Let's study the annuity formula on the example of fixed-rate mortgage in some more detail. Well, put it here, fixed-rate mortgage. So it has the following parameters, let's say that the mortgage amount is P sub 0. Well, this is the overall amount of the loan. If you buy house and you put a down payment, this is the remainder. For some special mortgages that have zero down payment, this is the whole amount. Now, mortgages are very long-term loans, so t is the time over which this mortgage exists. Normally, the mortgages are 30 years. And historically, they make the payments monthly, so that translates to 360 months. Now, What else do we have? We also have the rate on this mortgage. And again, normally, it goes like this, that the rate is quoted as annual. Although, because the payments are monthly, we have to somehow, from this, we have to come up with a monthly rate. Well, historically, that was a special agreement that actually this r monthly = r cap/12. Well, this is because mortgages existed long before people started to use computers. So we will study the following example. We'll say that P sub zero = $500,000. Then this t = 360 months. And then we'll study two rates. One rate will be 3% a year, which is close to what people observe right now, and that is 0.25% monthly. And the other will be, let's say, I will put it with R, Cap, 6%, and that will be equivalent to 0.5% monthly. Well, we know that the formula was put up on the previous chart. And then I'll just come up with some results. Well, we can say that the equation looks like this. P sub zero (r) = C/r (1- 1/(1+r) to the t-th power), this is the formula. So the answers, I did the calculation, so we know this. We know r, we know t, we have to find C. Well, now, the answer is that C(3%) annual = $2,108, and then C(6%) = $2,998. Now, is this good or bad, a lot or not a lot? We can compare that, how much money we will have to pay out over the life of this mortgage. And we assume that we hold it for 30 years, and we make all the payments. Now, you take this amount, 2,108, and you multiply that by 360, and I will put not this total. By the way, the total amount here will be, I will put total. It's $758,887, but for us, the most important is the ratio of total / P sub 0. And again, for 3%, I will put it in black, this = 1.52. And if we did the same set of calculations, now the total for the 6% mortgage will be over $1 million. And again here, the total / P sub 0 = 2.16. So you can see that, actually, if you take out a mortgage, you can enjoy living in this house right now. But you take a liability, and the overall amount of money that you'll have to pay over the life of this obligation is much higher. It's 50% higher at a low rate, and more than double at a higher rate. And again, 6% is, for now, in the developed country, is a high rate. But there were times when these rates were in double digits. In some developing countries, they're in double digits, and then this coefficient will go sharply up. And oftentimes, people, especially people who are not very familiar with that, say, well, this is not fair, to pay so much money. But the other option is to save all this amount and then buy this house for cash. But while you're saving this significant amount of money, you have to live somewhere, you pay rentals. And therefore it's an open question, what is fair and what is not fair? Corporate finance approach tells you that it is fair to pay this amount of money if you can take this mortgage. Now, another important angle at which we can take a look at that is what happens to our balance over the life of the mortgage? The general formula says that at the moment k, the amount of credit outstanding, Pk = P sub 0(1+r) to the kth power- c/r. And here in this comes [(1 + r) to kth power -1]. So this is a little bit more cumbersome formula. Well, it can be sort of interpreted like this is accrual of interest on the principal of the mortgage. And this is how these overall payments, they are actually combined together. Because you can see that this is sort of like a finite sum of these payments that take the form of a geometric progression. But, well, I did some calculations, and let's see. So here, that will be for r =3%, in black. And this will be for r = 6%, in blue. So first of all, I put P60, so what happens after five years? So for this one, the remaining amount will be equal to 444,533. And here it will be higher, it will be 465,255. So you can see that you've been paying for five years. And the amount of outstanding does fall, but doesn't fall much. Now, what happens after ten years, which is P120? Here you have some reduction. Here you have 418,388, so even after ten years, you've been paying for a third of the life of a mortgage. And still you will have paid off less than a fifth of the outstanding balance. And then, finally, P240, which is after 20 years, or two-thirds of the life of the mortgage. Here you will get 218,319, so at least you've jumped over the threshold, because this is less than half. But here you will still have 269,903 outstanding. So it's kind of, well, I wouldn't say funny. Unfortunately, you can see that you've been paying for 20 years. That's two-thirds of the life of the mortgage. And still, at 6%, your remaining balance is still more than half. So that shows to you the power of discounting or, if you will, the need to pay all these interest payments. The story is very simple, again, it can be analyzed in this formula. So let's say you have 500,000 outstanding. Then you make the first payment of whatever, close to $3,000 in this case. And this 3,000 is almost all interest. So after the first payment, the balance changes just a tiny bit. So the next time you accrue interest on this. And then slowly, step by step, you start to eat out pieces from this principal. And then by, let's say, in two-thirds of the life, you will end up at about half of that. And then, then the mortgage balance will go down progressively faster. So these things are used in actual calculations in many organizers. Now, on the Internet, you can find all these mortgage calculators that show to you the amount of payments, monthly payments as a function. You feed with the rate, you feed with the you feed with the life of the mortgage, and it produces you with c. And some of these things can be easily found too. Well, in the real world, some mortgages, especially now, they are more advanced. They are not fixed rate, they are adjustable rate or they are graduated payment. But all these things are analyzed with the use of the NPV approach. It's much more cumbersome, but the general idea is the same. Now, we are wrapping up this episode. And in the next one, which is the final in the first week, we will come back to the overall idea of NPV. And we will see what we already know and how we will proceed from there, again within the paradigm of fast takeoff. We will see exactly that although we studied just a little bit, but we already made some advancement. And we will be able to proceed very quickly in some important areas. Namely, in the areas of bonds and stocks.
