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Yield Curve

flat descending (or inverted)ascending (includes steep and normal) humped

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Definitions

• Term structure of interest rates: relationship between the yields on bonds and their terms to maturity.

• Yield curve: graphical portrayal of the term structure of US Treasuries.

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Factors Influencing Bond Yields

• General level of interest rates• Default risk• Term to maturity• Tax treatment• Marketability• Call or Put features

– Call: issuer can retire bond early – Put: holder can retire bond early

• Convertibility (for instance to stock)

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Example 1: Geometric Average

Over the past 4 years your investment advisor says he grew your money at 10%, 50%, -60%, 40%. Should you be happy?

10 50 60 40Arith Ave 10%

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Example 1: Geometric Average

Over the past 4 years your investment advisor says he grew your money at 10%, 50%, -60%, 40%. Should you be happy?

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10 50 60 40Arith Ave 10%

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Geom Ave (1 .10)(1 .50)(1 .60)(1 .40) 1

(1.1)(1.5)(.4)(1.4) 1

.98043 1

1.967%

Always true that:

Geometric Average ≤ Arithmetic Average

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U-3 and U-6 Unemployment Rates

Civilian Labor Pool = 157 million

U-3 = 5.1% of civilian labor pool (5.9 last year, 7.2 year before). Those without jobs, who are available to work and who have actively sought work in the prior four weeks.

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U-6 = 10.0% (11.8 last year, 13.6 year before) Includes “marginally attached workers” (1) neither working nor looking for work, but say they want

a job, and (2) want to work full-time but are working part-time

because that is best they can find.

11/7

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(1) Expectation Theory

• Shape of yield curve determined by expectations about future rates.

• This theory assumes investors are indifferent between a long-term security and a series of short-term securities.

First of four theories used to explain shape of yield curve is expectation theory:

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Term Structure Formula

0 1 1 1 2 1 1 1 0

01 1

0 1 1 1 2 1

0 1

01 1

0 1 0 1 0 1

1 1 1 1 1

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1 1 1

substituting -1 times in denominator the geometric mean

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1 1 1

n

n n n

n

nn

n

n

n

nn

n n n

R f f f R

Rf

R f f

n R

Rf

R R R

• Long-term interest rates are the geometric average of future period rates.

where: 0Rn observed YTM on n-year bond t fq forward rate on q-year bond that starts at

time t (where t = 0 is now)

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Implied 1-Year Forward Rate Formula

01 1 1

0 1

11

1

n

nn n

n

Rf

R

Example of how to apply:

1. Want implied yield of a 1-year security that starts 6 years from now.

2. Look up yields on 6-year security and 7-year security.

3. Use formula above with n = 7.

This results in the implied forward rate formula for the n-th period coming up

Example 2: Calculating Forward Rates

Assume following Treasury security quotes:yrs to

maturity YTM

1 2016 11-Nov 0.8953

2 2017 11-Nov 1.3725

3 2018 11-Nov 1.8770

4 2019 11-Nov 2.3172

5 2020 11-Nov 2.6626

Find the 1-year implied forward rates during nth period (where n = 2,3,4,5)

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Example 3: Another Example

find the 1-year implied forward rate for the period that begins 2 years from now where

1-year Treasury bill 1.9%2-year Treasury note 2.4%3-year Treasury note 2.7%

01 1 1

0 1

11

1

n

nn n

n

Rf

R

When doing, note for example: 4th period starts 3 years from now, and ends 4 years from now.

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(2) Liquidity Premium Theory Says…

• Long-term securities have greater price risk, and generally less marketability.

• Liquidity premiums contribute to an upward tendency of a yield curve.

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(3) Market Segmentation Theory Says…

• Market participants may have strong preferences for particular maturities, and buy and sell securities consistent with these preferences.

• Can theoretically lead to discontinuities in yield curve.

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(4) Preferred Habitat Theory Says…

• Preferred Habitat Theory (an extension of Market Segmentation Theory) allows market participants to trade outside of their preferred maturities if adequately compensated.

• Preferred Habitat Theory allows for humps in the yield curve.

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Which Theory is Right?

• Each has its point.

• Day-to-day changes in the term structure seem consistent with the Preferred Habitat Theory.

• Many economists also feel that Expectations and Liquidity Premiums are important, too.

• Market Segmentation Theory appears to be least realistic of the four.

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Bond RatingsFitch, too

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NRSROs• Nationally Recognized Statistical Rating Organizations

• NRSROs are credit-rating agencies authorized by the SEC and banking regulators. Currently 10 (best known Moody’s, Standard & Poor’s, Fitch).

• BBB- (Baa3) and above are investment-grade, below are speculative-grade or “junk.”

• Issuers pay to have their bonds rated. Banks, insurance companies, pension funds, many mutual funds can only hold investment-grade bonds.

• As conditions change, rating agencies change their ratings. Bad when an issue’s rating drops below cutoff.

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Default Rates

AAA 0.52%AA 1.31A 2.32BBB 6.64BB 19.52B 35.76CCC 54.38

Non-mortgage bond default history, when initially rated:

History with recent mortgage securities entirely different.

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Default Risk

• Investors require a default risk premium.

• Default risk premiums tend to increase in periods of recession (when people scared) and decrease in periods of economic expansion (when people overconfident).

• “flight-to-quality”

• Bond ratings are only for default risk.

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Call Options, Put Options • Call option permits the issuer to call (refund) an obligation,

under certain conditions, before maturity.

• Issuers will likely “call” in a bond from investors if interest rates decline.

• Difference in yields between callable and noncallable contracts is call interest premium.

• Put option permits a investor to terminate a contract, under certain conditions, before maturity.

• Investors are likely to “put” a bond back to its issuer if interest rates rise.

• Difference in yields between putable and nonputable contracts is put interest discount.

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Conversion Options• Permits the investor to convert a security contract

into another security

• Conversion yield discount is the difference between the yields on convertibles relative to nonconvertibles.