0. Interest Rates and Bond Valuation
Chapter
Seven
Interest Rates and Bond Valuation

1. Chapter Outline Bonds and Bond Valuation More on Bond Features
Bond Ratings
Some Different Types of Bonds
Bond Markets
Inflation and Interest Rates
Determinants of Bond Yields

2. Bond Definitions 7.1
Bond – a long term debt obligation issued by a corporation
Par value (face value) – the amount of money that will be repaid at the maturity date of the bond
Coupon rate – the percentage of the bond’s face value that will be paid in interest every year
Coupon payment – the dollar value of the interest that is paid
Maturity date – the point in time when the bond will be redeemed by the issuer. The investor will receive cash equal to the face value of the bond.
Yield or Yield to maturity – also known as the Internal Rate of Return (IRR) on the bond. It is the return to the investor that is derived from both the coupons received plus any capital gain or loss.

3. Present Value of Cash Flows as Rates Change
The market price of a bond is simply the present value of the bond’s future cash flows
Bonds have two types of future cash flows
Interest annuity (the stream of coupons)
Face value at maturity
When interest rates go up, the market value of the bond will go down
When interest rates go down, the market value of the bond will go up
Therefore, there is an inverse relationship between the interest rate (YTM) and the market value of the bond (price)

4. The Bond-Pricing Equation
Where:
C = the periodic interest paid by the bond (payment)
r = the yield-to-maturity for the bond
t = the number of time periods to maturity
This formalizes the calculations we have been doing.

5. Valuing a Bond with Annual Coupons
Consider a bond with a $1,000 face value, a coupon rate of 10%, paid annually and 5 years to maturity. The yield to maturity is 11%. What is the market value of the bond?
Formula Approach
Calculator Approach
1,000 FV
100 PMT
5 N
11 I
PV $963.04
Remember the sign convention on the calculator. The easy way to remember it with bonds is we pay the PV (-) so that we can receive the PMT (+) and the FV(+).
Slide 6.8 discusses why this bond sells at less than par

6. Valuing a Bond with Annual Coupons
Now consider a second bond, also with a $1,000 face value and a 10% coupon, paid annually, but with 20 years to maturity and a yield to maturity of 8%. What is the price of this bond?
Formula Approach
Calculator Approach
1,000 FV
100 PMT
20 N
8 I
PV $1,196.36

7. Graphical Relationship Between Price and Yield-to-Maturity
The important concept to note is the inverse relationship between price & YTM
Based on a 10 year, $1,000 bond with an 8% coupon
Bond characteristics: Coupon rate = 8% with annual coupons; Par value = $1000; Maturity = 10 years

8. Bond Prices: Relationship Between Coupon and Yield
If YTM = coupon rate, then face value = market price
If YTM > coupon rate, then face value > market price
The bond is selling at a discount, called a discount bond
If YTM < coupon rate, then face value < market price
The bond is selling at a premium, called a premium bond
There are the purely mechanical reasons for these results.
We know that present values decrease as rates increase. Therefore, if we decrease our yield below the coupon, the present value (price) must decrease below par. On the other hand, if we increase our yield above the coupon, the present value (price) must increase above par.
There are more intuitive ways to explain this relationship. Explain that the yield-to-maturity is the interest rate on newly issued debt of the same risk and that debt would be issued so that the coupon = yield.Then, suppose that the coupon rate is 8% and the yield is 9%. Ask the students which bond they would be willing to pay more for. Most will say that they would pay more for the new bond. Since it is priced to sell at $1000, the 8% bond must sell for less than $ The same logic works if the new bond has a yield and coupon less than 8%.
Another way to look at it is that return = “dividend yield” + capital gains yield. The “dividend yield” in this case is just the coupon rate. The capital gains yield has to make up the difference to reach the yield to maturity. Therefore, if the coupon rate is 8% and the YTM is 9%, the capital gains yield must equal 1%. The only way to have a capital gains yield of 1% is if the bond is selling for less than par value. (If price = par, there is no capital gain.)

9. Example – Semiannual Coupons
Most bonds in Canada make coupon payments semi-annually.
Suppose you have an 8% semi-annual pay bond with a face value of $1,000 that matures in 7 years. If the yield is 10%, what is the price of this bond?
Formula Approach
Calculator Approach
1,000 FV
80 ÷ 2= 40 PMT
7 x 2 =14 N
10 ÷ 2 =5 I
PV $901.01

10. Interest Rate Risk Price Risk Reinvestment Rate Risk
Change in price due to a change in interest rates
Long-term bonds have more price risk than short-term bonds
Bonds with low coupons have more price risk than bonds with high coupons
Reinvestment Rate Risk
Uncertainty concerning the interest rate at which future cash flows can be reinvested
Long-term bonds have more reinvestment rate risk than short-term bonds
Bonds with high coupons have more reinvestment rate risk than bonds with low coupons

11. Figure 7.2 – Interest Rate Risk and Time to Maturity
This graph shows the impact on price as the YTM changes for both a one-year bond and a thirty-year bond. Both bonds have a $1,000 face value and a 10% coupon. Note how much more sensitive the long bond is to a change in the YTM.

12. Computing Yield-to-Maturity
Yield-to-maturity is the discount rate implied by the current bond price
Finding the YTM requires trial and error if you do not have a financial calculator and is similar to the process for finding r with an annuity
If you have a financial calculator, enter N, PV, PMT and FV, remembering the sign convention (PMT and FV need to have the same sign, PV the opposite sign)

13. Example – Finding YTM - 950 PV
For example, assume that a ten year, $1,000 bond with a 6% coupon, paid annually, is currently trading at $950 in the market. What is the bond’s yield-to-maturity (YTM)?
Formula Approach
The YTM is the discount rate that makes the equality true in the bond pricing formula.
Solve for it using the function keys on the calculator or trial & error if using algebra
Calculator Approach
1,000 FV
60 PMT PV
10 N
I %

14. Example – Finding the YTM
Consider another bond with a 10% annual coupon rate, 15 years to maturity and a face value of $1000. The current price is $
Will the yield-to-maturity be more or less than 10%?
Calculator Approach
1,000 FV
100 PMT PV
15 N
I 11%
The students should be able to recognize that the YTM is more than the coupon since the price is less than par.

15. Understanding YTM
The YTM can always be decomposed into its two component parts. These are:
Coupon Yield
Capital gain or loss
Assume that you buy a 10 year, $1,000 bond with an 8% coupon, priced to yield 6%.
Step #1: First calculate the price of the bond.
Calculator Approach
1,000 FV
80 PMT
10 N
6 I
PV $1,147.20

16. Understanding YTM
In Step #2, we calculate the new price after one year has passed, assuming all else remains equal.
Step #3: Calculate the coupon yield by dividing the annual coupon payment by the beginning price
Calculator Approach
1,000 FV
80 PMT
9 N
6 I
PV $1,136.03

17. Understanding YTM Step #4: Calculate the capital gain or loss
Step #5: Calculate the Yield to Maturity

18. YTM with Semiannual Coupons
Suppose a 20 year, $1,000 bond with a 10% coupon, paid semi-annually, is selling for $
Is the YTM more or less than 10%?
What is the semiannual coupon payment?
How many periods are there?
Since the coupon is paid twice a year, the rate you calculate is a semi-annual yield. Double it to obtain the quoted YTM.
Calculator Approach
1,000 FV
100 ÷ 2 =50 PMT
- 1, PV
20 x 2 =40 N
I 4 x 2 = 8%

19. Bond Pricing Theorems
Bonds of similar risk (and maturity) will be priced to yield about the same return, regardless of the coupon rate
If you know the price of one bond, you can estimate its YTM and use that to find the price of the second bond
This is a useful concept that can be transferred to valuing assets other than bonds

20. Differences Between Debt and Equity 7.2
Not an ownership interest
Bondholders do not have voting rights
Interest is considered a cost of doing business and is tax deductible
Bondholders have legal recourse if interest or principal payments are missed
Excess debt can lead to financial distress and bankruptcy
Equity
Ownership interest
Common shareholders vote for the board of directors and other issues
Dividends are not considered a cost of doing business and are not tax deductible
Dividends are not a liability of the firm and shareholders have no legal recourse if dividends are not paid
An all equity firm can not go bankrupt

21. The Bond Indenture
Contract between the company (the issuer of the bond) and the bondholders and includes
The basic terms of the bonds
The total amount of bonds issued
A description of property used as security, if applicable
Sinking fund provisions
Call provisions
Details of protective covenants

22. Bond Classifications Registered vs. Bearer Bonds Security Seniority
Registered – can only be sold by the registered owner
Bearer – similar to cash; possession implies ownership
Security
Collateral – secured by financial securities & assets pledged as a secondary source of repayment
Mortgage – secured by real property, normally land or buildings
Debentures – unsecured debt with original maturity of 10 years or more
Notes – unsecured debt with original maturity less than 10 years
Seniority
Senior versus Junior debt – refers to preference in position with respect to other lenders (Senior debt is paid first; junior debt paid last)
Subordinated debt – indicates that it has a lower priority than other, more senior debt
This is standard terminology in the US – but it may not transfer to other countries. For example, debentures are secured debt in the United Kingdom

23. Bond Classifications Continued
Repayment
Sinking Fund – Account managed by the bond trustee for early bond redemption. Reduces default risk & improves marketability.
Call Provision – allows the company to repurchase all or a portion of the issue prior to the original maturity
Call premium – amount above the face value the borrower agrees to pay, should they call the bond before its original maturity
Deferred call -
Call protected -
Canada plus call – Protects the investor against a call by providing compensation equal to the foregone interest, should a call occur
The issuer usually cannot call the bond during the years immediately after the issue date.

24. Covenants Protective Covenants
Negative covenants – things the borrower agrees not to do
Agrees to limit the amount of dividends paid
Agree not to pledge assets to other lenders
Agree not to merge with, sell to or acquire another firm
Agree not to buy new capital assets above $x in value
Agree not to issue new debt
Positive covenants – things the borrower agrees to do
Maintain a minimum current ratio
Provide audited financial statements
Maintain collateral in good condition

25. Bond Characteristics and Required Returns
The coupon rate depends on the risk characteristics of the bond when issued
Debenture: secured debt is less risky because the income from the security is used to pay it off first
Subordinated debenture: will be paid after the senior debt
Bond without sinking fund: company has to come up with substantial cash at maturity to retire debt and this is riskier than systematic retirement of debt through time
Callable – bondholders bear the risk of the bond being called early, usually when rates are lower. They don’t receive all of the expected coupons and they have to reinvest at lower rates.

26. Credit Ratings: Investment Grade
Credit Risk
Moody’s
Standard & Poors
Fitch
Duff & Phelps
Highest quality
Aaa
AAA
High quality
(very strong) Aa AA
Upper Medium (strong) A
Medium grade
Baa
BBB

27. Credit Ratings: Speculative or Junk Debt
Credit Risk
Moody’s
Standard & Poors
Fitch
Duff & Phelps
Lower Medium Ba BB
Low grade (Speculative) B
Poor Quality
Caa
CCC
Most speculative Ca CC

28. Standard & Poor's Bond Rating Scale
AAA - An obligor rated 'AAA' has EXTREMELY STRONG capacity to meet its financial commitments.
AA - An obligor rated 'AA' has VERY STRONG capacity to meet its financial commitments. It differs from the highest rated obligors only in small degree.
A - An obligor rated 'A' has STRONG capacity to meet its financial commitments but is somewhat more susceptible to the adverse effects of changes in circumstances and economic conditions than obligors in higher-rated categories.
BBB - An obligor rated 'BBB' has ADEQUATE capacity to meet its financial commitments. However, adverse economic conditions or changing circumstances are more likely to lead to a weakened capacity of the obligor to meet its financial commitments.
BB - An obligor rated 'BB' is LESS VULNERABLE in the near term than other lower-rated obligors. However, it faces major ongoing uncertainties and exposure to adverse business, financial, or economic conditions which could lead to the obligor's inadequate capacity to meet its financial commitments.
B - An obligor rated 'B' is MORE VULNERABLE than the obligors rated 'BB', but the obligor currently has the capacity to meet its financial commitments. Adverse business, financial, or economic conditions will likely impair the obligor's capacity or willingness to meet its financial commitments.
CCC - An obligor rated 'CCC' is CURRENTLY VULNERABLE, and is dependent upon favorable business, financial, and economic conditions to meet its financial commitments.
CC - An obligor rated 'CC' is CURRENTLY HIGHLY VULNERABLE.
R - An obligor rated 'R' is under regulatory supervision owing to its financial condition. During the pendency of the regulatory supervision the regulators may have the power to favor one class of obligations over others or pay some obligations and not others. Please see Standard & Poor's issue credit ratings for a more detailed description of the effects of regulatory supervision on specific issues or classes of obligations.
SD and D - An obligor rated 'SD' (Selective Default) or 'D' has failed to pay one or more of its financial obligations (rated or unrated) when it came due. A 'D' rating is assigned when Standard & Poor's believes that the default will be a general default and that the obligor will fail to pay all or substantially all of its obligations as they come due. An 'SD' rating is assigned when Standard & Poor's believes that the obligor has selectively defaulted on a specific issue or class of obligations but it will continue to meet its payment obligations on other issues or classes of obligations in a timely manner. Please see Standard & Poor's issue credit ratings for a more detailed description of the effects of a default on specific issues or classes of obligations.
Note: Obligors rated 'BB', 'B', 'CCC', and 'CC' are regarded as having significant speculative characteristics. 'BB' indicates the least degree of speculation and 'CC' the highest. While such obligors will likely have some quality and protective characteristics, these may be outweighed by large uncertainties or major exposures to adverse conditions. Plus (+) or minus (?): Ratings from 'AA' to 'CCC' may be modified by the addition of a plus or minus sign to show relative standing within the major rating categories.
Source: Standard & Poors

29. Bond Ratings – Investment Quality
High Grade
DBRS’s AAA – capacity to pay is exceptionally strong
DBRS’s AA – capacity to pay is very strong
Medium Grade
DBRS’s A – capacity to pay is strong, but more susceptible to changes in circumstances
DBRS’s BBB – capacity to pay is adequate, adverse conditions will have more impact on the firm’s ability to pay

30. Bond Ratings - Speculative
Low Grade
DBRS’s BB, B, CCC, CC
Considered speculative with respect to capacity to pay.
Very Low Grade
DBRS’s C – bonds are in immediate danger of default
DBRS’s D – in default

31. Stripped or Zero-Coupon Bonds 7.4
Make no periodic interest payments (coupon rate = 0%)
All cash flows occur on the maturity date
Sometimes called zeroes, or deep discount bonds
Bondholder must pay taxes on accrued interest every year, even though no interest is received (thus are best held in a tax-deferred account, such as an RRSP)
Market price is the PV of the face value at maturity

32. Zero Coupon or Stripped Bonds
Assume that you want to purchase a 30 year stripped bond with a $100,000 face value. If the appropriate YTM is 8%, how much will you have to pay today to buy this bond?
Calculator Approach
100,000 FV
0 PMT
30 N
8 I
PV $9,937.73

33. Floating Rate Bonds Coupon rate floats depending on some index value
There is less price risk with floating rate bonds
The coupon floats, so it is less likely to differ substantially from the yield-to-maturity
Coupons may have a “collar” – the rate cannot go above a specified “ceiling” or below a specified “floor”

34. Other Bond Types
Disaster (CAT) bonds – payout to the investor is dependent on the occurrence of some major catastrophic event
Income bonds – coupons are tied to firm profitability
Convertible bonds – bonds may be converted into common stock
Real Return bonds – the bond is adjusted for inflation
Put bond (retractable bond) – the investor may sell the bond back to the issuer at a fixed price
LYON (Liquid Yield Option Note) - created by Merrill Lynch. Bond is a callable, puttable, convertible, zero coupon, subordinated note
It is a useful exercise to ask the students if these bonds will tend to have higher or lower required returns compared to bonds without these specific provisions.
Disaster bonds – issued by property and casualty companies. Pay interest and principal as usual unless claims reach a certain threshold for a single disaster. At that point, bondholders may lose all remaining payments Higher required return
Income bonds – coupon payments depend on level of corporate income. If earnings are not enough to cover the interest payment, it is not owed. Higher required return
Convertible bonds – bonds can be converted into shares of common stock at the bondholders discretion Lower required return
Put bond – bondholder can force the company to buy the bond back prior to maturity Lower required return

35. Bond Markets 7.5
Primarily over-the-counter transactions with dealers connected electronically
Extremely large number of bond issues, but generally low daily volume in single issues
Makes getting up-to-date prices difficult, particularly on small corporate issues. One alternative for the retail investor is
Treasury securities are an exception

36. Bond Quotations From the Financial Post, October 20, 2005
Canada Jun 01/
The issuer is the Government of Canada
The coupon rate is 10% (assumed to be semiannual)
The maturity date is June 1, 2008
The quoted price can be interpreted as either the price per $100 of face value or as a percentage of face value
The yield to maturity is 3.55%
Issuer
Coupon
Maturity
Price
YTM
It is useful to have the students look at Figure 6.3 or have them look at a current issue of the Wall Street Journal. You can generate some interesting discussion as they see not only the quote you are looking at, but all of the ones that surround it.
Company: AT&T
Coupon rate: 6%; coupon payment per year = $60
Bond matures in 2009
Current yield = 6.4%; computed as annual coupon divided by current price
Bonds traded: 177
Quoted price: 93 7/8% of face value, so if face value is 1000, the price is $ It is important to emphasize that bond prices are quoted as a percent of par, just as the coupon is quoted as a percent of par.
Price change: increase by ¼ percent, so the dollar change is .0025(1000) = $2.50

37. Inflation and Interest Rates 7.6
Nominal rate of interest – quoted rate of interest, includes compensation for deferring consumption and expected inflation
Real rate of interest – compensation for deferring consumption
Expected Inflation – the expected fall in purchasing power of the dollar, due to rising prices
Be sure to ask the students to define inflation to make sure they understand what it is.

38. The Fisher Effect
The Fisher Effect defines the relationship between real rates, nominal rates and inflation
Exact relationship
Where:
R = the nominal interest rate
r = the real interest rate
h = the expected future inflation rate
Approximation of the above relationship is:
The approximation works pretty well with “normal” real rates of interest and expected inflation. If the expected inflation rate is high, then there can be a substantial difference.

39. Example – Fisher Effect
If we require a 4% real return and we expect inflation to be 6%, what is the nominal rate?
Therefore, the nominal rate is 10.24%
If both inflation and the real return are low, we can safely use the approximation, which would give us a nominal rate of 10%

40. Term Structure of Interest Rates 7.7
Term structure is the relationship between time to maturity and yield-to-maturity, all else equal
The term structure is always derived using government bonds, as they are the only issuer with sufficient bonds in all maturities with the identical default risk (zero).
The terms yield curve and term structure may be used interchangeably. Both depict a graphical representation of the relationship between term and yield
The term structure typically has one of four shapes
Upward-sloping - long-term yields are higher than short-term yields
Downward-sloping - long-term yields are lower than short-term yields
Flat – yields of all maturities are the same

41. Figure 7.4 – Upward-Sloping Yield Curve

42. Figure 7.4 – Downward-Sloping Yield Curve

43. Figure 7.5 – Government of Canada Yield Curve November 29, 2002
www: Click on the web surfer to go to Bloomberg to get the current Treasury yield curve

44. Factors Affecting Required Return
Default risk – the probability that the issuer will not be able to repay the bond as contractually obligated to do (bond rating)
Liquidity premium – liquidity refers to the ability to:
Convert to cash
At or near face value
Short bonds with high coupons that are more frequently traded have greater liquidity & hence a lower required return
Call features – since a call feature allows the bond to be redeemed early, it increases risk to the bond investor
Anything else that affects the risk of the cash flows to the bondholders, will affect the required returns