# The basics of Time Value of Money

Updated on May 7, 2011

Many people strive to be rich.  Many never achieve it.  Part of the problem is that folks don't quite understand the future value of money, i.e. making your money work for you.  This hub is written in an attempt to illustrate some of the formulas needed to show how your money can work for you.  I've given a brief description of some of the major points of this subject, and have verbally illustrated how and why the equations work.  After reading this, I hope you get your calculators out and start crunching numbers to see how significant this can be...and then put your money to work for you!

## Time Lines

Time lines are simply a graphical representation used to show the timing of cash flows.  The line contains different intervals from the current value up to the end of the time frame.  The intervals can represent different time periods such as quarters or months, but usually years are used in financial time lines.  Each tick mark on the line represents the end of one period and the beginning of the next period.

To set up a time line to represent the future value of \$1,000 over 12 years at 8% interest rate with no additional funds added other than interest payments, first create a horizontal line.  Place 13 tick marks on the line from zero years (representing present value) all the way up to 12 years (representing the future value of the investment).  Note the initial investment of \$1,000 under the zero tick mark.  Add the interest rate of 8% on the line between the zero and the one year tick mark.  At the 12 year tick mark, place the future value of \$2,518.17 to indicate the total amount accumulated after 12 years with no additional capital added.

## Future Values

Future values are the amount that a cash flow or cash flows will grow to over a period of time when compounded at a given interest rate.  In simple terms, a dollar today will be worth more next year if properly invested.  The formula to calculate this is fairly simple.  The future value is equal to the present value, multiplied by one plus the interest rate raised to the amount of years of compounding.  The filled in equation would look like this:

FV12 = \$1000(1+.08)12 = \$2,518.17

Essentially, \$1,000 compounded at 8% interest over a 12 year period would equal \$2,518.17

## Present Values

Present value is the value today of a future cash flow or cash flows.  To find the present value, simply reverse the process of finding the future value.  Take the future value and divide it by 1 plus the interest rate raised to the amount of years compounded. The formula for this is:
PV = FVN / (1+I)N
This equation is handy when analyzing opportunity costs.  Opportunity cost is the rate of return one could earn on an alternative investment of similar risk.  Assuming that a person was deciding between purchasing a treasury bond or investing in a certificate of deposit, he would use the present value formula to calculate the bond and the future value formula to calculate the CD.  He would then compare the results and make the highest yielding investment.

## Finding the Interest Rate

If the variables of future value, present value, and compounding periods are known, finding the interest rate of an investment is possible.  Finding the interest rate can be done by hand, but not as simply as previous formulas.  The best solution would be to use a financial calculator to solve this for you.  In the calculator enter N= compounding period (12 years); PV= -1,000; PMT= any additional capital added (0); and FV= \$2,518.17.  Select solve and the calculator will give you the interest rate of 8%.

## Finding the Number of Years

To find the time that is required to reach a certain financial amount, an equation needs to be used to solve for the compounding period (N). Like solving for the interest rate, it is easiest to use a calculator to solve for N. Using the calculator, enter I= 8.0; PV= 1000; PMT= 0; and FV= 2,518.17. Select solve and the calculator will reveal that it takes 3 years to reach the desired amount of \$2,518.17. These variables can be plugged back into the original FV equation and it will equal out.

FV = PV(1+1)N = \$1000(1.08)3 = \$2,518.17

## Annuities Defined

An annuity is a series of equal payments at fixed intervals for a specified number of periods.  There are two basic types of annuities: ordinary annuity and annuity due.  Ordinary or deferred annuities are annuities whose payments occur at the end of each period.  An annuity due’s payment occurs at the beginning of each period.  Ordinary annuities are the most commonly used in the finance world.

"The most powerful force in the universe is compound interest" - Albert Einstein

## Calculating the Future Value of an Ordinary Annuity

The future value of an ordinary annuity can be calculated by the use of a formula, but for long term annuities a financial calculator is the best option. The formula for a \$100, three year annuity at a 8% interest rate is:

FVAN = PMT(1+I)N-1 + PMT(1+I)(N-2) + PMT(1+I)(N-3)
FVAN = \$100(1.08)2 + \$100(1.08)1 + \$100(1.08)0
FVAN = \$324.64

## Calculating the Future Value of an Annuity Due

An annuity due earns an additional interest payment over an ordinary annuity thus making the future value of an annuity due higher.  To work the equation out by hand, take the value computed from ordinary annuity formula and multiply it by 1+I:

Working out the formula shows that a similar annuity due nets \$25.97 more than the ordinary annuity.

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