What are Net Present Value and Internal Rate of Return?

NPV and IRR Calculator

In investing and financial mathematics, the Net Present Value of an investment is the difference between the initial investment and the present values of the future dividends (series of cash flows). To calculate NPV, you need to know the interest rate (sometimes called the discount rate), the cash flow amounts, and the initial investment. If the NPV of a project or investment is positive, the investment is profitable.

The Internal Rate of Return is the interest rate that makes the net present value equal to zero. When you compare two investment opportunities, the one with the higher IRR is more profitable. In bond valuation, IRR is the same as YTM, yield to maturity.

NPV can be calculated directly with simple arithmetic. To determine IRR, you must find the solution to an equation in the variable R, the interest rate. Unfortunately, the equation for IRR is too complex to be solved by hand, so you must use a calculator with built-in solving capabilities, such as the convenient NPV and IRR Calculator below.

How to Compute NPV

Assume that the cash flow for Year 1 is C₁, the cash flow for Year 2 is C₂, and the cash flow for Year N is C_N. If r is the interest rate (a known quantity) and I is the initial investment, then the NPV is given by this equation:

NPV = C₁/(1+r) + C₂/(1+r)² + ... + C_N/(1+r)^N - I

For example, suppose you invest $1,000 in a project. The cash flow values are C₁ = $300, C₂ = $400, C₃ = $500, and C₄ = $600. If the discount rate is 5%, then r = 0.05 and the NPV is

300/1.05 + 400/1.05² + 500/1.05³ + 600/1.05⁴ - I
= 300/1.05 + 400/1.1025 + 500/1.1576 + 600/1.2155 - 1000
= 285.7143 + 362.8118 + 431.9281 + 493.624 - 1000
= $574.08

The net present value of this investment is $574.08, so you should undertake this project.

How to Compute IRR

The equation for IRR is similar to the equation for NPV, except you set the difference equal to zero and solve for R. For example, suppose you invest $1000 in a project, and the cash flows are C₁ = $300, C₂ = $400, C₃ = $500, and C₄ = $600. To find the internal rate of return, you must solve the following equation for R:

0 = 300/(1+R) + 400/(1+R)² +500/(1+R)³ + 600/(1+R)⁴ - 1000

This equation is too complex to be solved with simple algebra by hand, but using the calculator below, we find that the IRR for this investment is 24.89%.

Let's compare this to another investment. Suppose you input $1000 into a different project in which you receive $300 every year for 6 years. Then C₁ = C₂ = C₃ = C₄ = C₅ = C₆ = 300. Using the IRR calculator, we find that the internal rate of return for the new project is 19.9%. So the first project may be a better investment.

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