Net Present Value (NPV) Calculator

Description

Calculate the Net Present Value (NPV) of an investment or project, which helps evaluate its profitability by considering the time value of money.

Net Present Value (NPV) Formulas

Basic NPV Formula

The basic formula for Net Present Value (NPV) is:
$ \text{NPV} = \sum_{t=1}^{N} \frac{C_t}{(1 + r)^t} - C_0 $

Where:

$ C_t $ = Cash flow at time $ t $
$ r $ = Discount rate
$ N $ = Number of time periods
$ C_0 $ = Initial investment (cash outflow at time 0)

Alternative NPV Formula (Expanded)

An expanded form of the NPV formula is:
$ \text{NPV} = \frac{C_1}{(1 + r)^1} + \frac{C_2}{(1 + r)^2} + \dots + \frac{C_N}{(1 + r)^N} - C_0 $

NPV with Continuous Compounding

If cash flows are compounded continuously, the formula becomes:
$ \text{NPV} = \sum_{t=1}^{N} C_t \cdot e^{-rt} - C_0 $

Where:

$ e $ = Euler's number (approximately 2.71828)
$ -rt $ = Negative exponent of the discount rate times the time period

Net Present Value (NPV) Problems and Solutions

Problem 1

Calculate the NPV of a project with an initial investment of $500,000 and cash flows of $150,000, $180,000, $210,000, and $250,000 over the next 4 years. The discount rate is 8%.

Solution:
$ \text{NPV} = -500,000 + \frac{150,000}{(1 + 0.08)^1} + \frac{180,000}{(1 + 0.08)^2} + \frac{210,000}{(1 + 0.08)^3} + \frac{250,000}{(1 + 0.08)^4} $
After calculation, $ \text{NPV} = \$121,203.44 $.

Problem 2

A project has an initial investment of $200,000 with cash flows of $60,000, $80,000, $100,000, and $120,000 over the next 4 years. If the discount rate is 10%, should you invest in this project?

Solution:
$ \text{NPV} = -200,000 + \frac{60,000}{(1 + 0.10)^1} + \frac{80,000}{(1 + 0.10)^2} + \frac{100,000}{(1 + 0.10)^3} + \frac{120,000}{(1 + 0.10)^4} $
After calculation, $ \text{NPV} = \$11,348.59 $. Since the NPV is positive, you should invest in this project.

Problem 3

Calculate the NPV of a project with an initial investment of $1,000,000 and a series of irregular cash flows: $250,000, $300,000, $400,000, and $500,000 over the next 4 years. The discount rate is 7%.

Solution:
$ \text{NPV} = -1,000,000 + \frac{250,000}{(1 + 0.07)^1} + \frac{300,000}{(1 + 0.07)^2} + \frac{400,000}{(1 + 0.07)^3} + \frac{500,000}{(1 + 0.07)^4} $
After calculation, $ \text{NPV} = \$140,223.95 $.

Problem 4

A project requires an initial investment of $800,000 and will generate cash flows of $100,000 annually for 10 years. Calculate the NPV if the discount rate is 5%.

Solution:
$ \text{NPV} = -800,000 + \sum_{t=1}^{10} \frac{100,000}{(1 + 0.05)^t} $
Using the formula for the present value of an annuity, we find $ \text{NPV} = \$61,358.56 $.

Problem 5

Calculate the NPV of a project with an initial investment of $300,000 and cash flows of $75,000 for 6 years. The discount rate is 6%.

Solution:
$ \text{NPV} = -300,000 + \sum_{t=1}^{6} \frac{75,000}{(1 + 0.06)^t} $
After calculation, $ \text{NPV} = \$24,388.87 $.

Problem 6

A project has an initial cost of $1,500,000 with a 4-year cash flow stream of $400,000, $450,000, $500,000, and $600,000. The discount rate is 9%. Calculate the NPV.

Solution:
$ \text{NPV} = -1,500,000 + \frac{400,000}{(1 + 0.09)^1} + \frac{450,000}{(1 + 0.09)^2} + \frac{500,000}{(1 + 0.09)^3} + \frac{600,000}{(1 + 0.09)^4} $
After calculation, $ \text{NPV} = -\$12,050.47 $.

Problem 7

Consider a project with an initial investment of $250,000, generating cash flows of $50,000, $75,000, $100,000, and $125,000 over 4 years. If the discount rate is 8% with continuous compounding, calculate the NPV.

Solution:
$ \text{NPV} = -250,000 + \sum_{t=1}^{4} 50,000 \cdot e^{-0.08 \cdot t} $
After calculation, $ \text{NPV} = \$15,205.54 $.

Problem 8

Calculate the NPV of a project with a cost of $600,000, cash flows of $150,000, $180,000, and $200,000 for 3 years, and a discount rate of 12%.

Solution:
$ \text{NPV} = -600,000 + \frac{150,000}{(1 + 0.12)^1} + \frac{180,000}{(1 + 0.12)^2} + \frac{200,000}{(1 + 0.12)^3} $
After calculation, $ \text{NPV} = -\$16,526.72 $.

Problem 9

A project requires an initial outlay of $1,200,000 and produces cash flows of $300,000 each year for 5 years. Calculate the NPV if the discount rate is 7%.

Solution:
$ \text{NPV} = -1,200,000 + \sum_{t=1}^{5} \frac{300,000}{(1 + 0.07)^t} $
After calculation, $ \text{NPV} = \$52,396.80 $.

Problem 10

Calculate the NPV of a project with an investment of $400,000 and annual cash flows of $100,000 for 8 years with a discount rate of 6%.

Solution:
$ \text{NPV} = -400,000 + \sum_{t=1}^{8} \frac{100,000}{(1 + 0.06)^t} $
After calculation, $ \text{NPV} = \$157,393.32 $.