Albert Einstein said that compound interest may be one of the most powerful forces in the universe, true story. Let's kind of look at why this is so important in a little section I like to call, the Time Value of Money. What will $500 placed in an account earning 3% interest be worth in five years or in ten years for that matter? How does that work? Well in five years it's going to be about $580, and in ten years it's going to be about $672. How do we get to those numbers, though? Because that's not a straight forward application of just x percent per year times the number of years. It's a little more complicated than that. And that's what we're going to refer to as the magic of compound interest. Now why is that so much more? Compound interest refers to the idea that we put money somewhere and the money is going to earn money, right? So our interest earns interest, and then that interest earns interest. And you can see where this goes, right, it become a very big effect. In fact, mathematically we're going to see that this is an exponential function. The relationship then between present value and future value of a one time payment is what we've just been exploring. So, money today, right, times 1 plus a growth rate, which is going to be a decimal form, raised to the power, right? The number of years, and then that's going to be equal to our money tomorrow. So, that's what we want to think about then. That's that basic math function, that we call the Future Value of a lump sum. Now, this can work backwards too. Say, I want to have $1,000 in five years, and I can earn 5% on that money. How much do I have to put in today? So think of this as a goal we're trying to reach, and we need to operationalize that goal. We need to figure out how much I have to put in, how much of that money I have to defer to the future. How do we get there? Same math, right? $783.53, now how did we get there? We'll take a look, right, $1,000 divided by 1.05, which was our interest rate, right, 1 plus the interest rate raised to the power of five, because we said this was going to be when we want to have that money. So, the formula for that is the future value divided by 1 plus the percent, raised to the number of years. Same equation, just manipulated with a little bit of algebra. So this works forwards and backwards when we think of future values and present values of one time dollar amounts. So we have two key concepts that we're introducing right now then. The first is when we go forward. When we take money today and have it grow towards the future, we refer to that as compounding, right? So, in the first example we started the presentation with, we had compounding interest. When we started with the future and wanted to come back to the present, that's discounting, right? So, and either way we're talking about this multiplier or exponential effect of compound interest, whether it's compounding to the future, discounting back to today. We're going to use those terms a lot in this course. So just want to make sure we're very comfortable with them. So, when something earns interest, the interest earns interest. This increases the value exponentially. The fact that the interest was earned means that it needs to be factored into saying, what would that have taken today to get to that point. So right, again, in either case the math is the same, it's just a, it's just how we manipulate the equation to get from point A to point B. So we have three types of amounts that we're going to focus on in this course. It can get infinitely more complicated. It can compound. So the first is a lump sum. This is a one-time payment, right? We put something in, we get something out, we want something in the future, we have to put something in today, that's the idea. That's going to be contrasted with an annuity, which is a fixed recurring payment. We add money regularly, we receive money regularly, either pattern works, and we'll look at some graphics to help to shed some light on that. And then an annuity payment, at least for these purposes, the dollar amount is always going to be the same, okay? So this is the way when we talk about this mathematically. Let's not confuse when we use an annuity in this term, that we're talking about the insurance product also called the annuity. The mathematics certainly apply, but they're not synonymous by any stretch of the imagination. We also can have a series of cash flow, which means money taking place on a regular basis. But it's not going to be fixed amounts. That gets a little more fun and complicated as you can imagine. So let's look at a timeline. I like to use maps, that can be very helpful as we're moving forward. So we begin right with the present value, the amount of money we had today. Over time right, we have time and our interest rate per year, the amount that it's going to be compounding by, the amount of change every year and the number of years with which that compounding's going to take place. And then, of course, we're going to end up with our future value, the amount of money we have at the end, after compounding. So present value represents before compounding, future value represents, after compounding. Now, how's this work, algebraically? Because I'm sure, you have a burning question about, what the algebra looks like, and I'm happy to show you. So number one, we would have this again, the future value of a lump sum, is equal to the present value, times 1 plus the interest rate raised to the number of years or the number of periods for compounding. When we simply reverse that algebraically the present value of a lump sum equals the future value divided by one plus the interest rate raised to the number of years. So again algebraically right the equation is the same, it's a simple algebraic manipulation of the two. So, we can certainly look at those, and we'll continue to play with the numbers throughout the course, I promise. I also want to point out what an annuity might look like, so with the annuity, right? The big difference is we have a series of payments potentially going in, so this would be the future value of an annuity, dollars going in on a regular basis, time, right, over time, number of periods that we're having, so that would represent the number of payments that we're making, and of course the interest rate that we're having per year, and as always, ending up with a future value. Now, what would be an example of this? Regular savings plan If you're putting money into a 401k plan every paycheck. These are all examples of what we might think of as when future vie of annuity math may become relevant. The opposite of this or the other side of this right was the present value of an annuity and this would be that we have money that we start off with, right, and then it generates a series of payments. Now one way to think about this, although it's not money you're earning, is a loan. You get a bunch of money today, right, you've borrowed it, you get that amount of money. Then you're going to have to make payments on that money over time. So if you've ever had a car loan, that's a great example of this. You got money from a financial institution, to buy your car, they gave you money. And then, you're going to make those payments over time, and you're going to pay interest for it as well. So again, future value of annuity, present value of annuity. So, we have money, potentially upfront generating payments or we're putting money in, on a regular basis, to end up with an amount in the future. Either way, right, again, we can represent this algebraically, it's a little trickier, of an equation. Lets take a look. The future value of a payment, and we're going to use the term PMT, or payment, to represent our annuities. So the payment represent, right, times 1 plus r, raised to the power n, minus 1 divided by r. That big mess as I call it right is going to help us to understand what the future value would like of that stream of payments. The present value right again, a manipulation of the same equation. So it's the same math. You can memorize really two formulas for this if you want, right? Just be able to manipulate them algebraically. So the payment, right, would be taking the mess and simply subtracting it from 1. So 1 minus 1 over 1 plus r raised to the power n, right? Again, all times the payment divided by r. So, algebraically can get a little messy. We're going to have some practice with that too. And in this case as always, we have the payment, which represents the annuity, the recurring payment, we have the number of periods, and we have the interest rate, which is represented by r here. Now, we're going to often use a financial calculator in this class and for a lot of our demonstrations, I'll be showing time value of money calculator keys. So, just to orientate you with some of the keys we might have and I'm just going to share with you some examples here, too. Number one, you can go zenwealth.com for the t, time value of money calculator, I tested that out. That's a pretty good one. It's free. For my demonstrations, I'm going to use a calculator called TI BA II+ and we'll show an example of that in just a moment. But again just to familiarize yourself with the keys that you're going to see, the number of periods, the interest rates per year, the present value, the payments, the future value, and you'll see on the calculator itself, right? We have those various keys all right here, spelled out. So we'll simply be entering them in on the calculator and solving for the unknown. So again, you can do it either way. You can do it algebraically, you can use the online calculator. You can purchase a calculator, and you can purchase any one you like, but I only am going to probably demo just the one, just to keep it simple or you can download an app if you happen to have a smart phone. I do always recommend if you have a smart phone to go ahead and consider getting the app, and that's really because it's a fraction of the price and its the same darn calculator that you can buy in physical form. Plus, if it's on your smartphone you always have it with you, so you can always do your homework, isn't that great? Few others you can check out too, ultimatecalculators.com can be a good one. Investopedia has a pretty good set of calculators as well. So check out any of those if you want to. A few other tricks that we'll kind of learn if you're using a calculator, is when we have to tell our calculator a few tricky things. If I'm making monthly payments, for example, on a loan, I would have to change my calculator setting to 12 payments per year. If I'm making payments at the beginning of the period or the end of the period, I'm going to have to tell my calculator that to, why that one? Well, think about it for a moment. If I put money in at the beginning of the month versus the end of the month, what's the difference between the two? If it's in there at the beginning of the month, that's a whole month of compounding. If it's in there at the beginning of the year, that's a whole other year of compounding. So we have to kind of consider these basic things because that's going to be a big difference. If I put money in at the beginning of the year. And then it compounds over the next 20 years, versus putting it in at the end of the year. You'll actually see a pretty big difference between the two. Not astronomical, but big enough that makes you think, I want to put my money in upfront when I'm saving for the future. A few other things too, on the calculator side. We want to distinguish between cash flows. So, and this is good on your time value of money charts that we were looking at earlier, those money timelines if you will. Cash outflows we want to designate as being a, a minus. Now this could include an investment you're making, payments you're going to have to make, but we want to hit the plus minus key on the calculator. This isn't really saying that the number is less than zero it's an accounting term, right these are debits. These are monies that are leaving us for the moment, even if it's an investment we're still taking that money and putting it over here for now. Money we're receiving should be positive on the calculator. So we always want to make sure that we're distinguishing between the two right,positive. Versus negative. So think of the negative number as a, as an outflow for example or an investment. A positive number's the return. It's not about being more or less than zero. So when we're doing time value of money problems, which I promise you're going to get to practice, draw a cash flow chart. Think of the ones we were showing earlier. For present value of an annuity, future value of an annuity, thinking about future value and present value of a lump sum. Break the problem, when you see it, into one of those four types. Right, that's what we begin with this class to make sure we have a good, strong understand and foundation of time value math. So we break it into one of those and sometimes it's more than one of those. So just as a little hint for the assignment that you'll take a look at. Figure out what type of information you have. What's the interest rate? What types of things are you looking for? And always know what it is you're trying to solve for. Sometimes the wording can be tricky because life isn't always going to put things in present value and future value terms. So we sometimes have to think about, is that before or after compounding would take place. Remember, if it's before compounding would take place, present value. If it's after compounding would take place, future value. And we can solve for any variable this way. Right. So, Peter Parker bought a car in 1990 for $8,500. The same model of the car sells $18,500 in 2006. What was the annualized rate of price increase. We can set it up this way, right, and again try and solve for that algebraically. We can also use our calculator. $8500, remember, right, if that was money we would have spent, we hit the plus or minus key, future value of $18,500. No payment here, right, this is just going to be future value and present value. I like to zero out the payment key, just keeps me honest on a particular problem. The number of years, if you were paying attention to the problem 16. The last button you push on a calculator is going to be the thing you're solving for. So in this case, our interest rate per year or the rate of change between the two times is 4.98, almost a 5% rate of change. In this case, we're talking about, right, this is how much the car appreciate or the car value changed over those 16 years, right? This was about roughly a 5% rate of increase in the price. Now, [COUGH] in life we would beg the question, how much of that is attributable to inflation, right? Inflation, as we know, is a number that's used to track the change in overall prices that we face as a consumer. So what would be the rate of inflation in this case, or more importantly, what's the inflation adjusted rate. We call that real rate of return. So we're again going to use this as an example do distinguish between real and nominal. Real is really what we talk of is measuring change in purchasing power. So nominal would represent the change in actual dollars that we're talking about. So nominal is often going to be greater then real, often. Real is going to be after we've accounted for inflation. So our nominal rate was 4.98%, right? Meaning that that was the rate of change for the car irrespective of inflation. If we want to know the amount above and beyond the inflation of the rate of change for a car, we have to make a slight adjustment. And the way we do that is if inflation was running 2% during that period, then the real rate of return would look something like this, right? 1 plus little r, which is going to be right our real rate of return, that's going to be equal to 1.0498 which is the equivalent of saying one plus our nominal rate of return, divided by 1 plus our inflation rate. When we do all of that algebra, we solve and we end up with 0.0293, to put that into percent form, we have 2.93%. So in other words, while the nominal return or the nominal change in value for that car was 4.98. The inflation adjusted return on or, change on that car was 2.93. So let's have a few other examples here for a moment. Austin Danger Powers would like to have 1 million dollars someday, if he can 8% and deposit $250,000 today, when will he make his goal? Now, if you want, pause this for a moment and maybe try this as an example on your own. This one has some tricky things associated with it, right. We know how much money we have to have, we know what he put in there, we know what rate of return he can have, so what, when will he make this goal if this is what he can do. So we're solving for time, that's kind of an interesting thing. When can he get there? So how would we do that, mathematically? Well, on the calculator, right, again, we start off with $250,000, we end up with $1 million, we have no payment again that we talked about, the interest rate was given at 8% per year, and that when we solve for n, we're going to end up with 18 years. Now, a little caveat, if you don't put in the plus or minus sign on your calculator, the calculator is going to tell you, I have no solution for this. And that's because in order to develop, or calculate a rate of change, the calculator needs to be able to distinguish between the money that was going in and the money that is going out. So that plus minus thing isn't just for your own sanity's sake, it's so that your calculator can actually do it's job for you. Without that simple information it cannot identify a solution to the problem, even though we could have done it algebraically at the same time. So how does this stuff get tricky? Does that seem easy I'm sure for now. Several ways, we can have all of the variables operating at once. So far we've looked at examples that say, a present value or future value, annuity payments or not. So we can have problems that'll use all five variables. I promise you the practice problems set will have all five variables to use on one or two questions. You could also have uneven cash flows too. So, again, money occurring regularly, but not fixed payment amounts. Could be $500 this month, A $1,000 next month, $300 the third month. That's going to be a little bit different type of math that we'll delve into in another class. But let's take a look at one example here first. Jack and Diane, this is going to be a little ditty for you, have $10,000 in their money market account. If Jack and Diane plan to deposit $1,500 at the beginning of each year for the next 15 years, and the account earns 5.85% annually, how much will they have after 15 years? So let's think for a moment. What's going on in this question? Well, we've got a present value. They have $10,000 going on. They're putting money into the account regularly. That's a payment. That's an annuity. We know we're looking for how much money they're going to have, so we are solving for a future value. So we have the future value of, right, two things, present value and an annuity payment. So how's that going to look for us. Algebraically, right, we can simply break it down in to those two questions and apply the formulas. From the calculator standpoint, a few key things. First of all, we need to tell the calculator it's a beginning of the month payment. Remember, that means we're going to account for that extra period of compounding. We have to put in the $10,000 present value, again, referring to it as a cash outflow, even though the money is in our account, we have to think of it as an investment, so plus minus. The $1,500 we're going to add to that. Has to have the same sign because it all represents money that's going towards something. So, again, plus or minus again for that particular one. Interest rate we were told was 5.75 per year, so we go ahead and plug that in to the calculator. And we're doing this in that order in that sense. Type in the amount. Then push the button. 5.85 then hit interest rate per year. 15 was the number of years we were talking about and of course, the future value, if we're in beginning mode, is $59,999. If you were trying this on your own and you happen to get $57,979, check to see if you were doing it with beginning mode versus ending mode. Again, it's going to make all the difference in the world. In this case, it made about a $2,000 difference. So, do kind of keep that in mind that we want to account for that simple act of compounding at the beginning or at the end of the period as part of this. Another fun thing I like to share when we talk about the time value of money, I refer to this as the only party trick that we have in the repertoire. It's called the Rule of 72. This is a simple type of approximation, but it works like a charm. You can always verify it with your calculator or algebraically, you'll still get the same answer. This can be used to determine the simple length of time it takes for any investment to double. If you give the rate of return right, we can come up with it irrespective of the dollars involved. In fact the formula's very simple. The number of years it will take for any amount to double is simply equal to 72 divided by the interest rate. I know you're thinking there's no way this works, on contrary, it absolutely does. If Steve can earn 6.5% annually, how long's it going to take for $10,000 to double. The first thing I'll tell you is, does it matter that it's $10,000? And the answer's no, right? You're going to see that if we actually apply the rule of 72 formula the dollar amount isn't going to be used. But we can also do it in our calculator to double check. So we could do $10,000 plus or minus present value. We could do $22,000 as a future value, that's the doubling. Six and a half percent per year, which we were told in the question, and if we solve for that we get N equals 11, or simply take 72 divided by 6 and a half and you get 11. Same answer, and you can try it all the time until you're exhausted, you'll still keep getting the same answer. I told you, it's a pretty fun little party trick. Impress all your friends.
