The General NPV Formula in Sigma Notation
In corporate finance and financial economics, the formal mathematical definition of Net Present Value is written using summation notation (Σ):
• CFt = Net cash flow received (or paid) at period t
• r = Discount rate per period (expressed as a decimal, e.g., 0.10 for 10%)
• t = Time index of the period (Year 1, Year 2, ..., Year n)
• n = Final period in the investment horizon
• CF0 = Initial cash investment required at time 0 (outlay)
Expanded Algebraic Form
Writing out the summation explicitly for an n-year investment illustrates how each year's cash flow is discounted by a progressively higher power of $(1 + r)$:
Understanding the Discount Factor: 1 / (1 + r)ᵗ
The term 1 / (1 + r)t is known as the Present Value Discount Factor (or Present Value Interest Factor, PVIF). It represents the present worth today of exactly $1 received t periods in the future.
Reference Discount Factor Table
| Period (t) | 5% Discount Rate | 8% Discount Rate | 10% Discount Rate | 12% Discount Rate | 15% Discount Rate |
|---|---|---|---|---|---|
| Year 1 | 0.9524 | 0.9259 | 0.9091 | 0.8929 | 0.8696 |
| Year 2 | 0.9070 | 0.8573 | 0.8264 | 0.7972 | 0.7561 |
| Year 3 | 0.8638 | 0.7938 | 0.7513 | 0.7118 | 0.6575 |
| Year 4 | 0.8227 | 0.7350 | 0.6830 | 0.6355 | 0.5718 |
| Year 5 | 0.7835 | 0.6806 | 0.6209 | 0.5674 | 0.4972 |
| Year 10 | 0.6139 | 0.4632 | 0.3855 | 0.3220 | 0.2472 |
Manual Calculation Walkthrough (By Hand)
Suppose an investor is evaluating a project with a $20,000 upfront outlay, a 10% discount rate (r = 0.10), and expected cash flows of $8,000 in Year 1, $10,000 in Year 2, and $12,000 in Year 3:
- Year 1 Present Value:
PV₁ = $8,000 / (1.10)¹ = $8,000 / 1.10 = $7,272.73 - Year 2 Present Value:
PV₂ = $10,000 / (1.10)² = $10,000 / 1.21 = $8,264.46 - Year 3 Present Value:
PV₃ = $12,000 / (1.10)³ = $12,000 / 1.331 = $9,015.78 - Sum of Present Values:
$7,272.73 + $8,264.46 + $9,015.78 = $24,552.97 - Subtract Initial Outlay:
$24,552.97 − $20,000 = +$4,552.97
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Frequently Asked Questions
Q1What is the standard Net Present Value (NPV) formula?
NPV = ∑ [ CF_t / (1 + r)^t ] − CF_0where CF_t is the cash flow in period t, r is the discount rate per period, t is the time index (1, 2, ..., n), and CF_0 is the upfront initial investment.
Q2What is a Discount Factor and how is it calculated?
DF_t = 1 / (1 + r)^t. Multiplying a nominal cash flow by its discount factor gives its present value (PV).Q3What is the Mid-Year Discounting Convention?
PV = CF_t / (1 + r)^(t − 0.5), discounting each cash flow to the exact midpoint of the period.Q4How does continuous discounting differ from discrete annual discounting?
e^(-r * t) where e ≈ 2.71828. Continuous discounting yields a slightly lower present value than discrete annual discounting because interest compounds infinitesimally.