IRR as a Decision Rule for Independent Projects

From an IRR to a Project Decision

The internal rate of return is a property of a project’s cash-flow pattern. A decision requires one more input: the hurdle rate, or required rate of return for that project. Comparing the two gives a practical accept-or-reject rule—but only when the cash flows satisfy the conditions that make the comparison reliable.

This tutorial focuses on conventional projects considered independently. “Independent” means that undertaking one project does not change the cash flows or availability of another. The decision is whether each project meets its own required return, not which of several competing alternatives is preferable. The rule is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: For a project with an initial investment $C_0>0$ at time 0 and subsequent cash flows $C_1,\ldots,C_n$, its internal rate of return (IRR) is a rate $r$ satisfying $$ -C_0+\sum_{t=1}^{n}\frac{C_t}{(1+r)^t}=0. $$ The project’s net present value (NPV) at a rate $k$ is $$ \operatorname{NPV}(k)=-C_0+\sum_{t=1}^{n}\frac{C_t}{(1+k)^t}. $$ A conventional project has one initial outflow followed by nonnegative inflows: $C_t\geq 0$ for $t=1,\ldots,n$, with at least one $C_t>0$.

Assume cash flows occur at the ends of equally spaced periods, and express both the IRR and the hurdle rate per period. For example, annual cash flows require annual rates. The formulas also require rates greater than $-1$, so each discount factor is defined and positive. A project with later outflows, such as a major clean-up cost, may not be conventional; the simple comparison rule should not be used without checking the resulting NPV profile and roots.

Why the Rule Works for Conventional Cash Flows

For a conventional project, increasing the discount rate lowers the present value of every positive future inflow. This makes NPV strictly decreasing in the rate. The consequence is that a project’s unique IRR divides discount rates that produce positive NPV from those that produce negative NPV.

Start with the NPV function.
For $r>-1$, write the initial investment and each discounted future inflow in one expression.

Check how NPV changes.
Differentiate with respect to $r$. Each positive future cash flow contributes a negative term to the derivative.

Compare the hurdle rate with the root.
Since NPV decreases and equals zero at the IRR, rates below the IRR give positive NPV; rates above it give negative NPV.

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

The derivative is strictly negative because $r>-1$, every $C_t\geq 0$, and at least one future cash flow is positive. Also, as $r$ approaches $-1$ from above, the present value of at least one positive inflow grows without bound; as $r$ grows without bound, those present values approach zero and NPV approaches $-C_0$. Thus the NPV function crosses zero exactly once for $r>-1$. This is the unique IRR for these conventional cash flows.

Theorem: For a conventional project with a unique IRR $r$ and a hurdle rate $k>-1$, the project has positive NPV at the hurdle rate exactly when $r>k$. Therefore, accept if $r>k$, reject if $r<k$, and be financially indifferent under the model if $r=k$. The equality case means NPV is zero; other practical considerations may still affect a real decision.

Applying the Rule to One Project

The decision process has two steps: find or verify the project’s IRR, then compare it with the hurdle rate. Alternatively, calculate NPV directly at the hurdle rate. For conventional cash flows, both methods give the same accept-or-reject result.

Worked Example: Accepting a Project Above Its Hurdle Rate

A hypothetical project requires $1,000 now and returns $200 at the end of year 1 and $1,200 at the end of year 2. The required annual return is 12%. Determine whether to accept the project. Use annual end-of-year cash flows and a constant annual discount rate.

Method: solve for the IRR and compare it with the hurdle rate. The cash-flow pattern is conventional: one initial outflow, followed only by positive inflows.

Set the NPV equal to zero and let $x=1+r$:

-1000+\frac{200}{1+r}+\frac{1200}{(1+r)^2}=0

Multiply by $x^2$:

-1000x^2+200x+1200=0, \qquad 1000x^2-200x-1200=0.

Divide by 200 and factor:

5x^2-x-6=0, \qquad (5x-6)(x+1)=0.

The roots are $x=1.2$ and $x=-1$. Since $x=1+r>0$ for $r>-1$, the valid root is $x=1.2$, so the IRR is $r=0.20=20\%$. Because $20\%>12\%$, accept the project by the IRR rule.

Check using NPV. At 12%, the year-1 present value is $200/1.12=\$178.5714$, and the year-2 present value is $1200/1.2544=\$956.6327$. Thus NPV is $-1000+178.5714+956.6327=\$135.2041$, which is positive. Rounded to the nearest cent, NPV is $135.20, confirming acceptance. The common mistake avoided here is comparing the IRR with zero rather than with the stated 12% hurdle rate.

Rejecting a Project Below Its Hurdle Rate

An IRR can be positive and still be too low to meet the required return. A positive IRR is not, by itself, a reason to accept. The relevant comparison is always with the hurdle rate appropriate to the project.

Worked Example: A Positive IRR That Does Not Meet the Requirement

A hypothetical project costs $1,000 at time 0 and pays $300 at the end of year 1 and $787.50 at the end of year 2. The required annual return is 8%. Assume annual end-of-year cash flows and a constant annual rate. Decide whether to accept.

Method: verify the IRR, then compare it with the hurdle rate. Let $x=1+r$. The IRR equation becomes

$$ 1000x^2-300x-787.50=0. $$

At $x=1.05$, the left side is $1000(1.1025)-300(1.05)-787.50=1102.50-315-787.50=0$. Therefore, $r=1.05-1=5\%$ is an IRR. The cash flows are conventional, so the IRR is unique. Since $5\%<8\%$, reject.

Check using NPV. At the 8% hurdle rate, the discounted inflows are $300/1.08=\$277.7778$ and $787.50/1.1664=\$675.1543$. Therefore,

\operatorname{NPV}(8\%)=-1000+277.7778+675.1543=-\$47.0679.

NPV rounds to $-\$47.07$, confirming rejection. The common mistake avoided is treating a positive 5% return as sufficient despite the project’s 8% requirement.

Making Separate Decisions for Independent Projects

If projects are independent, each project can be tested against its own hurdle rate. When there are no capital constraints or other dependencies, accept every project that passes its test; rejecting one project does not require accepting another instead. Their NPVs add, so accepting projects with positive NPV increases total value under the stated model.

Worked Example: Screening Two Independent Projects

Two hypothetical projects can be undertaken independently, with no budget limit or other constraint. Each has an 8% annual hurdle rate and annual end-of-year cash flows. Project A costs $1,000 and pays $400 in year 1 and $862.50 in year 2. Project B costs $800 and pays $100 in year 1 and $858 in year 2. Decide separately for each project.

Method: check each project’s IRR against the common hurdle rate, then verify the NPV signs.

For Project A, set $x=1+r$. Its IRR equation is $1000x^2-400x-862.50=0$. At $x=1.15$, the left side is $1000(1.3225)-400(1.15)-862.50=1322.50-460-862.50=0$. Thus its IRR is 15%, which exceeds 8%; accept A.

For Project B, the equation is $800x^2-100x-858=0$. At $x=1.10$, the left side is $800(1.21)-100(1.10)-858=968-110-858=0$. Its IRR is 10%, which also exceeds 8%; accept B.

NPV checks. The year-1 discount factor at 8% is $1/1.08$, and the year-2 factor is $1/1.1664$. For A, the present values are $400/1.08=\$370.3704$ and $862.50/1.1664=\$739.4547$, giving NPV $=\$109.8251$, or $109.83 to the nearest cent. For B, they are $100/1.08=\$92.5926$ and $858/1.1664=\$735.5967$, giving NPV $=\$28.1893$, or $28.19. Both NPVs are positive, consistent with both IRR decisions.

Because the projects are independent and unconstrained, the decision is to accept both, not to select only the one with the higher IRR. The common mistake avoided is treating separate accept-or-reject tests as if they automatically rank alternatives.

Conditions, Limits, and Common Traps

The IRR comparison is useful only when the decision setup matches its assumptions. Before applying it, check the cash-flow signs, rate period, and project relationship.

Conventional cash flows: the standard result requires an initial outflow followed by nonnegative inflows. Later outflows can create more than one IRR or make the NPV profile behave differently. In that situation, compare NPV at the hurdle rate rather than assuming one IRR gives the decision.
Consistent periods: annual cash flows must be compared with annual rates. If cash flows are monthly, use monthly rates or convert rates consistently; do not compare a monthly IRR directly with an annual hurdle rate.
Appropriate hurdle rate: the required return should correspond to the project’s risk and the period basis used for its cash flows. A single hurdle rate need not be appropriate for every project.
Independent, unconstrained decisions: taking one project must not change another project’s cash flows, and there must be no binding resource limit. Under a budget constraint, it may not be possible to take every project with positive NPV; an individual IRR screen does not resolve that allocation problem.
Acceptance is not ranking: an IRR above the hurdle rate answers whether a project passes its own required-return test. It does not establish that the project is preferable to a mutually exclusive alternative. Comparing competing projects requires a separate analysis.

Key takeaway: For conventional cash flows, compare a project’s unique IRR with its correctly specified hurdle rate: accept above the hurdle, reject below it, and identify zero NPV at equality. Apply this decision separately to independent projects only when there are no constraints or dependencies, and do not treat the rule as a ranking method for competing choices.

Check Your Understanding

For each problem, assume cash flows occur at the ends of the stated years and rates are annual. Use the IRR decision rule where its conditions apply.

A project costs $1,000 now and pays $100 at the end of year 1 and $1,100 at the end of year 2. Its hurdle rate is 9%. Should it be accepted? (Answer: IRR is 10%; accept.)
A project costs $2,000 and pays $500 at the end of year 1 and $1,680 at the end of year 2. Its hurdle rate is 7%. Should it be accepted? (Answer: IRR is 5%; reject.)
Two independent projects have an 8% hurdle rate. Project A costs $1,000 and pays $300 and $918.40 at the ends of years 1 and 2. Project B costs $1,500 and pays $200 and $1,473.40 at the ends of years 1 and 2. With no budget limit, which should be accepted? (Answer: A has IRR 12% and is accepted; B has IRR 6% and is rejected.)
A conventional project has an IRR equal to its hurdle rate. What is its NPV at that rate, and what is the decision under the financial model? (Answer: NPV is zero; the decision is financial indifference.)
Two independent projects each have positive NPV at their respective hurdle rates, but a binding budget allows only one to be undertaken. Does the independent-project IRR rule alone determine which one to choose? (Answer: No; both pass their individual screens, but the binding constraint requires a separate constrained decision analysis.)

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