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Five-year series −100,000 then +30,000 at 8 %: NPV, IRR, MIRR, paybacks, EAA, SIR and AIRR with the NIST Handbook 135 factors

Tool	Investment Appraisal (NPV, IRR, Payback)
Source	Kneifel, J., and Webb, D. (2020). Life Cycle Costing Manual for the Federal Energy Management Program. NIST Handbook 135, National Institute of Standards and Technology, Gaithersburg, MD (eqs. 3-6 to 3-8, 6-3, 6-6, 6-11, 12-4, 12-10, 12-12; Tables 3-1, 3-2, 6-3; Examples 5-1 and 7-1); Office of Management and Budget (1992). Guidelines and Discount Rates for Benefit-Cost Analysis of Federal Programs. Circular A-94, Washington, DC (Appendix B).
Level	Undergraduate

Problem

A project costs 100,000 at the base date and returns 30,000 at the end of each of five years. At a discount rate of 8 percent find the net present value, the internal rate of return and whether it is unique, the modified IRR with reinvestment at the discount rate, the simple and discounted payback periods interpolated within the year, the equivalent annual annuity, the profitability index, and the savings-to-investment ratio and adjusted internal rate of return of the FEMP method; show the NPV profile.

Inputs to Enter

Cash flows: period 0 outflow 100,000 (capital period); periods 1–5 inflow 30,000 each; discount rate 8 % as entered; end-of-period timing; reinvestment and finance rates blank (= 8 %)

Open the Investment Appraisal (NPV, IRR, Payback) tool (loaded with this example)

Answers

NPV = 30,000 (P/A, 8 %, 5) − 100,000 = 30,000 × 3.99271 − 100,000 = 19,781.30
IRR = 15.238 % (one sign change: Descartes' bound 1; cumulative series starts negative and changes sign once: Norstrøm's criterion, unique); NPV(IRR) = 0
MIRR = (30,000 (F/A, 8 %, 5) / 100,000)^(1/5) − 1 = (175,998 / 100,000)^0.2 − 1 = 11.970 %
Simple payback 3 + 10,000/30,000 = 3.333 years; discounted payback 4 + 636.19/20,417.50 = 4.031 years
EAA = NPV (A/P, 8 %, 5) = 19,781.30 × 0.25046 = 4,954.35 per year; PI = SIR = 1.1978; AIRR = 1.08 × 1.1978^(1/5) − 1 = 11.970 % (equals the MIRR when savings are reinvested at the discount rate, Handbook 135 Section 6.3.2)

Notes

The tool's default inputs are this series (illustrative, stated on the page); press Submit. The library is checked against Handbook 135 to the printed digit: SPV 0.863 / UPV 4.580 / UPV* 4.8562 (3 %, 5 years), Example 5-1 (NS 43,803, SIR 2.61, AIRR 8.06 %, SPB year 6 / DPB year 7 of Table 6-3), Example 7-1 (SIR 1.80, AIRR 6.07 %), the Chapter 17 formulas, and OMB A-94 Appendix B (NPV 36.01 at 7 %, 37.25 mid-year); IRRs on 20 seeded series and two multi-root series agree with numpy.roots (test_investment_appraisal.php, 957 checks).