculate its NPV at 10%, rounded to cents. (Answer: $-1600+600/1.10+0+1331/1.331\approx-\$54.55$; 10% is not an IRR.)
A project has a $1,250 outflow today, then receipts of $300 after six months, $500 after 18 months, and $600 after 30 months. Write the IRR equation using an unknown annual effective rate $i^*$. (Answer: $-1250+300/(1+i^*)^{0.5}+500/(1+i^*)^{1.5}+600/(1+i^*)^{2.5}=0$.)
A project has a $2,400 outflow today, an $800 receipt at year 1, no year 2 cash flow, and a $1,464.10 receipt at year 4. Find its NPV at 10%, rounded to cents, and state whether 10% satisfies the IRR equation. (Answer: $-2400+800/1.10+0+1464.10/(1.10)^4\approx-\$672.73$; no.)

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