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# Time Value of Money

## What It Means

The concept of the time value of money is the idea that cash received now is worth more than the same amount of cash received at a later date because money has the capacity to earn interest. A person who receives a sum of cash can put that money in a savings account and immediately begin to earn interest on that money. In this case interest refers to the fee the bank pays a client for depositing money.

To illustrate the principle of the time value of money, consider a scenario in which a person has the option of receiving \$100 one year in the future or receiving \$100 immediately and depositing the money in a bank account that pays a simple interest rate of 5 percent each year. Simple interest is calculated only on the initial sum of money. Thus, if the person chose to take the money immediately and deposit it into the account, he or she would have \$105 at the end of a year. At the end of the second year, the person would have \$110. A simple interest rate of 5 percent would add \$5, or 5 percent of \$100 (the original sum), each year. On the other hand, if that person waited a year to receive the money, he or she would only have \$100.

In this example the benefit between taking the money immediately and waiting is small, but that is because the original sum is small. If the original sum were \$1 million, the person would have \$1,050,000 at the end of a year if the account paid a simple yearly interest rate of 5 percent. In this particular case the person has an extra \$50,000 instead of only \$5 after a year. This simple interest rate would add \$50,000 to the original sum every year.

## When Did It Begin

The practice of charging interest on loans, which dates back to ancient societies, indicates that moneylenders have long had an intuitive, if not explicitly stated, understanding of the time value of money. Records dating as far back as 5000 bc indicate that the Mesopotamians, Hittites, Phoenicians, and Egyptians charged interest when they loaned such items as olives, dates, seeds, and animals. The time value of any loaned item was perhaps easiest to see with the loaning of seeds because any successfully planted seed would yield a plant that would produce more additional seeds. Thus, it was wise to get seeds in the ground, both to yield a healthy crop and to have more seeds for future plantings.

During the Middle Ages (from about 500 to about 1500) there is also evidence that people understood the time value of money. At trade fairs merchants and money changers in need of cash would often issue documents to others that could be redeemed at future trade fairs. If a person had accepted a document in exchange for a certain amount of cash, the merchant would often pay the person back with additional cash when he returned to redeem the document. Thus, both parties in the transaction understood that the original value of the loan would grow over time because the merchant was going to use the funds acquired in the loan to make more money.

## More Detailed Information

The time value of \$1 million from the earlier example increases more dramatically if, instead of being simple interest, the interest is compounded. With compound interest the interest is calculated not solely on the beginning principal but rather on that and then the new principal each time the interest is paid. Thus, with a compound interest rate of 5 percent, the person who received \$1 million would still have \$1,050,00 after the first year. However, in the second year, rather than adding another 5 percent of the original principal of \$1 million, 5 percent of the new principal of \$1,050,000 is added to the balance in the savings account. Thus, instead of having \$1,100,000 at the end of the second year in an account bearing simple interest, the person has \$1,102,500 if the money is in an account that pays compound interest. At the end of three years the person has \$1,157, 625 with compound interest and only \$1,150,000 with the simple interest.

The value of the \$1 million would increase even more dramatically if the account bearing compound interest paid that interest quarterly (four times a year) rather than annually (only once a year). In this case every three months the account would pay 1.25 percent (a quarter of the 5 percent annual rate), and at the end of a year the person would have \$1,050,945 instead of \$1,050,000. At the end of the second year, the person would have \$1,104,486 instead of \$1,102,500. The person who accepted the \$1 million immediately and placed the cash into such an account would have \$104,486 more than the person who waited one year to accept the \$1 million.

When considering the time value of money, people generally want to know two things: the future value of cash invested today (how much more it would be worth in the future because of interest) and the present value of cash received at a later date (how much one would need to put into an account now to have, as a result of interest, a certain amount, say \$1 million, at that later date). Although, in most real-world cases, determining these figures requires the use of more complex mathematical equations, the preceding examples provide a general idea. Say you wanted to know the present value of \$100 that you were going to receive in a year. If the simple interest rate were 5 percent, you have to determine how much to put into an account now for that money to be worth \$100 in a year. To do that you would divide \$100 by 1.05 (which represents the 5 percent interest per year), which is about \$95.24. Thus, given a simple interest rate of 5 percent, \$100 received a year from now would be worth only \$95.25 today. In the same manner, the present value of \$1 million received a year from now is \$952,380 today.

It is especially important to have a sense of the time value of money when considering annuities. An annuity is an equal, annual series of cash flows. Annuities may be equal annual payments of money or equal annual receipts of money. Thus, assuming there is fixed rate of interest a person’s mortgage (the loan taken to pay for a home) is an annuity because the person pays the same amount to the finance company each time she makes a payment. A person thinking about buying a house might have second thoughts upon discovering that she will be required to pay \$1,500 every month for 30 years in order to purchase the home. However, according to the principle of the time value of money, that \$1,500 is going to amount to a smaller sum of money each passing year because as time goes by that \$1,500 will hold less value. More than likely what seems like a large monthly payment in the early years of the mortgage will feel like a less significant monthly outlay of cash over the course of the loan.

## Recent Trends

Because real estate typically appreciates, or increases in value, one can see the time value of money when investing in a home. For young people coming out of school and entering the job market, it is usually wise to stop renting an apartment as soon as possible and buy a home, which over the course of a 30-year mortgage is very likely to increase in value. Increasingly adults in their early 20s have been doing that. According to the U.S. Census Bureau, homeownership among people under the age of 25 increased dramatically from 1994 to 2004. In 1994 only 14.9 percent of adults under 25 owned homes. Ten years later 25.2 percent of adults in this age group owned homes.

One of the reasons for this trend is the reduction of minimum down payments. In the past home buyers were often required to make a down payment for as much as 20 percent of the value of the home. For a \$100,000 home, which is a modest dwelling in many communities, one would need \$20,000 to spend on the home in order to secure a loan. Such requirements prevented young first-time buyers from purchasing homes. Many of these requirements have either vanished altogether or been greatly reduced. People can now purchase homes by putting zero to 5 percent down on the home. Such terms have allowed young people to enter the housing market, which in turn has allowed them to benefit from the time value of money—because of rising home prices, the \$100,000 house they buy today will likely be worth much more in the future.