0:00

In this second lecture, we're going to discuss the time value of money.

Â This is a critical component to understand how to calculate the net present

Â associated with the project.

Â So, one way to think about the time value of money

Â is to simply say that a dollar today is worth more than a dollar tomorrow.

Â 0:43

And there's a flow of cash associated with

Â each of those future periods and they're denoted by the letter F.

Â And there's a subscript on each one of those,

Â that denotes what period that cashflow is either going to be paid or receive.

Â So, Fs of one would be cashflow one period ahead.

Â Fs of three would be cashflow three periods ahead and

Â Fs of n would be cashflow that's n periods ahead.

Â 1:34

When we do these calculations,

Â there's going to be an interest rate that is going to govern this calculation.

Â And for now,

Â we're going to assume that that interest rate is the same every period.

Â Okay and we'll denote the interest rate by R.

Â Now, I'm going to start out by talking about future values,

Â because I think sometimes, there's more intuition associated with that.

Â So, if you wanted to know what the future value of cash that you have on hand is.

Â That you are going to say, deposit in a bank account.

Â The way that you would calculate what the value of that cash would be at period n.

Â Is to take the cash that you're going to deposit in the bank today.

Â The present value and multiply that by one plus r raised to the nth power.

Â So what would be an example of this?

Â Well, suppose that there was a bank account around and

Â that interest rate on that bank account was 10% per year.

Â And you wanted to know what amount of money you would have in the bank at

Â the end of one year.

Â If you deposited $1000 in the bank today and so,

Â applying that formula, what would we do?

Â Well, what we're trying to calculate is we're trying to calculate F sub one.

Â So, you would take the amount of money that you're depositing

Â in the account today $1,000.

Â And you would multiply it by one plus the interest rate of 10%,

Â which of course 10% is 0.1.

Â And so, you're taking a thousand and you're multiplying it by 1.1.

Â What that means is, at the end of the year, you would have $1,100 in the bank,

Â right.

Â So you'd deposit $1,000 today.

Â Over the year, you earn interest at 10% per year, and

Â at the end of the year you have $1,100.

Â That is the future value of that $1,000 a year from today.

Â When there's a bank account around that pays you at the rate of 10% per year.

Â Now, what if you wanted to ask the question of what amount of money would

Â you have in that same bank account if you left that money in there for two years?

Â 4:13

So notice what happened.

Â In the first example we got interest of $100 essentially

Â and in the second year we got interest of $110.

Â We didn't just get $100.

Â We got interest of $110 and

Â that's because in the second year, not only did we get interest on the $1000.

Â But we got interest on the $100 in interest that we got

Â in the first year and people often refer to this as compound interest.

Â 4:45

So again, if you left the money in a bank for a year, you'd have $1100.

Â If you left the money in the bank for two years, you'd have $1,210.

Â Now these examples are all using dollars, but

Â you can use any currency that you want to use.

Â You could do these calculations with absolutely any currency and

Â the examples I am going to use are just going to use dollars.

Â Let's talk about present values, because that's really what we're going to use for

Â evaluating projects.

Â The formula here for the present value just solves for

Â what the present value is from that other formula that I showed you previously here.

Â So all we're doing is, putting the present value on the left hand side and

Â we're solving for it.

Â 6:14

What we're solving for here is,

Â the present value of a $1000 a year from today.

Â Assuming there's a bank account pays you at the rate of 10% a year.

Â So to calculate this, what you do is you take the 1000,

Â which is f sub one ion this case.

Â Because it's a year away and you divide by 1+r.

Â The r again is 10% so you divide by 1.1 and you find out that

Â present value of $1000 a year from today.

Â When there's a bank account around that pays you at the rate of 10% is $909.09.

Â And you can see, that if I took that $909.09 and

Â left it in the bank and it earned 10%, at the end of the year I would have $1,000.

Â What's interesting here is,

Â to understand that what this calculation is basically telling you.

Â 7:30

And the reason we say they're economically equivalent,

Â because if you had nine hundred and nine dollars and nine cents today.

Â And you put it in a bank account that paid ten percent a year,

Â at the end of the year you would have a thousand dollars.

Â So those two things are economically equivalent.

Â 7:58

It's not enough for me to say I'm going to give you $5,000.

Â That really doesn't tell you anything.

Â You need to know when I'm going to give you the $5,000 dollars,

Â because of the time dimension associated with money.

Â Let's look at another example.

Â Here we're asking the question what amount of money would you have to put into

Â the bank today.

Â If you wanted to be able to withdraw $1,000 Two years from today.

Â 8:37

So in calculating the present value,

Â we take the $1,000, which is the amount we want.

Â But now we divide by 1 plus r To the 2nd power,

Â because we're going to let that money sit in the bank for two years.

Â 9:05

Another way to say that is $826.45 today,

Â is economically equivalent to $1,000 two years from today.

Â If there's a bank account around that pays you at the rate of 10% a year.

Â In terms, of the jargon that we're establishing here.

Â The present value of $1,000 two years from today

Â is $826.45.

Â Now you probably wouldn't be surprised if you took

Â that $826.45 and put it in the bank for one year at 10%.

Â You probably wouldn't be surprised that the amount you would have in

Â the bank would be $909.09.

Â And then of course, what would you do?

Â You'd leave that $909.09 in the bank account for

Â another year and it will grow to $1,000.

Â So now, let's put those two examples together.

Â Let's suppose now that you wanted to be able to withdraw $1,000

Â at the end of Year one, and $1,000 at the end of Year two.

Â And you wanted to know what amount of money

Â you would have to put in the bank today in order to be able to do that.

Â 10:43

And so, you can see what I'm doing in this equation is, I'm taking 1000 and

Â I'm dividing it by 1.1.

Â Which is the present value of $1, 000 a year from today.

Â And I'm adding to it $1, 000 divided by 1.1

Â sqaured which is the present value of $1, 000 two years from today.

Â 11:06

And of course, not surprisingly the answer is really the sum

Â of the two examples that we just did, right.

Â $909.09 is the present value of $1000 a year from today.

Â $826.45 is the present value of $1000 two years from today.

Â So if you wanted to be able to withdraw $1000 at the end of year one and

Â $1000 at the end of year two.

Â 11:35

You're simply adding those two calculations together and

Â you're getting $1,735.54.

Â What's key here is, that even though when you look at that equation

Â you see the $1,000 written twice.

Â You'll notice that I didn't take the first $1,000 and add it to the second $1,000 and

Â try to operate on the number $2,000.

Â And the reason I didn't do that is,

Â because even though those numbers look the same they're not the same.

Â And the reason they're not the same is,

Â because they're coming at different points in time.

Â 12:14

As I said, money has a time dimension.

Â And so, what I have to do in order to be able to calculate the present value,

Â is I have to first convert those into like terms.

Â And the way I'm doing that is, calculating the present value of each.

Â So I'm calculating the present value of $1,000 a year from today, and

Â calculating the present value of $1,000 two years from today.

Â And once I know those present values,

Â I can add them up because now they're stated at the same point in time.

Â They are in present value terms and then, I can add them together.

Â 12:54

And that's why I don't add the $1,000 to the other $1,000.

Â And try to do something with 2,000, because again, money has a time dimension.

Â And those two $1,000,

Â even though they look the same, are fundamentally different.

Â 13:32

And then of course, that would grow at 10%, and

Â at the end of the second year I would be able to withdraw the second $1,000.

Â And there would be nothing left in the bank account.

Â This is just another example and here, the cash flows are varying.

Â And so, the question here basically is.

Â What amount of money would you have to put in the bank today to allow you to

Â withdraw $1000 at the end of year one, $1500 at the end of year two,

Â and $2000 at the end of year three?

Â And this calculation is very much like the calculation we just did.

Â And you can see what I'm doing here is, I'm calculating the present value of

Â the $1,000 a year from today by taking $1,000 and dividing it by 1.1.

Â And to that, I'm adding the present value of $1,500 two years from today

Â 14:29

And to that I'm adding the present value of $2000 three years from today

Â by taking the $2000 and dividing it by 1.1 to the third.

Â So, let's use a spread sheet to calculate

Â the present value of the example we just did.

Â Where we had a cash flow of $1000 in the first year,

Â $1500 in the second year and and 2,000 in the third year.

Â 14:57

to calculate the present value of each of these cash flow.

Â So in this case, we could take c2 and

Â we could divide it by 1.1 because, remember,

Â the discount rate in this example is 10%.

Â And when we do that we get $909.09.

Â Then we're going to take the year two cash flow and

Â because it's the second year's cash flow, we're going to divide it by 1.1 squared.

Â To calculate 1.1 squared, you put 1.1 in and then you hit the carrot.

Â The carrot is usually above the six on most keyboards, and then you put squared,

Â and then that gives us the present value of that $1,500.

Â And then, the third year's cash flow is $2000 and

Â in order to calculate the present value of that.

Â we're going to take the $2000 were going to divide

Â it 1.1 now raised to the third power.

Â Again, we have to hit the carrot sign and

Â then the third, and that's going to give us a present value of that flow.

Â 16:10

So now, you've calculated the present value of each of the flow.

Â So, now to calculate the net present value of the three flows.

Â We can just sum up the present value of each of the flows for

Â years one, two and three.

Â So to do that, we would just take

Â the sum of in this case C3 to E3.

Â 16:34

And when we do that, we find out that the present value of

Â the sum of those three cash flows is 365139.

Â Now, there's another way to do that.

Â It's a shortcut and that is to use the NPV function in Excel.

Â So in this case, what we're goin to type in is, = NPV.

Â And then, the first argument that goes in the brackets

Â is the discount rate, which in our case is 10%.

Â And then, you put a comma and then,

Â you tell it the cash flows that you're discounting.

Â Which in this case, are the 1,000, the 1,500, and the 2,000.

Â So, we would put in c2 To E2 to calculate that and

Â you see that the answer is the same, $3651.39.

Â Now up until this point, we've

Â assumed that the interest rate on the bank account stays the same every year.

Â And so, we've just been talking about an r and

Â we've been assuming that the r stays the same through time.

Â 18:05

So, to calculate the present value when the interest rate varies through time.

Â You take that Fn, but

Â rather than dividing by one plus r raised to the nth power as we did before.

Â We now have to multiply the product of all the 1+r's together,

Â because the r's are different.

Â So in the denominator now, again, instead of having 1+r to the nth power.

Â We're going to have the product of 1+r 1 times 1+r 2 times 1+r 3 etc.

Â And so, we can certainly allow for the interest rate to vary through time and

Â still calculate a present value.

Â So a quick example of that is,

Â suppose you wanted to know what amount of money you have to put in the bank today.

Â To receive a thousand dollars two years from today if the interest rate for

Â year one is going to be 5 percent.

Â And the interest rate for year two is 15%.

Â Well to calculate that present value, you simply take 1,000 and

Â you're going to divide it by (1.05)(1.15).

Â And the present value of that $1000 would be $828.16.

Â