[MUSIC] So let's get started and take a look at what we will first define as an ordinary annuity. Now there are three conditions that define an ordinary annuity. First, equal amounts. Second, made in equal time intervals and third, occurring at the end of each period. So let's say, you deposit $10 at the end of each year for the next three years. And let's assume a 10% interest rate. So I want you to visualize this information of an ordinary annuity on a timeline. Okay, so we're going to draw that timeline. It involves three periods. 0, 1, 2, and 3. And we noted that you're expecting to deposit $10. Each year, of the following three years, okay. So these are, as you can see, equal amounts, in equal time intervals, at the end of each period. That's what defines this, as an annuity. And we also said the interest rate, we're going to assume, is 10%. Now, from one of the previous videos we did learn a particular formula that converts each amount into its future value. You will recall that formula for future value is simply present value into 1 + R raised to the power t. So if you take a look at each number here, this $10 can be converted into the future for two time periods. One and two. This $10 can be converted into the future for one compounding period, and the third amount is already in the future. So if we were to use this equation and plug the numbers in, what would we have? Well, we would have time periods, we would have amounts, we would have a compounding factor, so let me just call that compounding factor. And then we would have, of course, the future value, which we are after. So let's just use this particular example to understand how annuities work. So what do we have here? Well, we have three time periods I'm going to use a different color here. We have one, two, and three. So the first amount this $10. Okay. The compounding factor you can see were going to take that $10. Compound it s=for two periods, so its going to be one plus the interest rate 10% for, don't forget we're talking about two compounding periods for the first amount, two periods. And if you work this out this is 1.21. We multiply that by 10, we get the future value of this 10 to be $12.10. Let's do that again for the second deposit. We make the second deposit at the end of year 2. This time we compounded only for one year, (1.10) raise to the power 1 which is 1.10, multiply and we get $11. And finally the third amount, which is already in the future Has a compound factor of 1 and that of course is 10. And now we can add up our future sums, and what do we have? We have $33.10. That's the future value of this annuity, this ordinary annuity, at the end of the three years. Notice that the sum of these compound factors, if I add them up, it's 3.31. Now, what I want to do is show you a general formula where you can arrive at this compound factor. So we don't have to go through tables like these to come up with the final answer. And so we'll apply that formula to this particular problem. What is that formula? Well the formula says that the future value of an innuity is equal to the innuity, multiplied by one plus r, raised to the power of t, minus one over r. Focus for a moment this part of the formula. This is the future value annuity factor for R%, any R% and any T for time periods. If we apply this information in our formula here, what do we have? We have one plus the interest rate 10% for a three period annuity Minus one over the interest rate 10%, and not surprisingly this works out to be 3.31. Now you can see 3.31 multiplied by the annuity of 10, right, is going to give us. The answer of $33.10, which is exactly the same as what we arrived at the long way. So, [COUGH] At the end of the day, we can use any series of numbers as long as they are equal amounts and equal time intervals at the end of each period. That defines our annuity. If we have the interest rate, the time period, we can always calculate a future value. A quick note, there is a difference between what I defined an ordinary annuity to what is known as an annuity due. That is, the annuity due, the only difference is that the amounts occur at the beginning of each period rather than at the end of each period. So for this series, if the amounts had begun here, so the last one would not be here, this would still be an annuity, but it's a special kind known as an annuity due. What do we do for an annuity due in terms of coming up with the future value Same problem but we have a. And here's the very exciting answer. All we do is we adjust this particular formula with an extra compounding period, because now we have a extra compounding period You simply multiply this by one plus r. So if you wanted to take this result, $33.10, and simply multiply it by one plus r. So we would just be multiplying our final future value answer by one point one we would come up with The future value of $36.41. Clearly, it's more than the previous one because we have an extra compounding period. Now, I'd like to take the formula and apply it to a question. You probably heard again and again on many TV shows around the world and it's called who wants to be a millionaire. Well of course, most people want to be a millionaire and if time is on your side, the magic of compounding can help you in a very dramatic way. Let's just imagine for a moment that you are 20 years old and you work for another 50 years. How much would you have to save each day to become a millionaire? Okay, let's start with some assumptions. So remember, you are 20 years old, this is times 0, but you are 20 years old. And you're going to keep saving money until you become 50 years old, right? [COUGH] So there's Is a time period of 30 years and you're going to save every day so the time horizon is going to be daily. And lets assume an interest rate of 5%, and lets not forget that when we have daily compounding we introduce the variable m for frequency daily. There are 365 days in a year, so that's the value of m. Now, remember the interest rate is per year but the frequency is per day, so we must convert this on a daily basis. r divided my m takes us to .05 divided by 365 and that gives us a daily interest rate of .00014. What about the time period? It's no longer 50 years, it's 365 days multiplied by 50 years and that gives us a value of 18,000 250 periods. That is you're going to make some kind of a deposit every single day for the next 18,250 periods to arrive at the future amount which we like to be $1,000,000.00. How do we solve this problem? Well, the problem actually is really quite easy because as we said earlier on this happens to be an annuity. So, we're looking for this annuity, A all the way right up till here, and all we have to do is substitute in this wonderful formula we have. We know the future value. The future value is a million, and we're solving for a, multiplied by, let's not forget our formula, we're going to use this particular formula So we have 1 + the interest rate. We just specified that to be 0.00014 raised to the power, let's not forget the time periods, 18,250 minus 1. And this factor is divided by the interest rate. And we work this out, what do we get, we get a value for a which is $12.25. So, if you deposit $12.25 every single day, you become a millionaire after working for 30 years. Notice the result is extremely sensitive to the interest rate in this equation. Instead of using 5%, what if we used a different rate? I'm going to assume for a moment we are able to earn, 12%, okay. Why 12%? Well, there's a whole segment in our course where we're going to discuss about earning superior rates of return even in low interest rate environments. In the last 50 years for example is not difficult for investors to earn between 10 and 15 percent depending on how much risk they want to take on but that's another subject. If we could earn 12% and we plug in this rate into our formula here, you know what happens to your annuity? Your annuity goes down from $12.25 a day It goes down to just 82 cents, per day, less than a dollar. That's a price many young people tip other people, at least in North America so they don't have to carry around change. Now depending on your age, you can calculate how much you need to save each day. Or you can figure out how much you need to save every two weeks when you get paid, so that you can pay yourself first this amount for your retirement before you pay anybody else. That's not a bad takeaway from this example. Pay yourself first. Pay everybody else later on. You're more important than everybody else. So, do this automatically if you can. Take it out of your paycheck, put it away. If the compounding time periods and interest rates are on your side, small amounts results in huge future values. In this case, $1 million.