Time Value of Money

Now, let's discuss a specific type of annuities called perpetuity. Before proceeding, consider this scenario: Suppose you receive $1,000 at the end of each year indefinitely. This income will continue forever, raising the question of its future value. What is the present value of this infinite series of $1,000 payments? This question, along with the calculation of its net present value, will be addressed next.

Perpetuity refers to a type of cash flow that occurs at regular intervals with a fixed amount, similar to annuities, but continuing indefinitely. To determine the present value (PV) of a perpetuity, one must consider the future value of the cash flows. This involves calculating the net present value (NPV) of each individual cash flow stream, while also taking into account the infinite nature of such cash flows.

The formula above is straightforward to derive. Since perpetuity is a special case of an annuity where n equals infinity, substituting n as infinity in the annuity formula results in the perpetuity formula due to the principles of limit theory. 

Using the aforementioned formula, the answer to my previous question becomes clear. If you are receiving a perpetuity valued at $1,000 with a 10% interest rate, its present value is calculated as $1,000 / 0.10 = $10,000. This calculation highlights the concept of net present value, illustrating how future value can be assessed through present value. While this may seem counterintuitive, the sum of an infinite series can indeed be a finite number, showcasing the remarkable nature of mathematics.

We hope you have gained some understanding of the Present Value (PV) and Future Value (FV) of a cash flow stream. This is a critical concept to grasp, which is why we are dedicating significant time to this topic. Our objective is for you to master it both mechanically and visually. Essentially, any problem associated with the time value of money, including calculating net present value, can be easily solved if you can visualize moving back and forth in time. The following exercise should help clarify this point.

Assume you intend to pursue a PhD in 5 years' time. If you are already pursuing one, commendations to you—let us assume you wish to undertake another PhD. You estimate that you will require $1,000 each month for your course and living expenses. You anticipate earning an interest rate of 10% per annum on all your savings.

The problem is as follows: How much should you invest in your savings each year for the next 5 years (at the beginning of each year), so that when you commence your PhD program, the investments will cover your annual expenses of $12,000 for the entire duration of the course, which will last 4 years?

Try to devise a solution to this investment problem. We encourage you to think thoroughly about this problem before consulting our answer or methodology. Please refrain from moving to the next topic until you have given this problem sufficient consideration. The solution is provided at the end of the next topic. Therefore, take out pen and paper, draw a timeline, map out the cash inflows and outflows on the timeline, and attempt to resolve the situation.

We trust that you found the previous problem stimulating. Now, we will discuss one final concept in this section before presenting a potential solution to our question. 

Net Present Value (NPV) represents the present value of an investment or series of cash flows. 

The NPV of an investment is calculated as the present value of all expected cash inflows minus the present value of expected cash outflows. To determine the NPV, the project's cash flows must be discounted at the appropriate cost of capital, which helps in assessing the future value of the investment. 

NPV indicates the value of the investment as of the reference date, typically today or the present. It is the most widely used metric for investment decision-making by CEOs, CFOs, and other key decision-makers worldwide. Therefore, it is essential to understand this concept thoroughly. 

Steps to calculate NPV: 

1. Measure all project costs (cash outflows). 

2. Measure all project benefits (cash inflows). 

3. Decide on an appropriate discount rate (consult finance team). 

4. Calculate the present value of cash flows. Positive cash inflows increase NPV; negative cash outflows decrease NPV.

Consider the following example: A project requires a $30 million upfront investment. As a result of this investment, the project generates positive cash flows of $10 million, $25 million, $20 million, and $15 million at the end of each year. What is the value of this project on the inception date, assuming a discount rate of 12%? To determine the project's value, we need to calculate the Net Present Value (NPV), which is derived from the Present Value (PV) of the future cash flows. We will find the PV of all cash flows and sum them up to obtain the project NPV. 

NPV = -30 + 10 / (1.12) + 25 / (1.12)^2 + 20 / (1.12)^3 + 15 / (1.12)^4 

NPV = $35 million

Here is the solution to the problem discussed earlier. Let's summarize the issue: How much should be invested in savings each year for the next 5 years (at the beginning of each year) so that by the time you start your PhD program (beginning of the 6th year), the future value of the investments will cover annual expenses of $12,000 for the entire duration of the course, which is 4 years? Assume these expenses occur at the end of each year, and we need to consider the net present value of these future cash flows. First, draw a timeline to illustrate the cash flow requirements. The timeline should include all X values representing investments required in the first 5 years (at the beginning of each year) and all Y values representing the present value of the cash needed at the end of each year for spending needs.