1. Applying a general asset valuation model to bonds
Bond Valuation
Applying a general asset
valuation model to bonds

2. A General Valuation Model
The same basic approach can be used for valuing any financial (or real) asset. First we need to answer a basic question:
What do we mean by Value?
The most you should be willing to pay for the asset (given the required return and future cash flows).
Will this always be the same as the price?

3. A General Valuation Model
The basic components
Estimate the future cash flow stream from owning the asset
The required rate of return for each period based upon the riskiness of the asset
The value is then found by discounting each cash flow by its respective discount rate and then summing the PV’s (Basically the PV of an Uneven Cash Flow Stream)

4. The Inputs in the Valuation Model
Let rj represent the required rate of return on asset j based upon its level of risk (it is the opportunity cost of capital for asset j)
Let CFt be the cash flow expected to be received in period t.

5. The Formal Model The value of the asset should then be equal to:
Do you foresee any problems implementing the formula?

6. A Basic Bond
A bond is a debt contract issued by a corporation or government entity.
At issue the buyer is lending the issuer an amount of money (the par value).
The issuer agrees to pay interest at specified intervals (coupon payments) to the buyer, and return the par value at the end of the contract (the maturity date).
The bond can then be sold in the secondary market

7. Components of a bond: Par Value: The Initial issue amount
Coupon Payment: Interest payments on the par value.
Coupon Rate: The rate that determines the coupon payments.
Maturity Date: The point in time when the par value is returned and final coupon payment is made.
Call Provisions: The issuer may be able to “call” the bond prior to its maturity.
Market Price: The current price the bond is selling for in the market.

8. Applying the general valuation formula to a bond
What component of a bond represents the future cash flows?
Coupon Payments: The amount the holder of the bond receives in interest at the end of a specified period.
The Par Value: The amount that will be repaid to the purchaser at the end of the debt agreement.

9. The Cash Flows on a Time Line
Time n
CF CP CP CP CP MV
Value
The Value of the bond should equal the sum
of the PV of each of its cash flows.

10. Basic Bond Valuation Model
Given
kd: The interest rate or return paid on assets of similar risk (debt issued by firms of similar risk)
CP: The coupon payment
MV: The Par Value (or Maturity Value)
The value of the bond is represented as:

11. The coupon payment is the same every period so the first part is just an annuity. The formula can be rewritten as:

12. A Simple Example PVIFA9%,20 = 9.1285 PVIF9%,20=.1748 1,000(.09) = $90
A Bond with 20 years left until maturity, with a 9% coupon rate and a par value of $1,000. Assume for now that the discount rate is also 9%.
What are the relevant PVIFA and PVIF?
PVIFA9%,20 = PVIF9%,20=.1748
How much is the coupon payment?
1,000(.09) = $90

13. Applying the formula On a Financial Calculator:
20 N 9 I 90 PMT 1,000 FV PV = -1,000

14. A Second Example On a financial Calculator:
What if there are only 10 years left until the bond matures?
PVIFA9%,10= PVIF9%,10=.4224
On a financial Calculator:
10 N 9 I 90 PMT 1,000 FV PV = -1,000

15. The Discount Rate
So far we assumed that the interest rate is the same as the coupon rate. When this is true, the value of the bond equals the par value.
Are the two usually the same?
No, the discount rate should represent the current required return on assets of similar risk. This changes as the level of interest rates in the economy changes

16. Continuing our example
Assume that we bought our 9% yearly coupon bond with 20 years left to maturity and one year later the required return decreased to 7%.
What is the value of the bond?
19 N 7 I 90 PMT 1,000 FV PV =-1,206.71
PVIFA7%,19= PVIF7%,19= .2765

17. Why did the price increase?
New bonds of similar risk are only paying a 7% return. This implies a coupon rate of 7% and a coupon payment of $70.
The old bond has a coupon payment of $90, everyone will want to buy the old bond, (the increased demand increases the price)
Why does it stop at $1,206.71?
If you bought the bond for $ and received $90 coupon payments for the next 19 years you receive a 7% annual return.

18. Changes in Bond Value Over Time
Assuming that interest rate stay constant, what happens to the price of the bond as it gets closer to maturity?
N= # of Payments
I = Interest Rate 7% 7% 7%
PMT = Coup Pay
FV = Par Value
PV = Bond Value

19. Calculating Yield Two forms of income from the bond
Changes in Price: The capital gain (or loss)
Interest Income from the coupon payment:
The Current Yield (or interest yield)

20. Our Example
Assume that you purchase the 9% bond with one year remaining and held it until maturity.
Original price
Par value
Coupon payment 90.00
Change in bond value
Cap Gains Yield / =
Current Yield / =
Total Yield ( )/ =

21. Compare to a 7% Coupon bond
Assume that you purchase a 7% bond with one year remaining (let kd=7 %).
Original price
Par value
Coupon payment 70.00
Change in bond value 00.00
Cap Gains Yield / =
Current Yield / =
Total Yield ( )/ =

22. The PV of $20 for 1 year at 7% is equal to:
Note:
Comparing the 7% bond to the 9% bond.
You would pay more for the 9% bond
In one year you receive 1090 on the 9% bond and 1070 on the 7% bond. A difference of $20
In other words you would have paid $ more today to get $20 more in one year.
The PV of $20 for 1 year at 7% is equal to:
20/1.07=18.69

23. Total Yield or Total Return
The total yield for the year is equal to the capital gains yield plus the current yield.
The yearly total return will not always equal the required return at the beginning of the year.

24. Practice Problem
What is the value of for a 11% coupon bond that makes annual coupon payments for the next 10 years assuming that bonds of similar risk are paying 12% return?
What is the Coupon Payment?
1,000(.11) = 110
What is the value of the bond?
V = 110(PVIFA12%,10)+1,000(PVIF12%,10)
V =

25. Practice problem part 2
Assume that you bought the bond on the previous slide and held it for one year. During the year the required return decreased to 10% what is the total return (Total Yield) from owning the bond?
V = 110(PVIFA10%,9)+1,000(PVIF10%,9)
V = 1,
(1, )/ =
110 / =
Total Yield = or 23.75%

26. Yield to Maturity
Before, we were looking for the “value” of the bond given a required rate of return.
Now, given the current market price, we want to find the interest rate that makes the cash flows from the bond equal to its market price - this rate is known as the Yield to Maturity.
The YTM is the return you earn IF you buy the bond today and hold it until maturity and the bond does not default

27. Calculating YTM
To solve for YTM we are solving for the interest rate (rd) in the bond valuation formula:
We cannot solve for rd algebraically, only by trial and error, however a financial calculator solves this easily.

28. Calculating YTM Unfortunately calculating YTM is difficult:
The best method it using trial and error.
You can approximate it by looking in the PVIFA and PVIF tables
Solve for I on the Financial Calculator (make sure to enter both (-) and (+) CF’s)

29. YTM and Risk
What is the relationship between the YTM and the risk associated with the firms projects?
The YTM will change as the level of interest rates in the economy changes and as the risk associated with the firm and its projects change.
What does the YTM tell us about the cost of borrowing to fund future projects?
It represents the firm’s cost of debt (the return the firm must pay if it borrows money in the debt market)

30. Promised or Expected Return
You will earn the YTM if the bond does not default and you hold it to maturity.
The expected return should encompass that chance of default, probability the bond is called, and the possibility of interest rate fluctuations.
The YTM is only the expected return if the prob. of default is zero, the prob. of call is zero, and interest rates remain unchanged.

31. Quick Summary
If rd <Coupon rate the price of the bond is above the par value - it is selling at a premium
If rd >Coupon rate the price of the bond is below the par value - it is selling at a discount
If the level of interest rates in the economy increases the bond price decreases and vice versa

32. Semiannual Compounding
Most bonds make coupon payments twice a year, to account for this:
Divide the annual coupon interest payment by 2.
Multiply the number of periods by 2.
Divide the annual interest rate by 2

33. Example: Semiannual Compounding
What is the most you would be willing to pay for a 10% coupon bond that makes semiannual coupon payments, 10 years left to maturity and an annual required return of 12%
PMT = 1,000(.10) / 2 = 50 (each 6 months)
I = 12%/2 = 6% each six months
N = 10 (2) = 20
FV = 1,000
PV = ? =

34. Call Provisions
Many bonds have a provision that allows the issuer to call the bond prior to maturity.
If the bond is called the owner receives the par value, plus a call premium to compensate the owner for the early maturity of the bond.

35. Yield to Call
This is the same idea as the YTM only it is the return you would earn IF the bond is called.
In other words it is the interest rate that makes the PV of the future cash flows if the bond is called, equal to the current price.
When will the call be exercised?
Should the required return be greater or less on a callable bond compared to a similar noncallable bond?

36. Example
XYZ has an 11% coupon bond outstanding with 12 years left until maturity. Assume that the bond makes semiannual coupon payments and can be called in 6 years. If it is called the firm will pay a premium of $75. The bond current has a YTM of 12%. What is the value of the bond?
N = 24 PMT = 55 FV = 1,000 I = 6%
Vbond =

37. Pricing the bond
Whether or not the bond is called, the bond has the same price $
If the bond is called the cash flows will be different from the original case since the coupon payments will stop earlier and the call premium will be added.

38. Comparing Cash Flow Streams
If the Bond is Called
0.5 1 6 12 55 55 55
1,000 75
If the Bond is Not Called
0.5 1 6 12 55
1000 55 55 55

39. Pricing the bond
What is the Yield to Call of the bond? ( the return earned IF the bond is called)
N = 12 PMT = 55 FV = 1,075
Vbond = PV=
YTC = I = 6.7% every 6 months
or 13.4% per year

40. Bondholders and Risk Interest Rate Risk Reinvestment Rate Risk
As interest rates increase the bond value decreases (even if the industry and firm risk don’t change)
Reinvestment Rate Risk
If rates decrease you CP’s cannot be reinvested at the same rate
Default Risk
The most basic risk - what if the firm does not meet its obligations.

41. Interest Rate Risk
Keeping everything else the same: The longer before maturity of the bond, the larger the impact of a change in interest rates.
Keeping everything else the same: The larger the coupon rate on a bond, the larger the impact of a change in interest rates.

42. Impact of Maturity
Compare two 10% coupon semiannual bonds, both with a current yield of 12%. Let Bond A have 30 years to maturity and Bond B have 15 years. What is the associated price change if the yield changes to 11%? to 13%?
Bond A 30 Years Bond B 15 Years
Yield Price Change Price Change
11%
(8.87%) (7.54%)
12%
(7.62%) (6.75%)
13%

43. Different Coupons
Compare two 30 year semiannual coupon bonds, both with a current yield of 12%. Let Bond A have a coupon rate of 10% and Bond B have a coupon rate of 8%. What is the associated price change if the yield changes to 11%? To 13%?
Bond A 10% coupon Bond B 8% coupon
Yield Price Change Price Change
11%
(8.87%) (9.08%)
12%
(7.62%) (7.77%)
13%

44. Reinvestment Rate Risk
Reinvestment rate risk is closely tied to the risk that a bond will be called.
As interest rates decline, a bond’s coupons cannot be reinvested at the original YTM. It also increases the probability that the firm will exercise its call option. If the call option is exercised the holder looses the original coupon payment as well as any reinvestment.

45. Default Risk and Bond Ratings
Moody’s investors services and Standard and Poor’s Corporation provide ratings for corporate bonds based upon the quality of the bond.
The ratings allow investors to compare the safety of bonds to each other. A large part of the rating is based upon default risk.
The highest rating, AAA or Aaa, represents a very low probability of default.

46. Bond Ratings
As the probability of default increases, the rating drops from AAA to AA (or Aaa to Aaa).
After A the ratings go to BBB then BB etc.
Bonds rated below BB are considered high risk or “Junk Bonds”.

47. Summary of Bond Ratings
Moody’s
S&P
Fitch
Aaa
AAA AA
Maximum safety
Investment Grade Aa
Very High Grade A
Upper Med Grade
Baa
BBB
Lower Med Grade Ba BB
Low Grade Speculative
Low Credit Worthiness B
Highly Speculative
Caa
CCC
In poor standing
Substantial Risk close to default or in default Ca CC
May be in default C
More risky than CC D
In Default
In default
Fabozzi Bond Markets Analysis and Straegies 2004

48. Long Term Average Yearly Yields Over Time (Moody’s)

49. Yield Spreads Yield Spreads
The difference in required return between two assets, the difference in required return represents the difference in risk.
Often bonds that are the same except for the possibility of default are compared, implying that the yield spread is a measure of the default risk

50. Bond Ratings and Average Yield Spreads vs
Bond Ratings and Average Yield Spreads vs. US Treasuries (long term bonds)
Rating Spread Rating Spread
AAA .20% B+ 2.5%
AA .50% B 3.25%
A+ .80% B %
A 1.0% CCC 5.00%
A % CC 6.00%
BBB 1.5% C 7.5%
BB 2.0% D 10.0%

51. Yield Spread Monthly Data Jan 1919 – June 2004 (Moody’s)

52. Yield Spreads

53. Bond Rating Criteria Financial Ratios Mortgage Provisions
Guarantee Provisions
Sinking funds
Maturity
Stability
Regulation
Others

54. Bond Rating Criteria and Financial Ratios
AAA AA A BBB BB B
TIE
Int Cov
NetCF/TotDebt 90% 67% 50% 32% 20% 11%
FCF/TotDebt 41% 22% 17% 6% 1% -4%
ROC 31% 25% 20% 15% 13% 9%
LTDebt/TotCap 21% 29% 33% 41% 55% 69%
TotDebt/TotCap 32% 37% 39% 46% 59% 71%

55. Long Term Debt Terminology
Indenture: written agreement between the borrower and lender detailing the terms of issue
a)     Bearer form vs. registered form
b)     Total amount of bonds issued
c)      Collateral
d)     Repayment arrangements:
e) Sinking fund
e)      Call provisions
g)      Protective covenants 

56. Types of Bonds and embedded provisions
Corporate Bonds
Government Bonds 
Zero Coupon Bonds
Floating Rate Bonds:
Put provisions
floors and ceilings
Inflation linked bonds

57. Other Special Features
Convertible bonds
Income bonds
Put bonds