Let's talk about how it is that you value a stream of rents. Suppose that I own an apartment building complex and the apartment buildings in that complex generate rents every month and they can accumulate over time. The period of time that I own that, let's say from now for the next 20 years, that's what we call annuity. An annuity is a stream of rents that you receive, in this case literal rents, from some type of an asset. Now your asset may be, like as I said, an apartment complex. Or your asset might be a trucking company. Or your asset may be a fast food franchise. Or your asset maybe be just a bunch of stocks in a portfolio. If you're going to have a starting period and an ending period, and the assets generate revenue, then we call that annuity. If those assets generate revenues and they go on forever and ever and ever and ever and ever, we call that a perpetuity. There are not many perpetuities in the universe. There are a couple but for the most part, assets have lives and those are annuities. How do we value that stream of rents? The idea is that we have to discount the future stream of rents for now and for later. Let's consider a situation when talking about present values, a very simple situation. Would you loan me $100 today, in exchange for $100 tomorrow? And obviously I already know the answer to this, sure you would, I'm a trustworthy guy. Besides the value of $100 today is virtually same as that is tomorrow so you have no problem exchanging $100 today for $100 tomorrow. But would you be willing to loan me $100 today in exchange for $100 a year from now. Probably not. Not because I'm any less trustworthy but, because you know that $100 a year from now isn't really worth $100. It's worth less than $100. How do we know that? We know because you could put $100 in the bank today in some kind of an interest bearing asset. And it would just sit there and churn, churn, churn, churn, churn, and then 365 days later, you could take out all the money and it would be worth more than $100. You would get $100 and then some interest on top of that. So if you gave me $100 today, you would expect that I would return to you the same money that some type of a financial institution would return to you. This is the difference between what we call present value and future value, okay? So how much is a $100 given today worth in one year? And how much should I loan somebody if they could only pay me back $100 in one year? Two questions, but different sides of the same coin, okay. So to answer this question, how much is $100 today worth in one year? The answer is we need to calculate what's called the future value of a cash payment, all right. And someone says how much should I loan someone today if they could pay me $100, then we have to calculate the present value of a payment made in the future. Let's do each one of these things, okay. So here's the formula for future value and present value. Future value is present value times one plus the interest rate to the power of t. The interest rate r, in this case, is literally a rate. It's in terms of a percentage. So if, let's say your interest rate was 3%, r would be 0.03. If your interest rate was 5%, r would be 0.05, 7% then 0.07, etc. You get the idea. And t is the number of years from now, so let's suppose that you're going to give somebody $100, that's the present value and they were going to give you money in one year. Then t would be one and if you were going to loan it to them at 3%, then r would be 0.03 and then you could do the calculation right here. Let's suppose that we give somebody $100, they're going to keep it, we're going to charge them 5% interest. And we want our money back in one year. So we ask the question, what is the future value of a $100 loan, that's $100 right now, charged at 5% interest for one year? So we take 100 times 1.05 to the power of 1, and then we get $105. That is, that's the future value of a present value $100, that accumulates interest of 5%, that's going to be paid back in one year's time. So the question that this answers is, hey, if I want to put money in the bank at 5% for one year, what will I get back in one year? You'll get back $105. If I was going to loan somebody $100 and I was going to charge them interest of 5%, how much would they have to pay me back in one year? $105. So this is the relationship between future value knowing your present value. Now we can flip it around, okay, and ask what's the present value of your future payment? In this case, we have the future payment labeled as C, okay, the cash payment. And the present value of a cash payment made in t years is your cash payment divided by one plus your interest rate to the power of t. So let's suppose someone says, hey, I can give you $100 in one year. In this case I've got I can give you $1, one year from now. Okay, you're going to give $1 a year from now so I'd take my dollar as the future payment. One plus the interest rate, 1.05 to the power of 1, one year from now, and when I do this calculation I get $0.95. What does that mean? It means the present value, where it's valued at right now is $0.95. So the present value of a future cash payment is $0.95 if the future cash payment is $1. If this is $100, then the present value would be $95. What this means is, if someone says look, I can pay you $100 one year from now, what's that worth to you right now? You'd say it's worth $95. Say, I can pay you $1 one year from now, okay. That's worth $0.95 to me. That's the relationship between present value in future value.
