The economics of Money, Banking and Financial Markets Part 2

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Part II Financial Markets Ch a p ter 4 PREVIEW www.bloomberg.com /markets/ Under “Rates & Bonds,” you can access information on key interest rates, U.S. Treasuries, Government bonds, and municipal bonds. Understanding Interest Rates Interest rates are among the most closely watched variables in the economy. Their movements are reported almost daily by the news media, because they directly affect our everyday lives and have important consequences for the health of the economy. They affect personal decisions such as whether to consume or save, whether to buy a house, and whether to purchase bonds or put funds into a savings account. Interest rates also affect the economic decisions of businesses and households, such as whether to use their funds to invest in new equipment for factories or to save their money in a bank. Before we can go on with the study of money, banking, and financial markets, we must understand exactly what the phrase interest rates means. In this chapter, we see that a concept known as the yield to maturity is the most accurate measure of interest rates; the yield to maturity is what economists mean when they use the term interest rate. We discuss how the yield to maturity is measured and examine alternative (but less accurate) ways in which interest rates are quoted. We’ll also see that a bond’s interest rate does not necessarily indicate how good an investment the bond is because what it earns (its rate of return) does not necessarily equal its interest rate. Finally, we explore the distinction between real interest rates, which are adjusted for inflation, and nominal interest rates, which are not. Although learning definitions is not always the most exciting of pursuits, it is important to read carefully and understand the concepts presented in this chapter. Not only are they continually used throughout the remainder of this text, but a firm grasp of these terms will give you a clearer understanding of the role that interest rates play in your life as well as in the general economy. Measuring Interest Rates Different debt instruments have very different streams of payment with very different timing. Thus we first need to understand how we can compare the value of one kind of debt instrument with another before we see how interest rates are measured. To do this, we make use of the concept of present value. Present Value The concept of present value (or present discounted value) is based on the commonsense notion that a dollar paid to you one year from now is less valuable to you than a dollar paid to you today: This notion is true because you can deposit a dollar in a 61 62 PART II Financial Markets savings account that earns interest and have more than a dollar in one year. Economists use a more formal definition, as explained in this section. Let’s look at the simplest kind of debt instrument, which we will call a simple loan. In this loan, the lender provides the borrower with an amount of funds (called the principal) that must be repaid to the lender at the maturity date, along with an additional payment for the interest. For example, if you made your friend, Jane, a simple loan of $100 for one year, you would require her to repay the principal of $100 in one year’s time along with an additional payment for interest; say, $10. In the case of a simple loan like this one, the interest payment divided by the amount of the loan is a natural and sensible way to measure the interest rate. This measure of the socalled simple interest rate, i, is: i $10  0.10  10% $100 If you make this $100 loan, at the end of the year you would have $110, which can be rewritten as: $100  (1  0.10)  $110 If you then lent out the $110, at the end of the second year you would have: $110  (1  0.10)  $121 or, equivalently, $100  (1  0.10)  (1  0.10)  $100  (1  0.10)2  $121 Continuing with the loan again, you would have at the end of the third year: $121  (1  0.10)  $100  (1  0.10)3  $133 Generalizing, we can see that at the end of n years, your $100 would turn into: $100  (1  i)n The amounts you would have at the end of each year by making the $100 loan today can be seen in the following timeline: Today 0 Year 1 Year 2 Year 3 Year n $100 $110 $121 $133 $100  (1  0.10)n This timeline immediately tells you that you are just as happy having $100 today as having $110 a year from now (of course, as long as you are sure that Jane will pay you back). Or that you are just as happy having $100 today as having $121 two years from now, or $133 three years from now or $100  (1  0.10)n, n years from now. The timeline tells us that we can also work backward from future amounts to the present: for example, $133  $100  (1  0.10)3 three years from now is worth $100 today, so that: $100  $133 (1  0.10 )3 The process of calculating today’s value of dollars received in the future, as we have done above, is called discounting the future. We can generalize this process by writing CHAPTER 4 Understanding Interest Rates 63 today’s (present) value of $100 as PV, the future value of $133 as FV, and replacing 0.10 (the 10% interest rate) by i. This leads to the following formula: PV  FV (1  i )n (1) Intuitively, what Equation 1 tells us is that if you are promised $1 for certain ten years from now, this dollar would not be as valuable to you as $1 is today because if you had the $1 today, you could invest it and end up with more than $1 in ten years. The concept of present value is extremely useful, because it allows us to figure out today’s value (price) of a credit market instrument at a given simple interest rate i by just adding up the individual present values of all the future payments received. This information allows us to compare the value of two instruments with very different timing of their payments. As an example of how the present value concept can be used, let’s assume that you just hit the $20 million jackpot in the New York State Lottery, which promises you a payment of $1 million for the next twenty years. You are clearly excited, but have you really won $20 million? No, not in the present value sense. In today’s dollars, that $20 million is worth a lot less. If we assume an interest rate of 10% as in the earlier examples, the first payment of $1 million is clearly worth $1 million today, but the next payment next year is only worth $1 million/(1  0.10)  $909,090, a lot less than $1 million. The following year the payment is worth $1 million/(1  0.10)2  $826,446 in today’s dollars, and so on. When you add all these up, they come to $9.4 million. You are still pretty excited (who wouldn’t be?), but because you understand the concept of present value, you recognize that you are the victim of false advertising. You didn’t really win $20 million, but instead won less than half as much. Four Types of Credit Market Instruments In terms of the timing of their payments, there are four basic types of credit market instruments. 1. A simple loan, which we have already discussed, in which the lender provides the borrower with an amount of funds, which must be repaid to the lender at the maturity date along with an additional payment for the interest. Many money market instruments are of this type: for example, commercial loans to businesses. 2. A fixed-payment loan (which is also called a fully amortized loan) in which the lender provides the borrower with an amount of funds, which must be repaid by making the same payment every period (such as a month), consisting of part of the principal and interest for a set number of years. For example, if you borrowed $1,000, a fixed-payment loan might require you to pay $126 every year for 25 years. Installment loans (such as auto loans) and mortgages are frequently of the fixed-payment type. 3. A coupon bond pays the owner of the bond a fixed interest payment (coupon payment) every year until the maturity date, when a specified final amount (face value or par value) is repaid. The coupon payment is so named because the bondholder used to obtain payment by clipping a coupon off the bond and sending it to the bond issuer, who then sent the payment to the holder. Nowadays, it is no longer necessary to send in coupons to receive these payments. A coupon bond with $1,000 face value, for example, might pay you a coupon payment of $100 per year for ten years, and at the maturity date repay you the face value amount of $1,000. (The face value of a bond is usually in $1,000 increments.) A coupon bond is identified by three pieces of information. First is the corporation or government agency that issues the bond. Second is the maturity date of the 64 PART II Financial Markets bond. Third is the bond’s coupon rate, the dollar amount of the yearly coupon payment expressed as a percentage of the face value of the bond. In our example, the coupon bond has a yearly coupon payment of $100 and a face value of $1,000. The coupon rate is then $100/$1,000  0.10, or 10%. Capital market instruments such as U.S. Treasury bonds and notes and corporate bonds are examples of coupon bonds. 4. A discount bond (also called a zero-coupon bond) is bought at a price below its face value (at a discount), and the face value is repaid at the maturity date. Unlike a coupon bond, a discount bond does not make any interest payments; it just pays off the face value. For example, a discount bond with a face value of $1,000 might be bought for $900; in a year’s time the owner would be repaid the face value of $1,000. U.S. Treasury bills, U.S. savings bonds, and long-term zero-coupon bonds are examples of discount bonds. These four types of instruments require payments at different times: Simple loans and discount bonds make payment only at their maturity dates, whereas fixed-payment loans and coupon bonds have payments periodically until maturity. How would you decide which of these instruments provides you with more income? They all seem so different because they make payments at different times. To solve this problem, we use the concept of present value, explained earlier, to provide us with a procedure for measuring interest rates on these different types of instruments. Yield to Maturity Of the several common ways of calculating interest rates, the most important is the yield to maturity, the interest rate that equates the present value of payments received from a debt instrument with its value today.1 Because the concept behind the calculation of the yield to maturity makes good economic sense, economists consider it the most accurate measure of interest rates. To understand the yield to maturity better, we now look at how it is calculated for the four types of credit market instruments. Simple Loan. Using the concept of present value, the yield to maturity on a simple loan is easy to calculate. For the one-year loan we discussed, today’s value is $100, and the payments in one year’s time would be $110 (the repayment of $100 plus the interest payment of $10). We can use this information to solve for the yield to maturity i by recognizing that the present value of the future payments must equal today’s value of a loan. Making today’s value of the loan ($100) equal to the present value of the $110 payment in a year (using Equation 1) gives us: $100  $110 1i Solving for i, i $10 $110  $100   0.10  10% $100 $100 This calculation of the yield to maturity should look familiar, because it equals the interest payment of $10 divided by the loan amount of $100; that is, it equals the simple interest rate on the loan. An important point to recognize is that for simple loans, the simple interest rate equals the yield to maturity. Hence the same term i is used to denote both the yield to maturity and the simple interest rate. 1 In other contexts, it is also called the internal rate of return. CHAPTER 4 Study Guide Understanding Interest Rates 65 The key to understanding the calculation of the yield to maturity is equating today’s value of the debt instrument with the present value of all of its future payments. The best way to learn this principle is to apply it to other specific examples of the four types of credit market instruments in addition to those we discuss here. See if you can develop the equations that would allow you to solve for the yield to maturity in each case. Fixed-Payment Loan. Recall that this type of loan has the same payment every period throughout the life of the loan. On a fixed-rate mortgage, for example, the borrower makes the same payment to the bank every month until the maturity date, when the loan will be completely paid off. To calculate the yield to maturity for a fixed-payment loan, we follow the same strategy we used for the simple loan—we equate today’s value of the loan with its present value. Because the fixed-payment loan involves more than one payment, the present value of the fixed-payment loan is calculated as the sum of the present values of all payments (using Equation 1). In the case of our earlier example, the loan is $1,000 and the yearly payment is $126 for the next 25 years. The present value is calculated as follows: At the end of one year, there is a $126 payment with a PV of $126/(1  i); at the end of two years, there is another $126 payment with a PV of $126/(1  i)2; and so on until at the end of the twenty-fifth year, the last payment of $126 with a PV of $126/(1  i)25 is made. Making today’s value of the loan ($1,000) equal to the sum of the present values of all the yearly payments gives us: $1,000  $126 $126 $126 . . .  $126  2  3  1i (1  i ) (1  i ) (1  i )25 More generally, for any fixed-payment loan, LV  where FP FP FP . . .  FP  2  3  1i (1  i ) (1  i ) (1  i )n (2) LV  loan value FP  fixed yearly payment n  number of years until maturity For a fixed-payment loan amount, the fixed yearly payment and the number of years until maturity are known quantities, and only the yield to maturity is not. So we can solve this equation for the yield to maturity i. Because this calculation is not easy, many pocket calculators have programs that allow you to find i given the loan’s numbers for LV, FP, and n. For example, in the case of the 25-year loan with yearly payments of $126, the yield to maturity that solves Equation 2 is 12%. Real estate brokers always have a pocket calculator that can solve such equations so that they can immediately tell the prospective house buyer exactly what the yearly (or monthly) payments will be if the house purchase is financed by taking out a mortgage.2 Coupon Bond. To calculate the yield to maturity for a coupon bond, follow the same strategy used for the fixed-payment loan: Equate today’s value of the bond with its present value. Because coupon bonds also have more than one payment, the present 2 The calculation with a pocket calculator programmed for this purpose requires simply that you enter the value of the loan LV, the number of years to maturity n, and the interest rate i and then run the program. 66 PART II Financial Markets value of the bond is calculated as the sum of the present values of all the coupon payments plus the present value of the final payment of the face value of the bond. The present value of a $1,000-face-value bond with ten years to maturity and yearly coupon payments of $100 (a 10% coupon rate) can be calculated as follows: At the end of one year, there is a $100 coupon payment with a PV of $100/(1  i ); at the end of the second year, there is another $100 coupon payment with a PV of $100/(1  i )2; and so on until at maturity, there is a $100 coupon payment with a PV of $100/(1  i )10 plus the repayment of the $1,000 face value with a PV of $1,000/(1  i )10. Setting today’s value of the bond (its current price, denoted by P) equal to the sum of the present values of all the payments for this bond gives: P $100 $100 $100 . . .  $100  $1,000  2  3  1i (1  i ) (1  i ) (1  i )10 (1  i )10 More generally, for any coupon bond,3 P where C C C F C ...   2  3  n  1i (1  i ) (1  i ) (1  i ) (1  i )n (3) P  price of coupon bond C  yearly coupon payment F  face value of the bond n  years to maturity date In Equation 3, the coupon payment, the face value, the years to maturity, and the price of the bond are known quantities, and only the yield to maturity is not. Hence we can solve this equation for the yield to maturity i. Just as in the case of the fixedpayment loan, this calculation is not easy, so business-oriented pocket calculators have built-in programs that solve this equation for you.4 Let’s look at some examples of the solution for the yield to maturity on our 10%coupon-rate bond that matures in ten years. If the purchase price of the bond is $1,000, then either using a pocket calculator with the built-in program or looking at a bond table, we will find that the yield to maturity is 10 percent. If the price is $900, we find that the yield to maturity is 11.75%. Table 1 shows the yields to maturity calculated for several bond prices. Table 1 Yields to Maturity on a 10%-Coupon-Rate Bond Maturing in Ten Years (Face Value = $1,000) Price of Bond ($) 1,200 1,100 1,000 900 800 3 Yield to Maturity (%) 7.13 8.48 10.00 11.75 13.81 Most coupon bonds actually make coupon payments on a semiannual basis rather than once a year as assumed here. The effect on the calculations is only very slight and will be ignored here. 4 The calculation of a bond’s yield to maturity with the programmed pocket calculator requires simply that you enter the amount of the yearly coupon payment C, the face value F, the number of years to maturity n, and the price of the bond P and then run the program. CHAPTER 4 Understanding Interest Rates 67 Three interesting facts are illustrated by Table 1: 1. When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate. 2. The price of a coupon bond and the yield to maturity are negatively related; that is, as the yield to maturity rises, the price of the bond falls. As the yield to maturity falls, the price of the bond rises. 3. The yield to maturity is greater than the coupon rate when the bond price is below its face value. These three facts are true for any coupon bond and are really not surprising if you think about the reasoning behind the calculation of the yield to maturity. When you put $1,000 in a bank account with an interest rate of 10%, you can take out $100 every year and you will be left with the $1,000 at the end of ten years. This is similar to buying the $1,000 bond with a 10% coupon rate analyzed in Table 1, which pays a $100 coupon payment every year and then repays $1,000 at the end of ten years. If the bond is purchased at the par value of $1,000, its yield to maturity must equal 10%, which is also equal to the coupon rate of 10%. The same reasoning applied to any coupon bond demonstrates that if the coupon bond is purchased at its par value, the yield to maturity and the coupon rate must be equal. It is straightforward to show that the bond price and the yield to maturity are negatively related. As i, the yield to maturity, rises, all denominators in the bond price formula must necessarily rise. Hence a rise in the interest rate as measured by the yield to maturity means that the price of the bond must fall. Another way to explain why the bond price falls when the interest rises is that a higher interest rate implies that the future coupon payments and final payment are worth less when discounted back to the present; hence the price of the bond must be lower. There is one special case of a coupon bond that is worth discussing because its yield to maturity is particularly easy to calculate. This bond is called a consol or a perpetuity; it is a perpetual bond with no maturity date and no repayment of principal that makes fixed coupon payments of $C forever. Consols were first sold by the British Treasury during the Napoleonic Wars and are still traded today; they are quite rare, however, in American capital markets. The formula in Equation 3 for the price of the consol P simplifies to the following:5 P C i (4) 5 The bond price formula for a consol is: P C C C   ... 1i (1  i )2 (1  i )3 which can be written as: P  C (x  x 2  x 3  . . . ) in which x  1/(1  i). The formula for an infinite sum is: 1  x  x2  x3  . . .  and so: PC 1 1x for x1 1  x  1  C c1  1(1  i )  1 d 1 1 which by suitable algebraic manipulation becomes: PC  i 1i  i i  i C 68 PART II Financial Markets where P = price of the consol C = yearly payment One nice feature of consols is that you can immediately see that as i goes up, the price of the bond falls. For example, if a consol pays $100 per year forever and the interest rate is 10%, its price will be $1,000  $100/0.10. If the interest rate rises to 20%, its price will fall to $500  $100/0.20. We can also rewrite this formula as i C P (5) We see then that it is also easy to calculate the yield to maturity for the consol (despite the fact that it never matures). For example, with a consol that pays $100 yearly and has a price of $2,000, the yield to maturity is easily calculated to be 5% ( $100/$2,000). Discount Bond. The yield-to-maturity calculation for a discount bond is similar to that for the simple loan. Let us consider a discount bond such as a one-year U.S. Treasury bill, which pays off a face value of $1,000 in one year’s time. If the current purchase price of this bill is $900, then equating this price to the present value of the $1,000 received in one year, using Equation 1, gives: $900  $1,000 1i and solving for i, (1  i )  $900  $1,000 $900  $900i  $1,000 $900i  $1,000  $900 i $1,000  $900  0.111  11.1% $900 More generally, for any one-year discount bond, the yield to maturity can be written as: i where FP P (6) F  face value of the discount bond P  current price of the discount bond In other words, the yield to maturity equals the increase in price over the year F – P divided by the initial price P. In normal circumstances, investors earn positive returns from holding these securities and so they sell at a discount, meaning that the current price of the bond is below the face value. Therefore, F – P should be positive, and the yield to maturity should be positive as well. However, this is not always the case, as recent extraordinary events in Japan indicate (see Box 1). An important feature of this equation is that it indicates that for a discount bond, the yield to maturity is negatively related to the current bond price. This is the same conclusion that we reached for a coupon bond. For example, Equation 6 shows that a rise in the bond price from $900 to $950 means that the bond will have a smaller
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