In this video, we examine the important topic of time value of money. The basic concept of time value of money is that money in hand today has greater value than money in the future. The time value of money is the notion that money today is worth more than the same amount tomorrow. Money doesn't just sit in a vault. Money is usually deployed in surplus-producing enterprises for surplus can mean profit or benefit or whatever. In other words, invested money grows into the future, and future money that we receive in the future is worth less than receiving that money today. As an example, would you rather receive $1,000 today or $1,000 in a year? Why? What's the rationale for your answer? Think about it for a moment. Let's examine the future value of money. The future value of money is the value tomorrow or some period of time in the future of money invested today. Let's take a simple example. Suppose we invest $1,000 in a renewable energy project. Assuming it's successful, you expect its value to increase to 10 percent in a year. How much will your investment be worth after one year? Let's do some simple arithmetic. If P is the principal, that's the amount you invest, r is the interest rate and FV is the future value of the investment, we can quickly figure out what you'll get back as the principal times 1 plus the interest rate. When we do the arithmetic, we find that we get $1,100 back in a year. In one year, our money will grow to be $1,100. The future value of this investment is $1,100. Now let's look at the future value of money over many periods. Here's another example. You invest $1,000 in a project of renewable energy project today. You expect this value to increase 10 percent in a year every year. How much will your investment be worth in 10 years? Well, here's the same notation we use before. P for principal, r for interest rate, and FV the future value of the investment in year, and the subscript n means some period of time in the future in years. The formula for calculating the future value is P times 1 plus r to the nth power. The reason for the nth power is that every year we get interest on the interest. To understand that we have to raise it to a power. In this case, the feature value in 10 years is equal to 1,000 plus 1 plus 0.1, which is a 10th to the 10th power. We get $2,594. In 10 years, our money will be worth $2,595. Again, what's going on here is that assuming we don't take out the interest payments, we start earning interest on the interest. Now let's talk about the present value of money received in the future. In other words, we're going to get a sum of money paid to us at some point in the future, and we want to know what it's worth to us today. As an example, suppose we invest in a renewable energy project that will pay $20,000 in one year. You could earn 10 percent interest elsewhere, so what's your investment worth today? Well, that's what we want to figure out. Because if someone offers you $15,000 today to buy your investment, would you take it? How about $20,000? Let's figure it out. Again r is interest rate, FV is the future value of money, and PV is the present value. We know from previous slides that the future value is equal to the present value or principal times 1 plus r. Rearranging, we have the present value is equal to the future value divided by 1 plus r again, divided by not multiplied by. We get 20,000 divided by 1.1. When we do the arithmetic, we get $18,182. That's what our investment or the receipt of money a year from now $20,000, is worth to us today, $18,182. If somebody offers is $15,000 today to buy our investment, would we take it? The answer is probably no because we know that $20,000 a year from now is worth more than $15,000 to us today given a 10 percent interest rate. How about somebody offers is $20,000? Well, we might want to take it because we'd get $20,000 today instead of $20,000 a year from now. The present value of the year from now 20,000 is only 18,000, so $20,000 today looks like a good deal. In summary, $20,000 in one year is only worth $18,182 today. This is an important concept. We'll be using it over and over again. Well, let's expand on our example. Let's say that we're looking out 10 years and we want to know what's our investment worth today. Instead of getting $20,000 in one year, we get 10 percent. We won't get that for 10 years. Again, we have an alternative investments or investments that we could earn 10 percent interest on. What's that $20,000 10 years out worth today? Again, if somebody offered you $5,000 to buy your investment, would you take it? How about 10,000? Let's answer those questions. Same notation as before, r is the interest rate, as in values, PV is the present value today, and FV_n is the future value in year n. The formula we want to use is FV_n is equal to FV, or principal times 1 plus r to the nth power. That gives us the future value. When we solve for the principal or present value, we have the future value in year n, in this case 10, divided by 1 plus the interest rate, raised to the nth power. When we plug in the numbers, we get $7,711. So $20,000 in 10 years is worth only $7,711 to us today. Note that this is substantially less than one year from now. Well, that makes sense. If we have to wait 10 years for the 20,000, it's going to be worth a lot less to us than if we get it in a year. Now answering the questions, somebody offered us $5,000 today, do we buy the AP investment? Probably not because the present value of that investment is greater than 5,000, it's $7,711. If somebody said, hey, I'll buy that from you for $10,000 today, it's a sure, I'll do that deal most likely, because the present value of the investment to us is worth a lot less than $10,000 today. Again, this is a concept we'll be coming back to over and over and over again. Now let's talk about how we calculate interest. We're going to start with simple interest, which is interest calculated on the loan principal only. As an example, we borrow $10,000 from a generous aunt at 10 percent. She says, yeah, I'll lend you $10,000 and I'll charge you 10 percent simple interest on the loan. Here's what she means. How much do you owe her after two years and after 10 years? Let's do the calculations. Again, r is equal to the interest rate, FV is the future value, PV is the present value, the principal, the loan amount, and n is the length of the loan, in this case, one year. The interest rate is easy to calculate, it's just the future value. The future value is equal to the present value plus 1 plus n times the interest rate. In this case, the future value after two years is $10,000 plus 1 plus 2 times 0.1, the interest rate, and that equals $12,000. After five years, we can do the calculation of the loan. If we don't pay her back within five years or at five years, we will over $10,000 times 1 plus 10 times 0.1, which is equal to $20,000. This is an easy calculation. It's the interest rate times the number of years plus the principal, and we come up with our solution. Clearly, we're going to owe our aunt more money after five years than after two years because more interest is accrued. That brings us to compound interest. Compound interest is interest charged on the principal plus the accrued interest, and this is an important concept. Your aunt, if she lends you $10,000, could put that money in a bank and earn 10 percent on it, let's say, and every year that money would grow, it would grow to $11,000 the 1st year, and something like $12,100 in the 2nd year because that interest remains in the bank and interest accrues on the interest, so the interest earned one year is charged interest in the next year and so forth. That's called compounding interest. How much will you owe your aunt after one year, two years, three years, and 10 years if she wants compounded interest? Again, the same notation we used before, and the future value in n years is equal to the principal or the present value of the loan times 1 plus r to the nth power. Doing the calculations, after one year, we'll owe on $11,000, and after two years, we'll owe on $12,100, and after three years we'll owe on $13,310. You can see that the amount of interest paid with compounding is more than simple compounding. Most loans, most banks, most debt deal takeout or if loan money would be compound interest, probably compounded monthly. That's the norm is that loans are compounded monthly of mortgages, car payments and so forth. This example shows compounding every year in simplicity, but understand that most compounding is by the month, not by the year. After 10 years, we can calculate that the loan that we'll owe $25,937 to our aunt. Compare that to the $20,000 with simple interest payments. It's quite a bit more. The concept of time value of money is quite simple. Money in hand today has greater value than money in the future. The value of money changes with time. Money received today has more value than money in the future, and money in the future received has less value today than that money will have in the future. Future value of money is the value of an investment in the future. We make every an investment today, it'll presumably grow into some value in the future. What is that value? The present value of money is that we receive money in the future sometime, what would that money be worth as today if we could receive it? Simple interest is interest charged on principal only, not on accrued interest. Compound interest is interest charged on both principal and accrued interest, assuming the interest remains in the account. In the next video, we'll take a look at perpetuities and annuities, two other very important concepts that we'll be using. We'll see you there.
