The Structure of Interest Rates (a) The Term Structure of Interest Rates(b) Risk Premiums and Quality Spreads

Blackwell, Griffiths and Winters, Chapter 3, and other material

1

The Yield Curve and the Term Structure of Interest Rates

Yield

to

Maturity

Maturity

2

Expectation HypothesisThe Yield Curve for Default Free, Very Liquidity Instrument

with Virtually No Information CostsYield

to

Maturity

Maturity

Real Risk-Free Rate

Effect of Expectations over a horizon of “M” Months

Maturity in Months 3

4

Expectations Theory

Current long-term interest rates are based on investors expectations of future interest rates…

which means that Current long-term rates are combinations of current and expected

future short-term rates.

Expectations Theory and Slopes of the Yield Curve

The yield curve plots yield against time to maturity for default-free securities

Expectations theory can explain three shapes of the yield curve: upward-sloping, downward-sloping, and flat.

1. Rates expected to rise in the future, then yield curve is upward sloping.

2. Rates expected to fall in the future, then yield curve is downward sloping.

3. Rates expected to remain constant in the future, then the yield curve

is flat.

5

6

Definition of Long-term Rate underExpectations Theory

Under expectations theory, a long-term rate is the geometric average of current and expected future short-term rates.

7

Formula for the Long-term Rate

1 1 1 1 10 0 1 1 2 2 3 1 r r f f ft

t

t t

wherer = spot rate

f = one-period expected future short-term ratet = number of time periods in the long-term rate

andprescript is beginning of time period covered by the rate

postscript is ending of the time period

8

Example for a three-year rate

1 1 1 10 33

0 1 1 2 2 3 r r f f

000 1 2 3

0r3

0r1 1f2 2f3

Timeline of three-year rate

9

Solving for an Expected Future Short-term Rate

11

1 1 1

1

11

0

0 1 1 2 2 1

0

0 11

t t

tt

t t

tt

ttf

r

r f f

r

r

1 1 1 1 10 0 1 1 2 2 3 1 r r f f ft

t

t t

then, solving for t-1ft is

10

Example: Finding an expected future rate

Using data from May 2000 term structure (Figure 3.7 ), the expected one-year rate for Year 2 is:

1

1

1

1 0 6 8 1

1 0 6 3 31 0 7 2 91 2

0 22

0 11

2

2

fr

rf

.

.. so is 7 .2 9 % .1

Graphic Portrayal of the Liquidity Premium Hypothesis

Yield

to

Maturity

Maturity

Term Premium

Observed Yield Curve

Yield Curve Excluding Term Premium

11

Graphic Portrayal of the Liquidity Premium Hypothesis

Yield

to

Maturity

Maturity

Term Premium

Yield Curve Excluding Term Premium

Observed Yield Curve

12

Graphic Portrayal of the Liquidity Premium Hypothesis

Yield

to

Maturity

Maturity

Term Premium

Yield Curve Excluding Term Premium

Observed Yield Curve

13

Graphic Portrayal of the Segmented Market Hypothesis(Explaining the Slope of the Yield Curve)

Yield

to

Maturity

Maturity

14

Graphic Portrayal of the Preferred Habitat Hypothesis

Yield

to

Maturity

Maturity

Term Premium

Observed Yield Curve

Yield Curve Excluding Term Premium

Other premiums

15

Preferred Habitat Theory

The preferred habitat theory is a corollary of the market segmentation theory.

Market segmentation theory assumes that investors have specific maturity preferences that they will not leave regardless of the additional compensation. This assumption is too restrictive. The preferred habitat theory assumes that investors have a desired time to maturity for investment (a preferred habitat), but if adequately compensated for the additional risk of moving from their desired maturity, they will move to other maturities.

The preferred habitat theory is the only theory that allows for ‘humps’ or ‘twists’ in the yield curve. In other words, it allows for a change in the direction of the slope of the yield curve.

This is important because this frequently occurs in plots of market yields.

16

The Basic Rate:The Nominal Risk Free Rate

The nominal risk free rate is the rate on a relatively short-term asset that has essentially two components:

– The real risk free rate (krf,t), and

– The inflation premium (IPm,t) for an investment maturity of m periods,

This may expressed as

For the risk free rate, the maturity m would be a very short horizon and the inflation risk premium would be close to zero

The nominal risk free rate can be approximated by a very short-term Treasury bill rate

– It has no credit risk

– It has a very high degree of liquidity

– It has no term premium

The superscript T in the previous equation represents a Treasury security

tmtrfT

tm IPkk ,,,

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Beyond the Risk Free Rate

However, once one considers an asset longer than a very short-term Treasury bill rate, other forms of risk emerge

– Some affect all debt contracts in an equal manner for a given maturity--general risks

• General risk seem to have a term structure

– This means that its size varies with an assets maturity

– However, for a given maturity, it affects all debt contracts by the same magnitude

– Other risks depend upon the specific characteristics of the debt contract--instrument specific risks

• Its size depends upon the asset under consideration; and the magnitude may also have a term structure

Maturity risk premium

– Term Structure

– Impact of Business Cycle:

Inflation premium

– Term Structure

– Impact of Business Cycle:

Inflation risk premium

– Term Structure

– Impact of Business Cycle:

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General Risks and the Yield on Treasury Securities

The yield on a Treasury security (an instrument lacking instrument specific risks) may be expressed as

The nominal risk free rate would have a zero maturity risk premium and no inflation risk premium

Supply premium (SPm,t): the shortage of Treasury securities could lead to a scarcity value for Treasury securities

tmtmtmtrfT

tm IRPIPMRPkk ,,,,, T

tmtmtmtmtrfT

tm SPIRPIPMRPkk ,,,,,,

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20

The Term Structure of Inflation ForecastsSurvey of Professional Forecasters

1

2

3

4

5

6

7

8

1985 1990 1995 2000 2005 2010

1-Year Epected Inflation10-Year Epected Inflation

The Composition of Treasury Yields and the Four Theories of the Yield Curve

Pure Expectations

The expectation embodied in the yield curve at time t is the inflation premium (expected inflation rate) over a horizon of m periods

Liquidity Premium Hypothesis

The pure expectation hypothesis is expanded by adding the maturity risk premium demanded for an instrument with a maturity of m periods

Preferred Habitat

Segmented Market Hypothesis

tmtrfT

tm IPkk ,,,

tmtmtrfT

tm IPMRPkk ,,,,

Ttmtmtmtmtrf

Ttm SPIRPIPMRPkk ,,,,,,

Ttmtrf

Ttm SPkk ,,,

21

Instrument Specific Risks

The risk that is unique to the asset under consideration

– These risks affect each instrument differently

– It is generally assumed that Treasury securities posses general risks but are free of the instrument specific risks

– As a result, Treasury securities are often used as a benchmark against which the magnitude of instrument specific risk might be gauged

Instrument Specific Risks– Default/Credit Risk– Liquidity Risk– Supply shifts– Management Risk– Call Risk– Convertibility Risk– Political Risk– Sovereign– Industry Risk– Legal Risk

– Operations Risk

22

More on Default Risk

Default on a debt contract occurs when the borrower violates any of the conditions of the contract.

When examining default risk, a lender is concerned with how a default will impair the expected cash flows from the debt contract.

So, the lender is concerned about the expected losses from a default and builds those expected losses into the debt contract rate through a default risk premium.

There is a wide range of possible losses from default, so we need a way to measure default risk.

Bond Rating Proxies for Default RiskCredit rating agencies provide bond ratings The two primary rating agencies are Moody’s and Standard & Poor’s.A bond rating is the agency’s opinion on a company’s ability to meet its financial obligations.To determine a company’s bond rating, the rating agency does an in-depth analysis of the company, which includes a complete analysis of the company’s financial statements.

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Bond Ratings (Figure 3-11)

Moody’s Rating S&P Rating Explanation

Investment Grades

Aaa AAA Extremely strong, best quality

Aa AA Very strong, high quality

A A Strong, upper-medium grade

Baa BBB Adequate, medium-grade

Speculative Grades

Ba BB Judged to have speculative elements

B B Lack the characteristics of a desirable investment

Caa CCC Currently vulnerable, of poor standing

Ca CC Currently highly-vulnerable, marked shortcomings

C C Extremely poor prospects

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Cumulative Default Rates by Bond Rating

Years 1 5 10 15 20

Aaa 0.00% 0.20 1.09 1.89 2.38

Aa 0.08 0.97 3.10 5.61 6.75

A 0.08 1.37 3.61 6.13 7.47

Baa 0.30 3.51 7.92 11.46 13.95

Ba 1.43 10.04 19.05 25.95 30.82

B 4.48 20.89 31.90 39.17 43.70

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Taxes

Investors look at after-tax returns because taxes are a cash outflow for investors.

Investors pay federal taxes on interest received from corporate debt, but not on interest received from municipal debt.

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Taxes (cont.)

Accordingly, municipal bonds can compete with corporate bonds of similar risk while paying lower interest rates.

The formula used to make the comparison is:

r rm u n ic ip a l co rp o ra te 1 in v es to r ' s m arg in a l fed e ra l tax ra te

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Example: Municipal debt interest rate

r rm u n ic ip a l co rp o ra te 1 in v es to r ' s m a rg in a l fed e ra l tax ra te

Assume a AAA corporate bond has a rate equal to 4.65% and an investor has a marginal tax rate of 28%. Then

what rate would a AAA muni have to pay to compete?

3.35%or 0335.028.01*0465.0 municipalr

Marketability (Liquidity)

Less marketable securities are harder to sell, so investors demand higher interest rates.

Long-term debt securities are often assumed to be less marketable than similar short-term debt securities, but we feel it is dangerous assumption to make.

Marketability refers to the speed and cost with which an investor can sell a security.

The costs to consider are:

•Price concession necessary for the sale

•The cost of executing the trade

•Search costs

•Information costs

29

Embedded Options in Debt Contracts

Embedded call option allows the issuing firm (borrower) to retire the debt before maturit

An embedded call provides a valuable right to the borrower and creates additional risk for the investor (lender), so callable debt pays higher interest rates than similar non-callable debt.

Embedded put option allows the investor to sell the bond back to the issuing firm (borrower) before maturity.

An embedded put provides a valuable right to the investor and creates additional risk for the issuer, so putable debt pays lower interest rates than similar non-putable debt.

Embedded conversion option allows the investor to convert the

debt contract into other type of security (typically stock) from the issuing firm (borrower) before maturity.

The investor acquires the valuable right by paying a higher price for the convertible debt than similar non-convertible debt, which results in a lower yield on the convertible debt.

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Required ReturnConventional Bonds

Represented by corporate bonds

DPc,m,t default premium on the corporate (c) security at maturity m.

LPc,m,t liquidity premium on the corporate bond at maturity m

OPc,m,t all other premiums assigned to this security at maturity m

CRPm,t the call risk premium

SPc,mt supply premium for the corporate bond

mt,cmt,ct,m,ct,m,c

t,m,cT

t,mt,mt,mt,mt,rft,m,c

SPCRPOPLP

DPSPIRPIPMRPkk

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Quality SpreadsCorporate Bonds

The quality for any given non-Treasury security is measured as the spread between that security and the Treasury security with the corresponding maturity

Tt,mt,m,ct,m,c kkspread

Substituting the equations for both the Treasury and the corporate bond yields

Ttmmtctmctmctmctmctmc SPCRPOPLPDPSPspread ,,,,,,,,,,,,

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The Magnitude of Instrument-Specific Risk

The magnitude of instrument specific risk faced by securities may be affected by

– Term structure of instrument specific risk

– The business cycle and instrument specific risk

– Industry characteristics instrument specific risk

– Market segmentation

– Event risk

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Credit or Default Risk

0

4

8

12

16

20

55 60 65 70 75 80 85 90 95 00 05 10

10-Year Treasury NoteAAA Corporate Bond RateBAA Corporate Bond Rate

34

The Term Structure of Credit/Default Risk

-1

0

1

2

3

4

5

55 60 65 70 75 80 85 90 95 00 05 10

AAA Less 10-Year Treasury Note3-Mo Commercial Paper Less 3-Mo Treas Bill

35

The Business Cycle and Credit/Default Risk

-1

0

1

2

3

4

5

6

7

55 60 65 70 75 80 85 90 95 00 05 10

AAA Less 10-Year Treas NoteBAA Less 10-Year Treas Note

36

The Business Cycle and Credit/Default Risk

37

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

55 60 65 70 75 80 85 90 95 00 05 10

BAA Less AAA Corp Bond

Industry Risk

2

4

6

8

10

12

14

16

18

55 60 65 70 75 80 85 90 95 00 05

Industria Bond YieldsUtility Bond Yields

38

Sovereign RiskAn Illustration

0

4

8

12

16

20

55 60 65 70 75 80 85 90 95 00 05 10

3-Mo EuroDollar3-Mo CD

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

55 60 65 70 75 80 85 90 95 00 05 10

3-Mo ED Less 3-Mo CD

Sp

rea

d

39

Sovereign RiskContinued

Care must be exercised in measuring sovereign risk

–Previous example used instruments denominated in dollars held in US and in the UK

Same conclusions cannot be reached using yields ob assets expressed in different currencies

–Difference factor other than sovereign risk

–See illustration in right panel bloomberg.com August 18, 2014

investing.com August 19, 2014

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  U.S. Germany Brazil

3-Month 0.02% 0.01%  

6-Month 0.05% 0.01%  

12-Month 0.08% 0.02% 11.10%

2-Year 0.42% 0.00% 11.47%

3-Year   0.00% 11.44%

5-Year 1.54% 0.22% 11.53%

7-Year   0.49%  

10-Year 2.36% 0.99% 11.65%

30-Year 3.18% 1.84% 

LiquidityThe TED Spread

0

1

2

3

4

5

6

55 60 65 70 75 80 85 90 95 00 05 10

3-Mo Euro Dollar Deposit Rate Less 3-Mo Treas Bill Yield

41

Tax Exempt SecuritiesImplicit Tax Rate

20

24

28

32

36

40

44

55 60 65 70 75 80 85 90 95 00 05 10

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