Hello, I'm Professor Brian Buchet. Welcome back. This will be the first of three videos in which we talk about time value of money. Time value of money is a very important concept that we use in a lot of applications, basically the concept is that the value of a dollar today is not the same as the value of a dollar in the future or the value of a dollar in the past. In this video we'll do a general overview of the concept and then we'll work on what are called future value calculations. Let's get to it. We say that money has time value because the value of a dollar today is not the same as the value of a dollar in the past or in the future. >> I'll say when I was a kid, I could fill up my truck with gas using a $5 bill and have enough left over for a grape Nehi. Back then, a dollar was really worth something. >> [LAUGH] So, we all fall into this trap of thinking that, because something cost less in the past, it must have been a better deal than we're getting today. Not realizing that the value of a dollar in the past is not same as the value of a dollar today. I even fall into this trap myself and I should know better. So I was reading an article a few months ago about the minimum wage which I think is now $7.25. Well when I first started working in 1984 the minimum wage was $3.35. And so I thought wow, it's so much higher than when I was working. But the fact of the matter is that a dollar in 1984 is not worth the same as a dollar today. And in fact, if you translate $3.35 of 1984 dollars into today's dollars, it would be $7.53, which is basically what the minimum wage is. So if you learn nothing else from these videos. And by the way, I do hope you that you learn other stuff [LAUGH]. But if you learn nothing else, keep in mind that we can't make these comparisons of dollars today and dollars in the past, because of this time value of money. Also, I should note, everything in these videos will be dollars, because I'm an American, and I'm used to talking about dollars, but obviously it works the same whatever currency you're using. Whether it's, Euros or Kroner or Yen, or whatever. [SOUND] So anyway, the reason why money has time value is because of factors like inflation, interest and risk. So inflation is that prices tend to rise over time. Interest is that banks charge you money for having loans outstanding. And so it costs more to pay something back in the future than it is to borrow it today. And risk, just having a dollar in your hand today is safer than getting a dollar in the future, just because you don't know what's going to happen. There's a lot of uncertainty, there could be another financial crisis. So all these factors combine to determine what's called the discount rate or the rate of return. Because of these factors, whenever we will receive or pay cash in the future we have to adjust the cash flows to the same value, usually today's value, if we want to compare them. So, just like you'd have to adjust foreign currencies to the US dollar, you would never directly compare Euros to dollars, you would have to translate them either into dollars or into Euros. Or you would never directly compare liters to gallons if you were looking at liquids. You'd have to either convert everything to gallons, or everything to liters. What we have to do is adjust future dollars or past dollars to today's value, to make across-time comparisons. So almost think of future dollars or past dollars as a different currency than today's dollars. And so we have to translate them into today's dollars to make some comparisons. So let's do an example. So, just proving that you can find any type of information on the internet. I went and found some historical gas prices. I want us to try to figure out who paid the highest price for gas. Was it in May, 2011, when in Fort Worth, Texas, it cost $4.15 per gallon. Also in May 2011, in Saskatoon, Saskatchewan, in Canadian dollars it was $1.04 per liter, and then if we go back to May of 1980, in Fort Worth, Texas, gas was $1.53 per gallon. Who do you think paid the highest price for gas? >> I don't know anything about these leaders, but I actually filled up with gas in Fort Worth in 1980. I tell you, gas was obviously much cheaper then. >> I know about liters. There are 3.79 liters in a each gallon. Let me research the exchange rate between Canadian dollars and U.S. dollars on my phone. >> Not necessary. I'll give you all the data you need to do the translations, but thank you for giving us the exchange rate on liters to gallons. 'Kay, so the way we're going to do this is translate everything back to May, 2011, U.S. dollars per gallon and then we can compare these three data points. So first, for May, 2011, in Saskatoon. We have to do two translations. We have to translate litres into gallon. And then we have to translate Canadian dollars into U.S. dollars, then we compare the price of gas in Saskatoon to the price of gas in Fort Worth, Texas. So the first step is to convert litres to gallons and as Veena helpfully pointed out, there's 1 gallon equals 3.79 liters. So if we've got a gas price of 1.04 Canadian dollars per liter, if we take that times 3.79, so that's 3.79 liters per gallon. The liters cancel out, and we get $3.94 Canadian per gallon. Now we have to translate Canadian dollars into US dollars, so if you go back to May of 2011 you can look this up on the internet. At that time, C$1 was equal to1.05 in US dollars. So what we need to do is take our Canadian dollar price of 3.94 times US$1.05 per Canadian dollar, and we come up with a price of US$ 4.14 per gallon, which is almost identical to what the price was in Fort Worth Texas. Now we want to look at May 1980 for Fort Worth. So here it's going to be in gallons, so we don't have to translate. But the problem is, a dollar in 1980 is not worth the same thing as a dollar in 2011. Again, think of it like a different currency and we need an exchange rate to get from 1980 to 2011. I went on to the web and if you just Google consumer price index, you can find these calculators which translate the value of a dollar over time. And basically $1 in 1980 is equivalent to $2.72 in 2011 so because of inflation a dollar in 1980 would buy $2.72 in 2011 terms. So, now we can take the $1.53 per gallon in 1980 dollars times 2.72 which is 2011 dollars to 1980 dollars. And we end up with $4.16 per gallon in 2011 terms. So even though gas seemed so much cheaper in 1980, if you think about what a 1980 dollar is worth compared to today's dollars, and you translate it all into today's terms. Basically it's the same price of gas. So whether you're in Canada or whether you were looking back in 1980 in Fort Worth Texas, gas prices have been the same in today's dollars. The key to see this though was to do the translations of both liters to gallons, Canadian dollars to U.S. dollars, and then of course, 1980 dollars to 2011 dollars. Next we're going to talk about compound interest, which is one of the big factors that creates this time value of money. There is an urban legend that Albert Einstein was asked, he's the famous physicist, Theory of Relativity. He was asked what is the most powerful force in the universe, and he reportedly said, compound interest because it makes the money grow so fast. So this slide will show you what Einstein was reportedly talking about. So in this example, we're going to invest $100 in a certificate of deposit that earns 8% interest per year. >> Dude, where do you do your banking? My bank only pays about 1% interest on its CDs. 8% would be really gnarly. >> Dude, where do you do your banking? I can't even get 1% interest rate at my bank. So the interest rates that we're going to use throughout these videos are not necessarily going to bear any resemblance to current interest rates. I tend to use fairly large numbers so that we can look at big effects. And, you know, maybe if these videos are around long enough, interest rates will eventually climb up to 8% or later, we're even going to look at 15%. So don't try to think about the rates as matching today's rates. Just think of this as just, in general, these are examples of interest rates and what the effects would be. So at the end of the first year, we're going to have our original $100 investment in the CD plus we've earned $8 of interest. So that's 8% of a 100 is $8 which means we have a total of $108 worth of value in our Certificate of Deposit. Another way to write this is it's a $100 times 1 plus .08. So basically that $100 is going to grow at a rate of 1 plus the interest rate, gives you $108. Now, I switched that notation because it's going to help us down the road. But it's just saying that at the end of each year you have your original principle. That's the 100 times 1 plus it grows by the interest rate of 8%, so that's the 100 times 0.08. At the end of the second year. We're going to have $108 times 1.08 equals $116.24. Here's the compound interest, where in the second year you get interest not only on your original $100. But you earn interest on the interest. So as long as you leave the interest in there from the first year, you get interest on that interest in the second year. So that's the compound interest, that interest starts applying to interest and things grow much faster. Because, what we don't have now, be careful here, we don't have a $100 plus two years of $8 interest equals $116. That's not compound interest because that would be only applying interest to your original principle each year. But compound interest, which is what we always have, applies to whatever interest and principle stays in the CD. And notice we get an extra $0.64 in the first year by getting the compound interest. [NOISE] Another way to write this is the original investment in a CD of a $100 grows at 1.08 in the first year, and then 1.08 in the second year, giving us a $116.64, which is the same as a $100 times 1.08 squared equals 116.64. So that little 2 which is called an exponent, what it means is that you take 100 times 1.08 twice, you multiply it twice when you see that exponent over two. And at the end of the third year, you have 116.64. So what you have at the end of the second year grows by 1.08 in the third year getting you up to $125.97. Or another way to write that would be it's your original investment of 100, grows by 1.08 three times so there's an exponent of three for a total of $125.97 at the end of the third year. >> Are we going to have to use exponents for this class? I haven't studied those since 1966. Don't tell me that this is going to be a highfalutin math class now! >> Yeah sorry. I am going to use formulas which look like highfalutin math in some of these slides. But even if you're not comfortable with that highfalutin math you can still solve these problems. So I'll give you the formulas and if you're comfortable with exponents you can punch those into your calculator. But I'll also give you other ways to solve these problems that will involve looking up numbers in present value tables or using Excel where you don't have to know anything about exponents or complex formulas. And you can still do these calculations. So you're going to see this done a number of different way, a number of different ways as we go through the examples. So now we're going to make this idea more general. So, we've gone out through three years what if you wanted to go out through n years. So n being whatever number you want it to specify. Well then the Future Value in n years which we're going to call FV so anytime you see FV that stands for Future Value n years from now, that's going to equal to a 100 times 1.08 raised to the nth power. So a way to think about that is if you wanted to do this for 10 years. It would be 100 times 1.08 ten times. You'd multiply it time, ten times over. If n was 20, it'll be 100 times 1.08, 20 times. So you can put in whatever period you want in the exponent. [SOUND] If the CD paid r% interest instead of 8% interest, then the future value is going to be 100 times 1 plus r raised to the n. So now we caould try this with 3% or 10% or 1%. If it was 1%, then it would be 1.01. And then if it was five years it would be raised to the fifth power. So you can specify whatever interest rate you want, whatever peerage you want. And then if your initial investment was $PV instead of a $100, here PV is going to stand for Present Value or today's value. Then we have future value equals present value, times 1 plus r to the n, and that gets us to the general formula that we can use for any level of in, initial investment, any interest rate, or any number of years or number of periods that we're going to have the money invested. So again, [NOISE] the terminology, Present Value, PV, of what you invest today is going to grow at an interest rate r, to earn a Future Value, FV, n years from now, using this formula FV equals PV times 1 plus r raised to the n power. >> Okay, this makes sense. Will there also be a formula to compute past values? Or does this formula just work going into the future? >> No we won't have a separate formula for past values, because as it turns out you can use the same formula to calculate values in the past. So I'll talk about on the next slide the terminology, but what you'll see is. Present value doesn't necessarily need to mean today, it could mean 20 years ago, and future value could be today. So just think of it as, present value and future value as before and after we have things grow or compound, as I'll shown on the next slide. So just to summarize this terminology in, in one handy slide. Whenever we do these Time Value of Money Calculations there's going to be four elements that we need to have in our formula. There's going to be PV or present value which is the value before effects of interest or discounting. FV is the future value. The value after the effects of interest or discounting. Again as we just talked about, present value doesn't have to mean today with the future value being some time in the future. It could be the future value was today and the present value is the value in say, 1980. So, even though we use the terms present value and future value. It really doesn't have to be today and then in the future. It just has to be the present value is the value before you do any interest or discounting. And the future value is the value afterwards. Present value could be 1980. It could be 1681. All it says is that it's going to be the value before you apply the interest rate, r is going to be the interest rate or discount rate and rate of return. Those are all synonyms that people use to describe the process of money growing from present value up to future value and then we use n for the number of periods between the present value and future value. So let's do an example of Future Value Calculation. So to do this calculation we could either plug the items into the formula. Or I'm going to show you a table in a second, and that table will allow you to just take present value times the number from the table for number of periods in interest rate to get future values. So you don't have to deal with exponents. All you have to do is find the number and multiply it times PV. Or you can calculate this using formulas in Excel. So you could put in the formula FV and then fill in the interest rate, the number of periods. Zero for payment which we'll talk about later in present value, and Excel will calculate the answer for you. The only thing to watch out for here is that if the interest rate is 10% you need to enter it as .10. Now, and I'll show you examples of doing the calculations with the tables, and with Excel in a little bit. So here's the first question. If you invested $10,000 in the stock market today, how much money would you have at retirement? Now to do this we need to assume that it's 20 years to retirement, and that the expected rate of return in the stock market is going to be 15% which is compounded annually. >> Dude, you must be like used to teaching old people if you assume it is only like 20 years to retirement. >> Hey, who are you calling old? >> Now it's mainly just wishful thinking because I'm hoping that I can retire in 20 years. Okay, let's so, let's try to solve this problem using the different methods. So the first would be, if you wanted to use the formula method, so we're trying to figure out the future value, the value 20 years from now, the present value is going to be $10,000. That's the amount you're putting in the stock market today, which is the amount before you start growing it by the interest rate. The r, the rate of return, is 15% and n is 20, for the number of periods. So you could fill those all in the formula, future value equals 10,000 times 1.15 raised to the 20th power. And it grows to 163,665. Now if you're not comfortable using the formula or punching this in your calculator, then you can also use this table approach. So I told you that future value equals present value, which again is $10,000 times the factor from table one for 20 years, and 15%. So let me show you the table one. So here is the table one that I'm talking about. And this will be part of the PDF file with the slides that you can download along with the video. So if you want to print it out for handy reference you can do so. So we're looking for 20 years, and so the rows are the period, so you can look down the rows until you get to 20. 15%, so we've got interest rates in the columns, so you look across the columns until you get to 15%. You look at the intersection of the 20 row, the 15% column, and you see 16.3665. So we can take this number times 10,000 to calculate the future value. So coming back to our slides, we've got future value equals 10,000, which is the present value, times that 16.3665 that we saw in table one for 20 periods and 15%, we get the same answer. The future value's going to be a $163,665, so this shows you the power of compounding. That basically 10,000 grows into a 163,000 over 20 years as long as it grows 15% per year. And of course, that 15% is not only based on the original investment, but whatever return has compounded up until that point. >> Before you go on, can you show me how to do this in Excel? >> Absolutely, let me bring up a spreadsheet right now. >> So let's take a look at how to do this in Microsoft Excel. So there's a little function button here, f with a small x. If we push that, then we can type in fv, what we're looking for, and sure enough, it's right at the top. So we hit OK for fv. Then we enter the rate so that was 15% which we enter in as 0.15. Number of periods is 20 periods. Payment is zero. We'll come back to what that means in a later video. The present value is 10,000. So remember we have $10,000 today that we're investing for 20 years at 15%. For type, you can always leave it blank. This is, has to do with whether the payment comes at the beginning or ending of the period. You could, for almost everything I could think of, you could leave this blank and you'll get the right answer. So then we hit OK, and you get negative $163,665. The reason why it shows up as a negative is Excel says if you put a positive number in, you get a negative number back, so it's almost like you're receiving $10,000 today, you have to pay 163,665 in the future. What I always do is I amend the formula by putting a little negative sign in front if it, and then I get a positive number. And the positive number is the same as what we got using the other two methods. Now let's play around with this a little bit and try varying these assumptions. So what if the expected return was only 5% instead of 15%. So, now for our $10,000 present value we still have 20 years to retirement but we have to change r to 5%, I'm going to go ahead and bring up the pause sign and ask you to stop the video and try to do it, but before that, let me bring up the Table 1, in case you want to use this method. Okay, before we leave this slide, I just want to point out the factor that you should have grabbed if you're using the table is 20 years, so the 20 row, 5% column, those intersect at 2.6533. So now if we go back to the problem, you could have done it using the formula, 10,000 times 1.05 raised to the 20th punched in your calculator. Or take that 10,000 times the 2.6533 that we just saw in the table. And you end up with a future value of 26,533, much lower present value when we, or much lower future value when we have only the 5% rate of return. Let's try another one. What if you planned to retire in 10 years? So we're still going to invest $10,000 present value. But now we're going to assume it's 10 years to retirement, so our N is going to be 10. And we'll take the expected rate of return back to 15%. So why don't you go ahead and try to calculate this one. So again before we leave the table, if you're using this approach, we've got 10 periods so we go down to the 10 row. We go across to the 15% column. The number is 4.0456. So to calculate this we can either do the formula, 10,000 times 1.15 raised to the 10th power. Or 10,000 times that factor we just saw in table 1. Which is 4.0456 and we get a future value of 40,456 which again is much lower than the 163,000 we saw originally. So if you are going to have fewer years to retirement, you're going to have less money when you retire. >> Am I to infer from these examples that if the number of years got bigger, or the rate of return got bigger, then the future value would be bigger? >> Great insight. As the number of periods and the rate of return got smaller, the future value got smaller. So it's gotta be the case that as those number of periods and rate of return get bigger. The future value will get bigger as well. Yeah, so we can generalize this then go you know how about 30 years, so go more years than 20. How about 25%, go higher than 15%. So this is something I did in Excel, where I, I tried different. Interest rates are, or rate of returns are, I tried different number of periods to retirement and, and what you can see is if you look at the 25% rate of return as the number of periods to retirement grows then the future value grows. Or if you look at the same number of periods, if you look at all the 30's. That as we go from 5% to 15% to 25% the future value grows. So the general rule is, that future value is positively related to the rate of return and the number of periods. So as the rate of return gets bigger, as the number of periods gets bigger the future value grows, and if you see this, and it, as we see in this example if you put $10,000 in stock market today and happen to get 25% return over the next 30 years, it would grow to over $8 million. Now that we've seen how to run the time machine forward, to take a present value and figure out what it would be worth in the future, the future value, in the next video, we're going to run the time machine backward. We're going to start with the future value, and see if we can calculate a present value, or in other words, the amount before applying the interest rate or the discount rate. I'll see you then. >> See you next video.