## Time Value of Money

This article will explain the financial concept of time value of money. The overview provides an introduction to the principles at work when money grows in value over time. These principles include future value of money, present value of money, simple interest and compound interest. In addition, other concepts that relate to factors that can impede the growth in value of money over time are explained, including risk, inflation and accessibility of assets. Basic formulas and tables have been provided to assist in calculating various formulations of time value of money problems. Explanations of common financial dealings in which the time value of money is an important consideration, such as annuities, loan amortization and tax deferral options, are included to help illustrate the concept of the time value of money in everyday life.

The time value of money is a fundamental financial principle. Its basic premise is that money gains value over time. As a result, a dollar saved today will be worth more in the future, and a dollar paid today costs more than a dollar paid later in time. The reason for the increasing value in money over time is that money can be invested to earn interest, and the gain in interest can be significant over time. This is also why a dollar paid today costs more than a dollar paid in the future. Money expended today cannot be invested for the future and thus the loss is essentially two-fold -- the money is spent on the payment and any earning potential it could have had in an investment vehicle is forgone.

The concept of time value of money is an important consideration in any long-term, and even short-term, investment or financial obligation. Financial managers and advisers frequently use time value of money formulas to determine the true costs of various investment opportunities. In addition, people consider the time value of money concept-perhaps without even realizing it-in making common financial decisions, such as considering whether to take out a loan or mortgage, sign a lease, deposit money in a savings account or an annuity or perhaps even to spend the money or pay off bills.

Although calculating the changing value of money over time requires formulas and mathematical computations, the underlying principle that money in hand is more valuable than money down the road is almost self-evident. Most people, if given a choice as to whether they would rather have money today or in the future, would instinctively choose money today. Ready money, or money that is presently accessible, is available to be invested in a range of vehicles that can return the money-plus interest-down the road. The sooner the money is invested, the sooner it can begin earning interest, and the longer the money is invested, the more capacity it has to grow in value. However, money that is not readily available but is to be paid in the future will only then become available for investment upon receipt, and thus it lacks present interest-earning potential.

To understand the economics of the time value of money, it is important to first grasp its underlying concepts of future value of money and present value of money. The future value of money is the value that money will grow to when invested at a given rate for a specified period of time. The present value of money is the amount that an investment earned in the future is worth today. For instance, if a person invests one dollar for one year at a 6% annual interest rate, the dollar accumulates six cents in interest while invested and thus is worth $1.06 at the end of the year. Since the time value of money is measured according to the future value and the present value of an investment of money, the future value of the person's dollar is $1.06 at a 6% interest rate for a one-year period. The present value of the $1.06 that could be earned at the end of one year is $1.

The following sections provide a more in-depth explanation of these concepts.

### Basic Financial Concepts

### Future Value of Money

The future value of money is the value of a sum of money, invested at a given interest rate for a defined period of time, at a specified date in the future and that is equivalent in value to a specified sum today. The future value of money can be calculated if given the interest rate of the investment, the length of time of the investment and the amount of the initial deposit. The calculation can determine the future value of a single sum investment that is deposited at the beginning of the duration of the investment. Or, if an investment consists of a series of equally spaced payments, generally known as an annuity, the future value of this investment can also be calculated. When calculating the future value of money, we commonly assume that the future value of an investment will be greater than its present value, and we use mathematical formulas to solve for the exact increase of an investment over time. The rate that money gains in value over time depends on the number of compounding periods that an investment is allowed to grow and the interest rate that the investment is earning. In other words, assume you have $10, 000 to invest today. If you spend it, the money will be gone and thus no interest will be earned on it and the money has no future value. If instead you decide to invest the $10, 000, you can increase the future value of your money over time because you will have the $10, 000 plus any interest that investment has earned. If you invest the money at 5% interest for one year, you would multiply $10, 000 by 5% to determine annual interest earned of $500. Thus, the future value of your investment is $10, 500. This calculation is explained in more detail below.

### Calculating Future Value

Investors frequently calculate the future value of their investment options to determine the most profitable way to grow their money. Suppose an investor sets aside $100 to deposit in her money market account at her local bank, which is paying an annual interest rate of 10% on money market savings accounts. If she keeps her money in her money market account for one year, at the end of that year she will be able to withdraw both her initial $100 deposit plus the $10 she earned in interest. The following formula illustrates this concept:

Original deposit + Interest on deposit = FV

$100 + (10%) ($100) = $110

Thus, the investor can calculate the future value of her money by plugging in her deposit of $100 plus the 10% interest rate that her bank is paying on savings accounts to solve for the future value of her original deposit. When she performs the calculations, she will find that the future value of her $100 after one year is equal to $110 ($100 plus 10). While this calculation is relatively straightforward, another investor may want to calculate how much money he would have if he invested his money in a retirement plan and left it there to earn interest for 20 years. Luckily, there is an easy formula that he could use to determine the future value of his investment:

FV = P(1 + R)N

FV = future value

P = principal (initial deposit)

R = annual rate of interest

N = number of years

This equation can be used for any number of years. The investor must simply have the amount of the principal, the annual interest rate of the investment vehicle he is considering and the number of years the money will be invested. Once an investor has these figures accessible, he can solve for the future value of his investment. For instance, here is how an investor could compare his earning potential of two different investment options by solving for the future value of a $100 deposit at a 10% interest rate for one year and again for two years:

1 Year on DepositFV = P(1 + R)1

FV = $100(1 + .10)

FV = $100(1.10)

FV = $110

2 Years on DepositFV = P(1 + R)2

FV = $100(1 + .10)2

FV = $100(1.10)(1.10)

FV = $121

While these calculations illustrate the growth in the value of money over a one and two-year time period, the same formula can be used to calculate the time value of money for any number of years. For instance, if an investor wanted to figure the amount of her account if she invested $100 at a 10% annual interest rate for 10 years, she would need to multiply $100 by 1.10 and then repeat that calculation using the sum of each calculation for a total of ten times. If she performed this calculation, she would determine that (1.10)10 equals 2.594, and multiplying $100 by 2.594, she would discover that the future value of her initial $100 investment at 10% annual interest rate for 10 years would be $259.40.

Solving future value of money calculations can become more complicated the longer the money is invested. This is because almost all investments earn interest every year, and this interest is added to the principal at the end of each year, so that the next year interest is earned on both the principal plus all of the interest that has been earned. This financial concept, known as compound interest, is described in more detail below. Because making these calculations grows increasingly complicated the longer the investment, most investors and money managers use future value tables as a shortcut to solve these calculations. These tables have been completed with the calculation of one dollar invested at various annual interest rates for certain given periods of time. Thus, if an investor knows the annual rate of interest of an investment option and the length of time he plans to invest, he can quickly find the factor he will need to use to multiply by his principal, or initial deposit, in order to determine the future value of his investment.

The following future value table contains factors that are used in determining various future values. To read the future value table, you simply identify the column that represents the annual interest rate of your investment and the row that represents the duration of your investment. The cell where the column and row meet contains the factor that you will use to multiply by your original investment. For example, suppose you wish to find the future value of an original investment of $1, 000 over a 5-year period at 10% interest. Look up the factor (1.61), and multiply it by the original investment: $1, 000(1.61) = $1, 610. Thus, the future value of $1, 000 invested for five years at a 10% annual interest rate is $1, 610. You can also calculate how fast your investment will grow over the five years you invest it. To do this, simply deduct 1.00 from the factor and you get the total percentage increase (1.61 -- 1.00 = .61, or 61%). In other words, a $1, 000 investment that grows to $1, 610 in five years represents an increase in value of 61%.

Interest Rate Periods 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 1.0700 1.0800 1.0900 1.1000 1.1 100 1.1200 2 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 1.1664 1.1881 1.2100 1.2321 1.2544 3 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 1.2250 1.2597 1.2950 1.3310 1.3676 1.4049 4 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 1.3108 1.3605 1.4116 1.4641 1.5181 1.5735 5 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 1.4026 1.4693 1.5386 1.6105 1.6851 1.7623 6 1.0615 1.1261 1.1941 1.2653 1.3401 1.4185 1.5007 1.5869 1.6771 1.7716 1.8704 1.9738 7 1.0721 1.1487 1.2299 1.3159 1.4071 1.5036 1.6058 1.7138 1.8280 1.9487 2.0762 2.2107 8 1.0829 1.1717 1.2668 1.3686 1.4775 1.5939 1.7182 1.8509 1.9926 2.1436 2.3045 2.4760 9 1.0937 1.1951 1.3048 1.4233 1.5513 1.6895 1.8385 1.9990 2.1719 2.3580 2.5580 2.7731 10 1.1046 1.2190 1.3439 1.4802 1.6289 1.7909 1.9672 2.1589 2.3674 2.5937 2.8394 3.1059 11 1.1157 1.2434 1.3842 1.5395 1.7103 1.8983 2.1049 2.3316 2.5804 2.8531 3.1518 3.4786 12 1.1268 1.2682 1.4258 1.6010 1.7959 2.0122 2.2522 2.5182 2.8127 3.1384 3.4985 3.8960 13 1.1381 1.2936 1.4685 1.6651 1.8857 2.1329 2.4098 2.7196 3.0658 3.4523 3.8833 4.3635 14 1.1495 1.3195 1.5126 1.7317 1.9799 2.2609 2.5785 2.9372 3.3417 3.7975 4.3104 4.8871 15 1.1610 1.3459 1.5580 1.8009 2.0789 2.3966 2.7590 3.1722 3.6425 4.1773 4.7846 5.4736

You can also use future value tables to determine annual rates of compound interest and the number of years you must invest your principal in order for it to grow by a certain percentage at a known interest rate. For instance, to calculate the annual rate of compound interest that applies to an investment of $1, 000, which is expected to grow 61% in five years, you would calculate the factor by adding .61 + 1.00 to get 1.61. You would then go to the fifth year row since you plan to invest for five years, and move along the row until you find 1.61. You would then trace up the column to find the annual interest rate of 10%. Thus, you would know that the rate of interest for a $1, 000 investment expected to grow 61% in five years is 10%. Finally, if you want to find out how many years it will take for an investment growing at 10% annually to increase 61%, you would find the column representing 10% and trace down to find the factor of 1.61. You would then follow the row across the table until you reach the Periods column, and you would then find that you must invest your $1, 000 at a 10% interest rate for five years in over for your investment to increase 61%. Thus, the future value formulas and tables provide investors with a relatively quick and easy method to solve for the future value of a range of investment opportunities to decide which options are the most profitable.

### Present Value of Money

While the future value of money calculates the value of an investment in the future, the present value of money represents the amount of money that would be required today to equal a desired future sum of money that has been discounted by an appropriate interest rate. The future amount of money could be the result of a long-term investment or the payment of a single sum that will be received or paid at a set date in the future or a series of payments of equal amounts paid or received at equal durations over a period of time. A series of equivalent, equally spaced payments is known as an annuity. Annuities will be discussed in more detail in the sections below. If the principle of time value of money holds that money today is more valuable than money tomorrow, the inverse of this principle is also true. Money to be received in the future is less valuable than money received today. Thus, the longer an investor must wait to receive her money, the less value she will obtain from the asset.

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