Two investments can change rank when the discount rate changes

An investment comparison can change its preferred option as the discount rate changes. A project returning more cash later may lead at one rate and fall behind an alternative that returns cash sooner at another. The point where their net present values are equal is a ranking boundary. It is not automatically a rate at which either project becomes worthwhile.

Start by explaining why the alternatives compete. They might occupy the same site, serve the same contract or represent incompatible designs. OpenStax describes choosing between mutually exclusive projects by their contribution to value, while Damodaran's discussion examines interactions and constraints that can change the choice. The comparison below isolates cash timing for two hypothetical alternatives.

Use the same valuation date, currency, service horizon and treatment of tax. The example also assumes comparable risk so one appraisal rate can be used for both options. If their risk differs materially, a common-rate sensitivity table may be an incomplete representation of the actual decision.

Suppose Option A requires $100,000 today and returns $80,000 after one year and $40,000 after two years. Option B requires the same $100,000 today, then returns $20,000 and $110,000 on those dates. Both schedules end after year two. These are illustrative cash flows, excluding tax and any unlisted costs or benefits.

At a 10% annual appraisal rate, A has an NPV of approximately $5,785: negative $100,000 plus $80,000 divided by 1.10 plus $40,000 divided by 1.10 squared. B has an NPV of approximately $9,091. B leads by about $3,306 because its extra later receipt more than compensates for its smaller first-year receipt at this rate.

At 20%, A's NPV is approximately negative $5,556 and B's is approximately negative $6,944. A now ranks above B, but neither has a positive NPV under these assumptions. Reporting only that A wins would omit the separate question of whether management should undertake either investment.

Subtract A's cash flow from B's at every date. The initial difference is zero. At the first year-end, choosing B gives up $60,000; at the second year-end, it gains $70,000. This incremental schedule explains the ranking more clearly than comparing two unexplained return percentages.

The NPVs are equal when negative $60,000 divided by (1 + r), plus $70,000 divided by (1 + r) squared, equals zero. Multiplying through by (1 + r) squared gives $60,000 multiplied by (1 + r) equals $70,000. The crossover is therefore approximately 16.67%. Use the unrounded value of one-sixth when reproducing the table.

At that exact rate, both projects have an NPV of about negative $2,041. Equality describes their relative position; it does not mean their individual NPVs are zero. Keep the crossover and each project's own approval threshold separately labeled so a board paper does not turn one calculation into the other.

For a longer schedule, calculate the incremental NPV over the relevant rate range and identify where it changes sign. More complex cash patterns can create several crossings or no useful crossing. A formula returning one root is not evidence that the whole ranking relationship has been examined.

A crossover is useful for asking how robust a preference is to uncertainty in the appraisal basis. If the supported rate range lies well below the boundary, the ranking may be stable within that range. If the supported range straddles it, management needs to understand the timing tradeoff rather than receiving one confident recommendation based on an arbitrary point estimate.

Do not select a rate because it makes a preferred option win. Document the basis for the rate independently, including the currency and risk assumptions. A higher rate can reduce the importance of later receipts, but that mathematical effect does not justify changing the company's appraisal policy for a particular proposal.

Nor is the rate a substitute for correcting the receipts. If a later payment is uncertain because a customer may not renew, examine that commercial uncertainty directly through appropriate cash-flow scenarios. Simply increasing a common discount rate can leave the underlying contract assumption invisible to the person approving the project.

Ask operations and commercial owners to support the payment dates. A promised first-year receipt may depend on installation, acceptance, customer financing or collection terms. Compare contractual cash timing with the model's dates, and distinguish invoicing from collection. Moving revenue into year one does not create an early cash receipt if the customer still pays later.

Review whether accelerating receipts has a cost. A customer deposit may require a price concession, a performance obligation or a guarantee. Include relevant cash consequences in the same alternative rather than presenting the early receipt as a free improvement. Preserve the underlying agreement so a later reviewer can understand why the timing changed.

Check the ending date as carefully as the first receipt. An omitted closure payment, maintenance commitment or working-capital shortfall can change both the NPV and the crossover. Keep supported residual proceeds and final obligations on the schedule even when they complicate a neat comparison.

A useful recommendation states the chosen appraisal basis, each option's NPV, the incremental advantage and the conditions under which that advantage changes. In the example, B leads by about $3,306 at 10%, but the schedules cross near 16.67%. The recommendation should also explain why 10%, or another rate, is appropriate for the actual analysis.

Assess affordability and alternatives separately. If neither project has acceptable economics at the supported rate, consider whether doing neither, changing the design or renegotiating terms is feasible. Where a mandatory service must continue, compare the full set of realistic ways to provide it instead of treating the two initial proposals as the only possibilities.

Retain the incremental schedule with the approval record and update it when dates or amounts change. A shift in ranking can then be traced to a changed receipt, an additional obligation or a revised appraisal basis. Management receives an explanation of the decision rather than a percentage that changes without a visible cause.