R. GLENN HUBBARD ANTHONY PATRICK O’BRIEN

R. GLENNHUBBARDANTHONY PATRICKO’BRIEN Money,Banking, andthe Financial System

Interest Rates and Rates of Return 3 C H A P T E R LEARNING OBJECTIVES After studying this chapter, you should be able to: Explain how the interest rate links present value with future value 3.1 Distinguish among different debt instruments and understand how their prices are determined 3.2 Explain the relationship between the yield to maturity on a bond and its price 3.3 Understand the inverse relationship between bond prices and bond yields 3.4 Explain the difference between interest rates and rates of return 3.5 Explain the difference between nominal interest rates and real interest rates 3.6

Interest Rates and Rates of Return 3 C H A P T E R • BANKS IN TROUBLE • During the financial crisis, the number of insolvent banks increased sharply. • With the collapse of the housing market, increasing numbers of homeowners had stopped making payments on their mortgage loans. Banks that held these loans saw their value drop. • Mortgage loans that were turned into mortgage-backed securities, similar to bonds, declined by 50% or more during 2008 and 2009. • Banks had badly misjudged both the default risk and the interest-rate risk on these bonds. • An Inside Look at Policy on page 78 discusses the performance of the bond market through 2010.

Key Issue and Question Issue: During the financial crisis, soaring interest rates on assets such as mortgage-backed securities caused their prices to plummet. Question: Why do interest rates and the prices of financial securities move in opposite directions?

3.1 Learning Objective Explain how the interest rate links present value with future value.

The Interest Rate, Present Value, and Future Value Why Do Lenders Charge Interest on Loans? • The interest rate on a loan should cover the opportunity cost of supplying credit, particularly, the costs associated with three factors: • Compensation for inflation: if prices rise, the payments received will buy fewer goods and services. • Compensation for default risk: the borrower might default on the loan. • Compensation for the opportunity cost of waiting for the money to be paid back.

The Interest Rate, Present Value, and Future Value Most Financial Transactions Involve Payments in the Future The importance of the interest rate comes from the fact that most financial transactions involve payments in the future;the interest rate provides a link between the financial present and the financial future.

Compounding and Discounting Future value The value at some future time of an investment made today. If: i = the interest rate Principal = the amount of your investment (your original $1,000) FV = the future value (what your $1,000 will have grown to in one year) then we can rewrite the expression as: Compounding for More Than One Period CompoundingThe process of earning interest on interest as savings accumulate over time. If you invest $1,000 for n years, where n can be any number of years, at an interest rate of 5%, then at the end of n years, you will have: The Interest Rate, Present Value, and Future Value

3.1A Solved Problem Comparing Investments • Suppose you are considering investing $1,000 in one of the following bank CDs: • First CD, which will pay an interest rate of 4% per year for three years • Second CD, which will pay an interest rate of 10% the first year, 1% the second year, and 1% the third year • Which CD should you choose? The Interest Rate, Present Value, and Future Value

3.1A Solved Problem Comparing Investments Solving the Problem Step 1Review the chapter material. Step 2Calculate the future value of your investment with the first CD. Step 3 Calculate the future value of your investment with the second CD and decide which CD you should choose. Principal = $1,000, i = 4%, n = 3 years FV = $1,000 x (1 + 0.04)3 = $1,124.86 Principal = $1,000, i1= 10%, i2= 1%, i3= 1%, n = 3 years FV = $1,000 x (1 + 0.10) x (1 + 0.01) x (1 + 0.01) = $1,122.11 Decision: You should choose the investment with the highest future value, so you should choose the first CD. The Interest Rate, Present Value, and Future Value

An Example of Discounting Funds in the future are worth less than funds in the present, so they have to be reduced, or discounted, to find their present value. Present value The value today of funds that will be received in the future. Time value of money The way that the value of a payment changes depending on when the payment is received. DiscountingThe process of finding the present value of funds that will be received in the future. The Interest Rate, Present Value, and Future Value

Some Important Points about Discounting 1. Present value is sometimes referred to as “present discounted value.” 2. The further in the future a payment is to be received, the smaller its present value (See Table 3.1). 3. The higher the interest rate used to discount future payments, the smaller the present value of the payments (See Table 3.1). 4. The present value of a series of future payment is simply the sum of the discounted value of each individual payment. The Interest Rate, Present Value, and Future Value

3.1B Solved Problem Valuing a Contract Jason Bay played the 2009 baseball season with the Boston Red Sox. When he became a free agent, the Red Sox offered him a contract for $15 million per year for four years. The New York Mets offered him a contract that would pay him a total of $66 million. According to sportswriter Buster Olney: “The Mets’ offer to Jason Bay is heavily backloaded, to the point that the true value of the four-year [contract] falls to within the range of the offer he turned down from the Red Sox.” What does Olney mean by the payments in the Mets’ contract being “backloaded”? What does he mean by the “true value” of the contract? How would backloading the payments affect the true value of the contract? The Interest Rate, Present Value, and Future Value

3.1B Solved Problem Valuing a Contract Solving the Problem Step 1Review the chapter material. Step 2Explain what Olney means by “backloaded” and “true value.” A “backloaded” contract means that the Mets offered Jason Bay lower salaries in the first years and higher salaries in the later years of the contract. The “true value” probably refers to the present value of the contract. Step 3 Explain how backloading affects the value of the contract. We know that the present value of payments is lower the further away in time those payments are made. So, if the Mets’ contract pays Bay most of the $66 million in the third and fourth years of the contract, it could have a present value similar to the Red Sox contract that paid $60 million spread out as four annual $15 million payments. The Interest Rate, Present Value, and Future Value

A Brief Word on Notation This book will always enter interest rates in numerical calculations as decimals. For instance, 5% will be 0.05, not 5. Discounting and the Prices of Financial Assets Discounting gives us a way of determining the prices of financial assets. By adding up the present values of all the payments, we have the dollar amount that a buyer will pay for the asset. In other words, we have determined the asset’s price. The Interest Rate, Present Value, and Future Value

3.2 Learning Objective Distinguish among different debt instruments and understand how their prices are determined.

The price of a financial asset is equal to the present value of the payments to be received from owning it. Debt instruments(also known as credit market instrumentsor fixed income assets) Methods of financing debt, including simple loans, discount bonds, coupon bonds, and fixed payment loans. Equity A claim to part ownership of a firm; common stock issued by a corporation. Debt Instruments and Their Prices

Loans, Bonds, and the Timing of Payments In this section, we discuss four basic categories of debt instruments: Simple loans Discount bonds Coupon bonds 4. Fixed-payment loans Debt Instruments and Their Prices

Simple Loan Simple loan A debt instrument in which the borrower receives from the lender an amount called the principal and agrees to repay the lender the principal plus interest on a specific date when the loan matures. After one year, Nate’s would repay the principal plus interest: $10,000 + ($10,000 × 0.10), or $11,000. Debt Instruments and Their Prices

Discount Bond Discount bondA debt instrument in which the borrower repays the amount of the loan in a single payment at maturity but receives less than the face value of the bond initially. The lender receives interest of $10,000 - $9,091 = $909 for the year. Therefore, the interest rate is $909/$9,091 = 0.10, or 10%. Debt Instruments and Their Prices

Coupon Bonds Coupon bond A debt instrument that requires multiple payments of interest on a regular basis, such as semiannually or annually, and a payment of the face value at maturity. • Terminology of coupon bonds: • Face value, or par value, is the amount to be repaid by the bond issuer (the borrower) at maturity. • The coupon is the annual fixed dollar amount of interest paid by the issuer of the bond to the buyer. • The coupon rate is the value of the coupon expressed as a percentage of the par value of the bond. • The current yield is the value of the coupon expressed as a percentage of the current price. Debt Instruments and Their Prices

Coupon Bonds • Maturity is the length of time before the bond expires and the issuer makes the face value payment to the buyer. • For example, if IBM issued a $1,000 30-year bond with a coupon rate of 10%, it would pay $100 per year for 30 years and a final payment of $1,000 at the end of 30 years. The timeline on the IBM coupon bond is: Debt Instruments and Their Prices

Fixed-Payment Loan Fixed-payment loan A debt instrument that requires the borrower to make regular periodic payments of principal and interest to the lender. • For example, if you are repaying a $10,000 10-year student loan with a 9% interest rate, your monthly payment is approximately $127. The time line of payments is: Debt Instruments and Their Prices

Making the Connection Do You Want the Principal or Do You Want the Interest? Creating New Financial Instruments • Back when the U.S. Treasury offered only short-term discount bonds, investors were seeking to benefit from longer terms knowing the exact return if they held the bonds to maturity. • In 1982, Merrill Lynch created the TIGR (Treasure Investment Growth Receipt), which is a discount bond that works like a Treasury Bill. • Two years later, the Treasury introduced its own version called STRIPS (Separate Trading of Registered Interest and Principal Securities). These bonds allowed investors to buy each interest payment and the face value of the bond. • Individuals can obtain long-term discount bonds as well as the regular Treasury coupon bonds, thereby increasing their options for investment. Debt Instruments and Their Prices

3.3 Learning Objective Explain the relationship between the yield to maturity on a bond and its price.

Bond Prices • Consider a coupon bond with i = 6%, FV = $1,000, n = 5 years. The expression for the price, P, of the bond is the sum of the present values of the six payments: • Below is a general expression for a bond that makes coupon payments, C, has a face value, FV, and matures in n years: Bond Prices and Yield to Maturity

Yield to Maturity Yield to maturity The interest rate that makes the present value of the payments from an asset equal to the asset’s price today. Whenever participants in financial markets refer to the interest rate on a financial asset, the interest rate is the yield to maturity. Yields to Maturity on Other Debt Instruments Simple Loans • Consider a $10,000 loan required to pay $11,000 in one year. • Value today = Present value of future payments • Solving for i: Bond Prices and Yield to Maturity

Discount Bonds • Consider a $10,000 one-year discount bond with a value today of $9,200. • Value today = Present value of future payments • Solving for i: • A general equation for a one-year discount bond that sells for price, P, with face value, FV. The yield to maturity is: Bond Prices and Yield to Maturity

Fixed-Payment Loans • Consider a $100,000 loan with annual payments of $12,731. • Value today = Present value of future payments • In general, for a fixed-payment loan with fixed payments, FP, and a maturity of n years, the equation is: Perpetuities • A perpetuity does not mature. The price of a couponbond that pays an infinite number of coupons equals: • So, a perpetuity with a coupon of $25 and a price of $500 has a yield to maturity of i = $25/$500 = 0.05, or 5%. Bond Prices and Yield to Maturity

3.3 Solved Problem Yield to Maturity for Different Types of Debt Instruments For each of the following situations, write the equation that you would use to calculate the yield to maturity. You do not have to solve the equations for i; just write the appropriate equation. a) A simple loan for $500,000 that requires a payment of $700,000 in four years. b) A discount bond with a price of $9,000, which has a face value of $10,000and matures in one year. c) A corporate bond with a face value of $1,000, a price of $975, a coupon rateof 10%, and a maturity of five years. d) A student loan of $2,500, which requires payments of $315 per year for 25years. The payments start in two years. Bond Prices and Yield to Maturity

3.3 Solved Problem Yield to Maturity for Different Types of Debt Instruments Solving the Problem Step 1Review the chapter material. Step 2Write an equation for the yield to maturity for each of the following debt instruments. A simple loan for $500,000 that requires a payment of $700,000 in four years. A discount bond with a price of $9,000, which has a face value of $10,000 and matures in one year. Bond Prices and Yield to Maturity

3.3 (continued) Solved Problem Yield to Maturity for Different Types of Debt Instruments Solving the Problem Step 1Review the chapter material. Step 2Write an equation for the yield to maturity for each of the following debt instruments. A corporate bond with a face value of $1,000, a price of $975, a coupon rate of 10%, and a maturity of five years. A student loan of $2,500, which requires payments of $315 per year for 25 years. The payments start in two years. Bond Prices and Yield to Maturity

3.4 Learning Objective Understand the inverse relationship between bond prices and bond yields.

The Inverse Relationship between Bond Prices and Bond Yields • Coupon bonds may be sold many times in a secondary market. • The issuer of the bond is no longer involved in these transactions. What Happens to Bond Prices When Interest Rates Change? • If new bonds are issued at a higher interest rate, holders of bonds that pay lower rates would have to adjust the price at which they are willing to sell their bonds. • To calculate the new price, we need to use the same yield to maturity of the newly issued bonds. • Because the yield to maturity is higher, the bond’s market price will fall below its face value. • As interest rates rise, bond prices fall. If the price of an asset increases, it is called a capital gain. If the price of the asset declines, it is called a capital loss.

Making the Connection Banks Take a Bath on Mortgage-Backed Bonds • Many mortgage-backed securities are similar to long-term bonds in that they pay regular interest based on the payments borrowers make on the underlying mortgages. • In the secondary market for mortgage-backed securities, as borrowers began to default on their payments, buyers required much higher yields to compensate for the higher levels of default risk. • Higher yields on these securities meant lower prices. By 2008, the prices of many mortgage-backed securities had declined by 50% or more. • By early 2009, U.S. commercial banks had suffered losses on their investments of about $1 trillion. • Banks had relearned the lesson that soaring interest rates can have a devastating effect on investors holding existing debt instruments. The Inverse Relationship between Bond Prices and Bond Yields

Bond Prices and Yields to Maturity Move in Opposite Directions • If interest rates on newly issued bonds rise, the prices of existing bonds will fall. • If interest rates on newly issued bonds fall, the prices of existing bonds will rise. • In other words, yields to maturity and bond prices move in opposite directions. • The reason, as noted earlier, is that if interest rates rise, existing bonds issued when interest rates were lower become less desirable to investors, and their prices fall. If interest rates fall, existing bonds become more desirable, and their prices rise. • This relationship should also hold for other debt instruments. The Inverse Relationship between Bond Prices and Bond Yields

Secondary Markets, Arbitrage, and the Law of One Price • An investor in a financial market buys securities to earn a return. A trader buys and sells securities to profit from small differences in prices. • During the period before bond prices fully adjust to changes in interest rates, there is an opportunity for arbitrage. • The prices of financial securities at any given moment allow little or no opportunity for arbitrage profits. • The prices of securities should adjust so that investors receive the same yields on comparable securities. For example, bonds with 8% coupon rates will have the same yield as bonds with 6% coupon rates. • This rationale follows the principle called the law of one price, which states that identical products should sell for the same price everywhere. Financial arbitrageThe process of buying and selling securities to profit from price changes over a brief period of time. The Inverse Relationship between Bond Prices and Bond Yields

Making the Connection Reading the Bond Tables in the Wall Street Journal Treasury Bonds and Notes • Bond A matures on August 15, 2015, and has a coupon rate of 4.250%, so it pays $42.50 each year on its $1,000 face value. • Prices are reported per $100 of face value. For Bond A, 112:08 means “112 and 08/32,” or a price of $1,122.50 for this $1,000 face value bond. • The bid price is the sell price; the asked price is the price to buy the bond. • For Bond A, the bid price rose by 8/32 from the previous day. The Inverse Relationship between Bond Prices and Bond Yields

Making the Connection Reading the Bond Tables in the Wall Street Journal Treasury Bonds and Notes • The current yield equals the coupon divided by the price: $42.50/$1,122.50, or 3.79% for Bond A. • The current yield of Bond A is well above the yield to maturity of 1.7066%. • This illustrates that the current yield is not a good substitute for the yield to maturity for instruments with a short time to maturity because it ignores the effect of expected capital gains or losses. The Inverse Relationship between Bond Prices and Bond Yields

Making the Connection Reading the Bond Tables in the Wall Street Journal Treasury Bills • Treasury bills are discount bonds, not coupon bonds. • Treasury notes and bonds quote prices, while Treasury bills quote yields. • The bid yield is the discount yield for sellers. The asked yield is for buyers. • The dealers’ profit margin is the difference between the asked bid yields. • The yield to maturity, in the last column, is useful for comparing investments. The Inverse Relationship between Bond Prices and Bond Yields

Making the Connection Reading the Bond Tables in the Wall Street Journal New York Stock Exchange Corporation Bonds • A bond’s rating shows the likelihood that the firm will default on the bond. • Prices are quoted in decimals. • The last time this Goldman Sachs bond was traded that day, it sold for a price of $1,048.68. The Inverse Relationship between Bond Prices and Bond Yields

3.5 Learning Objective Explain the difference between interest rates and rates of return.

Return The total earnings from a security; for a bond, the coupon payment plus the change in the price of the bond. Rate of return, RThe return on a security as a percentage of the initial price; for a bond, the coupon payment plus the change in the price of a bond divided by the initial price. For example, for a bond with a $1,000 face value and a coupon rate of 8%: If the end-of-year price was $1,271.81, then, the rate of return for the year was: If the end-of-year price was $812.61, then, the rate of return for the year was: Interest Rates and Rates of Return

A General Equation for the Rate of Return A general equation for the rate of return on a bond for a holding period of one year is: Three important points to note: 1. For the current yield, the calculation uses the initial price. 2. If you sell the bond, you have a realized capital gain or loss. If you do not sell the bond, your gain or loss is unrealized. 3. Neither the current yield nor the yield to maturity may be a good indicator of the rate of return because they ignore your capital gain or capital loss. Interest Rates and Rates of Return

Interest-Rate Risk and Maturity Interest-rate risk The risk that the price of a financial asset will fluctuate in response to changes in market interest rates. • Bonds with fewer years to maturity will be less affected by a change in market interest rates. • At the end of one year, the yield to maturity on similar bonds has risen to 10%. The table shows that the longer the maturity of your bond, the lower (more negative) your return after one year of holding the bond. With a maturity of 50 years, your rate of return for the first year of owning your bond will be -33.7%. Interest Rates and Rates of Return

3.6 Learning Objective Explain the difference between nominal interest rates and real interest rates.

Nominal interest rate An interest rate that is not adjusted for changes in purchasing power. Real interest rate An interest rate that is adjusted for changes in purchasing power. • Inflation causes the purchasing power of both the interest income and the principal to decline. • Because lenders and borrowers don’t know what the actual real interest rate will be during the period of a loan, they must estimate an expected real interest rate. • The expected real interest rate, r, equals the nominal interest rate, i, minus the expected rate of inflation, e. • Therefore, the nominal interest rate equals the real interest rate plus the expected inflation rate: i = r + e. p p Nominal Interest Rates versus Real Interest Rates

Nominal Interest Rates versus Real Interest Rates

Nominal and Real Interest Rates, 1981–2010 Figure 3.1 In this figure, the nominal interest rate is the interest rate on three-month U.S. Treasury bills. The actual real interest rate is the nominal interest minus the actual inflation rate, as measured by changes in the consumer price index. The expected real interest rate is the nominal interest rate minus the expected rate of inflation as measured by a survey of professional forecasters. When the U.S. economy experienced deflation during 2009, the real interest rate was greater than the nominal interest rate.• Nominal Interest Rates versus Real Interest Rates

It is possible for the nominal interest rate to be lower than the real interest rate. For this outcome to occur, the inflation rate has to be negative, meaning that the price level is decreasing rather than increasing. Deflation A sustained decline in the price level. • In January 1997, the U.S. Treasury started issuing indexed bonds to address investors’ concerns about the effects of inflation on real interest rates. With these bonds, called TIPS (Treasury Inflation Protection Securities), the Treasury increases the principal as the price level increases. Nominal Interest Rates versus Real Interest Rates

R. GLENN HUBBARD ANTHONY PATRICK O’BRIEN