[MUSIC] Now, having seen this simple case of a perpetuity, let's try something even a little harder. Suppose that instead of an asset generating the same amount of income each year, the asset generates a different amount of income each year. And further suppose, that instead of generating a stream of income from now to eternity, like our indestructible apartment building, the asset generates a stream of income over a fixed period of time. Maybe five years, maybe ten years, maybe 50 years. Here's an example of just such an investment. Suppose you are the chief executive officer of a textile company, and that your company is considering replacing your old mechanical looms with a set of highly computerized looms. These new machines won't come cheap. The price tag is a cool $2 million. Note however, that your chief economist forecasts that these new machines will increase revenues by $500,000 for each of the five years of the service life of the looms. Also, at the end of five years, the machines will have a salvage value of another $500,000. Now from this data, it may seem pretty obvious that the company should make the investment. After all, while the machines will cost $2 million, they will generate an even cooler $3 million in revenues and salvage value over the five-year period. But wait. Let's not forget about the time value of money. And here's the formula you would use to calculate the net present value of this investment. In this equation, I0 is the initial investment at time period 0. I is the one period market interest rate and for this example, we'll assume that that interest rate is constant at 15%. Now, N sub 1 is the net receipts from the investment in the first period or year. And N sub 2 is the net receipts in the second period during the year, and so on. Then, the sum of the initial investment and the stream of payments, N sub 1, N sub 2, and so on, will have the present value, NPV, given by the formula. Now let's use this formula to calculate the net present value of our loom investment. Here's the math. We start off with our initial $2 million investment, which has a negative sign, because it is an outlay. Then, we must discount the income stream of $500,000 per year, plus the $500,000 the company will receive at the end of the fifth year as salvage value. This gives us an NPV of $1,924,664 for the income stream. Subtracting that from our initial $2 million dollar investment, we find that the actual NPV of the total investment is a negative sum, minus $75,336. Now, because the net present value of the investment is negative, your company should not make the investment. And note, that this is despite the fact that over the life of the investment, the investment will generate an undiscounted sum a full half million dollars greater than the initial investment. Now, what about if the interest rate were 5%. Would your company now make the investment? Yes, your company would make the investment. This is because the value of that investment in net present value terms is positive at the prevailing cost of borrowed funds. Put another way, by making the investment, you would increase your profits. From an intuitive perspective, what's happening is this, the higher the interest rate, the more one has to discount the revenue stream. This, in turn, reduces the present value of the revenue stream and vice versa.
