[MUSIC] Let's now turn to reversing this problem by calculating the present value of an annuity. Again we can visualize this three ordinary annuity. It can be actually any number of periods you wish. I'm just choosing three for simplicity. We had an annuity defined as equal amounts in equal time intervals at the end of each period. So there we have it, but this time we're discounting these back, we're bringing them back into today's dollars, or the present value. And we know, that to bring each number back, we can use a formula that we've already been accustomed to, and that is, you take the future amount, future value, and you divide that by 1 + r raised to the power t. So if we did that for every single number that you see here, again, we can go the long way, right. So we have time period, we have amounts, we have the present value factor, also sometimes known as the discount factors. And then, of course we have the present value. Okay? So let's plug the numbers in. We have three time periods and we have an annuity three amounts. So we divide each one of these by the factor, and the factor is as you can see, is going to be one over one plus r raised to the power t or one divided by 1.1. Note I'm bringing this back for one period. Okay, the factor, if you would do this calculation, works out to .9091, multiply it by 10. And I get $9.09. Let's do that again, this time for two periods. I'm going to take the second $10 expectation divided by one over one plus r raised to the power two. Work this factor out. It is .8261 multiplied by ten dollars to get $8.26. Again, do it the third time, one over 1.1 raided to the power of three. This time it works out to .7513. Multiply it through, and I have $7.51. If I add these up, I get my present value. It works out to $24.86. That's the present value of receiving $10 each year for the next three years. What's interesting here is not so much the Individual calculations I did but in fact to show you that if you sum these factors up once again if we sum them up we're going to get 2.486. What we can do is generalize this whole process with one particular formula which is the formula for the present value of an annuity. And that is simply equal to the annuity multiplied by one minus one over one plus r to the power t divided by r. And this here is the present value annuity factor for r% and t periods. Okay, let's apply that to our particular problem. We have, of course, an annuity of 10, and we have a factor which is based on 1 minus 1 over 1.1, raised to the power of 3, divided by .1. Not surprisingly, this factor works out to 2.486, which multiplied by 10, gives us the result 24.86, which exactly is the same answer as we did the long way, right? Let's not forget the difference between an ordinary annuity and an annuity due. Last time we said that an ordinary annuity is one like this but an annuity due is one which begins at the start of each period right. So you still have a three period annuity but it's an annuity due. How do you adjust for that? Well you can see the numbers are closer to present value so the value will be higher and like before you simply multiply this by one plus the interest rate. And if you do that you're going to get an answer of $27.36 which of course is higher than $24. Let's now apply this present value annuity tool for let's say purchase that you have in mind. You going to go shopping, you go to a mall. Let's take a universal example of a product that every young person in lining up to buy. And let's go to the most valuable company on Earth. Could that company be Apple, and could the product be an iPhone? Let's just assume the price of the iPhone is $1100. It's fully loaded. But you can't afford that. However, when you notice a sale sign at another store for $1000 you march right in and you want that iPhone. Now the sales person takes one look at you and somehow figures out you can't come up with $1000 right now, but they make a sales pitch as every sales person does. And they ask, well what about, what if you pay less than a dollar a day to have the phone? Will you go for it? Notice a moment ago that if we can put about a dollar day for our future, we can accumulate a million dollars. So now, of course, you're out there to spend money, not save money. So, let's come back to our problem. You ask the salesperson how long will you have to make these $1 daily payments, and you quickly learn about this method. Known as payments by installments. Most retail outfits use this kind of method. So in this particular example what information do we have? We have the information that the phone is worth a $1,000 right now. So that's the present value. It's $1,000, right? Let's take a time period. Let's say that the salesperson offers you a three-year repayment plan. And let's just assume for simplicity the interest rate is 10%. How is the payment going to be computed? First of all Typically these payments are made monthly, so let's not forget we have a frequency now of 12 months in a year, so m is equal to 12. How is the payment computed? The payment is based on, of course, the loan plus the interest you owe, divided by the number of payments. So let's put the data into this particular equation. Sounds very logical I'm going to take out a loan for 1,000 and my interest is going to be, let's not forget 10% of 1,000. So I'm going to owe $100 in interest. But I am taking this out for three years so I'm going to owe $300 in interest, and there are 12 payments a year for a total of three years. That is 36 payments, let's do the math. And we come up with an answer that is equal to $36.11 per month. So, this calculation is quite useful because a lot of countries use this methodology. And, you know what? In fact, you're ending up paying a lot more than what you could have had you worked with annuity. Let's see why. So here we're looking at an amortized loan. And in an amortized loan what you have, again, is an annuity. You have fixed amount in fixed time periods. But in this particular example that we're working with, we can try and find out what those amounts are. Notice on the timeline that we have the present value of the loan right here at times zero, that's the $1,000. And then we have our time period, there was a three year problem with M equal to 12, so we have 36 periods, that's on the timeline. And, the interest rate, let's not forget, is monthly rate, an annual rate of 10%, divided by m of 12. So we use our formula, the annuity formula that we're familiar with, plug the numbers in, and we get our monthly payment of $32.27. That's this particular column that you see on this table. We started with a loan balance of one thousand. We made a payment of 32 dollars and 27 cents. Now if we want to know what portion of this payment is interest we simply can multiply the monthly rate by the beginning balance and you will see that result is $8.33. Of course, if you subtract the interest from the total payment, you get the repayment of your principle which in the first period. In this case, a month, is $23.93 and that leaves us an ending balance. Since we have paid back 23 from our 1000, we are left with $976 as the beginning balance for the next period. And so, 976 becomes the next amount from which we calculate our interest, use our monthly interest And calculated for the second year. Take the difference between the two. Notice that your total payment is always equal to interest plus the principle. So, you do that again, you get the difference. That produces your ending balance, okay, and we continue to do this for 36 times. So what do you notice in this amortization table is of course that your interest payments keep on decreasing over time. That's because your balance is getting lower and lower and your principle repaid is increasing over time. If you add up the 36 payments that you have made It will be equal to $1,000 which was your loan, so you've paid back your entire loan. There you have it, this is commonly how any amortization payment schedule is figured out, whether it's a short term loan in this case for three years or whether it's a very long term loan for your home for 30 years, more or less the calculations are the same. So with the annuity calculation, you can see that the amortized loan payments of $32 that we computed in the table, is in fact, that 32 is lower than the $36 that we calculated using the installment payment method. So you might be saying, well, there's a $4 difference there and that's not probably such a big deal. But in fact, it is a big deal. And I want to show you how by demonstrating the effect of the differential payment on the interest rate, okay? So what I'm going to do now is use our trusted formula, but plug in the installment number to see what that implies in terms of the interest rate that you are actually paying in this long. So let's put the numbers in. I've a got a loan for a 1,000. And I know my payments, according to the installment sales person, is $36.11. And what I'm looking for is the interest rate. So I've got 1 minus 1/1 + R. I'm looking for R. I know that this works out to be 36 periods divided by r. So I've got an equation with one unknown. With my trusted calculator, I can figure out what that rate is and that rate actually is going to be 1.493%. This is per month. Remember if you want to annualize it, you need to multiply it by m = 12, and that works out to be 16.6% annually. Now this is the nominal rate you're paying. You're not paying 10%, you're actually paying 16.6%. And you might say, hey wait a minute, there's another formula you showed us earlier on. And that formula, if I remind you, is taking into account the compounding within a year. That is the effective annual rate formula. And that was 1 plus the nominal rate, R, divided by M raised to the power M minus 1. Hey, what if we plug these numbers in our formula what do we get then? Well what we get is an effective rate of one plus the nominal rate,16.6 divided by m which is 12. Raise to the power m, minus one. Figure this out and what do you get? You get pretty close to almost 18%. The exact rate is 17.92 or about 18%. Now, mortgage payments on financing a home are typically computed using the same amortization procedure as I have described. The difference is the size of the loan and the time period. For a house you typically borrow 100 times more than an iPhone, and your loan would therefore be outstanding for much much longer, maybe ten times longer. Instead of three years, it would be lasting for 30 years. We saw earlier that the key variable that is constantly changing and affects the value of the transaction is the interest rate. So if you look at this amortization schedule very small changes in the interest rates will have a very large impact on your payment schedule. This is why home owners have recently been on kind of a spending spree in many parts of the world as interest rates or mortgage loans have historically never been lower. Some banks in Canada, for example, are lending at about 2 percent on a thirty year loan, and you can actually fix this rate for five years. Can you imagine that? Historically we've never seen this happen before. Rates are even lower if you have a variable rate loan. Let's step back for a moment and take a look at the big picture... The housing and commercial real estate market is huge not only in modern economies but around the world. And it implicates so many of us. In a later video on risk we will revisit the idea of these very low interest rates that were offered to everyone and how this led to this devastating American subprime mortgage crisis. In 2007, the effects of this crisis was felt around the world and as we will see we seem to have recreated condition of an completely overvalued sector where some experts think real estate may be 30 or 40% higher than it should be. If you believe that it is overvalued then while you might save yourself some money with lower payments as you shop around, these savings are completely going to be wiped out if the value of homes decrease significantly. So looking at the big picture is just as important as shopping for low interest rates.