1. Chapter 4 Valuing Bonds
Professor Thomson
Fin 3013

2. Quote from Gyourko (UPenn)
"Basically we've been growing rich people in the U.S.," he said. In 1940, less than 1% of families earned $100,000 in today's dollars. By 1970 that income level applied to about 5% of families. By 2000 it applied to about 12% of families.
From Knowledge at Wharton

3. Bonds Bonds are debt instruments (i.e. loans)
Government Bonds include:
US Treasury
Federal Agency
Municipal (Government bonds issued by non Federal entities such as States and Local Governments
Corporate Bonds – are private Bonds

4. Bond Markets
The U.S bond market has grown from $250 billion in 1950 to $22 trillion in 2004
-Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity.
The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate.
When P0  Par, the YTM will differ from the coupon rate.
YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM
Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43

5. Bonds
Bonds typically make “interest only” payments, and repay the amount borrowed at maturity.
New bonds may be issue to repay an issue that is maturing.

6. Par Value or Face Value
The Par or Face Value is the amount that will be refunded at the maturity date.
The par value is usually $1000 (especially for corporate bonds).
Assume $1000 par value unless another value is provided.

7. Coupon Rate
The coupon rate is the annual rate of interest (APR) paid on the bonds par value. E.g % or 6 1/8
Bond coupons historically were in 1/8% increments, but that is becoming less common

8. Coupon Payment
The coupon payment is the periodic interest payment made to bond holders.
Payments per year, P/YR, is the number of coupon payments per year
P/YR = 2 for a typical (semiannual) bond
If P/YR not stated, assume P/YR=2

9. Show bond quote from WSJ

10. Example 4.1: Semi Annual Coupon Bond
A 6% semi annual coupon bond has 10 years until maturity. The market (discount) rate is 5%. What is its price?
What would be the price of a similar annual coupon bond?

11. Example 4.2:
Consider a 7% semiannual coupon bond with 28 years until maturity. The market rate = 8%. What is the price of the bond?

12. Bond Price depends on the Market rate (or discount rate or YTM)
Example 4.3: Consider 2 Bonds
A: 7% Bond with 28 years to maturity
B: 7% Bond with 4 years to maturity
Compute the market price of these bonds as the market rate changes from 3-10%

13. Example 4.4: What is the YTM of the Wachovia Bond?
The Wachovia Capital trust bond with a coupon which matures Mar 15, 2011 has a price quote of What is its YTM? (Assume today is March 16, 2006)

14. Example 4.5 Bond Prices
You purchased a 5% semi annual coupon bond, with 10 years until maturity, one year ago when the market rate was 7%. The market rate is now 6%. What price did you pay for your bond, and what could you sell it for today?

15. Inflation
What matters is not how many $ you have, but what you can purchase with the $ you have.
Measures of inflation compute how many $ it takes over time to purchase the same basket of goods

16. Inflation: Some Notation
NR = Nominal Rate, which is the quoted rate or yield on a bond
RR = Real Rate, which measures your increase in purchasing power over time
IN = INflation rate, which is the rate at which the cost of goods goes up

17. Fisher Effect
The Fisher effect shows the relationship among the nominal rate [NR], the inflation rate, [IN], and the real rate [RR].
Exact relationship is:
(1+NR) = (1+IN)*(1+RR)
Approximate relationship (works best if inflation rate is low) is:
NR = IN+RR

18. Example 4.5: Calculating the real rate of return on a Treasury Bill
If Treasury bills are currently paying 5 percent and the inflation rate is 3.4 percent, what is the approximate real rate of interest? The exact real rate?

19. What affects Bond Yields?
Inflation – the higher the inflation rate, the higher the bond yield (bond yields were much higher in early 1980’s than today because inflation was higher)
Default risk – Riskier bonds have higher yields (Thus, Treasury bonds have lower yields than corporate bonds, and investment grade bonds have lower yields than “junk bonds”

20. Investment-grade bonds
Bond Ratings
Bond ratings: grades assigned to bond issues based on degree of default risk
Investment-grade bonds
Moody’s Aaa to Baa3 ratings
S&P and Fitch AAA to BBB- ratings
-Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity.
The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate.
When P0  Par, the YTM will differ from the coupon rate.
YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM
Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43
Junk bonds
Moody’s Ba1 to Caa1 or lower
S&P and Fitch BB to CCC+ or lower

21. Bond Yields (continued)
Liquidity – highly liquid bonds such as Treasuries have lower yields than safe, but less traded bonds.
Taxes – Bonds that are tax free (most Municipal Bonds) have lower yields than taxable bonds

22. Bond Yields (continued)
Maturity – Other things equal, bonds with a longer time until maturity usually have a higher yield than bonds near their maturity date (when this is not true, we say we have an inverted yield curve)
The Treasury Yield Curve plots the yield on Treasury Bonds versus their maturity, and it is typically upward sloping. (see Wall Street Journal)

23. Term Structure of Interest Rates
Relationship between yield and maturity is called the Term Structure of Interest Rates
Graphical depiction called a Yield Curve
Usually, yields on long-term securities are higher than on short-term securities.
Generally look at risk-free Treasury debt securities
Yield curves normally upwards-sloping
Long yields > short yields
Can be flat or even inverted during times of financial stress
The relationship between nominal (observed) and real (inflation-adjusted) interest rates and expected inflation called the Fisher Effect (or Fisher Equation)
Fisher said the nominal rate (r) is approximately equal to the real rate of interest (a) plus a premium for expected inflation (i).
If real rate equals 3% (a = 0.03) and expected inflation equals 2% (i = 0.02):
r  a + i   0.05  5%
The true Fisher Effect is multiplicative, rather than additive:
(1+r) = (1+a)(1+i) = (1.03)(1.02) = ; so r = 5.06%
What do you think a Yield Curve would look like
graphically?

24. Yield Curves U.S. Treasury Securities16
May 1981 14 12 10
Interest Rate %
January 1995 8
August 1996 6
October 1993 4 2 1 3 5 10 15 20 30
Years to Maturity

25. See Smart Animation

26. Example 4.7
Example Zero coupon Treasury strips with 3 years until maturity have a yield of 5%, while similar 2-year strips yield 6%. According to the expectations theory, what yield will one year strips have two years from now?

27. Bond Indenture Are “the rules” I.e. is the bond contract
It states the provisions of the bond including the coupon rate and maturity date

28. Possible Bond Provisions
Call Provision
Allows bond to be refunded (I.e. called) prior to its maturity date
Put Provision
Allows the bond purchaser to sell the bond back to the issuer prior to its maturity date

29. Possible Bond Provisions
Convertible Bond
Allows the bond to be swapped for stock
This is an “option” as it gives you the right but not the obligation to do the swap

30. Covenants to Protect Bondholders
May have a limit on the dividends the firm is allowed to pay, to keep cash in the firm for paying bondholders coupons
Seniority Rule
Most senior (I.e. first issued) have their coupon paid before junior issues, and has first claim on firm assets in default

31. Covenants to Protect Bondholders
Sinking Fund – various ways to ensure that the bond will be repaid at maturity, such as making deposits into a dedicated account for this purpose, or repurchasing some bonds over time
Collateral – may be required to pledge certain assets of the firm as surity
Debenture – a bond where no collateral is pledged. Most bonds are debentures

32. Zero Coupon Bonds
These bonds are bought at a deep discount and redeemed for par
A disadvantage is the you must pay taxes each year on the imputed interest earned, even though no coupon interest is paid to you.
More popular for tax sheltered investments such as IRA or insurance

33. Other Bond Types
Variable Coupon – The coupon paid each period depends on an interest rate index
For example, the coupon on Series EE savings bond is set to 90% of the yield on 5 year Treasury Bonds
TIPS (Treasury Inflation Protected Securities)
These Treasury Bonds pay a real coupon rate, on a Face Value that adjusts with inflation thus providing a real rate of return for investors

34. End of lecture notes
The slides which follow are alternate ways to state what has been covered thus far, and some additional information for those that may be interested.

36. Corporate Bond Quotations
Company
(Ticker)
Coupon
Maturity
Last Price
Last Yield
Estimated Spread
UST
Est $ Vol (000s)
SBC Comm
(SBC)
5.875
Aug 15,2012
4.836 80 10
73,867
Corporate prices are quoted as percentage of par, without the 32nds of a dollar quoting convention
-Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity.
The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate.
When P0  Par, the YTM will differ from the coupon rate.
YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM
Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43
Yield spread: the difference in yield-to-maturities between a corporate bond and a Treasury bond with same maturity
The greater the default risk, the higher
the yield spread

37. U.S. Treasury Bond Quotations
RATE
MATURITY
MO/YR
BID
ASKED
CHG
ASK
YLD
Government Bonds & Notes
5.500
May 09n
107:13
107:14 3
3.83
Rate
Coupon rate of 5.5%
Bid prices
Ask prices
(percentage of par value)
Bid price: the price traders receive if they sell a bond to the dealer. Quoted in increments of 32nds of a dollar
Ask price: the price traders pay to the dealer to buy a bond
Bid-ask spread: difference between ask and bid prices.
-Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity.
The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate.
When P0  Par, the YTM will differ from the coupon rate.
YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM
Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43
Ask Yield
Yield to maturity on the ask price

38. Valuation Fundamentals
Present Value of Future Cash Flows
Link Risk & Return
Expected Return
on Assets
Valuation

39. The Basic Valuation Model
P0 = Price of asset at time 0 (today)
CFt = Cash flow expected at time t
r = Discount rate (reflecting asset’s risk)
n = Number of discounting periods (usually years)
This model can express the price of any asset at
t = 0 mathematically.
Value of any financial asset is the PV of future cash flows
Bonds: PV of promised interest & principal payments
Stocks: PV of all future dividends
Patents, trademarks: PV of future royalties
Valuation is the process linking risk & return
Output of process is asset’s expected market price
A key input is the required [expected] return on an asset
Defined as the return an arms-length investor would require for an asset of equivalent risk
Debt securities: risk-free rate plus risk premium(s)
Required return for stocks found using CAPM or other asset pricing model
Beta determines risk premium: higher beta, higher reqd return
Marginal benefit of owning the asset: right to receive the cash flows
Marginal cost: opportunity cost of owning the asset

40. Valuation Fundamentals: Example
Company issues a 5% coupon interest rate, 10‑year instrument with a $1,000 par value
Assume annual interest payments
Investors in company’s financial instrument receive the contractual rights
$50 coupon interest paid at the end of each year
$1,000 principal at the end of the 10th year
The simplest debt instruments to value are U.S. Treasury securities since there is no default risk.
Instead, the discount rate to use, r, is the pure cost of borrowing.
Assume you are asked to value two Treasury securities, when rf is 1.75 percent (r = ):
A (pure discount) Treasury bill with a $1,000 face value that matures in three months, and
A 1.75% coupon rate Treasury note, also with a $1,000 face value, that matures in three years.
For the T-Bill, three months is one-quarter year (n=0.25)
For 3-year bond, n = 3
Most U.S. corporate bonds:
Pay interest at a fixed coupon interest rate
Have an initial maturity of 10 to 30 years, and
Have a par value (also called face or principal value) of $1,000 that must be repaid at maturity.
Bond’s value has two separable parts:
(1) PV of stream of annual interest payments, t=1 to t=10
(2) PV of principal repayment at end of year 10.
Can thus also value bond as the PV of an annuity plus the PV of a single cash flow using PVFA and PVF from tables.
P0 = C x (PVFA5%,10yr) + Par x (PVF5%,10yr)
= $50 (7.7220) + $1,000 (0.6139) = $1,000.00
Bonds with a few cash flows can be valued with Eq 4.1; for bonds with many cash flows, use PVFA/PVF factors, calculator or Excel.

41. Yield to Maturity (YTM)
Estimate of return investors earn if they buy
the bond at P0 and hold it until maturity
The YTM on a bond selling at par will always equal
the coupon rate.
-Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity.
The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate.
When P0  Par, the YTM will differ from the coupon rate.
YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM
Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43
YTM is the discount rate that equates the
PV of a bond’s cash flows with its price.

42. Bond Premiums and Discounts
What happens to bond values if required return is not equal to the coupon rate?
The bond's price will differ from its par value
P0 < par value
r > Coupon Interest Rate
DISCOUNT =
When r is greater than the coupon interest rate, P0 will be less than par value, and the bond will sell at a discount
For Sun, if r >5%, P0 will be less than $1,000
For practice:
Value Sun Company, 10-year, 5% coupon rate bond if required return, r =6% and again if r = 4%.
Premiums & discounts change systematically as r changes.
P0 > par value
r < Coupon Interest Rate
PREMIUM =

43. Semi-Annual Interest Payments
An example....
Value a T-Bond
Par value = $1,000
Maturity = 2 years
Coupon rate = 4%
r = 4.4% per year
= $992.43
Most bonds pay interest semi-annually rather than annually
Can easily modify basic valuation formula; divide both coupon payment (C) and discount rate (r) by 2, as in Eq 4.3:
In Eq 4.3, C is the annual coupon payment, so C/2 is the semi-annual payment.
r is the annual required return, so r/2 is the semi-annual discount rate.
n is the number of years, so there are 2n semi-annual payments.

44. Factors that Affect Bond Prices
Time to maturity: bond prices converge to par value (plus final coupon) with passage of time.
Interest rates: bond prices and interest rates move in opposite directions.
-Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity.
The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate.
When P0  Par, the YTM will differ from the coupon rate.
YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM
Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43
Changes in interest rates have larger impact on long-term bonds than on short-term bonds.

45. Interest Rate Risk What does this tell you about the relationship
Whenever r is different from the coupon interest rate, the time to maturity affects bond value (even if the required return remains constant until maturity)
The shorter is n, the less responsive is P0 to changes in r. Assume r falls from 5% to 4%
For n=8 years, P0 rises from $1,000 to $1,067.33, or 6.73%
For n=3 years, P0 rises from $1,000 to $1,027.75, or 2.775%
Same relationship if r rises from 5% to 6%, though percentage declines in price less than increases (maximum decline is 100%, increase unlimited)
For n=8 years, P0 falls from $1,000 to $937.89, or 6.21%
For n=3 years, P0 falls from $1,000 to $973.25, or 2.675%
Even if r doesn’t change, premiums and discounts will decline towards par as bond nears maturity.
What does this tell you about the relationship
between bond prices and yields for bonds with
different maturities?

46. Primary vs. Secondary Markets
Primary market: the initial sale of bonds by issuers to large investors or syndicates
Secondary market: the market in which investors trade with each other
-Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity.
The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate.
When P0  Par, the YTM will differ from the coupon rate.
YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM
Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43
Trades in the secondary market do not raise any capital for issuing firms.

47. Bonds by Issuer Corporate Bonds Municipal Bonds Treasury Bonds
Usually with par $1000 and semi-annual coupon
Bonds if maturity > 10 years; notes if maturity < 10 years
Municipal Bonds
Issued by local and state government
Interest on municipal bonds tax-free
Treasury Bonds
If maturity < 1 year: Treasury Bills
If 1 year < maturity < 10 years: Treasury Notes
Maturity > 10 years: Treasury Bonds
Used to fund budget deficits
-Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity.
The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate.
When P0  Par, the YTM will differ from the coupon rate.
YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM
Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43
Agency Bonds
Issued by government agencies: FHLB, FNMA (Fannie Mae), GNMA (Ginnie Mae), FHLMC (Freddie Mac)

48. Bonds by Features Fixed vs. Floating Rates Secured vs. Unsecured Bonds
Floating-rate bonds: coupon tied to prime rate, LIBOR, Treasury rate or other interest rate
Floating rate = benchmark rate + spread
Floating rate can also be tied to the inflation rate: TIPS, for example
Secured vs. Unsecured Bonds
Unsecured bonds (debentures) are backed only by general faith and credit of issuer
Secured bonds are backed by specific assets (collateral)
Mortgage bonds, collateral trust bonds, equipment trust certificates
-Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity.
The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate.
When P0  Par, the YTM will differ from the coupon rate.
YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM
Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43

49. Bonds by Features (Continued)
Zero-Coupon Bonds
Discount bonds or pure discount bonds
Sell below par value
Treasury Bills (Tbills)
Treasury STRIPs
Convertible and Exchangeable Bonds
Convertible bonds, in addition to paying coupon, offers the right to convert the bond into common stock of the issuer of the bond
Exchangeable bonds are convertible in shares of a company other than the issuer’s
-Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity.
The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate.
When P0  Par, the YTM will differ from the coupon rate.
YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM
Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43

50. Bonds by Features (Continued)
Callable and Putable Bonds
Callable bonds: bond issuer has the right to repurchase the bonds at a specified price (call price).
Firms could retire and reissue debt if interest rates fall.
Putable bonds: the investors have the right to sell the bonds to the issuer at the put price.
-Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity.
The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate.
When P0  Par, the YTM will differ from the coupon rate.
YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM
Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43
Protection from Default Risk
Sinking fund provisions: the issuer is required to gradually repurchase outstanding bonds.
Protective covenants: requirements the bond issuer must meet
Positive and negative covenants

51. Bond Valuation
Bond price equals present value of its coupons and principal.
Bond prices are inversely related to interest rates.
Bonds could have a number of features: such as convertibility, callability.