Lecture 2

Lecture 2 Economists' Meaning of Money 1. Basic Definition Money is anything that is generally accepted in payment for goods and services and for the repayment of debts, as a matter of social custom. It follows that money is defined more by its function (what purposes it serves) than by its form (coin, paper, gold bars, etc.). Moreover, the stress on "generally accepted" in this definition indicates that money is largely a social convention in the sense that what actually constitutes money in a society depends on what people in the society are generally willing to accept as money. An interesting question is how this "general acceptance" comes to be established! Note on Terminology: Money must be distinguished from both "wealth" and "income." The wealth of an agent at any given point in time is the current market value of the total collection of assets currently owned by that agent. Money holdings might constitute part of an agent's wealth, but the agent would presumably own other types of assets as well (e.g., land, equipment,...). On the other hand, income is a flow of value accrued over some specified period of time. Example: A student works part time as a teaching assistant, earning $900 per month, and has a checking account balance of $400. He also owns a car worth $1100 and books worth $500. Consequently, ignoring for simplicity the student's "human capital" (e.g., his embodied labor skills, valued by estimating the capitalized stream of all of his potential future wage earnings), one has: • • •

Money holdings = $400 Wealth = Market value of his asset holdings consisting of (money holdings, car, books) = ($400 + $1,100 + $500) = = $2,000 Income = $900 per month

As illustrated by this example, income is a flow variable in the sense that it measures an amount of value accrued over a specified period of time (e.g., a month). In contrast, money and wealth are both stock variables in the sense that they measure an amount of value at a given point in time.

2. Types of Money •

Commodity Money: Commodity money is any commodity (economic good) that is used as money, i.e., that is generally accepted as a means of payment for goods and services and for the repayment of debts. Commodity Money Examples from the Past: Cattle, skins, furs, corn, gold, silver, and copper.

•

Fiat Money: Fiat money is any paper money that is "unbacked legal tender." A paper money is unbacked if it is not collateralized by any valuable commodity. That is, no one is obliged by law to convert the paper money into coins, precious metals, or any other type of physical good or service. A paper money is legal tender for a country if, by law, the citizens of the country must accept the paper money for repayment of debts. 1

Fiat Money Example: In the United States, the Federal reserve notes (dollar bills) issued by the Federal Reserve System (the central bank of the US) are unbacked legal tender and hence are fiat money. The general acceptance of dollars in the US as payment for goods and services depends upon the persistence of a widely shared trust among citizens that any person who accepts dollars now in exchange for goods or services will be able to exchange these dollars later for other goods and services. •

Electronic Means of Payment (EMOP): A means of payment that permits payments to be transmitted using electronic telecommunications. EMOP Examples: One example is the Fed's use of Fedwire, a telecommunications system that permits all financial institutions that maintain accounts with the Fed to wire (transfer) funds to each other without having to send checks. Other examples include private EMOP systems such as CHIPS and SWIFT (used by banks, money market mutual funds, securities dealers, and corporations to wire funds) and ACHs (automatic clearing houses) used for smaller wire transfers, e.g., from employers to their employees. Interesting EMOP observation: As noted by Mishkin, in the United States, even though an EMOP is used by fewer than 1 percent of the number of payments made, over 80 percent of the dollar value of payments made is through EMOP transfers.

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Electronic Money (e-Money): E-money is money that is stored electronically rather than in paper or commodity form. Once established, e-money cuts way down on transactions costs; but it can be expensive to set up an e-money system, and concerns have been raised about record-keeping, security, and privacy (as well as the elimination of "float" for consumers!). e-Money Examples: •

• • •

Debit Cards: Charged expenses are immediately deducted from some corresponding bank account -- there is no float (time between purchase and deduction) as with credit cards and paper checks; Stored-Value Cards: Charged expenses are immediately deducted from a fixed amount of digital cash stored on the card; Electronic Cash: A form of e-money that can be used to purchase goods and services on the Internet; Electronic Checks: A process by which users of the Internet can pay their bills directly over the Internet without having to send a paper check.

Functions of Money Money performs three basic functions in an economy: (1) It serves as a unit of account; (2) it serves as a medium of exchange; and (3) it serves as a store of value. •

Unit of Account: A unit in terms of which a single price for every good and service can be quoted. Example: In the US, the price of an apple is given as dollars per apple, the price of a gallon of milk is given as dollars per gallon of milk, etc. That is, each good or service on sale at an outlet is 2

generally offered at a single quoted "dollar price" -- that is, a price quoted in terms of dollars. In reality, however, any particular good or service (e.g., apples) has a huge array of different prices that could be quoted for it, one for each other good or service in the economy (e.g., pounds of bread per apple, cans of beer per apple, hours of doctor visits per apple, etc.) Without a money unit to provide a single accepted unit of account, sellers would have to quote prices of items in terms of whichever goods or services they were willing to accept in return at the time the items were purchased. That is, as clarified further below, the payment system would be a "barter" payment system. •

Medium of Exchange: An accepted means of payment for trade of goods and services. As noted above, the existence of a money unit permits each item for sale to have a single price quoted for it in terms of the money unit. But this is not enough to ensure the item will actually be sold to buyers for money units. Sellers have to be willing to accept the money units from buyers in return for giving up the item, which requires a trust on the part of sellers that others will in turn be willing to accept these money units from them at a later time in return for goods and services. That is, the money units have to act as a medium of exchange in the economy before one can conclude that they indeed constitute money in the economy.

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Store of Value: A repository of purchasing power for future use. Money can be held for future use, allowing for the ability to save (store value) over time. All assets act as stores of value to some extent, but money by definition is the most liquid, i.e., the most easily converted into a medium of exchange, since by definition it already is a medium of exchange! On the other hand, money is by no means a risk-free store of value. The real purchasing power of money depends on the inflation rate, that is, on the rate at which the general price level is changing. If the inflation rate is positive (prices are increasing), any money held loses purchasing value through time. If the inflation rate is negative (prices are decreasing), any money held gains purchasing power over time. To the extent that the inflation rate is unpredictable, inflation reduces the ability of money to act as a reliable store of value and as a method of deferred payment in borrowing-lending transactions. A positive inflation rate is bad for lenders and good for borrowers since the dollars lent out are worth more than the dollars later paid back. Conversely, a negative inflation rate is good for lenders and bad for borrowers. In extreme cases in which the inflation rate exceeds 50 percent per month -- a situation referred to as hyperinflation -- the entire monetary system generally breaks down and is replaced by barter. This has devastating effects on an economy.

Basic Types of Debt Instruments There are four types of credit instruments •

Simple Loan Contracts: Under the terms of a simple loan contract, the borrower (contract issuer) receives from the lender (contract holder) a specified amount of funds (the principal) for a specified period of time (the maturity). The borrower agrees that, at the end of this period of time -- referred to as the maturity date -- the borrower will repay the principal to the lender together with an additional payment referred to as the interest payment. 3

• • • • • • • • • •

Borrower Receives:

Lender Receives:

Principal | START |___________________________ MATURITY DATE | | Principal + Interest Payment

The annual borrowing fee for a simple loan with a principal P, an interest payment I, and a maturity of N years is measured by the simple interest rate given by I divided by [P times N]. IMPORTANT NOTE: Mishkin (Chapter 4) always implicitly assumes that the maturity N on simple loans is one year (N=1). As will be clarified further below, his assertion on page 71 -- "for simple loans, the simple interest rate equals the yield to maturity" -- is only true if N=1 is assumed for the simple loans and the yield to maturity is calculated as an annual rate. Example of a Simple Loan Contract: A borrower receives a loan on January 1, 1999, in amount $500.00, and agrees to pay the lender $550.00 on January 1, 2001. Thus, the principal is $500.00, the maturity is two years, the maturity date is January 1, 2001, and the interest payment is $50.00. The simple (annual) interest rate for this loan is then $50/[$500*2] = .05, or 5 percent. Fixed-Payment Loan Contracts: Under the terms of a fixed-payment loan contract, the borrower (contract issuer) receives from the lender (contract holder) a specified amount of funds -- the loan value -- and, in return, makes periodic fixed payments to the lender until a specified maturity date. These periodic fixed payments include both principal (loan value) and interest, so at maturity there is no lump sum repayment of principal. • • • • • • • • • •

Borrower Receives:

Loan Value LV | MATURITY START |__________________________________ DATE | | | | | | Lender Fixed Fixed Fixed Receives: Payment FP Payment FP Payment FP

Example of a Fixed-Payment Loan Contract: Joe arranges a 15-year installment loan with a finance company to help pay for a new car. Under the terms of this loan, Joe receives $20,000 now to finance the purchase of a new car but must make payments of $2000 every year for the next 15 years to the finance company. •

Coupon Bond:

Note: For simplicity, the case of a newly issued coupon bond is considered below, so that the seller is the borrower (issuer of the bond) and the buyer is the initial lender. Coupon bonds can also be resold in secondary markets, in which case the seller is not the borrower and the buyer is not the original lender. 4

Under the terms of a coupon bond, the borrower (bond issuer) agrees to pay the lender (bond purchaser) a fixed amount of funds (the coupon payment) on a periodic basis until a specified maturity date, at which time the borrower must also pay the lender the face value (or "par value") of the bond. The coupon rate of a coupon bond is, by definition, the amount of the coupon payment divided by the face value of the bond. As will be clarified in the next section, below, the purchase price of a coupon bond depends on the "present value" of the stream of anticipated coupon payments plus the final face value payment promised by the bond. Coupon bonds that sell above their face value are said to sell at a premium, and those that sell below their face value, at a discount. Borrower Receives:

Purchase Price Pb | MATURITY START |_______________________ /\/\/\ _____ DATE | | | | | | Lender Coupon Coupon ... Coupon Receives: Payment C Payment C Payment C + Face Value F

Example of a Coupon Bond: Suppose a coupon bond has a face value of $1000, a maturity of five years, and an annual coupon payment of $60. Then, at the end of each year for the next five years, the borrower (bond issuer) must pay the lender (bond holder) a coupon payment of $60. In addition, at the end of five years (the maturity date), the borrower must pay the lender the face value of the bond, $1000. The coupon rate for this coupon bond is $60/$1000 = .06, or 6 percent. •

Discount Bond (or Zero-Coupon Bond):

Note: For simplicity, the case of a newly issued discount bond is considered below, so that the seller is the borrower (issuer of the bond) and the buyer is the initial lender. Discount bonds can also be re-sold in secondary markets, in which case the seller is not the borrower and the buyer is not the original lender. Under the terms of a discount bond, the borrower (bond issuer) immediately receives from the lender (bond holder) the purchase price Pd of the bond, which is typically less than the face value F of the bond. In return, the borrower promises that, at the bond's maturity date, he will pay the lender the face value F of the bond. Borrower Receives:

Lender Receives:

Purchase Price Pd | START |_________________________ MATURITY DATE | | Face Value F

Important Cautionary Note: The above definition of a discount bond follows the definition used in Mishkin. Some other authors refer to zero-coupon bonds as PURE discount bonds, labelling as a "discount bond" any bond that sells at a discount in the sense that its market price is less than its face value. Also, Mishkin asserts that discount bonds make no interest payments. While this is literally true, in the sense that only a face value payment is made, it is NOT true that the interest 5

RATE on discount bonds is zero. Indeed, as will be seen below, the most basic measure of interest rates in use today is the annual "yield to maturity" i. For a one-year discount bond, the formula for calculating i reduces to i = [F-Pd]/Pd, hence i is only zero in the highly unlikely event that F=Pd. Discount Bond Example: On January 1, 1999, a borrower gives a lender a discount bond with a face value of $200 and a maturity of 2 years, and the lender gives $150 to the borrower. The borrower must then pay the lender $200 on January 1, 2001.

The Concept of Present Value Suppose someone promises to pay you $100 in some future period T. This amount of money actually has two different values: a nominal value of $100, which is simply a measure of the number of dollars that you will receive in period T; and a present value (sometimes referred to as a present discounted value), roughly defined to be the minimum number of dollars that you would have to give up today in return for receiving $100 in period T. Stated somewhat differently, the present value of the future $100 payment is the value of this future $100 payment measured in terms of current (or present) dollars. The concept of present value permits debt instruments with different associated payment streams to be compared with each other by calculating their values in terms of a single common unit: namely, current dollars. Specific formulas for the calculation of present value for future payments will now be developed and applied to the determination of present value for debt instruments with various types of payment streams. Present Value of Payments One Period Into the Future: If you save $1 today for a period of one year at an annual interest rate i, the nominal value of your savings after one year will be

(1)

V(1)

=

(1+i)*$1

,

where the asterisk "*" denotes multiplication. On the other hand, proceeding in the reverse direction from the future to the present, the present value of the future dollar amount V(1) = (1+i)*$1 is equal to $1. That is, the amount you would have to save today in order to receive back V(1)=(1+i)*$1 in one year's time is $1. Notice that this calculation of $1 as the present value of V(1)=(1+i)*$1 satisfies the following formula: (2)

Present Value of V(1)

=

V(1) -------(1+i)

.

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Indeed, given any fixed annual interest rate i, and any payment V(1) to be received one year from today, the present value of V(1) is given by formula (2). In effect, then, the payment V(1) to be received one year from now has been discounted back to the present using the annual interest rate i, so that the value of V(1) is now expressed in current dollars. Present Value of Payments Multiple Periods Into the Future: If you save $1 today at a fixed annual interest rate i, what will be the value of your savings in one year's time? In two year's time? In n year's time? If you save $1 at a fixed annual interest rate i, the nominal value of your savings in one year's time will be V(1)=(1+i)*$1. If you then put aside V(1) as savings for an additional year rather than spend it, the nominal value of your savings at the end of the second year will be (3) V(2) = (1+i)*V(1) = (1+i)*(1+i)*$1 = (1+i)2*$1 . And so forth for any number of years n. (4)

START --------------------------------/\/\/\-------->YEAR | 1 2 n |

Nominal Value of Savings:

$1

2 (1+i) *$1

(1+i)*$1

n (1+i)

* $1

Now consider the present value of V(n) = (1+i)n*$1 for any year n. By construction, V(n) is the nominal value obtained after n years when a single dollar is saved for n successive years at the fixed annual interest rate i. Consequently, the present value of V(n) is simply equal to $1, regardless of the value of n. Notice, however, that the present value of V(n) -- namely, $1 -- can be obtained from the following formula:

(5)

Present Value of V(n)

=

V(n) -----------n (1+i)

.

Indeed, given any fixed annual interest rate i, and any nominal amount V(n) to be received n years from today, the present value of V(n) can be calculated by using formula (5). Present Value of Any Arbitrary Payment Stream: Now suppose you will be receiving a sequence of three payments over the next three years. The nominal value of the first payment is $100, to be received at the end of the first year; the nominal value of the second payment is $150, to be received at the end of the second year; and the nominal value of the third payment is $200, to be received at the end of the third year.

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Given a fixed annual interest rate i, what is the present value of the payment stream ($100,$150,$200) consisting of the three separate payments $100, $150, and $200 to be received over the next three years? To calculate the present value of the payment stream ($100,$150,$200), use the following two steps: •

•

Step 1: Use formula (5) to separately calculate the present value of each of the individual payments in the payment stream, taking care to note how many years into the future each payment is going to be received. Step 2: Sum the separate present value calculations obtained in Step 1 to obtain the present value of the payment stream as a whole.

Carrying out Step 1, it follows from formula (5) that the present value of the $100 payment to be received at the end of the first year is $100/(1+i). Similarly, it follows from formula (5) that the present value of the $150 payment to be received at the end of the second year is (6)

$150 ---------2 (1+i)

Finally, it follows from formula (3) that the present value of the $200 payment to be received at the end of the third year is

(7)

$200 ---------3 (1+i)

Consequently, adding together these three separate present value calculations in accordance with Step 2, the present value PV(i) of the payment stream ($100,$150,$200) is given by (8)

PV(i) = $100 + $150 + $200 (1 + i)1 (1 + i)2 (1 + i)3 More generally, given any fixed annual interest rate i, and given any payment stream (V1,V2,V3,...,VN) consisting of individual payments to be received over the next N years, the present value of this payment stream can be found by following the two steps outlined above. In particular, then, given any fixed annual interest rate, and given any debt instrument with an associated payment stream paid out on a yearly basis to the lender (debt instrument holder), the present (current dollar) value of this debt instrument is found by calculating the present value of its associated payment stream in accordance with Steps 1 and 2 outlined above. Consequently, regardless how different the payment streams associated with two such debt instruments may be, one can calculate the present values for these debt instruments in current dollar terms and hence have a way to compare them. Technical Note:

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The above procedure for calculating the present value of a debt instrument whose payments are paid out on an annual basis to a lender can be generalized to debt instruments whose payments are paid out at arbitrary times to lenders. To do this, one needs to transform the annual interest rate used to discount each payment so that its period matches that of the payment. For example, to convert an annual interest rate to a monthly interest rate, you divide the annual interest rate by 12 (the number of months in a year). Thus, for example, if the annual interest rate is .12 (i.e., 12 percent), this is equivalent to a monthly interest rate of .01 (i.e., 1 percent). Consequently, if a payment V is to be received at the end of the next month, and the annual interest rate is .12, then the present value of this payment V is V/(1+.01). Similarly, to convert an annual interest rate to a quarterly interest rate, you divide the annual interest rate by 4. Thus, a 12 percent annual interest rate is equivalent to a 3 percent quarterly interest rate.

Measuring Interest Rates by Yield to Maturity By definition, the current yield to maturity for a marketed debt instrument is the particular fixed annual interest rate i which, when used to calculate the present value of the debt instrument's future stream of payments to the instrument's holder, yields a present value equal to the current market value of the instrument. Mishkin (pages 70-75) discusses and illustrates the calculation of the yield to maturity for the four basic types of debt instruments introduced in the first section of these notes, above. Below we review this calculation for two of these debt instrument types: fixed-payment loan contracts and coupon bonds. Yield to Maturity for Fixed-Payment Loan Contracts: Recall from previous discussion the general form of a fixed-payment loan contract: Borrower Receives:

Loan Value LV | MATURITY START |__________________________________ DATE | | | | | | Lender Fixed Fixed Fixed Receives: Payment FP Payment FP Payment FP

Consider a particular fixed-payment loan contract with a loan value LV = $5000, annual fixed payments FP = $660.72, and a maturity of N = 20 years. What is the yield to maturity for this loan contract? The first question that must be answered is what is the current value of this loan contract at the date of its issuance? The borrower (contract issuer) receives from the lender (contract holder) the $5000 loan value at the date the loan contract is issued. This $5000 loan value, then, constitutes the current value of the loan contract. It is, in effect, the price paid by the lender to purchase the loan contract from the borrower. 9

By definition, then, the yield to maturity of this fixed-payment loan contract is the particular fixed annual interest rate i which, when used to calculate the present value of the loan contract, results in a present value that is exactly equal to $5000, the current value of the loan contract. Using the discussion in the previous section, given any fixed annual interest rate i, the present value of the fixed-payment loan contract at hand -- that is, the present value of the payment stream to the lender generated by this loan contract -- is found as follows. The payment stream to the lender generated by this loan contract consists of twenty successive yearly fixed payments, each having the nominal value FP=$660.72. Using formula (3), given any year n, n = 1,...,20, and any fixed annual interest rate i, the present value of the particular fixed payment FP = $660.72 received at the end of year n is FP/(1+i)n . Given any fixed annual interest rate i, the present value PV(i) of the loan contract is then given by the sum of all of these separate present value calculations for the fixed payments FP received by the lender (debt instrument holder) at the end of years 1 through 20, i.e., (9) PV(i) = FP/(1+i) + FP/(1+i)2 + ... + FP/(1+i)20. Since the current value of the loan contract is $5000, the desired yield to maturity is then found by solving the following equation for i: (10) $5000 = PV(i) . Because the present value PV(i) depends in a rather complicated way on i, the determination of i from formula (10) is not straightforward. To make life easier, tables have been published that can be used to determine yields to maturity for various types of fixed-payment loan contracts once the current value and fixed payments of the loan are known. For example, using such tables, it can be shown that the solution for i in equation (10) above is approximately i = .12. That is, the yield to maturity i for a fixed-payment loan contract with a current value of $5000, with annual fixed payments of $660.72, and with a maturity of twenty years, is approximately 12 percent. Yield to Maturity for Coupon Bonds: Recall from previous discussion the basic contractual terms of a coupon bond:

Borrower Receives:

Purchase Price Pb | MATURITY START |_______________________ /\/\/\ _____ DATE | | | | | | Coupon Coupon ... Coupon Lender Payment C Payment C Payment C Receives: + Face Value F

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Consider a coupon bond whose purchase price is Pb=$94, whose face value is F = $100, whose coupon payment is C = $10, and whose maturity is 10 years. By definition, the coupon rate for this bond is equal to C/F = $10/$100 = .10 (i.e., 10 percent). The payment stream to the lender generated by this coupon bond is given by (11) ( $10, $10, $10, $10, $10, $10, $10, $10, $10, [$10 + $100] ). For any given fixed annual interest rate i, the present value PV(i) of the payment stream (11) is given by the sum of the separate present value calculations for each of the payments in this payment stream as determined by formula (5). That is, (12) PV(i) = $10/(1+i) + $10/(1+i)2 + ... + $10/(1+i)10 + $100/(1+i)10 . The current value of the coupon bond is its current purchase price Pb = $94. It then follows by definition that the yield to maturity for this coupon bond is found by solving the following equation for i: (13) Pb = PV(i) . The calculation of the yield to maturity i from formula (13) can be difficult, but tables have been published that permit one to read off the yield to maturity i for a coupon bond once the purchase price, the face value, the coupon rate, and the maturity are known. For example, using such tables, it can be shown that the yield to maturity i for the coupon bond currently under consideration, which has a purchase price of $94 per $100 of face value, a coupon rate of 10 percent, and a maturity of 10 years, is approximately equal to 11 percent. More generally, given any coupon bond with purchase price Pb, face value F, coupon payment C, and maturity N, the yield to maturity i is found by means of the following formula: (14a) Pb = PV(i) , where the present value PV(i) of the coupon bond is given by (14b) PV(i) = C/(1+i) + C/(1+i)2 + ... + C/(1+i)N + F/(1+i)N . Some Final Important Observations on Yield to Maturity: For any coupon bond with a fixed coupon payment C and a fixed face value F, the purchase price Pb of the bond is equal to the face value F if and only if the yield to maturity i for the bond is equal to the coupon rate C/F. 11

This observation follows directly from the structure of a coupon bond. When the purchase price equals the face value, the coupon bond essentially functions as a bank deposit account into which a principal amount (the face value) is deposited by a lender, earns a fixed annual interest rate (the coupon rate) for some number of years, and is then recovered by the lender. Illustration for a One-Period Coupon Bond: For a one-period coupon bond with coupon payment C, face value F, and purchase price Pb, the formula Pb = PV(i) for determining the yield to maturity i can be written as (15)

Pb

F + C ---------(1+i)

=

.

Dividing each side of formula (15) by the face value F, one obtains (16)

Pb/F

=

1 + C/F ---------(1+i)

.

Given C and F, formula (16) implies that Pb equals F (i.e., the left-hand side equals 1) if and only if i equals C/F (i.e., the right-hand side equals 1). More generally, given any coupon bond with a fixed coupon payment C and a fixed face value F, the purchase price Pb of the bond is lower (higher) than F if and only if the yield to maturity i is higher (lower) than the coupon rate C/F. This follows directly from formula (14) for determination of the yield to maturity, using the previously noted fact that the purchase price Pb is equal to F if and only if the yield to maturity i is equal to the coupon rate C/F. For example, suppose formula (14) holds with Pb = F and i = C/F. Taking C and F as given, consider what happens if the yield to maturity i now increases, so that i exceeds the coupon rate C/F. Since C and F are given, PV(i) decreases, which implies that Pb must also decrease. Since F is given, and Pb was originally equal to F, this implies that Pb must now be lower than F. Moreover, for any given coupon bond with given C and F, the yield to maturity i of the bond is inversely related to the purchase price Pb of the bond. That is, the higher the yield to maturity i, the lower the purchase price Pb, and conversely. This inverse relationship also follows directly from formula (14). To see this, consider what happens when i increases in formula (14), keeping C and F fixed. When i increases, the denominator (1+i) of the discounted coupon payment C/(1+i) appearing in PV(i) in formula (14) increases, implying that the ratio C/(1+i) is smaller than before, and similarly for each of the other discounted coupon payments that are summed to obtain PV(i) in (14). Consequently, PV(i) decreases. It then follows from formula (14) that Pb also decreases. This inverse relationship between the yield to maturity of a debt instrument and its purchase price actually holds in general. For any debt instrument with any given payment stream, when the yield to maturity for the debt instrument rises, the purchase price of the debt instrument must fall, and vice versa. This follows directly from the general definition for the yield to maturity, applicable to all debt instruments.

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