Chapter 6

The Structure of Interest Rates

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From One Interest Rate to Many Term to Maturity The Expectations Theory Necessary Modifications to the Expectations

Theory Credit Risk Taxability

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From One Interest Rate to Many

Numerous types of financial claims are traded in financial markets

Interest rates generally move up and down together. All rates may not move the same amount.

However, occasionally some rates may not even move in the same direction as the rest.

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The purpose of chapter 6

To study the factors that are primarily responsible for determining the relationship among interest rates.

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From One Interest Rate to Many

There are four primary determinants of the relationships among interest ratesterm to maturitycredit riskliquiditytax treatment

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Term to Maturity

The major characteristic distinguishing one type of Treasury security from another is the term to maturityTreasury bills have short terms to maturity of

one year or lessTreasury notes and bonds have long maturities

of one year or more

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Term to Maturity

The pattern of relationships among interest rates and the time to maturity are referred to in financial markets as the term structure of interest rates

利率的期限结构：收益率和到期时间之间的关系。

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The Yield Curve A yield curve visually represents the term

structure of interest ratesshows the relationship between interest rates

(yields) on particular financial instruments (securities) and their terms to maturity

Each type of asset is represented on a single yield curvecontrols for factors other than term to maturity收益率曲线：描述某种证券的利率（收益率）与其期限之间的关系的图形。

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Interest Rates on Treasury Securities

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Yield Curves

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The Expectations Theory

The expectations theory postulates thatthe yield curve is determined by borrowers’ and

lenders’ expectations of future interest rateschanges in the slope (shape) of the curve result

from changes in these expectations

预期理论：该理论假设长期利率是当前短期利率及预期短期利率的几何平均数。

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Example

Suppose you have funds to lend for a two-year period and the current yield on a one-year bill (i1) is 5% and the current yield on a two-year note (i2) is 5.99%

Suppose you expect that the yield on one-year securities (ie1) will be 7% one year from now

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You have two optionsbuy a one-year security today and another one

year from nowbuy a two-year security today

Which option will give you a higher expected rate of return?

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To calculate the expected return of the first option, we find the geometric average of the two rateswe assume that the interest earned during the

first year will earn interest during the second year

几何平均数：考虑了复利影响的平均数。

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The long rate (i2) can be calculated as

(1 + i2) = [(1 + i1)(1 + ie1)]1/2

i2 = [(1 + i1)(1 + ie1)]1/2 – 1

i2 = [(1.05)(1.07)]1/2 – 1 = 5.99%

This is the rate of the two-year security You will be indifferent between the two

options

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Hypothetical Yield Curve

Term to Maturity

Yield to Maturity (percent)

1

5

2

5.99yield curve

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The Expectations Theory

Suppose that expectations about future rates change so that ie1 rises from 7 to 9 percent

(1 + i2) = [(1 + i1)(1 + ie1)]1/2

i2 = [(1 + i1)(1 + ie1)]1/2 – 1

i2 = [(1.05)(1.09)]1/2 – 1 = 6.98%

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The Expectations Theory

You will no longer be indifferent between the two optionsyou and others will want short-term securitiesthose who own long-term securities will want to

sell them and buy short-term securities the price of long-term securities will fall and their

yields will rise this will continue until the long rate rises to 6.98%

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The Expectations Theory

The interest rate changed as a result of a change in the interest rate expectations of the lenderaffected the demand for securities

The expectations of the borrower are also important

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The Expectations Theory

When borrowers believe that the average rate of current and expected future short-term securities is greater than the rate on long-term securities, they will increase their supply of long-term securitiesthis will lower the price of existing long-term

securitiesthis will raise the long-term interest rate

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The Expectations Theory

If expectations about future rates change such that future rates are expected to be higher, the yield curve will shiftthe long rate will rise relative to the short rate

and the yield curve will get steeper

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New Hypothetical Yield Curve

Term to Maturity

Yield to Maturity (percent)

1

5

5.99original yield curve

2

new yield curve

6.98

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The Expectations Theory

We can solve for the interest rate expected to prevail in the future by looking at the current structure of interest rates

(1 + i2) = [(1 + i1)(1 + ie1)]1/2

(1 + i2)2 = {[(1 + i1)(1 + ie1)]1/2}2

(1 + i2)2 = (1 + i1)(1 + ie1)

(1 + i2)2/(1 + i1) = (1 + ie1)

ie1 = [(1 + i2)2/(1 + i1)] – 1

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Alternative Yield Curve Shapes

When a rising yield curve is observed in the market, the expected short-term interest rate (ies) is expected to rise above current short rates (is)

A flat yield curve implies that interest rates are expected to remain constant (ies = is)

A negatively sloped yield curve implies that interest rates are expected to decline (ies < is)

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Alternative Yield Curve Shapes

Yield to Maturity

Term to Maturity

Yield to Maturity

Term to Maturity

a. A Rise Expected in Interest Rates

(ies > is)

Yield to Maturity

Term to Maturity

b. No Change Expected in Interest Rates

(ies = is)

c. A Decline Expected in Interest Rates

(ies < is)

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Determining Interest Rate Expectations

Remember that the interest rate is determined by several supply and demand factors

This must mean that the expected short-term interest rate must be determined by expectations of Y, M, and pe

)p,M,Y(fi e

)p,M,Y(fi eeees

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Determining Interest Rate Expectations

A positively-sloped yield curve reflects expectations of rising interest ratesfuture increases in incomerising inflation expectationsa reduction in the future growth rate of the

money supply

Thus, a positively-sloped yield curve should occur at a business cycle trough and during the first half of the recovery

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Determining Interest Rate Expectations

A negatively-sloped yield curve reflects expectations of falling interest ratesfuture declines in incomefalling inflation expectationsan increase in the future growth rate of the

money supply

Thus, a negatively-sloped yield curve should occur at a business cycle peak and during the early part of a recession

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Necessary Modifications to the Expectations Theory

Over the last 50 years, yield curves have almost always been positively slopeddo financial market participants always expect short-

term interest rates to rise?

Borrowers and lenders may not be indifferent between short- and long-term securitiesmany borrowers and lenders have preferred habitats

习惯性偏好：修正了的预期理论，假定大部分借贷者都对预期有偏好，这在一定程度上将市场分成了短期和长期。

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Necessary Modifications to the Expectations Theory

Research suggests that investors may be willing to switch preferred habitats from short-term to longer-term financial claims if they are provided a liquidity premiuman extra return required to induce lenders to

lend long term rather than short term the size of the premium is presumed to rise with the

term to maturity流动性补偿：吸引贷方提供长期而非短期资金的额外报酬。

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Necessary Modifications to the Expectations Theory

Suppose that the current short rate and the expected short rate are both 5%according to the expectations theory, the yield

curve would be flat

If the issuer of long-term bonds must offer an interest premium to get investors to buy them, the yield curve is actually upward-sloping

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The Role of Liquidity Premiums

Term to Maturity

Yield to Maturity (percent)

Yield Curve Based on Expectations (ies = is)

Liquidity Premium

Observed Yield Curve = Expectations (ies = is) + Liquidity Premium

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Necessary Modifications to the Expectations Theory

We can summarize the relationship between long-term interest rates (il), short-term interest rates (is), and the liquidity premium (l)

),,( liifi essl

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Credit Risk Credit risk refers to the probability of a debtor not

paying the principal or interest due on an outstanding debt信用风险：未偿还债务的借方不归还本金和利息的可能性。

Three major credit-rating agencies evaluate a borrower’s credit risk and assign the borrower to a particular risk classStandard & Poor’sMoody’sFitch Investors Service

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Credit Risk

In the case of business firms, the credit-rating agencies examine several thingsthe pattern of revenues and coststhe degree of leverage (dependence on

borrowed funds)the past history of debt redemptionthe volatility of the industry

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Credit Risk

In the case of state and local governments, the credit-rating agencies examine several thingsthe tax basethe level of outstanding debtthe current and expected budget conditiongrowth in spending

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Credit Ratings

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Credit Risk

Investors are generally risk aversethey must be rewarded or compensated with

extra interest for accepting more riskthis extra return is called a risk premium

the size increases with the riskiness of the borrower

风险溢价：贷方承担风险的额外补偿或报酬。

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Taxability

Interest income earned from securities issued by state and local governments is exempt from federal income tax

after-tax yield = i – it = i(1 – t) where t is the marginal tax rate

the rate paid on the last dollar of income the taxpayer earns边际贡献：纳税人为其收入的最后一单位所支付的税收。

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Taxability

In general, we would expect the substitution of securities to result in the yield on municipals being approximately equal to the yield on a similarly rated taxable corporate bond minus the portion that is taxed away

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Taxability

The market will gravitate to the rate that makes the “average” tax payer indifferent between municipals and similarly rated corporate bondsindividuals in high tax brackets will be

especially attracted to municipals because they are subject to a tax rate above the average marginal rate

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Summary of Major Points

The yield curve is a graphical representation of the relationship between interest rates (yields) on a particular security and its term to maturitya visual depiction of the term structure of

interest rates

A unique curve exists for each type of financial asset

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Summary of Major Points

The most widely accepted explanation for the shape (slope) and position (level) of the yield curve is the expectations theorypostulates that the long-term rate is the

geometric average of the current short-term rate and the short-term rates expected to prevail over the term to maturity of the security takes compounding into effect

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Summary of Major Points

The slope of the yield curve depends on the interest rates expected to prevail on short-term securities in the futurea positively sloped curve reflects expectations

of a rise in future short-term ratesa negatively sloped curve reflects expectations

of a fall in future short-term rates

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Summary of Major Points

Expectations about future short-term rates depend on expectations about future income, the money supply and inflationas expectations about these variables change,

expected short-term rates will change, resulting in a change in the slope and level of the yield curve

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Summary of Major Points

The expectations theory is an incomplete explanation of the term structure of interest ratesit has been modified to take account of liquidity

premiums

Long-term rates will be determined by current short-term rates, expected short-term rates, and liquidity premiums

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Summary of Major Points

Credit risk refers to the probability of a debtor defaultingthe three major credit-rating agencies evaluate

a borrower’s probability of default and assign the borrower a risk classification

Since investors are risk averse, they must be offered a risk premiumthe size will rise with the riskiness of the

borrower

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Summary of Major Points

Financial investors care about the after-tax return of their investmentssince interest earned on municipal securities is

exempt from federal income tax, the yield on municipal securities are typically below the yields of other taxable securities with similar credit ratings and terms to maturity