Net present value

$NPV = \sum_{t=0}^{n} \frac{C_t}{(1+r)^t}$

The sum of every incremental cash flow, discounted at the cost of capital, including the negative one at $t = 0$.

The rule: accept if NPV is positive. That is the whole thing.

Why it is the right rule

Four properties, and no other appraisal method has all four.

It is denominated in value. An NPV of ₹18 crore is a claim that the project makes the firm ₹18 crore richer today. Not a ratio, not a percentage, not a period — rupees of value created, which is exactly the objective from chapter 1.

It uses all the cash flows. Nothing beyond an arbitrary cut-off is ignored.

It accounts for timing. A rupee in year one and a rupee in year ten are weighted correctly.

It accounts for risk, through the discount rate. A riskier project is discounted harder, so it must produce more to clear the bar.

And one more, which matters when comparing projects: NPVs add. The value of doing two projects is the sum of their NPVs. No ratio-based measure has this property, which is why NPV is the only rule that scales to a portfolio of decisions.

Working the problem

₹100 crore out, ₹40 crore a year for three years.

At 10%:

$\frac{40}{1.10} + \frac{40}{1.21} + \frac{40}{1.331} = 36.36 + 33.06 + 30.05 = 99.47$

NPV = 99.47 − 100 = −₹0.53 crore. Reject, narrowly.

At 15%:

$34.78 + 30.25 + 26.30 = 91.33$

NPV = −₹8.67 crore. Reject, clearly.

Both negative, which is the honest answer: ₹120 crore of nominal return on ₹100 crore over three years is not enough once timing is priced.

What the gap between the two tells you is the project's sensitivity to the discount rate — a swing of ₹8 crore for five percentage points. A project whose verdict flips over a plausible range of $r$ is a project whose verdict rests on an input nobody can observe directly, and chapter 9 is about how imprecise that input really is.

What NPV assumes

Three assumptions, each of which is sometimes false:

That the discount rate is known. It is estimated, with wide error bars (chapter 8). Two analysts can produce opposite recommendations from identical cash flows.

That the cash flows are known. They are forecasts, usually by the people who want the project approved.

That the decision is now or never. This is the one people miss. NPV compares doing it today against never doing it. It does not value the option to wait — and waiting has value when there is uncertainty that time will resolve. A project with an NPV slightly below zero may be worth keeping alive rather than killing.

Why NPV is not used as much as it should be

Not because managers do not understand it. Three real reasons:

It requires a discount rate, and arguing about $r$ is harder than arguing about a payback period.

It is not intuitive to non-finance audiences. "This creates ₹18 crore of value" invites "how do you know?" in a way that "it pays back in two years" does not.

It can be gamed. Since the forecaster chooses the cash flows, a determined sponsor can make almost anything positive. The defence is not a different rule — it is sensitivity analysis, and asking who produced the numbers.

The honest way to present one

A single NPV is a point estimate dressed as a conclusion. Three things make it usable:

• A range, from pessimistic through base to optimistic cash flows
• A sensitivity table across discount rates and the one or two assumptions that matter most
• The break-even assumption — what would volume, price or cost have to be for NPV to reach zero? That number is often far more persuasive than the NPV itself, because it can be judged against experience.

This is the same discipline as chapter 9 of the Quantitative methods subject: quote a range, not a point.

The point

NPV discounts every incremental cash flow at the cost of capital and accepts the project if the total is positive. It is denominated in value, uses all cash flows, prices timing and risk, and uniquely among appraisal rules it adds across projects. Its weaknesses are that the discount rate and the cash flows are both estimates, and that it values doing something now against never — not against waiting.