In this second lecture, we're going to discuss the time value of money. This is a critical component to understand how to calculate the net present associated with the project. So, one way to think about the time value of money is to simply say that a dollar today is worth more than a dollar tomorrow. But as it turns out, we can say something much more specific than that. So, here we have a timeline and you can envision us standing at time zero. And as you look, you can see that there are periods out in the future and in this graph they go from period one out through period n. And there's a flow of cash associated with each of those future periods and they're denoted by the letter F. And there's a subscript on each one of those, that denotes what period that cashflow is either going to be paid or receive. So, Fs of one would be cashflow one period ahead. Fs of three would be cashflow three periods ahead and Fs of n would be cashflow that's n periods ahead. We can talk about the amount of cash we have today as the present value of that cash. Or we could talk about the value of a future flow that we're going to receive some periods ahead. We could talk about what the present value of that is. When we do these calculations, there's going to be an interest rate that is going to govern this calculation. And for now, we're going to assume that that interest rate is the same every period. Okay and we'll denote the interest rate by R. Now, I'm going to start out by talking about future values, because I think sometimes, there's more intuition associated with that. So, if you wanted to know what the future value of cash that you have on hand is. That you are going to say, deposit in a bank account. The way that you would calculate what the value of that cash would be at period n. Is to take the cash that you're going to deposit in the bank today. The present value and multiply that by one plus r raised to the nth power. So what would be an example of this? Well, suppose that there was a bank account around and that interest rate on that bank account was 10% per year. And you wanted to know what amount of money you would have in the bank at the end of one year. If you deposited $1000 in the bank today and so, applying that formula, what would we do? Well, what we're trying to calculate is we're trying to calculate F sub one. So, you would take the amount of money that you're depositing in the account today $1,000. And you would multiply it by one plus the interest rate of 10%, which of course 10% is 0.1. And so, you're taking a thousand and you're multiplying it by 1.1. What that means is, at the end of the year, you would have $1,100 in the bank, right. So you'd deposit $1,000 today. Over the year, you earn interest at 10% per year, and at the end of the year you have $1,100. That is the future value of that $1,000 a year from today. When there's a bank account around that pays you at the rate of 10% per year. Now, what if you wanted to ask the question of what amount of money would you have in that same bank account if you left that money in there for two years? Well, in this case, we're calculating F sub 2. The 1000 is the same. That's the amount we're depositing today. But now, we're going to multiply that 1000 by 1 squared, not 1.1 to the first power. Because we're letting that money sit in the bank for two years. And you'll notice that when you do this calculation. What you'll find is, that you're going to end up with $1210 in the bank account at the end of two years. So notice what happened. In the first example we got interest of $100 essentially and in the second year we got interest of $110. We didn't just get $100. We got interest of $110 and that's because in the second year, not only did we get interest on the $1000. But we got interest on the $100 in interest that we got in the first year and people often refer to this as compound interest. So again, if you left the money in a bank for a year, you'd have $1100. If you left the money in the bank for two years, you'd have $1,210. Now these examples are all using dollars, but you can use any currency that you want to use. You could do these calculations with absolutely any currency and the examples I am going to use are just going to use dollars. Let's talk about present values, because that's really what we're going to use for evaluating projects. The formula here for the present value just solves for what the present value is from that other formula that I showed you previously here. So all we're doing is, putting the present value on the left hand side and we're solving for it. And when we do that, we find out that the present value of a future flow of cash f's of n. Is equal to that future flow of cash divided by 1+r raised to the n power on the left hand side. So let's see an example of this. Suppose that there's an interest rate around that pays you at the rate of 10% a year. And you want to know, what amount of money do you have to put in the bank today to allow you to withdraw $1000 a year from today? So, this is a different problem then we calculated before. What we're solving for here is, the present value of a $1000 a year from today. Assuming there's a bank account pays you at the rate of 10% a year. So to calculate this, what you do is you take the 1000, which is f sub one ion this case. Because it's a year away and you divide by 1+r. The r again is 10% so you divide by 1.1 and you find out that present value of $1000 a year from today. When there's a bank account around that pays you at the rate of 10% is $909.09. And you can see, that if I took that $909.09 and left it in the bank and it earned 10%, at the end of the year I would have $1,000. What's interesting here is, to understand that what this calculation is basically telling you. Is that a thousand dollars a year from today is economically equivalent to 909.09 today. If there's a bank account around that pays you with a rate of 10 percent. And the reason we say they're economically equivalent, because if you had nine hundred and nine dollars and nine cents today. And you put it in a bank account that paid ten percent a year, at the end of the year you would have a thousand dollars. So those two things are economically equivalent. The other thing that this really points out, and this is really key, is that money has a time dimension. It's not enough for me to say I'm going to give you $5,000. That really doesn't tell you anything. You need to know when I'm going to give you the $5,000 dollars, because of the time dimension associated with money. Let's look at another example. Here we're asking the question what amount of money would you have to put into the bank today. If you wanted to be able to withdraw $1,000 Two years from today. So, the only things that's changed here relative to the problem that we just did is, now we're saying we want the $1,000 two years from today. So in calculating the present value, we take the $1,000, which is the amount we want. But now we divide by 1 plus r To the 2nd power, because we're going to let that money sit in the bank for two years. And when you do this calculation, what you find out is, that the present value of $1,000 two years from today is $826.45. Another way to say that is $826.45 today, is economically equivalent to $1,000 two years from today. If there's a bank account around that pays you at the rate of 10% a year. In terms, of the jargon that we're establishing here. The present value of $1,000 two years from today is $826.45. Now you probably wouldn't be surprised if you took that $826.45 and put it in the bank for one year at 10%. You probably wouldn't be surprised that the amount you would have in the bank would be $909.09. And then of course, what would you do? You'd leave that $909.09 in the bank account for another year and it will grow to $1,000. So now, let's put those two examples together. Let's suppose now that you wanted to be able to withdraw $1,000 at the end of Year one, and $1,000 at the end of Year two. And you wanted to know what amount of money you would have to put in the bank today in order to be able to do that. So to solve that problem, what do we do? Well to solve that problem, essentially what we do is we calculate the present value of $1000 a year from today. And add to it the present value of $1000 two years from today. And so, you can see what I'm doing in this equation is, I'm taking 1000 and I'm dividing it by 1.1. Which is the present value of $1, 000 a year from today. And I'm adding to it $1, 000 divided by 1.1 sqaured which is the present value of $1, 000 two years from today. And of course, not surprisingly the answer is really the sum of the two examples that we just did, right. $909.09 is the present value of $1000 a year from today. $826.45 is the present value of $1000 two years from today. So if you wanted to be able to withdraw $1000 at the end of year one and $1000 at the end of year two. You're simply adding those two calculations together and you're getting $1,735.54. What's key here is, that even though when you look at that equation you see the $1,000 written twice. You'll notice that I didn't take the first $1,000 and add it to the second $1,000 and try to operate on the number $2,000. And the reason I didn't do that is, because even though those numbers look the same they're not the same. And the reason they're not the same is, because they're coming at different points in time. As I said, money has a time dimension. And so, what I have to do in order to be able to calculate the present value, is I have to first convert those into like terms. And the way I'm doing that is, calculating the present value of each. So I'm calculating the present value of $1,000 a year from today, and calculating the present value of $1,000 two years from today. And once I know those present values, I can add them up because now they're stated at the same point in time. They are in present value terms and then, I can add them together. And that's why I don't add the $1,000 to the other $1,000. And try to do something with 2,000, because again, money has a time dimension. And those two $1,000, even though they look the same, are fundamentally different. You probably wouldn't be surprised that if I took that $1735.54 and put it in the bank for a year at the rate of 10% a year. And withdrew $1000 at the end of that first year, I would have $909.09 left in the bank. And then of course, that would grow at 10%, and at the end of the second year I would be able to withdraw the second $1,000. And there would be nothing left in the bank account. This is just another example and here, the cash flows are varying. And so, the question here basically is. What amount of money would you have to put in the bank today to allow you to withdraw $1000 at the end of year one, $1500 at the end of year two, and $2000 at the end of year three? And this calculation is very much like the calculation we just did. And you can see what I'm doing here is, I'm calculating the present value of the $1,000 a year from today by taking $1,000 and dividing it by 1.1. And to that, I'm adding the present value of $1,500 two years from today by taking the $1,500 and dividing it by 1.1 squared. And to that I'm adding the present value of $2000 three years from today by taking the $2000 and dividing it by 1.1 to the third. So, let's use a spread sheet to calculate the present value of the example we just did. Where we had a cash flow of $1000 in the first year, $1500 in the second year and and 2,000 in the third year. One way in which we could do this would be to calculate the present value of each of these cash flow. So in this case, we could take c2 and we could divide it by 1.1 because, remember, the discount rate in this example is 10%. And when we do that we get $909.09. Then we're going to take the year two cash flow and because it's the second year's cash flow, we're going to divide it by 1.1 squared. To calculate 1.1 squared, you put 1.1 in and then you hit the carrot. The carrot is usually above the six on most keyboards, and then you put squared, and then that gives us the present value of that $1,500. And then, the third year's cash flow is $2000 and in order to calculate the present value of that. we're going to take the $2000 were going to divide it 1.1 now raised to the third power. Again, we have to hit the carrot sign and then the third, and that's going to give us a present value of that flow. So now, you've calculated the present value of each of the flow. So, now to calculate the net present value of the three flows. We can just sum up the present value of each of the flows for years one, two and three. So to do that, we would just take the sum of in this case C3 to E3. And when we do that, we find out that the present value of the sum of those three cash flows is 365139. Now, there's another way to do that. It's a shortcut and that is to use the NPV function in Excel. So in this case, what we're goin to type in is, = NPV. And then, the first argument that goes in the brackets is the discount rate, which in our case is 10%. And then, you put a comma and then, you tell it the cash flows that you're discounting. Which in this case, are the 1,000, the 1,500, and the 2,000. So, we would put in c2 To E2 to calculate that and you see that the answer is the same, $3651.39. Now up until this point, we've assumed that the interest rate on the bank account stays the same every year. And so, we've just been talking about an r and we've been assuming that the r stays the same through time. We can allow the interest rate to vary through time, but of course if we do that. Then we need to denote the fact that the interest rate for the first period is different than the interest rate for the second period, etc. And so, the r now is going to have a subscript, which is going to refer to what period it relates to. So, to calculate the present value when the interest rate varies through time. You take that Fn, but rather than dividing by one plus r raised to the nth power as we did before. We now have to multiply the product of all the 1+r's together, because the r's are different. So in the denominator now, again, instead of having 1+r to the nth power. We're going to have the product of 1+r 1 times 1+r 2 times 1+r 3 etc. And so, we can certainly allow for the interest rate to vary through time and still calculate a present value. So a quick example of that is, suppose you wanted to know what amount of money you have to put in the bank today. To receive a thousand dollars two years from today if the interest rate for year one is going to be 5 percent. And the interest rate for year two is 15%. Well to calculate that present value, you simply take 1,000 and you're going to divide it by (1.05)(1.15). And the present value of that $1000 would be $828.16.