Engineering Economics and Investment Appraisal

Net Present Value

Net present value (NPV) is the present value of all project inflows less the present value of all project outflows at the required discount rate.

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Caution

Key Formula — Net present value

$$\boxed{NPV=\sum_{t=0}^{n}\frac{CF_t}{(1+r)^t} =-C_0+\sum_{t=1}^{n}\frac{CF_t}{(1+r)^t}}.$$

Accept an independent project if $NPV>0$, reject it if $NPV<0$, and remain financially indifferent at $NPV=0$, subject to risk and non-financial criteria.

Internal Rates of Return

The internal rate of return (IRR) is a discount rate that makes project NPV zero:

$$0=\sum_{t=0}^{n}\frac{CF_t}{(1+IRR)^t}.$$

For a conventional independent project, accept when IRR exceeds the required rate of return. Non-conventional cash flows can produce multiple or no real IRRs; NPV remains the primary value criterion.

Measure	Perspective	Valuation basis	Included effects
FIRR	Investor or project entity	Financial cash flows at market/financial prices	Revenues, expenditures, taxes, financing effects as defined
EIRR	Economy or society	Economic/shadow prices after relevant transfer and distortion adjustments	Resource costs, social benefits, and material externalities

Financial and economic rates of return

Public infrastructure may have a weak FIRR but a satisfactory EIRR because connectivity, access, productivity, and other social benefits extend beyond project revenue.

Other Appraisal Measures

Caution

Key Formula — Benefit–cost ratio

$$\boxed{BCR=\frac{PV\text{ of benefits}}{PV\text{ of costs}}}.$$

For an independent project evaluated consistently, $BCR>1$ supports acceptance.

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Caution

Key Formula — Simple payback for equal inflows

$$\boxed{\text{Payback period} =\frac{\text{initial investment}}{\text{annual net cash inflow}}}.$$

For unequal inflows, accumulate cash flows until the investment is recovered. Simple payback ignores the time value of money and post-payback cash flows.

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Note

Example — NPV of an investment

A project costs Rs. 500,000 and returns Rs. 150,000, Rs. 200,000, Rs. 250,000, and Rs. 200,000 in years 1–4. At 12%:

Year	Cash flow (Rs.)	PV factor	Present value (Rs.)
0	$-500{,}000$	1.000	$-500{,}000$
1	150,000	0.893	133,950
2	200,000	0.797	159,400
3	250,000	0.712	178,000
4	200,000	0.636	127,200
		NPV	98,550

Note

Note

Because $NPV=\text{Rs.~}98{,}550>0$, the project adds value under the stated cash flows and 12% opportunity cost and is financially acceptable.

Method	Main advantage	Main limitation
NPV	Measures absolute value added and fully discounts cash flows	Requires a defensible discount rate and cash-flow forecast
IRR	Communicates a percentage return	Multiple IRRs, ranking conflict, and reinvestment assumptions can mislead
Payback	Simple indicator of recovery and liquidity exposure	Ignores value after payback; simple form ignores discounting
BCR	Useful for constrained or public-resource comparisons	Scale and classification of costs/benefits can cause ranking conflict with NPV

Investment-appraisal methods compared

For mutually exclusive alternatives, incremental NPV at a common discount rate normally gives the soundest wealth-maximizing decision.

Risk and Uncertainty Analysis

Method	What it reveals
Sensitivity analysis	Effect on NPV of changing one input at a time; identifies influential assumptions but ignores joint movement
Scenario analysis	Effect of internally consistent best, base, and worst or other combined states
Monte Carlo simulation	Distribution of outcomes generated by repeated sampling from specified input distributions and dependencies
Decision tree	Sequential choices, chance events, probabilities, and contingent values
Risk-adjusted discount rate	Applies a higher required return to riskier cash flows, though it can hide which risks drive the adjustment

Methods for examining investment risk

Note

Key Point — Separate forecast from decision rule

Cash-flow estimation, risk modeling, and the discount rate are distinct. Avoid double-counting risk by making cash flows pessimistic and also applying an unjustifiably high rate for the same uncertainty.