Now we know the basics of the discounted cash flow valuation method. Let's put it to the test and extend the example to a project. What analysts do is value firms' equity. What CFOs in corporations do is value investment projects for the firm. But there's no fundamental difference. We apply the same methodology. Both CFOs and analysts will assess whether the present value of future cash flows of the firm or the project are, in fact, worth an initial investment outlay. That could be, if it is the firm's value, the current share price, the price you need to pay to acquire a share in the firm. For the CFO, it would be the capital investment required to start the project. So the entitlements of investing in the firm or investing in the project are given, as we've seen before, a set of cash flows that happen after one year, two-year until the lifetime expires of the project or the firm n years. And we need to discount each of those cash flows back appropriately to a present value, which will tell us what the project or firm, are worth today. So to make that decision, and bring in the initial investment outlay, we go about that as follows. So both analysts and CFOs use, what we call, a net present value analysis. When net present value, NPV, is defined as the difference between the present value of the benefits and the present value of the costs. So we first compute the present value of the cash flow entitlements that the firm or the project delivers, and then we subtract from that the present value of the initial investment, the initial outlay. So for our entitlement of N cash flows discounted to the present, we subtract, in this simplified example, an initial investment at time period zero, the present. The decision rule is straightforward. Whenever the net present value is positive, that project of the firm would show that it increases shareholder wealth. Buying a share in the firm, investing in that project would be a good decision. On the other hand, if the net present value turns out negative, then we should be rejecting the project. We should not be investing in shares of the firm as they would reduce shareholder wealth. What happens if net present value is in fact zero? Well then, at least it doesn't destroy shareholder wealth, but it also doesn't add shareholder wealth. Let's put some numbers in. Consider a project with a three year life span, n equals 3. And we have a set of positive expected cash flow generated by this project of, in the first year 50 million, in the second year 50 million and in the third year $100 million entitlements. The project requires an upfront investment of $150 million to be paid today. The analysts, the CFO, decides that reasonable discount rate is 5% per annum. So the net present value of these cash flows of 50, 50, 100 are now easily determined as $50 million discounted at 5% for one year, plus $50 million discounted at 5% for two years, plus $100 million discounted at 5% for three years, and subtracting the $150 million of initial investment outlet, which doesn't have to be discounted to the present because it is already a present value. Adding up the present values of $47, $45 million and $86 million, and then subtracting the $150 million gives us a net present value for this particular project of $29.4 million. Positive. So the decision go, don't go into the project, is easily made. Go is the word. Just to illustrate that graphically. What I've done in this graph is to indicate with buzz, the future cash flows. There's the initial investment outlay of $150 million, the cash outflow, that's the blue bar on the left. Then, in year one and year two, there's a $50 million positive future cash flow. And in year three, there's a $100 million positive cash flow at the end of year three. If I discount each of those cash flows at 5% per annum, I get the red bars. So the $150 million remains $150 million. It already is a present value investment outlay. But, you can see that the gap between the blue bars and the red bars is increasing with time, as the further in the future the cash flows occur, the lower their present value will be. Adding the red bars together, we arrive at the green bar at time period zero, the net present value which is about $25 million, as we've seen just a moment ago. A positive number, hence the project adds shareholder wealth, the project should be invested in. Let's push the envelope a little. Because we know that investment projects for the cooperation are not risk-free. So 5% seems an entirely unrealistic discount rate. It turns out that the CFO is aware of this and decides to increase the discount rate to 20%. To capture the increased riskiness of these future cash flows. If I again compute the net present value of $50, $50, $100 million over the next three years and subtract the $150 million initial outlay, I now arrive at negative $15.7 million of net present value. So, all of a sudden, we see that the net present value has turned negative. So, the go, don't go decision now turns negative. This project is destroying shareholder wealth. You should not be investing upfront $150 million to be entitled to those cash flows. Clearly, the increased riskiness of the future cash flows had a major impact on its net present value. Turning it from a go decision into a don't go decision. Again, illustrating this with the bar chart, we can see that the gap between the blue cash flow bars and their present value, the red cash flow bars, is now a lot larger because of the higher discount rate. The more risk, the higher the discount rate. The lower the present value, the lower the positive red bars are going to contribute to the net present value, which is this negative net present value for this example at a discount rate of 20%. So clearly, there must be a discount rate somewhere between 5% and 20%, where we've crossed the line. Where a positive net present value turned into a negative net present value. At that point, it no longer pays off. The corporation should not be investing its $150 million in this project. So, what would be the trigger discount rate at which we break even? Well, you can solve this equation. And it will tell you, then, exactly where the crucial break even discount rate is. I've already indicated it's somewhere between 5 and 20%. So let's just do the analysis we've done for 5% and 20% individually for a whole sequence of discount rates. Slowly increasing from 5% to 20% and see what happens with the net present value. Well that's reflected in this graph. So here we plot the net present value on the vertical axis against increasing discount rates on the horizontal axis. We saw that at 5%, the net present value was close to $30 million positive. We saw it at a 20%, the net present value was negative at about $15 million, and we can see that probably somewhere in the area between 13 and 14% discount rate, we crossed the line, where net present value positive turns to net present value negative. That crucial rate is also known as the internal rate of return. You will hear much more about that in the third course.