# Time Value of Money

Time value of money is one of the most important concepts in finance. There is no way you could possibly understand finance without understanding the value of money across different time periods.

Generally, money is more valuable right now than at any other point in time in the future. If you were given a choice of taking $100 today or $100 in one year, it would be a smart decision to take it now. Why? Because even if you don’t need it, you can put that $100 in the bank and earn some interest on it (e.g. get $105 in a year) and end up getting a better deal than if you accepted $100 in a year ($105 > $100).

This example is pretty straight forward, and maybe too obvious. But what if you were offered $100 now, or $105 in a year? Which one would you choose now? Now, you should think about if you took $100 today, could you earn more or less than $5 over a one year period. If you can take $100 today, and earn $10 on it, of course that you will do that ($110 > $105). But if you can earn only $3 on your $100, then you’re better off just waiting and taking $105 in the future ($105 > $103).

In order to calculate time value of money, you have to know what you’re calculating. Are you interested in how much $100, $200, or $5M today is worth in one, two, or three years? Or maybe how much $100, $200, or $5M in three years is worth today? How about how valuable are yearly payments of $50 for five years? Or yearly payments of $50 forever? All these calculations are similar conceptually, but use different formulas.

#### Future Value of Money

Let’s look at how much $100 today is worth in two years. We will assume that if we invest money that we can earn 5% a year. Therefore, in one year, our $100 will become $105. Now, the tricky part is the second year. If we earn 5% again, we will earn that 5% on $105 instead of $100. So at the end of the second year, we will have $110.25. In the second year, we earned 5% on $100 plus 5% on $5 (our interest from the first year). Refer to the graphic below for visual explanation.

From this graph, one can see that in order to figure out the future value of a certain amount of money, he/she has to multiply it by (1 + r)^{n}, where *r* is the rate of return, and *n* is a number of compounding periods (compounding means earning interest on interest).

#### Present Value of Money

Present value is a reverse process of deriving the future value. So, for future value, we were trying to figure out how much will $100 be in two years and we got to the answer $110.25. Now, imagine you are given information that in year 2 you have $110.25, the rate is 5%, and you have to figure out present value. You already know from the graph above that the answer is $100. But let’s look at the process of deriving $100 from $110.25.

We can work backwards to go from $110.25 back to $105 back to $100. All we have to do is divide $110.25 by (1 + 5%) in order to get $105. After that, we divide $105 once again with (1+ 5%) in order to get $100. So, all we did was divide $110.25 twice with (1 + 5%). Refer to the graph below for a visual explanation.

From this graph, one can see that in order to figure out the present value of certain amount of money in year *n*, he/she has to divide it by (1 + r)^{n}, where *r* is the rate of return, and *n* is a number of compounding periods.

#### Present Value of Perpetuity

What is perpetuity? Perpetuity is when something goes on forever. In finance, it is those kinds of deals where you pay some amount today, and then you get, let’s say, $500 a year forever. So, how does someone figure out how much $500 a year forever is worth today? Actually, it is a very simple formula. All you have to do is divide your payment by the yearly interest rate. To make a sense out of it, look at the graph.

What’s going on here? Well, in order to calculate present value of future payments, you have to discount them to year 0. In this case, you have infinitely many payments. But if you look at the picture above, as you go farther in time, PV is getting smaller and smaller. Therefore, PV of $500 in year 1 equals $476.2, but those same $500 in year 300 equals $0.0002. So the sum of all of those present values is approaching $10,000, which is simply $500 divided by 5%.

#### Present Value of Annuity

Annuity is similar concept to perpetuity, but it is finite. So, payments are either paid or received for a set amount of time. Think about your parents’ mortgage or car loans. You make same payments every single month for a specified number of years.

In order to conceptually understand annuity, we have to play with the perpetuity formula. Check out the graph.

STEP 1:

Here, we calculated perpetuity for year 0, and year 5. They are both $10,000. However, in order to compare apples to apples, we have to discount $10,000 from year 5 to year 0. The difference will be present value of annuity.

STEP 2:

So, pretty much what we did is we took payments of $500 forever from year 0, and payments of $500 forever from year 5. If you subtract all payments of $500 forever from year 5, than you’re left over only with payments from years 1 through 5. Therefore, the formula for PV of annuity is: