NET PRESENT VALUE IS THE SUM OF ALL CASH FLOWS, EACH DISCOUNTED TO ITS PRESENT WORTH. NPV = Σ Cₜ / (1 + i)ᵗ for t = 0 to n where Cₜ is the NET cash flow in period t (negative for outflows) and i is the discount rate. WRITTEN OUT, with the investment separated: NPV = −C₀ + C₁/(1+i) + C₂/(1+i)² + … + Cₙ/(1+i)ⁿ THE DECISION RULE: NPV > 0 → ACCEPT. The project returns more than the discount rate demands; it CREATES value. NPV = 0 → INDIFFERENT. It earns exactly the required return — acceptable but marginal. NPV < 0 → REJECT. It earns less than the cost of capital. FOR MUTUALLY EXCLUSIVE PROJECTS: CHOOSE THE HIGHEST NPV. WHAT THE NUMBER MEANS, and this is the interpretation students most often cannot state: NPV IS THE AMOUNT BY WHICH THE INVESTOR'S WEALTH INCREASES, MEASURED IN TODAY'S RUPEES, AFTER PAYING FOR THE CAPITAL AT THE REQUIRED RATE. An NPV of Rs 13,724 does NOT mean "the project earns 13,724". It means: after recovering the full Rs 100,000 investment AND paying a 10% return on the capital employed for as long as it was employed, THERE IS Rs 13,724 LEFT OVER IN PRESENT-VALUE TERMS. A zero NPV project is not a failure — it earns exactly the 10% demanded. THE DISCOUNT RATE IS THE OPPORTUNITY COST OF CAPITAL — the return available on the best alternative of similar risk. In practice a firm uses its WEIGHTED AVERAGE COST OF CAPITAL (WACC), and a public body uses a SOCIAL DISCOUNT RATE set by policy.

PROBLEM 1 — UNIFORM CASH FLOWS, the running example. Investment 100,000; net inflow 30,000/yr for 5 years; i = 10%. Since the flows are uniform, use the P/A factor rather than discounting each year separately: NPV = −100,000 + 30,000 × (P/A, 10%, 5) = −100,000 + 30,000 × 3.7908 = −100,000 + 113,724 = + Rs 13,724 NPV > 0 → ACCEPT. PROBLEM 2 — UNEVEN CASH FLOWS WITH SALVAGE. A machine costs 250,000, returns 80,000, 90,000, 100,000 and 70,000 over four years, then sells for 30,000. i = 12%. YEAR FLOW P/F(12%) PRESENT VALUE ─────────────────────────────────────────── 0 −250,000 1.0000 −250,000 1 80,000 0.8929 71,429 2 90,000 0.7972 71,747 3 100,000 0.7118 71,178 4 100,000 0.6355 63,551 (70,000 + 30,000 salvage) ─────────────────────────────────────────── NPV = + Rs 27,905 ACCEPT. NOTE HOW THE SALVAGE VALUE IS SIMPLY ADDED TO THE FINAL YEAR'S FLOW BEFORE DISCOUNTING — a step frequently forgotten. PROBLEM 3 — COMPARING MUTUALLY EXCLUSIVE PROJECTS at i = 10%: PROJECT A: cost 100,000, returns 30,000/yr for 5 years NPV = −100,000 + 30,000(3.7908) = + Rs 13,724 PROJECT B: cost 150,000, returns 42,000/yr for 5 years NPV = −150,000 + 42,000(3.7908) = + Rs 9,214 BOTH ARE ACCEPTABLE, BUT A HAS THE HIGHER NPV, SO CHOOSE A. NOTE THAT B IS THE LARGER PROJECT AND RETURNS MORE IN ABSOLUTE RUPEES — yet it creates less value, because the extra 50,000 of capital does not earn its keep. THIS IS EXACTLY THE KIND OF COMPARISON PAYBACK AND SIMPLE PROFIT FIGURES GET WRONG. THE SENSITIVITY OF NPV TO THE DISCOUNT RATE, which is the most important practical caveat. For Problem 1's project: i = 5% → NPV = + 29,884 i = 10% → NPV = + 13,724 i = 15% → NPV = + 565 i = 20% → NPV = − 10,282 THE PROJECT IS EXCELLENT AT 5%, MARGINAL AT 15% AND REJECTED AT 20% — WITHOUT A SINGLE CASH FLOW CHANGING. THE DISCOUNT RATE IS THEREFORE THE MOST CONSEQUENTIAL ASSUMPTION IN THE WHOLE APPRAISAL, and any serious project report must state it explicitly and test the result against a range of values. THE RATE AT WHICH NPV REACHES ZERO — here just above 15% — IS THE IRR, which is the next topic.

THE ADVANTAGES, which make NPV the theoretically preferred rule: 1. IT USES ALL CASH FLOWS over the entire life — nothing is ignored, unlike payback. 2. IT ACCOUNTS FOR THE TIME VALUE OF MONEY correctly. 3. IT GIVES AN ABSOLUTE MEASURE OF VALUE CREATED, in rupees, which is directly meaningful. 4. IT IS ADDITIVE: NPV(A + B) = NPV(A) + NPV(B). This VALUE ADDITIVITY property is a genuine mathematical advantage that IRR does not possess, and it means a portfolio of projects can be evaluated by summing. 5. IT GIVES AN UNAMBIGUOUS RANKING for mutually exclusive projects. 6. IT ASSUMES REINVESTMENT AT THE DISCOUNT RATE, which is a realistic assumption — unlike IRR's. THE DISADVANTAGES: 1. IT REQUIRES THE DISCOUNT RATE TO BE KNOWN, and as the sensitivity table shows, the answer depends heavily on it. Estimating the cost of capital is itself difficult and contestable. 2. IT IS AN ABSOLUTE FIGURE, SO IT IS BIASED TOWARD LARGE PROJECTS. An NPV of 1 million on a 100 million investment looks better than 500,000 on a 1 million investment, though the second is far more efficient per rupee. WHEN CAPITAL IS RATIONED, RANK BY THE PROFITABILITY INDEX INSTEAD. 3. IT IS HARDER TO EXPLAIN to non-specialists than "it pays back in four years" or "it earns 15%". 4. It assumes the cash flows and the project life are known — forecasts that are often the weakest part of the analysis. 5. Comparing projects of UNEQUAL LIVES requires an adjustment (see below). THE RELATED MEASURES, which every project report reports alongside NPV: PROFITABILITY INDEX (PI), or BENEFIT-COST RATIO: PI = PV of inflows / PV of outflows Accept if PI > 1, which is exactly equivalent to NPV > 0. For Problem 1: PI = 113,724 / 100,000 = 1.137. USE IT WHEN CAPITAL IS RATIONED, because it measures value created PER RUPEE INVESTED and therefore ranks correctly when the budget, not the project list, is the constraint. EQUIVALENT UNIFORM ANNUAL WORTH (EUAW): EUAW = NPV × (A/P, i, n) Converts the NPV into an equivalent annual figure. For Problem 1: 13,724 × 0.26380 = Rs 3,620 per year. THIS IS THE CORRECT METHOD FOR COMPARING PROJECTS OF UNEQUAL LIVES, because an annual figure is life-neutral whereas a total NPV is not — a 10-year project naturally accumulates more NPV than a 5-year one of equal quality. NPV PROFILE — a graph of NPV against discount rate. It falls as i rises, and CROSSES ZERO AT THE IRR. Drawing it is the clearest way to show the relationship between the two measures, and to reveal the CROSSOVER RATE at which two projects' rankings reverse.