An annuity has equal payments, And there's lots of annuities around, right? Like a mortgage has this part of it in annuity. Most debt has as part of it an annuity, and then of course, there are actual annuities. So let's just do a quick example. How much, Would you pay for $100 a year for 3 years? Well, this is the same as how much you would pay for $100 in 1 year, Plus $100 in 2 years plus $100 in 3 years, and that's all there is to it. If we want to get fancy, we can think about a formula for an annuity that last 40 years. The present value is C over 1 + r + C over 1 + r squared, C is the cash flow, + now, we have an ellipsis because we have an arbitrary number of terms going out to some amount t. Now we could clearly plug this into a spreadsheet for any C and any r, and get a number. Well, we can also use the reasoning of geometric series and there's details in the notes, if you're interested to derive a formula for this. What it is, is C over r times 1-1 over 1 + r to the t. Sometimes it's useful to define something called an annuity factor. An annuity factor is simply 1 over r times 1-1 over 1 + r to the t. An annuity factor takes any cash flow C and tells you how much would you be willing to pay to get C every year 40 years. So by the way, in the example that we did, so let's say r i= 5% and t = 3. We're going to calculate the present value of $100. So the present value is 100 over 0.05 times 1-1 over 1.05 cubed and that is equal to $272.32. So let's do one more example with an annuity. Let's say you take out a 15 year mortgage. So we have 15 year mortgage, The loan amount, Is 0.5 million, the interest rate is 4%. Now here's the thing, in this case, the owner of the asset is the bank, the cash flows on the assets to C, that's your payments to the bank. So in solving this problem, what we are doing is we are setting C so that the value to the bank is correct. So the present value is the loan amount. Present value has to equal C times the annuity factor for 15 years at 0.04. Well, we can separately calculate what this annuity factor is, it's 11.12. So that tells us that C must be 0.5 million divided by 11.12 or $45,000 per year. Now suppose we wanted to calculate the future value of an annuity, that's not hard to do. Suppose we want to calculate the future value of anything. Suppose we want to calculate the future value of blob. Well, what we do is we take blob and we multiply blob by 1 + r to the t. So to calculate the future value of an annuity, all that is, is my present value, Times 1 + r to the t. So notice this present value, that's what I've substituted in for blob. So something very interesting happens to an annuity, when t goes to infinity. So remember the formula for the annuity? The formula, Looks like this. Remember what lies behind the formula? But we take the first cash flow and discount it back 1 year. The second cash flow and discounted back 2 years, and then dot dot dot, we go out to the last cash flow and discount it back t years. Okay, so that's what happens for our annuity formulas. So now we're interested in the limit as t goes to infinity. So let's think about this first equation. If r is bigger than 0, what does that tell us? That tells us that 1 over 1 + r is less than 1, we're actually discounting this. So as t goes to infinity, That implies 1 over 1 + r, so the t approaches 0, And we get as a result, one of the most beautiful formulas in all of finance, the present value of a console. A console is the limit of an annuity as t goes to infinity. Present value is simply C over r. Note what we assumed. For this formula, we really need r to be bigger than 0. For the annuity formula, you don't need r to be bigger than 0. It may look like you do but you actually don't, you can play around with it. But for this you need r to be bigger than 0, why is that? Let's remember what lies behind the formula? We're taking C discounted back 1 year, C discounted back 2 years, C discounted back 3 years. Well, I could just keep going forever and ever, but the key point is as long as our is bigger than 0, these terms are getting smaller and smaller. If r were equal to 0, they wouldn't be, and we'd get to infinity. Or if r were less than 0, they'd be getting bigger and bigger, and it would be even worse. So we do need r to be bigger than 0 to define a console. Let's do some examples of a console. Suppose that r is 10% and C is a $100. The question is how much money would you be willing to pay to get a $100 every year forever if the interest rate were 10%? Well, a $100 every year forever at an interest rate of 10%, that's a console. So we use the formula at C over r. In this case, it's a 100 over 0.1, or $1,000. What if the interest rate double to 20%? Well, in this case, the present value would be 100 over 0.2, or $500. So if the interest rate is 20%, $100 per year forever, that's only worth $500. 2 points to make about the console. First of all notice the interest rate doubles, the price halves. This is a general point about present value formula has, the higher is the interest rate, the lower is the present value. And that's simply because the higher the interest rate, the more you could earn in the bank, the greater is the time value of the money. And so the less you need to put aside today, to have a certain amount of money in the future. So this is a very simple example, but we can think about some of its real world consequences. For example, if the Federal Reserve announces that interest rates are going to be higher or at least at this target interest rate is higher. Sometimes we observe stock market declines, that shouldn't be mysterious. If we have higher interest rates, that just means that the time value of money is higher and so future cash flows are worth less. My next point is how relevant is this console formula anyway? Do we ever see console bonds? Well it might interest you that close to home, we saw something close to a console bond. So you might be interested to know that the University of Pennsylvania issued debt. And the maturity of that debt was a 100 years that's longer than the US government. So University of Pennsylvania, who knows what's going to happen to the US Government, but 100 years from now, we're still going to have the University of Pennsylvania. Now that's a 100 years not infinity, that's not a literal console. But there are in fact literal consoles. So the United Kingdom issued a console bond as a war bond just like the United States issued war bonds to pay for World War I. So the United Kingdom issued console bonds to pay for a war. In this case, the Napoleonic Wars. And in fact, those console bonds were still paying their coupons up until very recently until the government decided to buy them back because it just wasn't worth the bother anymore, so these consoles really do exist. As an approximation to the console, we can think about preferred stock on a company. Those cash flows tend to be known for sure as opposed to say common stocks, and we can think of them as going on for a long time.
