Finance Formulas

Time Value of Money

core

Net Present Value (NPV)

Builds onPresent Value of a Single Sum — if this page feels steep, start there.

NPV=−CF0+∑t=1nCFt(1+r)tNPV = -CF_0 + \sum_{t=1}^{n} \frac{CF_t}{(1+r)^t}

Reading the notation

NPVNPV
net present value — today's value of everything the project gives, minus what it costs
−CF0-CF_0
the minus sign marks money going OUT: the up-front cost at time zero
∑t=1n\sum_{t=1}^{n}
the big sigma means 'add up': one term for each year t, from 1 to n
CFtCF_t
the cash flow arriving at the end of year t
CFt(1+r)t\frac{CF_t}{(1+r)^t}
each cash flow divided by its own years of growth — later money is shrunk more

Why it must be true

NPV answers one question: is this project worth more than it costs? Discount every future cash flow to today — money's only common denominator — then subtract the up-front outlay. A positive NPV means the project creates value after paying investors their required return rr; the decision rule "accept if NPV>0NPV > 0" falls straight out of the definition.

The discount rate is doing the heavy lifting: it is the return the money could earn elsewhere at the same risk. NPV is therefore always a comparison against the next-best alternative, never an absolute number.

The derivation

Value each cash flow with the single-sum PV formula, then add. Present values can be added because they are all stated in the same units — today's dollars:

NPV=−CF0+CF1(1+r)1+CF2(1+r)2+CF3(1+r)3NPV = -CF_0 + \frac{CF_1}{(1+r)^1} + \frac{CF_2}{(1+r)^2} + \frac{CF_3}{(1+r)^3}

Nothing more is happening: NPV is repeated single-sum discounting plus arithmetic.

When to reach for it

An up-front outlay against a schedule of forecast cash flows, judged at a required return — the standard capital-budgeting accept/reject decision.

Listen for

“initial investment / outlay of …”“expected cash flows of … in years 1, 2, 3”“required return / hurdle rate / cost of capital”“should the firm accept the project”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

  • ≈

    First screen without discounting: sum the inflows and subtract the outlay. If that's already negative, NPV is negative — done, no arithmetic needed.

  • ≈

    That undiscounted number is also your ceiling: the true NPV must be below it. Often only one choice sits between zero and the ceiling.

  • ≈

    Rough discount: knock ~r% off per year of waiting. Mid-scheme flows (year 2 of 3) lose roughly 2 × r%.

Traps in applying it

  • ✗Leaving out the initial outlay — the sum of discounted inflows is gross PV, not NET present value.
  • ✗Discounting every flow one period instead of by its own horizon.
  • ✗Comparing raw NPVs of projects with different sizes or lives — scale and duration need separate treatment.

Limits & criticisms

NPV compresses the entire term structure into one flat discount rate for every period — year 1 and year 10 cash flows rarely deserve the same rate, so the single-r NPV is a deliberate simplification. It also treats the forecast cash flows as given, though they carry most of the real uncertainty, and it ignores managerial flexibility — the option to expand, delay or abandon — which is why real-options analysis exists as a correction.

Where it came from

Irving Fisher (1907) proved the deep result: whatever your personal preferences, you should take every project whose discounted cash flows exceed its cost — the market lets you rearrange the timing afterwards. But NPV stayed academic until Joel Dean's Capital Budgeting (1951) carried DCF into corporate boardrooms, displacing payback-period rules of thumb.

Today "accept if NPV > 0" is the closest thing corporate finance has to a law, used from factory investments to M&A models — always with the discount rate standing in for the next-best use of the money.

On the BA II Plus

Worked example: A firm can invest $8,250.00 now to receive $1,300.00 in year 1, $3,300.00 in year 2 and $3,600.00 in year 3. Using a required return of 9.5%, compute the net present value.

  1. 1.[CF] [2ND] [CLR WORK]open and clear the cash-flow worksheet
  2. 2.8,250 [+|-] [ENTER]CF0 — the outlay, negative
  3. 3.[↓] 1,300 [ENTER] [↓]C01 (leave F01 = 1)
  4. 4.[↓] 3,300 [ENTER] [↓]C02
  5. 5.[↓] 3,600 [ENTER] [↓]C03
  6. 6.[NPV] 9.5 [ENTER]I = discount rate, in percent
  7. 7.[↓] [CPT]NPV

→ -$1,568.60

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