# What Is the Present Value of a Perpetuity Formula and What Are Perpetuities?

First of all, what is the ordinary meaning of “perpetuity?” It simply method “forever.” So in finance, what do we average by this same concept? Well, imagine I gave you a piece of paper or certificate, and that paper promised that I would pay you a fixed amount every year, forever. That piece of paper is called a “perpetuity.” Simple! How is it different from, say, a promissory observe? It isn’t. We can say it’s a special kind of promissory observe which lasts forever, with regular payments every year (or every month or other time period).

Now the question is… if I tried to sell you this piece of paper, how much would you be willing to pay for it? If I said… buy this piece of paper for only \$100, and I’ll give you \$2 for the rest of your life, forever. Would you buy it? It sounds like a great deal, doesn’t it? After all, you just pay once, and then you’ll get money from me forever!

But think about it another way too… Let’s say the bank’s interest is 5% per year. If you put the same \$100 in the bank and left it there forever, how much would you get every year, forever? You’ll get \$5 per year! (\$5 is 5% of \$100). Much more than the \$2 per year you would get from me if you buy my piece of paper above for \$100! So, are you nevertheless willing to pay me \$100 to get \$2 per year forever? Or, would you rather use the same \$100 to place in the bank, and get a much higher \$5 per year instead?

Of course, you’ll prefer to put your money in the bank! However, you might nevertheless be willing to buy my piece of paper or perpetuity if I lower the price. How much should I lower it to make it worth your money? \$80? \$60? \$40? Naturally, we can’t simply use our feelings to guess the right price. So how do we find the exact amount? For this, we use the Present Value of a Perpetuity Formula. With this, you will find that the “fair value” in this case is \$40. The simplest formula, which assumes consistent annual cash flows, looks like this:

Present Value of a Perpetuity = (Yearly Cashflow)/(bank interest rate per year)

*Fair value = \$40 method that if you pay any more than that, you’re getting a sour deal… you’ll be in a better situation putting your money in the bank.

What is the logic behind this “fair value”? We go back to the IRR concept. At the fair value of \$40, the IRR of our perpetuity is exactly the same as the interest rate of your bank place. Meaning: \$40 earning \$2/year will have an IRR of 5%. A bank place of \$100 earning \$5 per year will also have an IRR of 5%; consequently making the returns the same or “fair.”