IRR And Payback Decision Workshop

From Calculations to a Decision

A capital-budgeting decision is not made by calculating a single measure in isolation. A project can look attractive under one rule and unattractive under another because the measures answer different questions. NPV estimates value added at a specified required return; IRR reports a break-even rate; payback measures how quickly an initial outlay is recovered. A sound recommendation identifies the decision being made and uses the rule appropriate to it.

This workshop treats annual cash flows as deterministic amounts received or paid at year-end, except when fractional payback is calculated. For fractional-year payback, assume cash flows accrue evenly within the recovery year. Let $C_t$ be the project cash flow at time $t$, with $C_0$ the initial cash flow, and let $k$ be the required return per year. The project NPV at $k$ is the present value of its cash flows, including the initial outlay.

Definitions: The internal rate of return (IRR) is a rate $r>-1$ satisfying $NPV(r)=0$. Simple payback is the time required for cumulative undiscounted cash inflows to recover the initial outlay. Discounted payback is the time required for cumulative cash inflows discounted at $k$ to recover that outlay. Payback rules require a stated cutoff to classify a project as acceptable or unacceptable.

Decision Rules and Their Proper Roles

For an independent project, the firm may accept or reject the project without having to choose it instead of another available project. At the required return $k$, the value-based rule is to accept when $NPV(k)>0$, subject to the project being feasible. For a conventional project, whose initial cash flow is negative and whose later cash flows are nonnegative with at least one positive inflow, the IRR rule gives the same accept-or-reject result when there is a unique IRR: accept if $r>k$, reject if $r<k$.

Payback is different. A simple or discounted payback cutoff is a screening policy, not a direct measure of value added. A project can recover its cost quickly but create little value overall, or recover slowly while producing substantial later inflows. Discounted payback accounts for the time value of money up to recovery, but it still ignores cash flows after the recovery date.

For mutually exclusive projects, only one project can be selected. If the projects have the same relevant life and risk basis, choose the feasible project with the greater NPV at the required return. IRR rankings can differ from NPV rankings, particularly when project scale differs. Payback rankings can differ as well. The recommendation should therefore state which measure governs the choice and why.

Why the IRR Rule Works for Conventional Cash Flows

The connection between IRR and NPV follows from how NPV changes with the discount rate. For a conventional project with cash flows $C_0<0$ and $C_t\geq 0$ for $t\geq1$, where at least one later cash flow is positive, write

NPV(q)=C_0+\sum_{t=1}^{n}\frac{C_t}{(1+q)^t}, \qquad q>-1.

Differentiating with respect to $q$ gives

\frac{d\,NPV(q)}{dq} =-\sum_{t=1}^{n}\frac{tC_t}{(1+q)^{t+1}}<0, \qquad q>-1.

For every $q>-1$, the denominators are positive. Each term in the sum is nonnegative, and at least one is strictly positive, so the derivative is strictly negative throughout that domain. Thus NPV decreases between any two rates greater than $-1$, not merely between nonnegative rates. If the project has an IRR $r>-1$, then $NPV(r)=0$. Consequently, for any required return $k>-1$, $k<r$ implies $NPV(k)>0$, and $k>r$ implies $NPV(k)<0$. This argument includes a negative IRR when the relevant rates lie in the allowed domain.

This monotonicity argument relies on conventional cash flows. When cash-flow signs change more than once, NPV need not decrease throughout the domain, and there may be multiple IRRs or no IRR. In those cases, evaluate NPV directly at the required return rather than applying the usual IRR comparison mechanically.

Independent Project Decisions

The following example uses all three measures for one independent project. It shows why NPV is the value-based acceptance test, while payback results depend on the chosen cutoff.

Worked Example: An Independent Two-Year Project

A hypothetical project costs $ \$1{,}000 $ now and pays $ \$600 $ at the end of each of the next two years. The required return is 10% effective annually. Assume no taxes, no additional interim cash flows, and even accrual within each year for fractional payback. Calculate NPV, IRR, simple payback, and discounted payback, and assess the project if the simple-payback cutoff is two years.

Method. Discount each year-end flow at the specified annual rate for NPV and discounted payback. Find IRR by setting NPV equal to zero. Use cumulative-flow recovery for simple payback.

NPV. The year-one present value is $600/1.10=\$545.4545$, and the year-two present value is $600/1.10^2=600/1.21=\$495.8678$. Therefore,

NPV(10\%)=-1{,}000+545.4545+495.8678 =\$41.3223 \approx \$41.32.

The positive NPV means the project is acceptable under the NPV rule.

IRR. Set NPV to zero and let $x=1+r$:

1{,}000=\frac{600}{x}+\frac{600}{x^2} \quad\Longrightarrow\quad 1{,}000x^2-600x-600=0.

The positive solution is $x=(600+\sqrt{2{,}760{,}000})/2{,}000=1.1306624$, so $r=0.1306624$, or 13.07% to two decimal places. Since 13.07% exceeds the 10% required return, the IRR rule agrees with NPV.

Simple payback. After year one, $ \$600 $ of the $ \$1{,}000 $ outlay has been recovered, leaving $ \$400 $. The year-two inflow is $ \$600 $, so the fraction of year two required is $400/600=0.6667$. Simple payback is $1+0.6667=\mathbf{1.67}$ years to two decimals. It meets the two-year cutoff.

Discounted payback. After year one, the discounted recovery is $ \$545.4545 $, leaving $1{,}000-545.4545=\$454.5455$. The discounted year-two inflow is $ \$495.8678 $, so the fraction of that year is $454.5455/495.8678=0.9167$. Discounted payback is $1+0.9167=\mathbf{1.92}$ years to two decimals, also within the two-year cutoff.

Check and common mistake. The positive NPV and IRR above the hurdle are consistent for these conventional cash flows. A common error is to use the undiscounted $ \$600 $ year-two inflow when finding discounted payback. The correction is to use its present value, $ \$495.8678 $; that is why discounted payback is longer than simple payback.

Mutually Exclusive Choices and Conflicting Rankings

When projects are mutually exclusive, a higher IRR does not automatically identify the better investment. A high percentage return on a small project can create less dollar value than a lower return on a larger project. The NPV comparison measures each project’s contribution at the same required return. Incremental analysis makes the scale difference explicit by asking whether the additional investment needed to move from one project to the other earns an adequate return.

Worked Example: Higher IRR Does Not Mean Higher NPV

A hypothetical organization can select only one of two one-year projects. Project A requires $ \$1{,}000 $ now and pays $ \$1{,}400 $ at year-end. Project B requires $ \$5{,}000 $ now and pays $ \$6{,}200 $ at year-end. The annual required return is 10%. Assume no other cash flows, taxes, or risk differences. Compare their IRRs, NPVs, and simple paybacks, then recommend one project.

Method. Calculate each project’s return and NPV on the same annual basis, then compare NPV because the projects are mutually exclusive. Check the incremental investment as a diagnostic.

Project A. Its IRR solves $1{,}000=1{,}400/(1+r)$, so $r=1{,}400/1{,}000-1=40\%$. Its NPV is

NPV_A(10\%)=-1{,}000+\frac{1{,}400}{1.10} =-1{,}000+1{,}272.7273 =\$272.73.

Simple payback is $1{,}000/1{,}400=0.7143$ years, or 0.71 years to two decimals.

Project B. Its IRR solves $5{,}000=6{,}200/(1+r)$, giving $r=6{,}200/5{,}000-1=24\%$. Its NPV is

NPV_B(10\%)=-5{,}000+\frac{6{,}200}{1.10} =-5{,}000+5{,}636.3636 =\$636.36.

Simple payback is $5{,}000/6{,}200=0.8065$ years, or 0.81 years to two decimals.

Decision and incremental check. Project A has the higher IRR and shorter payback, but Project B has the greater NPV, by $636.36-272.73=\$363.63$ using the displayed rounded values (the unrounded difference is $ \$363.6364 $). Moving from A to B requires an additional $ \$4{,}000 $ and produces an additional $ \$4{,}800 $ at year-end. The incremental IRR is $4{,}800/4{,}000-1=20\%$, above the 10% required return. Equivalently, incremental NPV is $-4{,}000+4{,}800/1.10=\$363.64>0$. Select B.

Check and common mistake. The NPV difference agrees with the positive incremental NPV, apart from rounding. Selecting A solely because 40% exceeds 24% would confuse percentage return with total value created. For this mutually exclusive choice, the higher NPV at the common required return governs.

Payback Can Disagree with Value

Payback places special weight on early recovery and does not account for all project cash flows in the way NPV does. Discounted payback corrects the undiscounted timing of inflows before recovery, but neither payback measure includes post-recovery cash flows in its reported time. The next example has equal initial outlays but different timing, allowing the ranking conflict to be seen directly.

Worked Example: Faster Recovery Versus Greater NPV

Two mutually exclusive hypothetical projects each cost $ \$1{,}000 $. Project X pays $ \$900 $ at the end of year one and $ \$300 $ at the end of year two. Project Y pays $ \$300 $ at the end of year one and $ \$1{,}100 $ at the end of year two. The required return is 10% effective annually. Assume conventional cash flows, no taxes, and even accrual within each recovery year. Find each project’s simple and discounted payback, NPV, and IRR. Which project should be selected?

Method. Calculate payback from cumulative inflows, discount each inflow for discounted payback and NPV, and solve each IRR equation. Use NPV to choose between the mutually exclusive projects.

Project X: simple payback. After the first year, $ \$900 $ has been recovered, leaving $ \$100 $. The second-year inflow is $ \$300 $, so payback is $1+100/300=\mathbf{1.33}$ years to two decimals.

Project X: discounted payback and NPV. Its first- and second-year present values are $900/1.10=\$818.1818$ and $300/1.21=\$247.9339$. The year-one shortfall is $1{,}000-818.1818=\$181.8182$, which is $181.8182/247.9339=0.7333$ of the second year. Discounted payback is $1.7333$, or 1.73 years. Its NPV is $818.1818+247.9339-1{,}000=\mathbf{\$66.12}$.

Project X: IRR. Setting NPV equal to zero and writing $x=1+r$ gives $1{,}000x^2-900x-300=0$. The positive solution is

x=\frac{900+\sqrt{2{,}010{,}000}}{2{,}000} =1.1588723, \qquad r=15.8872\%\approx\mathbf{15.89\%}.

Project Y: simple payback. After year one, $ \$300 $ has been recovered, leaving $ \$700 $. The second-year inflow is $ \$1{,}100 $, so payback is $1+700/1{,}100=1.6364$, or 1.64 years.

Project Y: discounted payback and NPV. The present values are $300/1.10=\$272.7273$ and $1{,}100/1.21=\$909.0909$. The remaining amount after year one is $1{,}000-272.7273=\$727.2727$. This is $727.2727/909.0909=0.8000$ of the second year, so discounted payback is 1.80 years. Its NPV is $272.7273+909.0909-1{,}000=\mathbf{\$181.82}$.

Project Y: IRR. With $x=1+r$, its IRR equation becomes $1{,}000x^2-300x-1{,}100=0$. The positive solution is

x=\frac{300+\sqrt{4{,}490{,}000}}{2{,}000} =1.2094810, \qquad r=20.9481\%\approx\mathbf{20.95\%}.

Decision and check. Project X recovers its cost faster under both payback measures, but Project Y has the greater NPV by $181.82-66.12=\$115.70$ using rounded amounts (the unrounded difference is $ \$115.7025 $). Y also has the higher IRR, but NPV is the governing rule for this mutually exclusive choice at the stated common required return. A 1.5-year simple-payback cutoff would accept X and reject Y; that policy outcome would not reverse the NPV recommendation.

Common mistake. It is incorrect to infer that the shorter payback identifies the value-maximizing project. Payback discards the value of later cash flows once recovery has occurred and does not rank projects by total value added. Use it only as a stated screening constraint or supplementary liquidity measure.

Practical Decision Sequence

A disciplined sequence keeps the calculations tied to the decision rather than allowing a convenient metric to determine the recommendation. Confirm first whether the projects are independent or mutually exclusive, and then check that cash-flow timing and the required-return basis are consistent.

Classify the choice.
For independent projects, assess each project separately. For mutually exclusive projects, identify that only one alternative can be selected.

Calculate NPV at the required return.
Use the same appropriate required return for the projects being compared. Positive NPV supports acceptance for an independent project; higher NPV guides a mutually exclusive choice under comparable assumptions.

Use IRR with its conditions in view.
For a conventional project with a unique IRR, compare that rate with the required return. For nonconventional flows or conflicting mutually exclusive rankings, do not treat the standalone IRR as decisive.

Apply payback as a separate policy test.
Calculate simple or discounted payback using the specified convention, compare with the actual cutoff, and state that the result is a screening measure rather than a complete value ranking.

A frequent exam trap is to calculate the requested measures correctly and then use the wrong one to make the final choice. Another is to assume the payback cutoff is an economic hurdle equivalent to the required return. They are distinct decision criteria: the required return enters NPV and discounted payback, while a payback cutoff is a separate policy limit.

Key takeaway: For independent conventional projects, NPV and IRR usually agree on acceptance when there is a unique IRR and the same required return is used. For mutually exclusive projects, use NPV at the common required return to rank value, checking incremental cash flows when useful. Treat payback as a supplementary measure or stated screening rule, not as a substitute for NPV.

Check Your Understanding

For each original problem, use annual year-end cash flows and the stated required return. For fractional payback, assume even accrual within the recovery year.

A hypothetical independent project costs $ \$800 $ and returns $ \$500 $ at each of the next two year-ends. The required return is 10%. Find NPV, IRR, and simple payback. (Answer: NPV $=\$67.77$; IRR $=16.26\%$; simple payback $=1.60$ years.)
A hypothetical independent project costs $ \$1{,}200 $ and returns $ \$500 $ at each of the next three year-ends. At an 8% required return, find NPV and discounted payback. (Answer: NPV $=\$88.55$; discounted payback $\approx2.78$ years.)
Two mutually exclusive one-year projects are available. M costs $ \$2{,}000 $ and pays $ \$2{,}600 $; N costs $ \$4{,}000 $ and pays $ \$5{,}100 $. At a 10% required return, compare their IRRs and NPVs, and select one. (Answer: M IRR $=30\%$, NPV $=\$363.64$; N IRR $=27.5\%$, NPV $=\$636.36$; select N by NPV.)
A project has cash flows $-\$100$, $+\$230$, and $-\$132$ at times zero, one, and two. Find all IRRs and NPV at a 15% required return. Explain why a simple IRR acceptance comparison is unsafe. (Answer: IRRs $=10\%$ and $20\%$; NPV at 15% is approximately $+\$0.19$; the two sign changes permit multiple IRRs, so evaluate NPV directly at the required return.)

What You'll Learn