Time Value of Money

The concept of time value of money is one of the cornerstones on which most financial theory is based.  The time value of money says that a dollar received today is worth more than a dollar received tomorrow.  This is true because a dollar received today can be invested to earn interest. The amount that is invested today is referred to as the present value, or principal, while the amount that will be received in the future is known as the future value.  This future value includes both the return of the investor’s principal and the interest earned on it.

The process of calculating a future value when a present value is known is called compounding, and the formula to calculate the future value is as follows:

FV = PV x (1 + i)n, where 

FV = future value;

PV = present value, or principal;

i = interest rate per period; and

n = number of periods

For example, assume you invest $1,000 today and can earn 10% a year on your money.  In five years, you would have $1,610.51 in your account:

FV = $1,000 x (1.10)5= $1,000 x 1.61051 = $1,610.51

One thousand dollars of this is your original principal, while $610.51 is the amount of interest you earned over the 5-year period.

Now let’s assume you have inherited $15,000.  You would like to blow some of it today on a nice vacation, but you are also fiscally responsible and would like to set aside enough in an account that pays 10% annually so that there will be $20,000 in it ten years from now when your eight-year-old graduates from high school.   In this case, you are looking for the amount you need to set aside today, or the present value.  This process is referred to as discounting. We can find the present value by a simple algebraic manipulation of the future value formula:

PV = FV ÷ (1 + i)n, so

PV = $20,000 ÷ (1.10)10= $20,000 ÷ 2.5937 ≈ $7,711

So, if you set aside just $7,711 of the $15,000 today in an account that earns 10% a year, there will be $20,000 in that account at the end of 10 years.

You may remember from algebra that if you know all the variables in an equation but one, you can solve for the missing variable.  Thus, you can use the same future value formula to determine an interest rate.  For example, let’s assume your Uncle Sly has agreed to lend you $3,000 today if you agree to pay him $4,320 at the end of two years, and you want to know what interest rate you will be paying for this loan.  In this instance, the present value is $3,000, the amount you are borrowing today; and $4,320, the amount you must repay in two years, is the future value:

FV = PV x (1 + i)n, so

$4,320 = $3,000 x (1 + i)2, thus

(1 + i)2 = $4,320 ÷ $3,000 = 1.4400.

             Solving, 1 + i = equation.pdf therefore

i = 1.20 – 1 = 0.20 = 20%.

Recall that i is the interest rate per period, so Uncle Sly is charging you 20% a year!

In the above examples, we have assumed a lump-sum deposit or withdrawal.  However, many times, we might want to know the future value or present value of a series of equal payments.  A series of equal payments is called an annuity.  The future value of an annuity can be determined by adding up the future values of each payment.  For example, assume you deposit $1,000 in an account that pays 5% at the end of this year and at the end of the following two years, and you want to know how much you will have in the account at the end of the third year.  

Since your first deposit will be made at the end of this year, it will be in there for two years to earn interest and will have grown to $1,102.50 at the end of the third year:

FV = PV x (1 + i)n

FV = $1,000 x (1.05)2 = $1,000 x 1.1025 = $1,102.50

Your second deposit that is made at the end of the second year will earn interest for only one year, so it will have grown to $1,050 by the end of the third year:

FV = $1,000 x (1.05) = $1,050

And your last deposit of $1,000 that is made at the end of the third year will not have earned any interest.  Thus, the total amount you will have in your account at the end of the third year will be $1,102.50 + $1,050 + $1,000 = $3,152.50.

You can imagine how laborious this would be if you wanted to determine how much you would have in your account after 20 years if you deposited $1,000 a year at the end of each year.  Of course, it is an easy calculation if you happen to have a financial calculator or a spreadsheet, like Excel, handy.  If you have neither, there is an equation that makes it easier:

FV = payment x (equation_1.pdf)

Applying this formula to the earlier problem, we get

FV = $1,000 x (equation_2.pdf) = $1,000 x 3.1525 = $3,152.50.

Sometimes we want to know how much we need to save each year to reach a specific goal.  For example, assume you hope to retire at the end of 25 years, and you want to have $1,000,000 accumulated in your retirement account at that time.  You want to know how much you will need to deposit at the end of this year and the remaining years in order to achieve that goal, assuming your money will earn 10% a year.  In this case, the future value is the $1,000,000 you want to have at the time you retire, and you are looking for the payments you have to make into the account to achieve this.

FV = payment x (equation_3.pdf), so

$1,000,000 = payment x (equation_4.pdf), so

$1,000,000 = payment x 98.3471.  Therefore,

Payment = $1,000,000 ÷ 98.3471 ≈ $10,168

Thus, if you deposit $10,168 in your account at the end of each year until the day you retire, you will have $1,000,000 in your account on the day you make that last deposit.

You can also find the present value of an annuity.  In other words, you can determine how much you have to have saved in order to make a series of equal withdrawals over a certain time frame.  There is a formula for that, too:

PV = payment x equation_5.pdf

To illustrate, assume you would like to be able to withdraw $20,000 a year for the next four years to help pay your child’s college expenses.  You won’t make the first withdrawal until the end of the year, but you want to set the money aside today in a separate account that pays 10% interest.  You just need to figure out how much you need to deposit in that account.  In this case, you are looking for the amount you need to deposit today, i.e., the present value.

PV = payment x equation_6.pdf

PV = $20,000 x equation_7.pdf = $20,000 x 3.1699 = $63,398

Thus, if you deposit $63,398 today in an account that pays 10% interest, you will be able to withdraw $20,000 a year at the end of each of the next four years.

The examples above are just a few of the applications of the concept of time value of money.

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