Hi, in the previous lecture, we studied the first two topics in comparing NPV and IRR techniques which were the multiple IRR problem and comparing projects with different sizes. In this lecture, we study the remaining topics which are comparing projects with different lives and reinvestment rate assumptions. Let's take a look at these two projects, A and B. The discount rate is 15% for both projects. As you can see, the two projects have different lives. Project A has the life of six years, and project B is a three year project. Which project do you think will have a higher NPV? I'll give you a hint here. Project A has a longer life and, therefore, has more cash inflows than project B. Not surprisingly, project A has a greater NPV than project B. On the other hand, the IRR is greater for project B. As you can imagine, a primary reason why project A had a greater NPV is because it has a longer life. Generally, the NPV of a longer life project will be greater than the NPV of a shorter life project. My question is is it really fair to compare projects with different lives using the NPV rule because we know that NPV rule will tend to prefer the longer life project? The answer can be either yes or no, depending on your situation. It is fair to use the NPV rule to compare projects with different lives if you believe that you have no other investment opportunities on the horizon after these two projects are undertaken. On the other hand, it would not be fair to use the NPV rule if you believe that investment opportunities similar to projects A or B can be easily found and undertaken in the future. Using the NPV rule will be inappropriate if you believe that the project can be rolled over for a long time, too. One solution to this problem is to use the equivalent annual cost method, or EAC. EAC is the value of the annual payment in the annuity that has the same present value as our original project. In other words, EAC calculates the average annualized net cash flow of the project. The annualized cash flow is not subject to the problem regarding different lives of projects, so we can compare EACs of projects to determine which project is better. To compute EAC, we need to know the project's NPV, its length, and the discount rate. We'll use PMT function in Excel to calculate EAC of a project. In PMT function, we'll need to enter the discount rate first and then the number of payments, which is just the length of the project, and then the present value of the project, which is the NPV of the project. There are situations where we also need to enter the fourth and the fifth input, which are [Future value] and the [Type] of cash flows. But here in EAC calculation, only the first three inputs are all we need. Now we'll go to the Excel spreadsheet to calculate EAC of the two projects. In the spreadsheet, cash flows of the two projects A and B, are already described in the table, and their NPVs and IRRs are already calculated as well. Now we try to calculate their EACs. To calculate the annualized net cash flow of project A, we use PMT function. The first input is the discount rate in cell C2, and we enter 6 as the number of payments. And we reference cell C8, which has the NPV of project A as the present value in this function. You may have noticed that I put the negative sign in front of C8. This is because, in Excel or financial calculators, the convention is that the present value and future period of payments have different signs. The EAC of project A is $82.92. It means that the project has the same value as an annuity, which pays you the equal amount of $82 per year for the next 6 years. Using the same way, we can calculate the EAC of project B. Note that we enter three for the number of payments this time, because B has the life of three years. The EAC of project B is $69.62. EACs of the two projects are just that, even in terms of the annualized value, project A has a greater value than project B. The last topic to study in this lecture is different assumptions on reinvestment rate of the NPV and the IRR. But first of all, I'd like to begin this section by telling you that both NPV and IRR assume that intermediate cash flows of a project are reinvested at a certain rate of return. NPV assumes that intermediate cash flows are reinvested at the actual discount rate or at the required rate of return. On the other hand, IRR method assumes that intermediate cash flows are reinvested at the IRR itself. About this issue, we first would like to see how we can verify those assumptions. And then we'll also see what the implications of those assumptions are. As the first step we can very easily verify the assumptions using Excel. We use project B in this exercise. We'll create imaginary projects B' and B'' by calculating future values of intermediate cash flows in years one and two and putting them altogether in the final year of the project, year three. So in projects B' and B'', the initial investment will be the same, but there will be only a single future cash inflow in year three. We assume that the intermediate cash flows are reinvested at the discount rate 15% in project B'. We then assume that intermediate cash flows are reinvested at the IRR of 26.7% in project B''. In both B' and B'', cash flows in the first 3 years are -800, 0, and 0. In project B', intermediate cash flows are reinvested at 15%, so we can calculate the single cash flow in year 3 as follows. For example, cash flow in year 1 is multiplied by 1 plus 15% squared, because it can be invested for 2 years until year 3. So project B’ has cash flows of -800, 0, 0 and 1,458. In project B'', cash flows are reinvested at the IRR 26.67%. Here we got a greater single cash flow than the one in project B' ,because we assumed a higher reinvestment rate. Then we calculate NPVs and IRRs of these two projects. Very interestingly, we got the same NPV when we assume that intermediate cash flows are reinvested at the discount rate, and we got the same IRR when we assume that intermediate cash flows are reinvested at the IRR. This is a direct way of verifying the reinvestment rate assumptions of NPV and IRR, respectively. Now, we'd like to know why the assumptions should matter. First, the NPV assumes that in the future the company can continuously find projects that can yield the rate of return which equals the discount rate of the company. But the implicit assumption of IRR is that, in the future, the company can continuously find projects that can yield the rate of return, which equals the IRR of this specific project. Which assumption do you think is more realistic? The NPV assumption is a reasonable and realistic one, because the discount rate of the company or the project is determined based on the overall risk and investment opportunities that the company has. On the other hand if you follow the reinvestment rate assumption of IRR, you are likely to overestimate the project value, especially if it has a high IRR. The fact that this specific project has the IRR of 26% when the discount rate is 15% does not mean that the company can continuously find an investing project with the rate of return of 26% in the future. Perhaps this is another reason why NPV is considered a better tool than IRR.