1. CHAPTER 8 Bonds and Their Valuation
Key features of bonds
Bond valuation
Measuring yield
Assessing risk

2. Key Features of a Bond
1. Par value: Face amount; paid at maturity. Assume $1,000.
2. Coupon interest rate: Stated interest rate. Multiply by par to get $ of interest. Generally fixed.

3. 3. Maturity: Years until bond
must be repaid. Declines.
4. Issue date: Date when bond
was issued.

4. How does adding a “call provision” affect a bond?
Issuer can refund if rates decline. That helps the issuer but hurts the investor.
Therefore, borrowers are willing to pay more, and lenders require more, on callable bonds.
Most bonds have a deferred call and a declining call premium.

5. What is a sinking fund?
Provision to pay off a loan over its life rather than all at maturity.
Similar to amortization on a term loan.
Reduces risk to investor, shortens average maturity.
But not good for investors if rates decline after issuance.

6. Sinking funds are generally handled in 2 ways
1. Call x% at par per year for sinking fund purposes.
2. Buy bonds on open market.
Company would call if kd is below the coupon rate and bond sells at a premium. Use open market purchase if kd is above coupon rate and bond sells at a discount.

7. Financial Asset Values1 2 n k
...
CF1
CFn
CF2
Value CF CF CF PV = 1 + 2 +
... + n . ( ) 1 ( ) 2 ( ) n 1 + k 1 + k 1 + k

8. The discount rate (ki) is the opportunity cost of capital, i. e
The discount rate (ki) is the opportunity cost of capital, i.e., the rate that could be earned on alternative investments of equal risk.
ki = k* + IP + LP + MRP + DRP.

9. What’s the value of a 10-year, 10% coupon bond if kd = 10%?
100 1 2 10
10%
,000
V = ?
... ( ) V k B d =
$100 $1 , 1
000 10 . +
= $ $ $385.54
= $1,000.

10. The bond consists of a 10-year, 10% annuity of $100/year plus a $1,000 lump sum at t = 10:
PVA = PMT * (1- (1+k)n ) k
PV = FV(1/(1+k)n)
V bond = PMT * (1- (1+kd)n + M(1/(1+kd)n) kd

11. 1
V bond = PMT * (1- (1+kd)n + M(1/(1+kd)n) kd
V bond = 100 * (1- (1+.1) (1/(1+.1)10) .1
PV annuity $614.46
PV maturity value
V bond = $
When kd = coupon interest rate, bond
par value.

12. With annual compounding, P/Y = 1, and End mode
With annual compounding, P/Y = 1, and End mode. Use all TVM keys in the calculation - you have an annuity and a future cash flow outside the annuity.
INPUTS
N I/YR PV PMT FV
-1,000
OUTPUT

13. Most bonds pay interest semiannually,
so m = 2. You must divide k by 2, multiply
n by 2. The payment will be $50 every 6
months.If the market rate of interest
increased to 12%: 1
V bond = PMT * (1- (1+kd/2)n*2+ M(1/(1+kd /2)n *2)
kd /2
V bond = 50 * (1- (1+.12/2)10* (1/(1+.12/2)10*2)
.12/2
V bond =50( )+1000(.3118) =
This bond trades at a discount to par.

14. What would happen if expected inflation rose by 2%, causing k =12%?.
P/Y=2, End mode
INPUTS
N I/YR PV PMT FV
OUTPUT
When kd rises, above the coupon rate, the bond’s value falls below par, so it sells at a discount.

15. If the market rate of interest decreased to 8%: 1
V bond = PMT * (1- (1+kd/2)n*2+ M(1/(1+kd /2)n *2)
kd /2
V bond = 50 * (1- (1+.08/2)10* (1/(1+.08/2)10*2)
.08/2
V bond =50( )+1000(.4564) =
This bond trades at a premium to par.

16. What would happen if inflation fell, and kd declined to 8%?
P/Y=2, End mode
N I/YR PV PMT FV
-1,135.90
INPUTS
OUTPUT
Price rises above par, and bond sells at a premium, if coupon > kd.

17. The bond was issued 20 years ago and now has 10 years to maturity
The bond was issued 20 years ago and now has 10 years to maturity. What would happen to its value over time if the required rate of return remained at 10%, or at 13%, or at 7%?

18. Years remaining to Maturity
Bond Value ($)
1,372
kd = 7%.
1,211
kd = 10%. M
1,000
837
kd = 13%.
775
Years remaining to Maturity

19. At maturity, the value of any bond must equal its par value.
The value of a premium bond would decrease to $1,000.
The value of a discount bond would increase to $1,000.
A par bond stays at $1,000 if kd remains constant.

20. What is the “yield to maturity”?
YTM is the rate of return earned on a bond held to maturity. Also called the “promised yield.”

21. What’s the YTM on a 10-year, 9% semiannual coupon, $1,000 par value bond that sells for $885.45?1 9 20 45
kd=?
1,000
PV1 .
PV20
PVM
885.45
Find kd that “works”!
...

22. If you don’t have a financial calculator, need to use
trial and error. Know the price of the bond has
dropped to $885.45, so yield must have gone up.
Try 11%, was a 9% coupon bond. 1
V bond = PMT * (1- (1+kd/2)n*2+ M(1/(1+kd /2)n *2)
kd /2
= 45 * (1- (1+.11/2)10* (1/(1+.11/2)10*2)
.11/2
=45( )+1000(.3427) =
Vbond needs to be higher, so yield needs to be
lower, but not much lower. Without a financial calculator,
an answer of 9% < k < 11% would be sufficient.

23. Find kd
INPUTS
N I/YR PV PMT FV
10.91
OUTPUT

24. If coupon rate < kd, discount.
If coupon rate = kd, par bond.
If coupon rate > kd, premium.
If kd rises, price falls.
Price = par at maturity.

25. If the price of the bond has increased to $1142.12, yield
(market rate of interest) must have gone down.
Try 7%, was a 9% coupon bond. 1
V bond = PMT * (1- (1+kd/2)n*2+ M(1/(1+kd /2)n *2)
kd /2
= 45 * (1- (1+.07/2)10* (1/(1+..07/2)10*2)
.07/2
=45( )+1000(.5026) =
Vbond is approximately equal, so the YTM is 7% .

26. Find YTM if price were $1,
INPUTS
N I/YR PV PMT FV
7.0
OUTPUT
Sells at a premium. Because coupon = 9% > kd = 7.0%, bond’s value > par.

27. Definitions
Annual coupon pmt
Current price
Current yield =
Capital gains yield =
= YTM =
Change in price
Beginning price
Exp total
return
Exp
Curr yld
Exp cap
gains yld

28. Find current yield and capital gains yield for a 9%, 10-year bond when the bond sells for $887 and YTM = 10.91%.
$90
$887
Current yield =
= = 10.15%.

29. YTM = Current yield + Capital gains yield.
Cap gains yield = YTM – Current yield
= 10.91% – 10.15%
= 0.76%.
Could also find value in Years 1 and 2,
get difference, and divide by value in
Year 1. Same answer.

30. What’s interest rate (or price) risk
What’s interest rate (or price) risk? Does a 1-year or 10-year 10% bond have more risk?
Interest rate risk: Rising kd causes bond’s price to fall. kd
1-year
Change
10-year
Change 5%
$1,048
$1,386
+4.8%
-4.4%
+38.6%
-25.1%
10%
1,000
1,000
15%
956
749

31. Value .
10-year
1,500 . . . .
1,000
1-year .
500 kd 0% 5%
10%
15%

32. What is reinvestment rate risk?
The risk that CFs will have to be reinvested in the future at lower rates, reducing income.
Illustration: Suppose you just won $500,000 playing the lottery. You’ll invest the money and live off the interest. You buy a 1-year bond with a YTM of 10%.

33. Year 1 income = $50,000. At year-end get back $500,000 to reinvest.
If rates fall to 3%, income will drop from $50,000 to $15,000. Had you bought 30-year bonds, income would have remained constant.

34. Long-term bonds: High interest rate risk, low reinvestment rate risk.
Short-term bonds: Low interest rate risk, high reinvestment rate risk.
Nothing is riskless!

35. Do all bonds of the same maturity have the same price and reinvestment rate risk?
No, low coupon bonds have less
reinvestment rate risk but more price risk than high coupon bonds.

36. A 10-year, 10% semiannual coupon,$1,000 par value bond
is selling for $1, with an 8% yield to maturity.
It can be called after 4 years at $1,050.
What’s the bond’s nominal yield to call (YTC)? The price of
the bond is now $ , need to find YTC. Try 6%. 1
V bond = PMT * (1- (1+kd/2)n*2+CP(1/(1+kd /2)n *2)
kd /2
= 50 * (1- (1+.06/2)4* (1/(1+.06/2)4*2)
.06/2
=50( )+1050(.7894) =
Need lower V bond = so YTC must be greater than 6%.

37. With a financial calculator, you are just
Solving for the interest rate, using the
Call premium as the future value and the
Time to possible call as n.
INPUTS
N I/YR PV PMT FV
3.568 x 2 = 7.137%
OUTPUT

38. kNom = 7. 137% is the rate brokers would quote
kNom = 7.137% is the rate brokers would quote. Could also calculate EFF% to call:
EFF% = ( )2 – 1 = 7.26%.
This rate could be compared to monthly mortgages, etc.

39. If you bought bonds, would you be more likely to earn YTM or YTC?
Coupon rate = 10% vs. YTC = kd = 7.137%. Could raise money by selling new bonds which pay 7.137%.
Could thus replace bonds that pay $100/year with bonds that pay only $71.37/year.
Investors should expect a call, hence YTC = 7.1%, not YTM = 8%.

40. In general, if a bond sells at a premium, then (1) coupon > kd, so (2) a call is likely.
So, expect to earn:
YTC on premium bonds.
YTM on par & discount bonds.

41. Disney recently issued 100-year bonds with a YTM of 7
Disney recently issued 100-year bonds with a YTM of 7.5%--this represents the promised return. The expected return was less than 7.5% when the bonds were issued.
If issuer defaults, investors receive less than the promised return. Therefore, the expected return on corporate and municipal bonds is less than the promised return.

42. Bond Ratings Provide One Measure of Default Risk
Investment Grade Junk Bonds
Moody’s Aaa Aa A Baa Ba B Caa C
S&P AAA AA A BBB BB B CCC D

43. What factors affect default risk and bond ratings?
Financial performance
Debt ratio
TIE ratio
Current ratio

44. Provisions in the bond contract
Secured vs. unsecured debt
Senior vs. subordinated debt
Guarantee provisions
Sinking fund provisions
Debt maturity

45. Other factors
Earnings stability
Regulatory environment
Potential product liability
Accounting policies

46. Top Ten Largest U.S. Corporate Bond Financings, as of July 1999
Issuer
Ford Motor Co.
AT&T
RJR Holdings
WorldCom
Sprint
Date
July 1999
Mar 1999
May 1989
Aug 1998
Nov 1998
Amount
$8.6 billion
$8.0 billion
$6.1 billion
$5.0 billion