# The Time Value of Money

The Time Value of Money

INTRODUCTION
As individuals, we often face decisions that involve saving money for a future use, or borrowing money for current consumption. We then need to determine the amount we need to invest, if we are saving, or the cost of borrowing, if we are shopping for a loan. As investment analysts, much of our work also involves evaluating transactions
with present and future cash flows. When we place a value on any security, for example, we are attempting to determine the worth of a stream of future cash flows.
To carry out all the above tasks accurately, we must understand the mathematics of time value of money problems. Money has time value in that individuals value a given amount of money more highly the earlier it is received. Therefore, a smaller amount of money now may be equivalent in value to a larger amount received at a future date.
The time value of money as a topic in investment mathematics deals with equivalence relationships between cash flows with different dates. Mastery of time value of money concepts and techniques is essential for investment analysts.

INTEREST RATES: INTERPRETATION
In this reading, we will continually refer to interest rates. In some cases, we assume a particular value for the interest rate; in other cases, the interest rate will be the unknown quantity we seek to determine. Before turning to the mechanics of time value of money problems, we must illustrate the underlying economic concepts. In
this section, we briefly explain the meaning and interpretation of interest rates.
Time value of money concerns equivalence relationships between cash flows occurring on different dates. The idea of equivalence relationships is relatively simple.
Consider the following exchange: You pay \$10,000 today and in return receive \$9,500 today. Would you accept this arrangement? Not likely. But what if you received the \$9,500 today and paid the \$10,000 one year from now? Can these amounts be considered equivalent? Possibly, because a payment of \$10,000 a year from now would probably
be worth less to you than a payment of \$10,000 today. It would be fair, therefore, to discount the \$10,000 received in one year; that is, to cut its value based on how much time passes before the money is paid. An interest rate, denoted r, is a rate of return that reflects the relationship between differently dated cash flows. If \$9,500
today and \$10,000 in one year are equivalent in value, then \$10,000 − \$9,500 = \$500 is the required compensation for receiving \$10,000 in one year rather than now. The interest rate—the required compensation stated as a rate of return—is \$500/\$9,500 = 0.0526 or 5.26 percent.

Interest rates can be thought of in three ways. First, they can be considered required rates of return—that is, the minimum rate of return an investor must receive in order to accept the investment. Second, interest rates can be considered discount rates. In the example above, 5.26 percent is that rate at which we discounted the \$10,000 future
amount to find its value today. Thus, we use the terms “interest rate” and “discount rate” almost interchangeably. Third, interest rates can be considered opportunity costs.
An opportunity cost is the value that investors forgo by choosing a particular course of action. In the example, if the party who supplied \$9,500 had instead decided to spend it today, he would have forgone earning 5.26 percent on the money. So we can view 5.26 percent as the opportunity cost of current consumption.

Economics tells us that interest rates are set in the marketplace by the forces of supply and demand, where investors are suppliers of funds and borrowers are demanders of funds. Taking the perspective of investors in analyzing market- determined interest rates, we can view an interest rate r as being composed of a real risk- free interest rate
plus a set of four premiums that are required returns or compensation for bearing distinct types of risk

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