1. Factors Affecting Bond Yields and the Term Structure of Interest Rates
Chapter 5 Pages only
Factors Affecting Bond Yields and the Term Structure of Interest Rates

2. Introduction
We have spent a lot of time discussing the required yield (interest rate) on a bond.
However there is an IMPORTANT point to remember: There is no single market yield. Every bond has it’s own yield.
In this chapter, we learn about the factors that affect bond yields.

3. Two Ways to Get Money
Buy it: Earn it in productive activities (equity).
Rent it: An interest rate is the price of renting money (debt).
Like all other prices, interest rates reflect supply and demand.
That is, the supply and demand for renting money.
There is an interest rate for each rental period.

4. Supply and Demand of Money
Interest rate (r)
Why is D$ downward sloping? r
• If r - less money demanded
• If r - more money demanded
Demand for $$ decreases as rates increase D$
Quantity of money

5. Bond Yield All bond yields can be expressed as:
Our first task is to understand the base rate (also called the benchmark rate)
Then we move to the spread or risk premium.

6. Base Rate
Minimum rate an investor would ever accept for investing in non-Treasury securities.
It is measured as the YTM of a comparable on-the-run Treasury security.
Example: US Treasury yields on October 7, 2005:
Maturity
1-Mo
3-Mo
6-Mo
1-Yr
2-Yr
3-Yr
5-Yr
10-Yr
30-Yr
Yield (%)
3.33
3.61
3.99
4.17
4.19
4.21
4.23
4.36
4.57
“On The Run” means most recently issued and most frequently traded Treasury Security.
Once a new issue is released, the previously issued security becomes the off the run bond or note
On the Run are most liquid, causing them to trade at slight premium
If you want to invest in a 10-year bond you would never accept less than a 4.36% yield.

7. Spread (Risk Premium)
Non-Treasury securities trade at a spread over a similar maturity Treasury security.
The spread is a risk premium that reflects the additional risks an investor faces by buying a bond riskier than a Treasury security.
How do we measure spreads?
Basis points (difference in yields) [most important]
Relative yield spread (% difference between yields)
Yield ratio (one yield divided by another)
Relative Yield Spread = (Ay – By)/By
Yield Ratio = Ay/By
See examples next slide

8. Example of Spreads The current 10-year US Treasury Note yields 3.60%.
A 10-year AAA rated corporate bond yields 4.96%.
What is the spread on the AAA rated bonds?
Basis points: = 1.36% or 136 bps
Relative yield:
Text Page 96
Yield ratio:

9. Factors Affecting Yield Spreads
Type of Issuer (market segments):
Different market segments (and sub-segments) have different ability to satisfy contractual obligations.
e.g., Municipals, corporates, agencies, etc.
Credit Worthiness of Issuer:
Spread between a Treasury and non-Treasury security that are identical in all respects except for credit quality is call a credit spread.
“Identical in all respects” means identical in terms of embedded options, liquidity, taxability, etc.
Otherwise the spread reflects the value of items other than default.
Exhibit 5-2 Page 97 shows yield spreads of corporate vs.Treasury

10. Factors Affecting Yield Spreads
Inclusion of Options (e.g., put and call provisions):
Callable bonds will have ________ spreads over Treasury rates.
Putable bonds will have ________ spreads over Treasury rates.
Expected Liquidity of Issue:
Bonds with lower liquidity trade with higher yields because they are more difficult to sell quickly for a fair price (hence are more risky).
Treasury securities have very high liquidity, although off-the-runs are less liquid than on-the-runs.
Financeability:
Treasury bonds can be used as collateral for loans. The more desirable a particular (usually on-the-run) T-bond is, the lower the rate the lender will charge for the loan. The reduced rate increases the spread between on-the-run and off-the-run Treasuries.
Callable – larger spreads / Putable – smaller spreads – Why?

11. Factors Affecting Yield Spreads
Term to Maturity:
Holding all other bond factors constant, the longer the time until maturity of the bond, the more risky (volatile) it will be when yields change.
Taxability of Interest:
Coupon payments are taxable at federal and state level.
Exception: Municipal bond interest is exempt at the federal level (and state level in some cases).
This means municipal securities will pay lower coupon rate (why?)
Because income from coupons is not taxable

12. Taxability of Bonds
To compare the yields on municipals with yields on taxable bonds, we need to look at after-tax yields:
Example:
A 10-year A-rated corporate bond has a yield of 5.26% and a 10-year A-rated municipal bond has a yield of 3.73%.
If your federal tax rate is 35% which bond would you prefer? (assuming all other features of the bonds are equal).
Answer:
After-tax muni yield is 3.73%.
After-tax corporate yield is 3.42% [= 5.26  (1 – 0.35)]
What if you are in the 15% TAX BRACKET?

13. Taxability of Bonds
We can also determine the yield that must be offered on a taxable bond to give the same after-tax yield as a tax-exempt issue.
This is called the equivalent taxable yield:
From the previous example, what taxable yield would offer the same after-tax yield as 3.73% at 35% tax rate:

14. Term Structure of Interest Rates
Is the relationship between yield and maturity on bonds that are identical in every way except maturity.
The term structure is an important tool in valuing bonds.
A graphical representation of the term structure is called the yield curve.

15. Term Structure of Interest Rates
Notice:
(1) Term structure can change over time.
(2) Short-term rates more volatile than long-term rates. r
Flat
Downward sloping
Upward sloping (most common)
Maturity (term)

16. Where Do We Get A Yield Curve?
It may seem a logical first step to use the YTM from bonds of different maturities:
However, YTM is accurate for yield curve construction only under the following conditions:
(1) Yield curve is flat.
(2) Coupons can be reinvested at a rate equal to YTM.
Otherwise YTM is incorrect. Why?
A bond is a portfolio of zero-coupon bonds.

17. Getting Yields for Yield Curve
A portfolio of zero-coupon bonds:
If each coupon is sold as a zero-coupon bond, then each should be discounted at a different rate (reflecting the maturity of the cash flow)
1 year coupon should be discounted at 1-year rate.
2 year coupon should be discounted at 2-year rate.
…and so on.
Such rates are called spot rates.
These are rates on zero-coupon bonds.

18. What Characteristics Should Spot Rates Have?
Spot rates should reflect the required yield for a single cash flow (i.e., a single maturity).
Therefore, bonds from which spot rates are determined ideally should have no intermediate cash flows.
Be default risk free:
Spot rates should reflect the pure supply and demand of loanable funds, not default risk.
Where do we get these rates?
From risk-free zero-coupon bonds.
The spot yield curve comes from risk-free zero-coupon rates.
To determine the value of each zero coupon instrument, it is necessary to know the yield on a zero coupon treasury with the same maturity
Problem – There are no Zero Coupon Treasury debt issues with a maturity greater than 1 year!!!
Cant get the rates from Market observations….

19. Where Do We Get Risk-Free Zeros?
Treasury coupon strips (logical starting point!):
Coupons are “stripped” from the bond and sold off separately as individual zero-coupon bonds.
Resulting securities are called STRIPS (Separate Trading of Registered Interest and Principal of Securities).
Problems with strips:
Liquidity of strips is less than the liquidity of the original Treasury securities. Thus, yields on strips reflect a liquidity risk premium.
Tax treatment of strips is different from that of original treasury securities. Accrued interest is taxed, creating a negative coupon (strip yields reflect this tax disadvantage).
Example – a 2 year note can be stripped apart to be 4 zeros – maturing in 6 months, 1 year, 1.5 yrs and 2 yrs.

20. Where Else Can We Get Risk-Free Zeros?
On-the-run Treasury issues.
On-the-run issues along with selected off-the-run Treasury issues.
All Treasury coupon security coupons and bills.
Note: most of the securities above are coupon paying bonds, not zero-coupon bonds!
So we will need some techniques to create the spot (zero-coupon) yield curve from coupon-paying bonds.
The resulting yield curve is called the theoretical spot rate curve.

21. On-The-Run Treasury Issues
We need to extract 60 spot (zero) rates from coupon-bearing Treasury bonds.
Why 60? There are 60 semi-annual coupon payments in 30 years.
Complication: There are usually only 6 or so on-the-run Treasuries available.
How do we estimate the remaining 54 yields?
Construct par yield curve – yield curve constructed from 6 coupon-bearing Treasuries assuming the bonds are priced at par (yield equals to the coupon rate). Use linear interpolation to fill in gaps.
“Convert” the par yield curve to theoretical zero coupon curve using a technique called bootstrapping.
Straight interpolation – Par Yield Curve - Draw yield curve with straight lines between points to illustrate
Par yield curve is based on securities with coupons

22. Par Yield Curve Construction
Suppose we have par yields two on-the-run Treasuries:
2-year: 6.0%
5-year: 6.6%
From these par yields we can interpolate the 2½, 3, 3½, 4, and 4½ year par yields using the following formula:
Use interpolation to fill in gaps on Par Yield Curve
Interpolated yields are:
2½ -year: 6.00% = 6.10%
3-year: % = 6.20%
3½-year: 6.20% = 6.30%
4-year: % = 6.40%
4½-year: 6.40% = 6.50%

23. Problems With Interpolation
There are large gaps between the 5-year and 10-year bonds and 10-year and 30-year bonds.
Using such long distances between maturities can reduce the accuracy of interpolation.
On-the-runs may be “special” in that they are desirable as collateral for loans. This can distort yields.
Solution?
In addition to the on-the-runs, use some selected off-the-run Treasuries to help fill in the gaps.
Usually the 20 and 25-year off-the-runs are used.
Show a curved line vs. the straight interpolation

24. Bootstrapping
Bootstrapping enables us to take (par) yields from coupon bonds and generate a spot yield curve.
Bootstrapping uses the concept that a coupon-bearing bond is a portfolio of zero-coupon bonds and should be priced accordingly.
Best illustrated by an example.
Convert a coupon paying security to a Zero – to get the Zero or Spot yield curve

25. Example of Bootstrapping
Suppose we have the following on-the-run Treasury securities (coupons paid semi-annually):
All bonds have a face value of $100.
The first 2 bonds are zero coupon bonds (why?).
Our goal: Extract zero-coupon yields.

26. Solution
Since the first 2 bonds are zero-coupon bonds (i.e., T-bills) their par rates are spot rates.
The next spot rate we need is the 1½ year rate:
Since it has a 5.75% coupon rate, the coupon on this bond should be $2.875 every six months.
The value of this bond is $100 since it is based on par yields.
STOP HERE - Example also on Page 106 – Check Math!!!!!
Solving for z3 we get:
Double the yield to get an annual spot yield: (or 5.76%).

27. Solution Now we find the spot yield for the 2-year maturity:
Since it has a 6.0% coupon rate, the coupon on this bond should be $3.00 every six months.
So, we can now find z4:
Solving for z4 we get:
Double the yield to get an annual spot yield: (or 6.02%).

28. Comments On Bootstrapping
This process continues until the entire spot yield curve is constructed.
The bootstrapped yields are yields the market would apply to zero-coupon Treasury bonds, if such securities existed.

29. All Treasury Securities
Using only on-the-run issues (even with selected off-the-run issues) fails to recognize all the information contained in Treasury security prices.
Some argue that all Treasury securities and T-bills be used to construct the theoretical spot yield curve.
If all securities are used, methodologies other than bootstrapping must be used because there may be more than one yield for each maturity:
The most common methodology is exponential spline fitting.

30. Revisiting the Theoretical Spot Curve
Let’s return to the base rate in the formula:
Earlier, we said the base rate came from the YTM of an on-the-run Treasury security.
We have to modify that explanation.
The base rate comes from the theoretical spot rate curve that we just learned to construct.

31. Forward Interest Rates
Borrowers and lenders often enter into agreements to make loans in the future.
This creates a need for a forward interest rate.
Forward rates are interest rates implied by current spot rates of interest.
A forward rate is often viewed as the market’s consensus for future interest rates.

32. Example of Forward Rates
Consider the following zero rates:
zT = The current T-period spot (zero) interest rate (i.e., annual interest rate that prevails from T0 to time T).

33. Forward Interest Rates
Consider the following two strategies:
(1) Invest $100 at 2.75% for 1 year (i.e., 2 six-month periods:
$100(1.0275)2 = $
(2) Invest $100 at 2.625% for 6 months and then reinvest the funds for one 6 more months at the prevailing rate:
At the end of 6 months we would have: $100( ) = $
At the end of 1 year we would have: $ (1 + f) = ?
We don’t know what f is (it’s a forward rate). However…
These must be equal
100(1.0275)2 = 100( )(1 + f):
Solving for f we get: f = 2.875%
To annualize, multiply: 5.75%
5.75% is the market’s consensus for the six-month rate six months from now.

34. Generalizing Our Result…
The relationship between the t-period spot rate, current six-month spot rate, and the six-month forward rates is:
where ft is the 6-month forward rate beginning t 6-month periods from now.
t = 0
t = 1
t = 2
t = 3
t = 4 z1 f1 f2 f3

35. Example Suppose we have the following rates: (or 5.02%) z1 = 0.0250
We can see how the 2-year spot rate is related to the various six-month forward rates:
(or 5.02%)

36. Another Generalization
Suppose given spot rates we want to find the forward rate.
We use the following formula:
Where:
zt = the annualized t-period spot rate.
zt+n = the annualized (t+n)-period spot rate.
ft,n = the an n-period forward rate beginning in period t.
Best to illustrate by example.

37. Another Example
Suppose we have the following annualized semi-annual yields:
Year
Period
Yield
0.50 1
3.0%
1.00 2
3.4
1.50 3
3.8
2.00 4
4.2
2.50 5
4.8
3.00 6
5.4
3.50 7
5.8
4.00 8
6.4
4.50 9
6.8
5.00 10
7.2
What is the two-year forward rate on a three-year bond?
That is, in two years, what will be the expected yield on a three-year bond?
We use the following formula:
Homework Problem Formula
(or 9.22%)