Chapter

Interest Rates

McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

9

9-2

Learning Objectives

It will be worth your time to increase yourrate of interest in these topics:

1. Money market prices and rates.

2. Rates and yields on fixed-income securities.

3. Treasury STRIPS and the term structure of interest rates.

4. Nominal versus real interest rates.

9-3

Interest Rates

• Our goal in this chapter is to discuss the many different interest rates that are commonly reported in the financial press.

• We will also:

– Find out how different interest rates are calculated and quoted, and

– Discuss theories of what determines interest rates.

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U.S. Interest Rate History, 1800-2009

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Money Market Interest Rates

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Money Market Rates, I.

• Prime rate - The basic interest rate on short-term loans that the largest commercial banks charge to their most creditworthy corporate customers.

• Discount rate - The interest rate that the Fed offers to commercial banks for overnight reserve loans.

• Federal funds rate - Interest rate that banks charge each other for overnight loans of $1 million or more.

• Banker’s acceptance - A postdated check on which a bank has guaranteed payment. Commonly used to finance international trade transactions.

• Call money rate - The interest rate brokerage firms pay for call money loans from banks. This rate is used as the basis for customer rates on margin loans.

• Commercial paper - Short-term, unsecured debt issued by the largest corporations.

• CDs - The interest rate on certificates of deposit, which are large-denomination deposits of $100,000 or more at commercial banks

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Money Market Rates, II.

• U.S. Treasury bill (T-bill) - A short-term U.S. government debt instrument issued by the U.S. Treasury.

• London Interbank Offered Rate (LIBOR) - Interest rate that international banks charge one another for overnight Eurodollar loans.

• Euro LIBOR refers to deposits denominated in euros—the common currency of 16 European Union countries.

• EURIBOR is an interest rate that also refers to deposits denominated in euros. However, EURIBOR is based largely on interest rates from the interbank market for banks in the European Union.

• HIBOR is an interest rate based on Hong Kong dollars. Hibor is the interest rate among banks in the Hong Kong interbank market.

• Eurodollars - U.S. dollar denominated deposits in banks outside the United States.

9-8

Money Market Prices and Rates

• A Pure Discount Security is an interest-bearing asset:

– It makes a single payment of face value at maturity.

– It makes no payments before maturity.

• There are several different ways market participants quote interest rates.

– Bank Discount Basis– Bond Equivalent Yields (BEY)– Annual Percentage Rates (APR)– Effective Annual Rates (EAR)

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The Bank Discount Basis• The Bank Discount Basis is a method of quoting interest

rates on money market instruments.

– It is commonly used for T-bills and banker’s acceptances.

• The formula is:

– Note that we use 360 days in a year in this (and many other) money market formula.

– The term “discount yield” here simply refers to the quoted interest rate.

YieldDiscount x

360

MaturitytoDays1 x ValueFacePrice Current

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Example: Calculating a Price Using a Bank Discount Rate

• Suppose a banker’s acceptance that will be paid in 71 days has a face value of $100,000.

• If the discount yield is 4%, what is the current price of the banker’s acceptance?

.$99,211.11

0.007889- 1 $100,000

0.04 36071

1 $100,000

yieldDiscount 360

maturitytoDays1 ValueFacePrice Current

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Treasury Bill Quotes (online at www.wsj.com)

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Treasury Bill Prices:November 24, 2009

• Figure 9.3 shows a T-bill that expires February 11, 2010.

– It has 79 days to maturity.– The bid discount is .028% (you use this to calculate

the bid price, i.e., the price you will receive for the T-bill).

– Prices are quoted for $1,000,000 face values.

6$999,938.5

0.00006144- 1 $1,000,000

0.00028 360

791 $1,000,000

yieldDiscount 360

maturitytoDays1 ValueFacePrice bill-TCurrent

Verify that the askprice is $999,956.11

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Bond Equivalent Yields• Bond Equivalent Yields (BEY) are another way to quote

an interest rate.• You can convert a bank discount yield to a bond

equivalent yield using this formula:

Note that this formula is correct only for maturities of six months or less. Moreover, if February 29 occurs within the next 12 months, use 366 days.

yielddiscount maturity xtodays360

yielddiscountx365BEY

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Example I: Bond Equivalent Yield• Figure 9.3 shows a T-bill that expires February 11, 2009.

– It has 79 days to maturity.– The bid discount is 0.028%.– What is the Bond Equivalent (bid) Yield?

– There is not much difference at this low interest rate.– Suppose, the bid discount was 6.00% for this 79-day T-

bill. What is the BEY in this case?

0.0284%. about or 0.000284,

0.00028793600.00028365

BEY

yieldDiscountx maturity to Days360 yieldDiscountx365

BEY

Remember to multiply before you subtract.

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Example II: Calculating T-bill Prices Using Bond Equivalent Yield

• We can calculate a Treasury bill asking price using the “asked” yield, which is a bond equivalent yield.

• Look at Figure 9.3 for the T-bill that expires on February 11, 2009.

– It has 79 days to maturity.– The reported ask yield is 0.020%.

1.$999,956.7

1.00004329$1,000,000

79/365 0.00020 1$1,000,000

365 Maturity / to Days Yield Equivalend Bond 1Value Face

Price AskBill

Note: The bill’s ask price differs from a previous slide by about $0.60 due to rounding of the reported asked yield.

The ask yield is really 0.0203% (verify using the BEY formula)

The error is bigger for higher interest rates.

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More Ways to Quote Interest Rates

• “Simple” interest basis - Another method to quote interest rates.

– Calculated just like annual percentage rates (APRs).

– Used for CDs.– The bond equivalent yield on a T-bill with less

than six months to maturity is also an APR.

• An APR understates the true interest rate, which is usually called the effective annual rate (EAR).

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Example: The BEY on a T-bill is Really Just an APR

• Earlier, using the bid discount rate, we calculated a bid price for an 79-day T-bill to be $999,938.56.

– At maturity, this T-bill will be worth $1,000,000.– You will earn $61.44 of interest on an investment of

$999,938.56 over 79 days, a return of 0.0061444%.– In a 365-day year, there are 365/79 = 4.62025 periods of 79

days in length.– 0.0061444% times 4.62025 is 0.2839%.

• This is the bond equivalent (bid) yield that we calculated before (that we rounded to 0.28%).

Note: The Wall Street Journal rounds bid and ask yields to 2 decimal places.

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Converting APRs to EARs

• In general, if we let m be the number of periods in a year, an APR can be converted to an EAR as follows:

• EARs are sometimes called effective annual yields, effective yields, or annualized yields.

m

m

APR1EAR1

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Example I: What is the EAR of this T-bill’s BEY (aka APR)?

too.case in this 0.02839% EAR theso,

1.0002839

71.00006144

4.62025

0.00028391 EAR 1

m

APR1EAR1

4.62025

4.62025

m

Note that the lower the interest rates, the closer is the APR to the EAR.

9-20

Example II: Converting Credit Card APRs to EARs

• Some Credit Cards quote an APR of 18%.

– 18% is used because 18 = 12 times 1.50– That is, the monthly rate is really 1.50%.– What is the EAR?

19.56%. EAR theso,

1.1956

1.015

12

0.181 EAR 1

m

APR1EAR1

12

12

m

Ouch.

9-21

Using Excel to Calculate T-bill Prices and Yields

$99.7638 =TBILLPRICE("2/23/2010","5/15/2010",0.0105)

1.067%

1.072% =EFFECT(B10,12)

Note: B10 is the TBILLEQ cell. B10=1.067%

=TBILLEQ("2/23/2010","5/15/2010",0.0105)

What is the effective annual rate (EAR) on this T-bill? Hint: Use the Excel function EFFECT.

Treasury Bill Price and Yield Calculations

A Treasury bill traded on February 23, 2010 pays $100 on May 15, 2010. Assuming a discount rate of 1.05 percent, what are its price and bond equivalent yield? Hint: Use the Excel functions TBILLPRICE and TBILLEQ.

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Rates and Yields on Fixed-Income Securities

• Fixed-income securities include long-term debt contracts from a wide variety of issuers:

– The U.S. government,– Real estate purchases (mortgage debt),– Corporations, and– Municipal governments

• When issued, fixed-income securities have a maturity of greater than one year.

• When issued, money market securities have a maturity of less than one year.

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The Treasury Yield Curve

• The Treasury yield curve is a plot of Treasury yields against maturities.

• It is fundamental to bond market analysis, because it represents the interest rates for default-free lending across the maturity spectrum.

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Example: The Treasury Yield Curve

9-25

The Term Structure of Interest Rates, I.

• The term structure of interest rates is the relationship between time to maturity and the interest rates for default-free, pure discount instruments.

• The term structure is sometimes called the “zero-coupon yield curve” to distinguish it from the Treasury yield curve, which is based on coupon bonds.

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The Term Structure of Interest Rates, II.

• The term structure can be seen by examining yields on U.S. Treasury STRIPS.

• STRIPS are pure discount instruments created by “stripping” the coupons and principal payments of U.S. Treasury notes and bonds into separate parts,which are then sold separately.

• The term STRIPS stands for Separate Trading of Registered Interest and Principal of Securities.

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U.S. Treasury STRIPS

• An asked yield for a U.S. Treasury STRIP is an APR, calculated as two times the true semiannual rate.

• Recall:

• Therefore, for STRIPS:

Nr1

valueFuturevalue Present

2M

2YTM1

ValueFacePriceSTRIPS

M is the number of years to maturity.

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Figure 9.5: U.S. Treasury STRIPS

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Example: Pricing U.S. Treasury STRIPS, I.

• Let’s verify the price of the November 2016 Strip.

– The ask quote is 80.975, or $80.975.– The ask YTM is 3.05%.– Matures in about 7 years from the time of the quote.

– Close (considering the two-decimal rounding of the ask YTM).

$80.91. 1.23601

100

20.03051

100

20.03051

100

2YTM1

Value FacePriceSTRIPS

72

722M

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Example: Pricing U.S. Treasury STRIPS, II.

• Let’s calculate the YTM from the quoted price.

• Close again (reported ask YTM was 3.05%--using the actual days to maturity. We used “about 7 years.”).

3.04%. or 0.03038, 11.234952

180.975

1002 1

Price STRIPSValue Face

2YTM

1/14

721

2M1

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Nominal versus Real Interest Rates

• Nominal interest rates are interest rates as they are observed and quoted, with no adjustment for inflation.

• Real interest rates are adjusted for inflation effects.

Real interest rate = nominal interest rate – inflation rate

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Real T-bill Rates

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Nominal versus Real Interest Rates

• The Fisher Hypothesis asserts that the general level of nominal interest rates follows the general level of inflation.

• According to the Fisher hypothesis, interest rates are, on average, higher than the rate of inflation.

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Inflation Rates and T-bill Rates

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Inflation-Indexed Treasury Securities, I.• Recently, the U.S. Treasury has issued securities that guarantee a fixed rate of

return in excess of realized inflation rates.

• These inflation-indexed Treasury securities:

– pay a fixed coupon rate on their current principal

– adjust their principal semiannually according to the most recent inflation rate

• Example: Suppose an inflation-indexed note is issued with a coupon rate of 3.5% and an initial principal of $1,000.

– Six months later, the note will pay a coupon of $1,000 × (3.5%/2) = $17.50.

– Assuming 2 percent inflation over the six months since issuance, the note’s principal is then increased to $1,000 × 102% = $1,020.

– Six months later, the note pays $1,020 × (3.5%/2) = $17.85.– Its principal is again adjusted to compensate for recent inflation.

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Inflation-Indexed Treasury Securities, II.

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Traditional Theories of the Term Structure

• Expectations Theory: The term structure of interest rates reflects financial market beliefs about future interest rates.

• Market Segmentation Theory: Debt markets are segmented by maturity, so interest rates for various maturities are determined separately in each segment.

• Maturity Preference Theory: Long-term interest rates contain a maturity premium necessary to induce lenders into making longer term loans.

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Problems with Traditional Theories• Expectations Theory

– The term structure is almost always upward sloping, but interest rates have not always risen.

– It is often the case that the term structure turns down at very long maturities.

• Maturity Preference Theory

– The U.S. government borrows much more heavily short-term than long-term.

– Many of the biggest buyers of fixed-income securities, such as pension funds, have a strong preference for long maturities.

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Problems with Traditional Theories

• Market Segmentation Theory

– The U.S. government borrows at all maturities.

– Many institutional investors, such as mutual funds, are more than willing to move maturities to obtain more favorable rates.

– There are bond trading operations that exist just to exploit perceived premiums, even very small ones.

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Modern Term Structure Theory, I.

• Long-term bond prices are much more sensitive to interest rate changes than short-term bonds. This is called interest rate risk.

• So, the modern view of the term structure suggests that:NI = RI + IP + RP

• In this equation:

NI = Nominal interest rate

RI = Real interest rate

IP = Inflation premium

RP = Interest rate risk premium

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Modern Term Structure Theory, II.

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Modern Term Structure Theory, III.• The previous equation showed the components of

interest rates on default-free bonds that trade in a liquid market.

• Not all bonds do.

• Therefore, a liquidity premium (LP) and a default premium (DP) must be added to the previous equation:

NI = RI + IP + RP + LP + DP

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Useful Internet Sites

• www.money-rates.com (for the latest money market rates)• www.bbalibor.com (learn more about LIBOR)• www.sifma.org (information on fixed income securities)• www.bloomberg.com (current U.S. Treasury rates)• www.smartmoney.com/bonds (view a “living yield curve”)• www.fanniemae.com (one of three mortgage security websites)• www.ginniemae.gov (one of three mortgage security websites)• www.freddiemac.com (one of three mortgage security websites)• www.publicdebt.treas.gov (information on STRIPS; other U.S. debt)

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Chapter Review, I.

• Interest Rate History and Money Market Rates

– Interest Rate History– Money Market Rates

• Money Market Prices and Rates

– Bank Discount Rate Quotes– Treasury Bill Quotes– Bank Discount Yields versus Bond Equivalent

Yields– Bond Equivalent Yields, APRs, and EARs

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Chapter Review, II.• Rates and Yields on Fixed-Income Securities

– The Treasury Yield Curve– Rates on Other Fixed-Income Investments

• The Term Structure of Interest Rates

– Treasury STRIPS– Yields for U.S. Treasury STRIPS

• Nominal versus Real Interest Rates

– Real Interest Rates– The Fisher Hypothesis– Inflation-Indexed Treasury Securities

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Chapter Review, III.

• Traditional Theories of the Term Structure

– Expectations Theory– Maturity Preference Theory– Market Segmentation Theory

• Determinants of Nominal Interest Rates: A Modern Perspective

– Problems with Traditional Theories– Modern Term Structure Theory– Liquidity and Default Risk