Solving IRR for Uneven Conventional Cash Flows

Why Uneven Receipts Require an IRR Equation

When an investment pays equal amounts each period, an annuity factor can compactly represent the receipts. If the amounts differ, each receipt must instead be discounted for its own time period. The internal rate of return (IRR) is the rate that makes the present value of all receipts equal the initial outlay. The essential work is to put each cash flow at the correct time and then solve the resulting equation.

This tutorial considers conventional cash flows: an outlay at time 0 followed by positive receipts at the ends of periods 1 through $n$. Assume equally spaced annual periods unless the problem says otherwise. The annual effective rate $i$ is measured per year, and the admissible domain for the discount factors is $i>-1$. In ordinary investment questions, the sought rate is typically nonnegative.

Definition: For an initial outlay $C_0>0$ at time 0 and positive receipts $C_1,\ldots,C_n$ at the ends of years $1,\ldots,n$, the IRR is a rate $i$ satisfying $$ -C_0+\sum_{t=1}^{n}\frac{C_t}{(1+i)^t}=0. $$ The receipts may be unequal. Each receipt is discounted according to its own time $t$.

The equation is the net present value (NPV) set equal to zero. At a specified trial rate, however, the same left-hand side is the NPV at that rate; it is not automatically zero. Distinguishing those two uses of the expression prevents a frequent setup error.

Why Each Receipt Has Its Own Discount Factor

Under an annual effective rate $i$, one dollar available at the end of year $t$ has present value $(1+i)^{-t}$ dollars. This follows by accumulating its present value for $t$ years: a present amount $P$ grows to $P(1+i)^t$, so setting that accumulated amount equal to the receipt $C_t$ gives $P=C_t/(1+i)^t$.

Present values of cash flows at the same valuation date can be added. The initial outlay is already at time 0, so its signed present value is $-C_0$. Adding that amount to the present values of the receipts gives NPV. At the IRR, the present value of the receipts exactly offsets the outlay:

\operatorname{NPV}(i)=-C_0+\frac{C_1}{1+i}+\frac{C_2}{(1+i)^2}+\cdots+\frac{C_n}{(1+i)^n}=0.

This result assumes that the stated cash flows occur at the specified period ends and that one effective rate per period is being used. If the periods are not annual, use a rate measured over the matching period; do not use an annual rate as though it were a monthly or quarterly rate. Irregularly timed cash flows require a separate timing treatment.

A Practical Solve-and-Check Process

For uneven receipts, use the full cash-flow equation rather than an annuity factor. A numerical solver is useful because the equation generally cannot be rearranged into a convenient closed-form expression for $i$. Trial rates help identify a sensible range and reveal whether the proposed rate is plausible.

Set the timeline.
Record the outlay at time 0 and each receipt at its stated period end.

Form NPV as a function of the rate.
Write one discounted term for every receipt, using exponent $t$ for a receipt at the end of period $t$.

Try rates or solve numerically.
Evaluate NPV at selected rates, then use a financial calculator or spreadsheet solver to find the rate that makes NPV zero.

Substitute the reported rate.
Recalculate the present values or NPV. A small nonzero residual is expected when the rate has been rounded.

Worked Example: Estimating an IRR from Trial Rates

A project requires an outlay of $1,000 now and pays $200, $300, and $700 at the ends of years 1, 2, and 3, respectively. Estimate its annual IRR. Use trial rates to assess the solution, then verify the numerical result.

Method. Write the zero-NPV equation using each receipt’s actual time, evaluate it at trial rates, and solve the equation numerically. This avoids treating the unequal receipts as a level annuity.

The IRR equation is

-1000+\frac{200}{1+i}+\frac{300}{(1+i)^2}+\frac{700}{(1+i)^3}=0.

At a 5% trial rate, the discount factors are $1.05$, $1.05^2=1.1025$, and $1.05^3=1.157625$. The receipt present values are

\frac{200}{1.05}=190.4762,\qquad \frac{300}{1.1025}=272.1088,\qquad \frac{700}{1.157625}=604.6863.

Their total is approximately $1,067.2714, so the NPV is approximately $1067.2714-1000=+\$67.27$. At a 10% trial rate, the present values are

\frac{200}{1.10}=181.8182,\qquad \frac{300}{1.21}=247.9339,\qquad \frac{700}{1.331}=525.9204.

The total is approximately $955.6724, giving an NPV of approximately $955.6724-1000=-\$44.33$. Thus the NPV changes from positive to negative over these trial rates. Entering the original zero-NPV equation in a numerical solver gives $i\approx0.079121$, or 7.9121%.

Check. Using the displayed rate of 7.9121%, the receipt present values rounded to cents are approximately $185.34, $257.62, and $557.04. They total $1,000.00, so the NPV is approximately zero at cent precision. The unrounded solver rate, rather than the rounded display, is what makes the equation zero more precisely.

Common mistake avoided: The third receipt is at the end of year 3, so its denominator is $(1+i)^3$, not $(1+i)^2$. Also, a trial-rate NPV such as $+\$67.27$ is a value to compare with zero, not the IRR equation’s right-hand side.

Solver Entry and Rate Interpretation

A solver needs the complete zero-NPV function, not a cash-flow total with no timing information. For the preceding example, a spreadsheet formula could be written as a function of a rate cell $r$:

-1000+\frac{200}{1+r}+\frac{300}{(1+r)^2}+\frac{700}{(1+r)^3}.

Use the solver to set that expression equal to zero by changing $r$. A financial calculator’s cash-flow worksheet can also be used: enter the initial amount as a negative cash flow and each receipt as a positive cash flow at its period. Confirm that the calculator is using annual periods for annual cash flows.

The numerical value of an IRR is a rate per cash-flow period. If payments are annual, the result from this equation is an annual effective rate. If cash flows were spaced every six months, the same form with one exponent per half-year would produce a half-year rate; conversion to an annual effective rate would then require $(1+i_{\text{half-year}})^2-1$. Do not label a periodic rate as annual without making the conversion.

Worked Example: Unequal Receipts with an Exact Solver Check

An equipment project costs $1,500 now. It produces receipts of $220, $605, and $1,064.80 at the ends of years 1, 2, and 3. Find the annual IRR using trial rates and a numerical solver, and check the result.

Method. Put the receipt in year $t$ over $(1+i)^t$. Trial values give a useful check on the equation and on the solver output.

The equation is

-1500+\frac{220}{1+i}+\frac{605}{(1+i)^2}+\frac{1064.80}{(1+i)^3}=0.

At 8%, the denominators are $1.08$, $1.1664$, and $1.259712$. The present values are approximately $203.7037, $518.6900, and $845.2726. Their total is $1,567.6663, so NPV is approximately $+\$67.67$. At 12%, the denominators are $1.12$, $1.2544$, and $1.404928$. The present values are approximately $196.4286, $482.3024, and $757.9038. Their total is $1,436.6348, so NPV is approximately $-\$63.37$.

A numerical solver applied to the equation returns $i=0.10$, or 10.00%. Check by substitution:

\frac{220}{1.10}=200,\qquad \frac{605}{1.10^2}=\frac{605}{1.21}=500,\qquad \frac{1064.80}{1.10^3}=\frac{1064.80}{1.331}=800.

The present values total $200+500+800=\$1,500$, exactly the outlay, so NPV is zero. The trial NPVs have opposite signs around the solver result, and direct substitution verifies it.

Common mistake avoided: The receipts are not equal, so replacing them with a single annuity amount would discard the cash-flow pattern. The correct equation retains all three separate receipt amounts and their separate exponents.

Common Exam Traps

The most frequent errors are timing errors, sign errors, and confusing a trial-rate calculation with the IRR equation. The following example illustrates why a plausible-looking rate must still be checked.

Worked Example: Correcting a False IRR Claim

A project costs $428.80 and pays $100, $250, and $400 at the ends of years 1, 2, and 3. A proposed solution claims that the IRR is 25.00%. Test the claim and estimate the actual annual IRR.

Method. Substitute the proposed rate into the NPV expression. If NPV is not zero, the claim is not an IRR; then solve the correctly formed equation.

The equation is

-428.80+\frac{100}{1+i}+\frac{250}{(1+i)^2}+\frac{400}{(1+i)^3}=0.

At 25%, the three present values are $100/1.25=\$80$, $250/1.25^2=250/1.5625=\$160$, and $400/1.25^3=400/1.953125=\$204.80$. Their sum is $444.80, so the NPV is $444.80-428.80=+\$16.00$. Therefore 25.00% is not the IRR.

Solving the original equation numerically gives $i\approx0.27028$, or 27.028% (27.03% to two decimal places). Substitution at the unrounded solver rate makes NPV approximately zero; rounding the rate to 27.03% leaves an NPV residual of only a few cents. The verified rate is therefore about 27.03%.

Common mistake avoided: A rate is not an IRR merely because it is suggested or because the calculation looks close. Compute the receipt present values at that rate and compare their sum with the outlay.

Checklist Before Reporting an IRR

For a conventional investment with unequal positive receipts, use this checklist:

Write the time-0 outlay as a negative amount.
Include every positive receipt once, at its stated period.
Use the period number as the discount exponent.
Make the rate’s period consistent with the cash-flow spacing.
Set NPV equal to zero only when writing the IRR equation; report the nonzero result when evaluating NPV at a trial rate.
Substitute the solver result and report a sensible rounding level.

A spreadsheet or calculator can solve the equation quickly, but it cannot detect every modeling mistake. In particular, it will still return a result if a receipt is omitted, entered with the wrong sign, or assigned to the wrong period. The timeline and the final NPV check remain essential.

Check Your Understanding

For each problem, use the stated periods and annual effective rate convention. Answers are provided so you can check the setup as well as the result.

A project costs $500 now and pays $110 at the end of year 1 and $484 at the end of year 2. Form the IRR equation and find the annual IRR. (Answer: $-500+110/(1+i)+484/(1+i)^2=0$; $i=10\%$.)
An investment has an initial outlay of $1,000 and receipts of $330 and $847 at the ends of years 1 and 2. Write the equation and verify its IRR by discounting both receipts. (Answer: $-1000+330/(1+i)+847/(1+i)^2=0$; $i=10\%$, with present values $300 and $700.)
A project costs $800 and pays $220, $302.50, and $465.85 at the ends of years 1, 2, and 3. Find the IRR and show the present value of each receipt at that rate. (Answer: $i=10\%$; the present values are $200, $250, and $350.)
A project costs $1,200 and returns $216, $466.56, and $755.8272 at the ends of years 1, 2, and 3. Form the IRR equation and verify the rate. (Answer: $-1200+216/(1+i)+466.56/(1+i)^2+755.8272/(1+i)^3=0$; $i=8\%$, with present values $200, $400, and $600.)
For cash flows of $-\$1,000$ at time 0 and receipts of $100, $200, and $800 at the ends of years 1, 2, and 3, write the IRR equation and calculate NPV at 10%. Do not set the trial-rate NPV equal to zero unless it actually is zero. (Answer: $-1000+100/(1+i)+200/(1+i)^2+800/(1+i)^3=0$; NPV at 10% is approximately $-\$142.75$.)

What You'll Learn