Hello, Professor Brian Buchet. Welcome back. In this video, we going to talk about computing present values. The formula that we're going to use I think is one of the most common formulas that gets used in financial applications. Plus, you could use the same formula to do. Radiocarbon dating of a fossil or figure out the spread of genetic mutations through a population? Very important formula. But, but for this video, we're just going to focus on financial applications and, at the end, we'll also talk about this concept of net present value. Let's get started. Now let's dive in and see how we would calculate present values. So what if we know the Future Value of something, but we don't know what the value is today. What's the Present Value? Well we saw in the last video that the Future Value was equal to the Present Value times one plus the discount rate or interest rate. Raised to the n power where n is the number of periods. We can use algebra to rearrange this so that the Present Value. Is equal to the future value divided by 1 plus the discount rate raised to the nth power. So to put this in a concrete example. How much would you have to invest today in a certificate of deposit or CD that earns 8% per year in interest to have $108 next year? So the present value is going to equal. A $108 which is the value next year divided by 1 plus that interest rate, so 1.08 going to give you an answer of $100. There's no exponent here because it's raised to the 1 power, so the exponent's 1 so we just drop it. And this is what we saw in the prior video with future values that if you had a $100 today. And invested it at 8% per year, it would grow to 108 at the end of the year, so we just reversed in that calculation. So similarly what if we wanted $125.97 in three years? Using the formula, the present value is going to equal the future value, $125.97 divided by 1 plus the discount rates, or interest rates. So 1.08 raised to the third power, because it's three years, present value is going to be $100. And again this is just the reverse of what we saw last video where we had $100 today grow to $125.97 in three years. We're just redoing the calculation the other way so if we want $125.97 in three years we have to invest $100 today at 8% interest. A more common way this happens is there's some kind of round number of. Cash that you want in the future. So what if we want 100 dollars one year from now? How much would we have to invest today? So the future value is going to be 100 dollars. We divide 1 plus the interest rate 1.08, we have to invest $92.59 today. To have it grow into $100 a year from now. And so what we usually say is the present value of $1 next year would be 93 cent at eight percent interest. Dude, like it looks like the present value is always smaller than the future value. Yes, you're almost always going to see the present value smaller than the future value. Because the interest rate, or discount rate, or rate of return is almost always positive. It would have to be a negative interest rate, or negative rate of return for the present value to be greater than the future value. So you will tend to see present value less than the future value, because we tend to have positive interest rates, or positive discount rates. So now let's practice doing some calculations with this present value formula. So again we'll do it three ways. One is you can plug the numbers into the formula in your calculator and calculate. Or there's going to be a new table, which I call Table 2. Where we can pull factors for number of periods and interest rate, multiply that times the future value to get the present value. Or we can also use Excel. Instead of the FV function, it's going to be the PV function. But otherwise it's going to work much the same. The first question we'll look at is, how much should you have invested in a savings bond 20 years ago to have $10,000 today? So last video we were going to use $10,000 and put it in stock market. Well, how do we get $10,000? One of the ways we could have gotten $10,000 today would be to invest money 20 years ago in a savings bond and let it grow. The question is how much would we have needed to invest. We're going to assume that savings bonds have no periodic interest payments. Which means that the interest is just added to the principal and compounded like we've been seeing in these examples. Later on we'll look at what happens when bonds do pay periodic interest and how we have to do those calculations a little differently. And we're going to assume interest on the bond was 15% compounded annually, So we'll take a brief look at the Present Value table. So this is Table 2. So if we're going to use this approach, what we need to find is the number of years. So we go down the rows until we get to 20. Go across to the 15% column. And the intersection of 20 and 15%, the factor is 0.0611. So, that's the factor that we can use, using the table approach to calculate the present value. So, if you want to do the formula. You could plug in $10,000 as the future value. Divide that by 1.15 raised to the 20th power so it's 15% interest rate over 20 years. Or we could calculate present value is 10,000 times the factor from the table, which we just saw was 0.0611. Either way, you come up with a present value of $611. Which means that you would have had to invest $611 20 years ago, at a 15% interest rate in a bond, to have it grow to be $10,000 today. And here's an example where the present value is really a past value. And the future value is really today. Don't get hung up on present value or future value. It's really the value before we start discounting and the value after we stop discounting. >> So you are telling me that if I had invested $611 20 years ago to get $10,000 today. And then invested that in the stock market, I could have $8 million 20 years from now? What? Is this some type of get rich quick scheme? >> Well, it's more like a get rich slow scheme, because it's going from $611 to $8 millions over 50 years. because it was 20 years of the bond and then 30 years in the stock market. The other reason it's so high, is we had fairly high rates of return, or interest rates. So we were earning 15% on the bond. And then 25 percent on the stock market. If you ever get and answer that seems unreasonable, go back and check your assumptions. It may be that you've assumed an interest rate or discount rate that's too high. When you put in something more reasonable, then you see a more reasonable return on your investment. Before we move on, let's quickly take a look at how we would solve this using Excel. So we hit our formula button, or function button again. We look for PV, which I already have up here, not surprisingly, and then we fill in the rate, so it's 15%. Number periods is 20, payment we ignore for now. The future value is $10,000. Again, we ignore type, we hit OK. We get negative 611, if you don't want to see the negative put a little minus sign in front of present value. And we get the same answer that if you invested $611 20 years ago at 15 percent. It would grow to a value today of $10,000. So let's play around with these a little bit and give you a chance to try some of them on your own. So what if the interest rate was only five percent? So we're still going to look for a future value of $10,000. We're still going to go back 20 years, but now the bond is only going to be paying 5% interest. I'm going to put up the paw sign, why don't you try to calculate this on your own and here's the present value table if you want to use it. Okay so, before we move off of this, let's look at the 20 year row again. But now we're only going to the five percent column. And we can see that the intersection is 0.3769. So if we put that in our calculations, we could either do it with a calculator, 10,000 divided by 1.05 raised to the 20th. Or 10,000 times that 0.3769 that we saw in table 2. You come up with an answer of 3,769. Now notice this present value is a lot bigger than the 611 in the first example because what happened is the money's not growing as fast. If you put in $611 and let it grow at 15% it gets up to $10,000. But if it would only grow at five percent, it's not going to make it. You have to have a higher present value, so a higher amount 20 years ago. So if we get up to $3,769 and let that grow up five percent, it will get us to the same place of 10,000. So a smaller interest rate means that you have a bigger present value to get to the same future value. What if we bought the savings bond ten years ago? So we're going to assume that the future value is still $10,000. We're going to go back to the 15% interest rate on the bond, but we're only going to buy the bond ten years ago, instead of 20 years ago. So I'll put up the pause sign and the present value table, and why don't you try to solve this one? So if we're using the table, we need to go down to the ten year row. And then across to the 15% column. The intersection is 0.2472, so that's the factor we would use if we were going to use the table. If you wanted to use the equation, then it's present value equals the future value of 10,000, divided by 1.15 raised to the 10th power. Or 10,000 times that 0.2472 factor that we saw on the table, which gives us a present value of 2,472. Again comparing back to original $611 the present values higher, what's going on is it ha, is the money has fewer years to grow. So when we can put in money and have it grow for 20 years, we can put in less money. If we're going to put in money and have it only grow for 10 years we have to put in more money initially which is why the present value is higher. So what happens is the more number of years we have compounding, the lower the present value's going to be to get to that same future value. >> I am starting to see a pattern again. Yes, the pattern that we're seeing here does generalize. So if we look at 30 years or 25%. So do the same kind of sensitivities we did before. What we find is that if you look at the five percent, as the number period gets bigger, the present value gets smaller. We look at 25%, same thing as the number of years goes up, the present value goes down. And then for a given number of years, like if you look at all the 30 year rows. As we go from five to 15 to 25 percent, the present value gets smaller. So the present value is inversely related to the rate of return and the number of periods, which means as r or n goes up. Present value comes down. Whereas r or n go down, present value goes up. And notice that if you got 25% on your bond and you invested 30 years ago you would have put in $12 to get $10,000 today. >> This is all well and good. But, I have a question, so what? >> Okay, well let me give you a so what. We'll look at an application where present value is useful in decision making. The application we're going to look at is calculating something called Net Present Value, which is often used to make decisions about. How you invest your money in different projects. So what we're going to do is lay out a timeline of the cash inflows and cash outflows. Then we're going to convert the cash flows in each period to their present values so that we can compare them. And we'll just add up the present values to figure out the net present value. Now, usually how this works is there's some kind of cash outflow in today's dollars. You're making an investment today, and then you get cash that comes in in future dollars. And then, of course, since they're future dollars we have to discount them back to present value, bring them back to present value. So that we can compare them to our original investment. So here's the example, let's say we have a four year horizon and two projects. So project A, we invest a $100 today. We don't get any cash that comes in for year one, year two, or year three, and then in year four we receive $200 cash. So we're investing a $100 today. With the hope of getting $200 back four years from now. In Project B, we invest the same $100 today, but we get some immediate payoff, so we get $80 next year. We get $50 the year after that, $30 the year after that, and then nothing in year four. Which of the two projects would you rather invest in? I want Project A. It gives me $200 back, whereas Project B only gives me $160 back. >> I want Project B. I like need cash now! I can't like wait four years dude! >> You two are clowns. Did you not pay attention to this video? We need to convert to present value before we can make any decisions. >> Well name calling aside, Rena's right that we have to put all of these dollars into present values, because we can't compare them directly. Again think about if these were different currencies. If you invested $100 and got a certain amount of Yen back, but another project you invested $100 and got a certain amount of Euros back. You couldn't just add up the numbers and decide which was better. You'd have to translate everything into dollars. Well, dollars in years one, two, three, and four are not the same as today's dollars, so first we have to translate everything into today's dollars. Then we can add it up, and decide which project is better. Yeah so let's convert these to present value at say 10% interest rate. So for project A we still pay $100 because that's today's dollars. But we don't receive $200 in terms of today's dollars. Will only receive $137 in terms of today's dollars. So 137 is the present value of 200, four years from now, at 10% interest. For Project B, we still pay out $100 in today's dollars, but instead of receiving 80, 50, and 30. In today's dollars, we're receiving the equivalent of 73, 41, and 23. So the 73 is the present value of 80 dollars one year from now at a 10 percent discount rate. 23 is the present value of 30 three years from now at a 10 percent discount rate. Now that we have all the cash flows in today's dollars. So we've removed the effect of inflation, or interest, or discounting. Now we can just add them up. So the net present value for Project A is, we pay 100 dollars out. We get the equivalent of 137 dollars of today's dollars back four years from now, which is a net present value of 37 dollars. For Project B, we pay 100 dollars out today. Then we get $73, $41, and $23 in the next three years. Now of course we're really getting 80, 50 and 30 in those future dollars. But the equivalent value of them today, the present value today is 73, 41 and 23. And lo and behold, it comes up with the same net present value of 37. So you would not care which one you would invest in. They're both going to give you the same net present value. They both have the same value, when you translate all of the cash flows into today's dollars. So hopefully that gives you a good sense of computing present values, and this concept of calculating net present values. And in the next video we're going to move on to the final topic of time value of money, which will be annuities. See you then. See you next video.