Reading: Net Present Value (NPV)

Example 3

A company can invest 4 million into a project that generates a payoff of 6 million in 5 years. The cost of capital is 10%. Should the company invest?

To find out, we can compute the NPV of the project: 

\( NPV = C_0 + \frac{C_5}{(1+R)^5}=-4 + \frac{6}{1.1^5} = -4 + 3.73 = -0.27 \)

The net present value is -0.27 million. The project destroys value, even though it generates a cash surplus of 2 million (= 6 - 4) over 5 years.

Why exactly does the project destroy value? Because an investment into the alternative asset at a rate of return of 10% would generate an even higher surplus over the next 5 years. The present value that we have computed above shows that, with the alternative asset, it only takes an investment of 3.73 million today to replicate the project's cash flow of 6 million in 5 years. With the replicating asset, we can therefore generate the same future cash flow 0.27 million cheaper. That's how much money we destroy today if we take the project.

Example 4

What's the maximum investment that you should be willing to accept in the project from Example 3 above?

Our computations above have shown that the present value of the project's future cash flow is 3.73 million. That's the maximum you should be willing to invest. With an investment of 3.73 million, the NPV of the project is just zero:

\( NPV = C_0 + \frac{C_5}{(1+R)^5}=-3.73 + \frac{6}{1.1^5} = -3.73 + 3.73 = 0 \)

In that case, the project leaves your net worth unaffected. By investing 3.73 million in the project, you can expect to earn the "fair" rate of return, given the riskiness of the project. You are therefore indifferent between investing in the project and investing in the alternative asset.

Consequently, If the required investment is smaller than the present value of all future cash flows (3.73 million), the project has a positive NPV and you should invest. In contrast, if the required investment exceeds the present value of all future cash flows, the project has a negative NPV and you are better off investing in the alternative asset.

Example 5

An innovator wants to sell her business idea for EUR 2 million today. To implement the idea, the buyer would have to invest an additional EUR 5 million today. Subsequently, the project would generate an annual cash flow of EUR 1 million forever (at the end of each year). The cost of capital is 15%. Based on this information, what is the value of the business idea and is it worthwhile to buy it?

We find the value of the business idea (PV) by computing the present value of all the cash flows that are associated with it. Note that the future cash inflows take the form of a level perpetuity, the valuation of which is discussed extensively in the the module Time Value of Money: 

\( PV = C_0 + \frac{C}{R} = -5 + \frac{1}{0.15} = -5 + 6.67 = 1.67 \)

Accordingly, the business idea has a value of EUR 1.67 million. However, this is NOT the value of the investment proposal, since a buyer has to pay the innovator EUR 2.00 million in exchange for that idea! The asking price therefore exceeds the value of the business idea by EUR 0.33 million. The investment proposal destroys value:

NPV = − Purchase price + PV of project cash flows = − 2.00 + 1.67 = − 0.33.

Based on our calculations, it is therefore not worthwhile to buy the business idea. The numbers also imply that the maximum price an investor should be willing to pay is 1.67 million. At that price, the investor will break even and can expect to earn the "fair" risk-adjusted rate of return.