Complete Guide To Corporate Finance
Time Value Of Money  Introduction To The Time Value Of Money
In addition to being able to understand financial statements, it's important to be able to estimate the value of an investment in the present and in the future.
The idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity is called the time value of money. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received. Thus, at the most basic level, the time value of money demonstrates that, all things being equal, it is better to have money now rather than later.
But why is this? A $100 bill now has the same value as a $100 bill one year from now, doesn't it? Actually, although the bill is the same, you can do much more with the money if you have it now because over time you can earn more interest on your money.
By receiving $10,000 today (Option A), you are poised to increase the future value of your money by investing and gaining interest over a period of time. If you receive the money three years down the line (Option B), you don't have time on your side, and the payment received in three years would be your future value. To illustrate, we have provided a timeline:
If you choose Option A, your future value will be $10,000 plus any interest acquired over the three years. The future value for Option B, on the other hand, would only be $10,000. So how can you calculate exactly how much more Option A is worth compared to Option B? Let's take a look.
Future Value Basics
If you choose Option A and invest the total amount at a simple annual rate of 4.5%, the future value of your investment at the end of the first year is $10,450, which is calculated by multiplying the principal amount of $10,000 by the interest rate of 4.5% and then adding the interest gained to the principal amount:
You can also calculate the total amount of a oneyear investment with a simple manipulation of the above equation:
If the $10,450 left in your investment account at the end of the first year is left untouched and you invested it at 4.5% for another year, how much would you have? To calculate this, you would take the $10,450 and multiply it again by 1.045 (0.045 +1). At the end of two years, you would have $10,920:
The above calculation is then equivalent to the following equation:
Think back to math class and the rule of exponents, which states that the multiplication of like terms is equivalent to adding their exponents. In the above equation, the two like terms are (1+0.045), and the exponent on each is equal to 1. Therefore, the equation can be represented as the following:
We can see that the exponent is equal to the number of years for which the money is earning interest in an investment. So, the equation for calculating the threeyear future value of the investment would look like this:
This calculation means that we don't need to calculate the future value after the first year, then the second year, then the third year, and so on. If you know how many years you would like to hold a present amount of money in an investment, the future value of that amount is calculated by the following equation:
For related reading, see Time Value Of Money: Determining Your Future Worth.
The idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity is called the time value of money. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received. Thus, at the most basic level, the time value of money demonstrates that, all things being equal, it is better to have money now rather than later.
But why is this? A $100 bill now has the same value as a $100 bill one year from now, doesn't it? Actually, although the bill is the same, you can do much more with the money if you have it now because over time you can earn more interest on your money.
By receiving $10,000 today (Option A), you are poised to increase the future value of your money by investing and gaining interest over a period of time. If you receive the money three years down the line (Option B), you don't have time on your side, and the payment received in three years would be your future value. To illustrate, we have provided a timeline:
If you choose Option A, your future value will be $10,000 plus any interest acquired over the three years. The future value for Option B, on the other hand, would only be $10,000. So how can you calculate exactly how much more Option A is worth compared to Option B? Let's take a look.
Future Value Basics
If you choose Option A and invest the total amount at a simple annual rate of 4.5%, the future value of your investment at the end of the first year is $10,450, which is calculated by multiplying the principal amount of $10,000 by the interest rate of 4.5% and then adding the interest gained to the principal amount:
Future value of investment at end of first year: = ($10,000 x 0.045) + $10,000 = $10,450 
You can also calculate the total amount of a oneyear investment with a simple manipulation of the above equation:
 Original equation: ($10,000 x 0.045) + $10,000 = $10,450
 Manipulation: $10,000 x [(1 x 0.045) + 1] = $10,450
 Final equation: $10,000 x (0.045 + 1) = $10,450
If the $10,450 left in your investment account at the end of the first year is left untouched and you invested it at 4.5% for another year, how much would you have? To calculate this, you would take the $10,450 and multiply it again by 1.045 (0.045 +1). At the end of two years, you would have $10,920:
Future value of investment at end of second year: = $10,450 x (1+0.045) = $10,920.25 
The above calculation is then equivalent to the following equation:
Future Value = $10,000 x (1+0.045) x (1+0.045) 
Think back to math class and the rule of exponents, which states that the multiplication of like terms is equivalent to adding their exponents. In the above equation, the two like terms are (1+0.045), and the exponent on each is equal to 1. Therefore, the equation can be represented as the following:

We can see that the exponent is equal to the number of years for which the money is earning interest in an investment. So, the equation for calculating the threeyear future value of the investment would look like this:

This calculation means that we don't need to calculate the future value after the first year, then the second year, then the third year, and so on. If you know how many years you would like to hold a present amount of money in an investment, the future value of that amount is calculated by the following equation: