## Money Markets

Although we have treated interest as a given known value, in reality there are many different rates each day. Different rates apply to different circumstances, different user classes, and different periods Most rates are established by the forces of supply and demand in broad markets to which they apply These rates are published widely; a sampling for one day is shown in Table 2,2. Many of these market rates are discussed

TABLE 2.2

Market Interest Rates

Interest rates (August 9, 1995)

U S Treasury bills and notes 3-month bill 6-month bill I-year bill

,3-year note (% yield) 10-year note {% yield) 30-year bond (% yield)

5 36

Fed funds rate Discount rate Prime rate Commercial paper

Certificates of deposit

1 month

2 months I year

Banker's acceptances (30 days) London late Eurodollars (1 month) London Interbank offered rate (I month)

5 88

Federal Home Loan Mortgage Corp (Freddie Mae) (30 years)

7 94

Many different rates apply on any given day This is a sampling more fully in Chapters 3 and 4 Not all interest rates are broad market rates There may be private rates negotiated by two private parties. Or in the context of a firm, special rates may be established for internal transactions or for the purpose of evaluating projects, as discussed later in this chapter

### 2.2 PRESENT VALUE

The theme of the previous section is that money invested today leads to increased value in the future as a result of interest. The formulas of the previous section show how to determine this future value.

That whole set of concepts and formulas can be reversed in time to calculate the value that should be assigned now, in the present, to money that is to be received at a later time. This reversal is the essence of the extremely important concept of present value.

To introduce this concept, consider two situations: (1) you will receive $110 in I year, (2) you receive $100 now and deposit it in a bank account for 1 year at 10% interest Clearly these situations are identical after 1 year—you will receive $110. We can restate this equivalence by saying that $110 received in 1 year is equivalent to the receipt of $100 now when the interest rate is 10%, Or we say that the $110 to be received in 1 year has a present value of $100. In general, $1 to be received a year in the future has a present value of $1/(1 ), where r is the interest rate

A similar transformation applies to future obligations such as the repayment of debt. Suppose that, for some reason, you have an obligation to pay someone $100 in exactly 1 year. This obligation can be regarded as a negative cash flow that occurs at the end of the year . To calculate the present value of this obligation, you determine how much money you would need /jou> in order to cover the obligation. This is easy to determine If the current yearly interest rate is r, you need $100/(1 + r) If that amount of money is deposited in the bank now, it will grow to $100 at the end of the year You can then fully meet the obligation The present value of the obligation is therefore $100/(1 +/ ).

The process of evaluating future obligations as an equivalent present value is alternatively referred to as discounting. The present value of a future monetary amount is less than the face value of that amount, so the future value must be discounted to obtain the present value The factor by which the future value must be discounted is called the discount factor. The 1-year discount factor is d\ — 1/(1 -f >'), where r is the 1-year interest rate So if an amount A is to be received in 1 year, the present value is the discounted amount d\A

The formula for present value depends on the interest rate that is available from a bank or other source If that source quotes rates with compounding, then such a compound interest rate should be used in the calculation of present value, As an example, suppose that the annual interest rate r is compounded at the end of each of m equal periods each year; and suppose that a cash payment of amount A will be received at the end of the /cth period Then the appropriate discount

2.3 PRESENT AND FUTURE VALUES OF STREAMS

factor is

The present value of a payment of A to be received k periods in the future is dkA.

### 2.3 PRESENT AND FUTURE VALUES OF STREAMS

The previous section studied the impact of interest on a single cash deposit or loan; that is, on a single cash flow. We now extend that discussion to the case where cash flows occur at several time periods, and hence constitute a cash flow stream or sequence First we require a new concept

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