## Time Value of Money

Anyone who uses a credit card knows that paying off debts over a long period of time costs more than if the debt is paid off more quickly. This is because there is a cost with taking more time to pay off a loan than if the loan had a shorter maturity. This increase in total payments is a function of the interest rate and the time taken to repay.

When money is not invested, such as if it is kept in a mattress or in a non-interest bearing savings account, the owner forgoes the opportunity to earn more money by investing. Essentially, the interest rate the owner could realize were the money invested represents the opportunity cost of not investing.

Suppose an investor buys a Certificate of Deposit (CD) today for $100. The CD is paying 4% annually. In one year, the value of the investment has grown to $104. The original $100 is known as the Present Value (PV), the $104 is called the Future Value (FV), and the rate of interest is simply r. This relationship can be generally shown with the following formula:

FV = PV * (1 + r)

However, this formula assumes one year of interest only. Suppose the investor rolls the CD over into another year. In other words the $104 is used to purchase a CD in year 2 at the same rate. We need a more general formula to account for the compounding of the interest over multiple years because interest rates are rarely expressed in any other form besides annually. The following formula is a more general representation of calculating FVs.

FV = PV * (1 + r)^{n}

The n in the formula above is the number of years the money is invested at the current interest rate, r. Suppose the investor rolls the CD over for 10 year. What is the expected FV of the investment?

$148.02 = $100 * (1 + 0.04)^{10}

By keeping the original investment in the CD for 10 years, the expected FV of the investment is $148.02 at the current rate of 4%. Sometimes an investor knows how much money is needed in the future and wants to know how much to invest at a constant rate to realize the FV needed. A little algebra helps. By solving for PV in the formula above we get:

PV = FV / (1 + r)^{n}

Suppose an investor needs $20,000 in 18 years to pay for his daughter’s college education. How much money (PV) does he need to invest at 5% to realize a $20,000 return in 25 years?

$8,310.41 = $20,000 / (1 + 0.05)^{18}

In, other words, the PV of the investment must be equal to $8,310.41 to grow to $20,000 at 5% in 18 years.

Of course, the formulas above do not account for ongoing payments. Usually, investors will continue to add to an investment such as a CD. This type of investment is known as an annuity.

The time value of money principle is an important lesson for investing in financial markets. Money today is not the same as money tomorrow. This is true for both sides of a transaction during the sale and purchase of an investment. By applying some basic mathematics, you can calculate the Present Value and Future Value of any investment.