Closed-Form IRRs for Two-Cash-Flow Projects

Introduction

When a project consists of one outlay now and one receipt later, its internal rate of return (IRR) can be found directly rather than by trial and error. The key is to account for the number of periods between the cash flows: the total growth from the outlay to the receipt occurs over all those periods, not just one.

This is the simplest cash-flow pattern for which an IRR can be expressed in closed form. It provides a useful benchmark for more involved IRR calculations and is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Consider a project with an outlay of $C_0>0$ at time 0 and a receipt of $C_n>0$ at the end of period $n$, with no other cash flows. Its IRR per period is the rate $i$ that makes its net present value equal to zero: $$ -C_0+\frac{C_n}{(1+i)^n}=0. $$ If the periods are years, $i$ is an annual effective rate. If the periods are quarters or another unit, the solved rate is per that period.

The Closed-Form Formula

At the IRR, the present value of the receipt exactly equals the initial outlay. Solving that equality for the rate gives the formula:

\begin{aligned} -C_0+\frac{C_n}{(1+i)^n}&=0,\\ \frac{C_n}{(1+i)^n}&=C_0,\\ (1+i)^n&=\frac{C_n}{C_0},\\ i&=\left(\frac{C_n}{C_0}\right)^{1/n}-1. \end{aligned}

The ratio $C_n/C_0$ is the total accumulation factor over the full $n$ periods. Taking the $n$th root converts that total factor into the accumulation factor for one period. Subtracting 1 gives the rate per period.

Theorem: For one positive outlay at time 0 and one positive receipt at the end of period $n$, where $n$ is a positive integer, the IRR per period is $$ i=\left(\frac{C_n}{C_0}\right)^{1/n}-1. $$ The formula assumes the cash flows occur at the stated period endpoints and that there are no other cash flows. It gives a rate greater than $-1$, as required for a positive accumulation factor.

This result also follows from the form of the equation. Since $C_n/C_0>0$, there is exactly one positive real $n$th root of that ratio. Also, $1+i$ must be positive for the discount factor to be defined, so $i>-1$. Thus, for this two-cash-flow pattern, the equation has exactly one admissible IRR. This conclusion is specific to this pattern; it should not be assumed automatically for cash flows with additional sign changes.

Interpreting the Result

The size comparison between the receipt and the outlay determines whether the IRR is positive, zero, or negative:

If $C_n>C_0$, then $i>0$.
If $C_n=C_0$, then $i=0$, regardless of the number of periods.
If $0<C_n<C_0$, then $-1<i<0$.

A negative IRR is mathematically valid: it means the project returns less than the initial outlay at the end of the stated period. The rate is still greater than $-100\%$, because the receipt is positive. For comparison with a required annual effective rate, both rates must refer to years. If the cash flows are spaced quarterly, first calculate the quarterly IRR and then convert it to an annual effective rate.

Identify the cash-flow amounts and dates.
Record the initial outlay $C_0$, the later receipt $C_n$, and the number $n$ of periods between them.

Use consistent periods.
Confirm whether the periods are years, quarters, or another unit. The formula returns a rate per that period.

Calculate the rate.
Divide the receipt by the outlay, take the $n$th root, and subtract 1.

Check the result.
Substitute the rate into the NPV equation. The discounted receipt should equal the outlay, apart from any rounding.

Worked Applications

Worked Example: A Larger Receipt After Three Years

A hypothetical project requires an outlay of $20,000 now and pays a single receipt of $26,620 at the end of year 3. Find its annual effective IRR. Then calculate its NPV at an annual effective discount rate of 9%.

Method. The cash flows are an outlay at time 0 and one receipt three annual periods later. Use the closed-form formula with $C_0=20{,}000$, $C_n=26{,}620$, and $n=3$.

First calculate the receipt-to-outlay ratio:

\frac{C_n}{C_0}=\frac{26{,}620}{20{,}000}=1.331.

Because $1.331=1.1^3$, its cube root is $1.1$. Therefore:

i=1.331^{1/3}-1=1.1-1=0.10.

The annual effective IRR is 10%. To check, accumulate the outlay for three years at 10%:

20{,}000(1.10)^3 =20{,}000(1.331) =26{,}620.

This equals the stated receipt, so the NPV at 10% is zero. At a 9% annual effective discount rate, the NPV is:

\begin{aligned} \operatorname{NPV}(0.09) &=-20{,}000+\frac{26{,}620}{(1.09)^3}\\ &=-20{,}000+\frac{26{,}620}{1.295029}\\ &\approx -20{,}000+20{,}555.52\\ &=\$555.52. \end{aligned}

The positive NPV is consistent with a 10% IRR being greater than the 9% discount rate. The common mistake to avoid is treating the three-year total growth factor, 1.331, as if it were a one-year factor. The annual factor is its cube root, 1.1.

Worked Example: A Receipt Smaller Than the Outlay

A hypothetical equipment investment costs $18,000 now and produces one receipt of $14,580 exactly two years later. Calculate the annual effective IRR. Also find the receipt that would have been needed at year 2 for the project to earn 5% annually.

Method. First solve the IRR equation. Then use accumulation at the stated 5% rate to find the required year-2 receipt.

For the IRR, the ratio is:

\frac{C_n}{C_0}=\frac{14{,}580}{18{,}000}=0.81.

Since $0.81=(0.9)^2$, the two-year root is $0.9$, and:

i=0.81^{1/2}-1=0.9-1=-0.10.

The annual effective IRR is $-10\%$. Check the result by accumulating the outlay at that rate for two years:

18{,}000(1-0.10)^2 =18{,}000(0.81) =\$14{,}580.

For a 5% annual effective return, the receipt would need to equal the accumulated value of $18,000 after two years:

18{,}000(1.05)^2 =18{,}000(1.1025) =\$19{,}845.

The actual receipt is $14,580, which is $5,265 less than the amount required for a 5% return. A negative IRR is therefore appropriate. The common mistake is to reject the result because it is negative; the formula allows a negative rate as long as it is greater than $-100\%$, and the check confirms it.

Worked Example: Quarterly Cash Flows and an Annual Effective Rate

A hypothetical project requires $10,000 immediately and pays one receipt of $13,727.86 after 16 quarters. Calculate the IRR per quarter and the equivalent annual effective rate. Round the final annual rate to three decimal places.

Method. Apply the formula using quarters as the periods. Then convert the quarterly rate to an annual effective rate by accumulating for four quarters.

The receipt-to-outlay ratio is:

\frac{13{,}727.86}{10{,}000}=1.372786.

The ratio is rounded in the displayed calculation because the receipt is stated to the nearest cent. Its 16th root is approximately $1.020000$, so:

i_{\text{quarter}} =(1.372786)^{1/16}-1 \approx 0.020000.

The quarterly IRR is approximately 2.000%. The annual effective rate is not four times the quarterly rate. Instead, accumulate for four quarters:

\begin{aligned} i_{\text{annual}} &=(1+i_{\text{quarter}})^4-1\\ &=(1.02)^4-1\\ &=1.08243216-1\\ &=0.08243216. \end{aligned}

Rounded to three decimal places, the annual effective IRR is 8.243%. As a check, the unrounded quarterly factor over 16 quarters is $(1.02)^{16}\approx1.372785705$, giving a receipt of approximately $13,727.86 from a $10,000 outlay. The common mistake is to label the quarterly result as an annual rate without converting its period.

Common Mistakes and Exam Checks

The closed form is short, but correct use depends on reading the timeline and rate unit accurately. Before finalizing an answer, check the following:

Count the intervals, not just the dates. A receipt at the end of year 4 is four periods after an outlay at time 0, so the exponent is 4.
Take the root before subtracting 1. The total accumulation factor is $C_n/C_0$; its $n$th root is the one-period factor.
Keep the rate unit explicit. If periods are quarters, the formula returns a quarterly rate. Convert it before comparing it with an annual effective hurdle rate.
Do not assume every IRR is positive. If the later receipt is less than the initial outlay, the IRR is negative, but still greater than $-100\%$.
Check by substitution. At the calculated IRR, the discounted receipt must equal the initial outlay.

This direct solution is useful when a project has exactly one outlay followed by exactly one receipt. With additional cash flows, the resulting IRR equation generally has a different form and may not have a closed-form solution. The formula here should not be applied by combining several receipts into one unless their equivalent value has first been established at a specified rate.

Key takeaway: For one outlay $C_0$ now and one receipt $C_n$ after $n$ equal periods, the IRR per period is $\left(C_n/C_0\right)^{1/n}-1$. The exponent counts periods, and the resulting rate uses the same period unit as the cash-flow spacing.

Check Your Understanding

For each problem, identify the period unit before applying the formula. Answers are provided for checking.

A project costs $16,000 now and pays $19,360 at the end of year 2. Find its annual effective IRR. (Answer: 10.000%)
A project costs $30,000 and pays $24,300 exactly three years later. Find its annual effective IRR. (Answer: approximately $-6.783\%$)
What single receipt at the end of year 4 would give a $12,000 outlay an annual effective IRR of 6%? (Answer: $15,149.72)
A project costs $40,000 now and pays $48,400 at the end of year 2. Find its IRR and determine whether its NPV is positive, zero, or negative at a 9% annual effective discount rate. (Answer: IRR $=10.000\%$; NPV is positive)
A project costs $25,000 now and pays $29,291.48 after eight quarters. Find the approximate quarterly IRR and equivalent annual effective IRR. (Answer: quarterly IRR $\approx2.000\%$; annual effective IRR $\approx8.243\%$)

What You'll Learn