Investment Appraisal (NPV, IRR, Payback)

Investment Appraisal (NPV, IRR, Payback)

Investment Appraisal: Discounting, Rates of Return, Payback and the Handbook 135 Measures

Cash flows and the discount rate

A project is a series of net cash flows CF_t (inflows less outflows) at the ends of periods t = 0, 1, … N, period 0 being the base date. Discounting at the rate d values a flow at t as CF_t/(1 + d)^t; the net present value is the sum. When the flows of a year come in as a steady stream rather than at its end, OMB Circular A-94 (Appendix B) advances them half a period, which multiplies every end-of-year present value by √(1 + d); the tool offers this as the mid-period timing. The rate must match the terms of the flows: constant-currency (real) flows take a real rate, current-currency (nominal) flows a nominal one, and the two are related by Fisher's equation (1 + d_nom) = (1 + d_real)(1 + f) with f the inflation rate (A-94 Section 6; NIST Handbook 135 Section 3.1). Federal benefit-cost analyses report NPV at a 7 percent real rate (A-94 Section 8.b), federal energy and water projects use the DOE real rate with its 3 percent floor (10 CFR 436; 3.0 percent in 2024), and a firm uses its cost of capital.

Internal rate of return

The IRR is the rate that makes NPV zero. In x = 1/(1 + r) the NPV is a polynomial Σ CF_t x^t, so Descartes' rule of signs bounds the number of positive-rate roots by the number of sign changes in the series: a conventional project (one outflow followed by inflows) has one; a project with a later capital outflow or a closing cost can have two or more, and then the IRR is not a decision criterion. Norstrøm's criterion is sufficient for uniqueness: if the cumulative cash flow starts negative and changes sign exactly once, there is one positive root. The tool scans the NPV from −99 to 1,000 percent, bisects every sign change and lists every root. A-94 itself says the IRR "does not generally provide an acceptable decision criterion" but is useful information; NPV at the required rate is the criterion.

The modified IRR removes the implicit assumption that interim receipts earn the IRR itself: the negative flows are discounted to the base date at a finance rate, the positive flows compounded to the last period at a reinvestment rate, and MIRR = (FV_pos/PV_neg)^1/N − 1. It is unique by construction.

Payback

The simple payback is the first period at which the cumulative net cash flow reaches zero, the discounted payback the same on the cumulative present values (Handbook 135 eq. 6-11: y is the smallest number of years with Σ_{t ≤ y} (S_t − ΔI_t)/(1 + d)^t ≥ ΔI₀). The handbook reports whole years; the tool interpolates linearly inside the crossing period and warns when a later outflow turns the cumulative sum negative again. Payback ignores everything after the crossing and is a screening measure, not a ranking one; a discounted payback shorter than the service period is the same statement as a positive NPV.

Ratios: profitability index, benefit-cost, SIR and AIRR

The capital periods named on the form split the outflows into investment (initial cost, capital replacements; a residual value is entered as a negative outflow in its capital period and reduces the investment) and operating costs; every inflow is a benefit or saving. The conventional benefit-cost ratio divides the present value of every inflow by that of every outflow; the modified ratio nets the operating costs from the benefits and divides by the investment alone. Handbook 135 writes the modified form as the savings-to-investment ratio, SIR = PV(operational savings)/PV(additional investment) (eq. 6-3 and 6-4, incremental to a base case), and the profitability index 1 + NPV/PV_I is the same quantity. The adjusted internal rate of return reinvests the savings at the discount rate r to the end of the study period N and is AIRR = (1 + r)·SIR^1/N − 1 (eq. 6-6); it exceeds the discount rate exactly when SIR exceeds 1, and 10 CFR 436 requires SIR or AIRR for ranking independent projects competing for a limited budget, funded in descending order (Handbook 135 Section 7.5). None of the ratios may be used to choose among mutually exclusive alternatives, where the lowest life-cycle cost or highest NPV decides.

Equivalent annual annuity and the factors

Spreading the NPV over the life with the capital recovery factor, EAA = NPV·(A/P, d, n), gives the uniform amount per period the project is worth; alternatives of unequal lives are compared by EAA on the assumption that each can be repeated at the same cost and return. The factor table lists the eight discrete-compounding factors P/F, F/P, A/P, P/A, A/F, F/A, P/G, A/G at the entered rate for every life up to the one entered, with their closed forms; Handbook 135 calls P/F the single present value factor SPV and P/A the uniform present value factor UPV, and adds the modified factors for an amount entered at base-date prices that escalates at e per year: SPV* = ((1 + e)/(1 + d))ⁿ and UPV* = (1 + e)/(d − e)·[1 − ((1 + e)/(1 + d))ⁿ] (eqs. 12-6 and 12-12). Its printed tables (3 percent: SPV 0.863 at 5 years, UPV 14.877 at 20 years, UPV* 4.8562 at 5 years and 2 percent escalation) are reproduced by the tool's test to the printed digit.

After-tax cash flows

With the after-tax option the operating net flow of each period (inflows less operating outflows) less that period's depreciation is taxed at the entered rate (a negative taxable amount is credited), and the capital outflows pass untaxed. Depreciation is straight line on basis less salvage, MACRS by the general-depreciation-system percentages of IRS Publication 946 Table A-1 with the half-year convention (read from the Depreciation tool's table), or an entered list. The tax on a disposal gain, bonus depreciation and the Section 179 election are not computed here; the Depreciation (MACRS) tool has them.

Two alternatives and the crossover rate

For mutually exclusive alternatives the incremental cash flow B − A is itself a project: its IRR is the crossover (Fisher) rate at which the two NPV profiles meet. Below the crossover the alternative with the larger later flows has the higher NPV, above it the other; the increment is worth taking when its IRR exceeds the discount rate, which is the same decision as comparing the NPVs at that rate. Ranking by the alternatives' own IRRs can contradict NPV (a smaller project with a higher IRR), which is why the incremental test exists. Lease against buy is the same comparison with both columns entered as costs; A-94 Section 13 compares the discounted life-cycle costs of leasing and of buying, with the asset's residual value deducted from the purchase price.

Replacement analysis

The economic service life of an asset is the holding period that minimizes its equivalent uniform annual cost, EUAC_k = (P − MV_k)(A/P, i, k) + i·MV_k + [Σ_{j ≤ k} OC_j(P/F, i, j)](A/P, i, k): capital recovery falls with k while the annual equivalent of rising operating costs climbs. The defender's first cost is its market value now (the opportunity cost of keeping it; its original price is sunk) and the challenger's is its purchase price. The defender is kept while its minimum EUAC, or the marginal cost of one more year (MV_k−1 − MV_k) + i·MV_k−1 + OC_k, stays below the challenger's minimum EUAC, and the comparison is repeated each year.

Not included

Continuous compounding, working-capital and inflation of individual cost categories at different rates (enter an escalated series or use UPV* per category), probabilistic cash flows (the Cost Risk Analysis tool), and real options: the value of waiting, abandoning or expanding is only named, and the Decision Analysis tool holds the decision tree.