1. Not To Be Naïve about Duration
The duration D we have been discussing also known as Macaulay duration.
First derivative (slope) of price-yield curve is
and is known as modified duration. Found in %ΔPB formula.
Convexity is second derivative (curvature) of price-yield curve. Is a complicated expression (not studied here).
10/29

2. Managing Interest Rate Risk
Duration is the holding period for which reinvestment risk exactly offsets price risk. Designed to give investor the YTM that was in effect at time bond purchased.
A way duration is used: If have a $5 million liability 7.5 years from now, buy a bond (or a portfolio of bonds) today that has a duration of 7.5 years. Then sell at the 7.5 year mark.

3. Example 1: Rebalancing Bond Portfolio
Consider the $20,000 portfolio ($4,000 in D = 5, $10,000 in D = 7, $6,000 in D = 9).
How much in D = 9 bonds should be sold, and how much in D = 5 bonds should be purchased, to reduce Portfolio D to 6.80?

4. Eliminating Interest Rate Risk
Zero-coupon approach (best way). Buy high quality “zeros” with maturity equal to desired holding period. Locks in YTM. No reinvestment risk because no coupons payments, no price risk when held to maturity.
Duration matching (next best way). Selecting a portfolio of bonds whose duration matches desired holding period. Theoretically perfect, but only approximately perfect in real world as per footnote 8 on p. 162.
Maturity matching (don’t use). That is, selecting bonds with terms to maturity equal to desired holding period. Don’t use. Doesn’t work for eliminating interest rate risk.

5. Yield Curve
Yield curve: graphical portrayal of the term structure of interest rates of US Treasuries.
flat, ascending (includes steep and normal)
descending (or inverted) humped

6. Factors Influencing a Bond’s Yield
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)
10/31

7. Example 2: Geometric Average
Over the past 4 years your investment advisor says he grew your money at 10%, 50%, -60%, 40%. Should you be happy?

8. Example 2: Geometric Average
Over the past 4 years your investment advisor says he grew your money at 10%, 50%, -60%, 40%. Should you be happy?
Always true that:
Geometric Average ≤ Arithmetic Average

9. Sep18 U-3 & U-6 Unemployment Rates
Civilian Labor Pool = 160 million
U-3 = 3.6% of civilian labor pool (4.1, 4.8, 5.1, 5.9 in years past). Those without jobs, who are available to work and who have actively sought work in the prior four weeks.
U-6 = 7.1% (8.6, 9.3, 10.0, 11.8 in years past) Includes “marginally attached workers”
neither working nor looking for work, but say they want a job, and
want to work full-time but are working part-time because that is best they can find.

10. (1) Expectation Theory
First of four theories used to explain shape of yield curve is 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.

11. Term Structure Formula
Long-term interest rate is the geometric average of the individual year interest 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)

12. Implied 1-Year Forward Rate Formula
This results in the implied forward rate formula for the n-th period coming up
Example of how to apply:
Want implied yield of a 1-year security that starts 6 years from now.
Look up yields on 7-year security and 6-year security.
Use formula above with n = 7.

13. Example 3: Calculating Forward Rates
Assume today is 2-Nov and that we have the following Treasury security quotes:
yrs to maturity
YTM 1
2019
2-Nov
1.8953 2
2020
2.1725 3
2021
2.3770 4
2022
2.5172 5
2023
2.7126
Find the 1-year implied forward rates during nth period (where n = 2,3,4,5)

14. Example 4: 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%
Note, for example: 3th period starts 2 years from now, and ends 3 years from now.

15. (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.

16. (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.

17. (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.

18. 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.

19. Bond Ratings
Fitch, too

20. 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.

21. Default Rates Moody’s 1970-2006 bond default history over 10-yr
holding period, when bond issue initially rated:
Aaa % Aa A
Baa Ba B
Caa-C
History with recent mortgage securities entirely different.

22. 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.

23. 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.

24. 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.