So the present value calculation that we've been working on so far has been assuming that you have an annuity. That your income stream starts at a period of time and then ends at some other period of time. But, what if your annuity never ends? Like the Georgia lottery offers a game called $1,000 a week for life. Suppose that you're going to live forever. It's not a great assumption but let's just suppose it. Then, you would have a perpetuity. That is you would get $1,000 per week forever and ever and ever and ever. Now, the calculation for a perpetuity is this, right here, it's a very simple calculation. Present value is equal to the cash flow divided by the interest rate. So, it doesn't seem like it's worth very much money, right? And so, let's suppose that I was going to get $52,000 a year, for every year for the rest of forever. And my interest rate was 10%, then this perpetuity will be worth $520,000. How does that work? Well our present value formula for the annuity is right here. Present value is equal to C over r times 1- 1 over (1 + r) to the power of t. If t goes to infinity, then what's here in the denominator gets incredibly large. And one divided by a very large number approximates zero. And so that terms drops out and then all we're left with is your cash payment over your interest rate. So what does this mean? It means that even if you have a payment that goes on forever and ever and ever and ever and ever, $1, 000 a week for life, 70 years from, now 100 years from now, 200 years from now, really isn't worth all that much. I mean, I can't spend it today. How do perpetuities work in reality? Let's talk about that. Several years ago, David Bowie issued what are called Bowie bonds. And what David Bowie did is he said, I'm going to sell the rights to my mechanical royalties. Essentially, the royalties that he gets when his song is played on the radio, or if somebody buys one of his CDs, or if one of his songs is used in a movie or a television show. He said, I'm going to sell the rights to those royalties. Even though I may die, my songs will never die. Ziggy Stardust will live forever. And so, for all intents and purposes, the royalties that accumulate from those will be a perpetuity. They will go on forever and ever and ever. And he said, I cannot use that money 100 years, or even 50 years from now, so he and his accountants got together and they identified the value of that perpetuity right now. And so he sold his future royalties for a lump sum payment at one time. Essentially, he identified the future cash payments of his songs on the radio and divided it by an interest rate which gave them a lump sum. And somebody exchanged that lump sum for the rights in the future. Let's suppose that those royalties were worth $10 million, they give him a lump sum of $10 million. If, in fact, there only were $10 million, why would somebody exchange a future royalties for a present lump sum? Well, the truth is that they must have a difference in opinion about what their discount rate is. David Bowie, who knows he's not going to live forever, has a higher discount rate, he doesn't value his future payments as much. But an investor, somebody who works for a large corporation and the corporation might be around for 50 years, 60 years, 70 years or 100 years, would have a lower discount rate. They see these future streams as being really tenable. Like they might actually have them at their possession at some point. David Bowie, God bless him, he never will. But some company will. As a result, differences in the discount rate will give one person a higher or lower present value and that's what sets up an opportunity for exchange. David Bowie wants the money now, somebody else is willing to take the royalties in the future.