# 007-Bonds

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```					           Bonds

 The Discounted Cash Flow
Valuation Model
 Basics of Bonds and their
Valuation
 Ref. Ch 7
1
Main Points

 important bond features and bond types
 bond values and why they fluctuate
 bond ratings and what they mean
 the impact of inflation on interest rates
 the term structure of interest rates and the
determinants of bond yields

2        2
Bond Definitions

Outset        1st Year         2nd Year   3rd Year
Bond CB         -\$1000        \$50              \$50        \$50+1000

   Bond: A bond is a security that represents a loan made by
investors to the issuer.
   Par value (face value): The issuer promises to pay the
face/par/maturity value of the bond when it matures.
   Coupon payment: The issuer may promise to pay the investor a
regular coupon payments every period until the bond matures.
   Coupon rate: Percentage of face value.
   Maturity date: Duration of the contract.
   Yield or Yield to maturity: Average rate of return.              3
Valuing Coupon Bonds

Value of a Level-coupon bond=
PV of coupon payment annuity + PV of face value

\$C       \$C            \$C      \$C  \$F

0      1        2            T 1       T

C      1        F
PV  1        T 

r  (1  r )  (1  r ) T

4
Present Value of Cash Flows as Rates
Change
 Bond Value = PV of coupons + PV of par
 Bond Value = PV annuity + PV of lump sum
 Remember, as interest rates increase present
values of future cash flows decrease
 So, as interest rates increase, bond prices
decrease and vice versa

5
Valuing a Par Bond with Annual Coupons

 Consider a bond with a coupon rate of 10% and
annual coupons. The face value is \$1000 and the
bond has 5 years to maturity. The yield to maturity is
10%. What is the value of the bond?

 Using the formula:
B = PV of annuity + PV of lump sum
B = 100[1 – 1/(1.10)5] / .10 + 1000 / (1.10)5
B = 379.08 + 620.92 = 1000

6
Valuing a Discount Bond with Annual
Coupons
 Consider a bond with a coupon rate of 10% and
annual coupons. The par value is \$1000 and the
bond has 5 years to maturity. The yield to maturity is
11%. What is the value of the bond?

 Using the formula:
B = PV of annuity + PV of lump sum
B = 100[1 – 1/(1.11)5] / .11 + 1000 / (1.11)5
B = 369.59 + 593.45 = 963.04

7
Valuing a Premium Bond with Annual
Coupons
 Suppose you are looking at a bond that has a 10%
annual coupon and a face value of \$1000. There are 5
years to maturity and the yield to maturity is 8%. What
is the price of this bond?
 Using the formula:
 B = PV of annuity + PV of lump sum
 B = 100[1 – 1/(1.08)5] / .08 + 1000 / (1.08)5
 B = 399.27 + 680.27 = 1079.54


8
Coupon Bond Principles

Let us summarize the findings from the
previous examples:
#1: For par bonds:
yield-to-maturity = coupon rate.
#2: for premium bonds (price > face value):
ytm < coupon rate.
#3: for discount bonds (price < face value):
ytm > coupon rate

9
Interest Rate Risk
   Price Risk
 Change in price due to changes in interest rates
 Long-term bonds have more price risk than short-
term bonds (same coupon rates)
 Low coupon rate bonds have more price risk than
high coupon rate bonds (same maturity)
   Reinvestment Rate Risk
 Uncertainty concerning rates at which cash flows can
be reinvested
 Short-term bonds have more reinvestment rate risk
than long-term bonds
 High coupon rate bonds have more reinvestment rate
risk than low coupon rate bonds                       10
Price Risk

11
Computing Yield-to-Maturity

 The yield-to-maturity is the discount rate that makes the
present value of the cash flows from the bond equal to the
current price of the bond.
 Finding the YTM requires trial and error if you do not have
a financial calculator.
 Example: What is the yield-to-maturity of a \$1,000 par
value, 10% coupon rate bond coming due in 3 years that
currently sells for \$1076?
\$1076 = \$100(PVAFr%,T) + (\$1,000)(1+r)-3
=> r = YTM = 7.10%

12
Yield-to-Maturity, what does it tell?

 They allow you to compare different kinds of
bonds – those with dissimilar coupons,
different market prices, and different
maturities.
 YTM will equal your total earnings if:
 You hold the bond to maturity,
 Coupons are reinvested at an interest rate
equal to YTM.
 So, It is a promised annual rate of return.
Why?

13
Current Yield and YTM

 The current yield of a coupon bond =
annual coupon payment / current price.
 Example: The current yield of the previous bond is:
= 100 / 1076 = 0.09293 or 9.293%
 YTM = Interest return + Capital Gain (Loss)
 Current yield: the portion of an investor’s return that
comes in the form of interest income.

14
Current Yield vs. Yield to Maturity

 Previous example: 10% coupon bond, face value of
1000, 3 years to maturity, \$1076 price
 Current yield = 100 / 1076 = .0929 = 9.293%
 Price in one year = 1052.36 USD, assuming no
change in YTM.
 Capital gain yield = (1052.36–1076) / 1076
= -.02197 = -2.197%
 Yield to maturity = current yield + capital gains yield
 YTM = 9.293 – 2.197 = 7.09%,
15
Bond Values with Semiannual Compounding

C
2N
F
P0          2
t
    2N
t 1     r  r
1   1  
 2  2

16
Semi-annual bonds -Example

   Suppose that the Genesco 15 year, 15% bond paid
interest semi-annually rather than annually. What
would be its price upon issue if current rates are 15%
on similar bonds?
C = \$1,000 x 0.15 = 150        C/2 = 150/2=75
N = 15  2N = 30
F = \$1,000
r = 15%  r / 2 = 15/2 = 7.5%

\$PV = \$75(PVAF7.5%,30) + (\$1,000)(1+0.075)-30
PV = \$1,000

17
YTM with Semiannual Coupons
 Suppose a bond with a 10% coupon rate and
semiannual coupons, has a face value of
\$1,000, 20 years to maturity and is selling for
\$1,197.93.
 Is the YTM more or less than 10%?
 What is the semiannual coupon payment?
 How many periods are there?
 1197.93 = \$50(PVAFYTM/2%,40) + (\$1,000)(1+0.05)-40
 YTM/2 = 4%  YTM=4%*2 = 8%
 Simple annualization is a convention!
18              18
Example: Valuing Bonds w. Semi-Annual Payments

Find the present value (as of January 1, 2002), of a 6.375%
coupon T-bond with semi-annual payments, and a maturity
date of December 2009 if the YTM is 5-percent.

\$31.875 \$31.875                  \$31.875       \$1,031.875

1 / 1 / 02   6 / 30 / 02   12 / 31 / 02       6 / 30 / 09    12 / 31 / 09

\$31.875         1      \$1,000
PV          1  (1.025)16   (1.025)16  \$1,049.30
.05 2                
19
Yield to Call

   Some bonds are calleable, they can be called
back by the issuer before the maturity.
Condional upon the market interest rates, the
issuer may prefer to use this option. Why?
   .....
   In this case, we compute the yield to maturity
of the bond as if you receive the call price
and the bond is called on its earliest date.

20
Yield to Call - Example

   Suppose that AZ Inc has a 10 year 8%
coupon bond outstanding that can be called
at the end of year 5 for a 5% premium.
   Further suppose that its current market price
is 112.42% (assume face value = \$100) and
that it has been outstanding for 2 years.
   If you buy this bond, what yield you would
probably obtain out of this investment
(assume annual coupon payments)?

21
Yield to call - Example

   Without call the yield is: PV=\$112.42, C=\$8, N=8,
F=\$100  r% = YTM = 6%,
   Since the bond has been outstanding for 2 years, the
bond can be called in 3 years.
   Since the bond is selling for premium, it is most likely
that the firm will call the bonds in 3 years. Why?
   Since the call premium is 5%, then its maturity value
will be \$100 x 1.05 = \$105 if called.
   So, what is the yield for: PV=\$112.42, C=\$8, N=3,
F=\$105
   The YTM between 5.5% and 5.75%. Find out the
number yourself.
   Conclusion: ..
22
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

23
Differences Between Debt and Equity
   Debt                              Equity
 Not an ownership interest         Ownership interest
 Creditors do not have             Common stockholders vote
voting rights                       for the board of directors
 Interest is considered a            and other issues
cost of doing business and        Dividends are not
is tax deductible                   considered a cost of doing
 Creditors have legal                business and are not tax
deductible
recourse if interest or
principal payments are            Dividends are not a liability
missed                              of the firm and stockholders
 Excess debt can lead to             have no legal recourse if
dividends are not paid
financial distress and
bankruptcy                        An all equity firm can not go
bankrupt
24
The Bond Indenture
 Contract between the company 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
25
Bond Characteristics and Required
Returns
 The coupon rate depends on the risk
characteristics of the bond when issued
 Which bonds will have the higher coupon, all
else equal?
 Secured debt versus a debenture
 Subordinated debenture versus senior debt
 A bond with a sinking fund versus one
without
 A callable bond versus a non-callable bond
26
Examples of Credit Ratings

•Moody's     S&P’s   Fitch’s   DCR’s   Definition
•Aaa         AAA     AAA       AAA     Prime. Maximum Safety
•Aa1         AA+     AA+       AA+     High Grade High Quality
•Aa2         AA      AA        AA
•Aa3         AA-     AA-       AA-
•A1          A+      A+        A+      Upper Medium Grade
•A2          A       A         A
•A3          A-      A-        A-
•Baa1        BBB+    BBB+      BBB+    Lower Medium Grade
•Baa2        BBB     BBB       BBB
•Baa3        BBB-    BBB-      BBB-
•Ba1         BB+     BB+       BB+     Non Investment Grade
•Ba2         BB      BB        BB      Speculative
•Ba3         BB-     BB-       BB-
•B1          B+      B+        B+      Highly Speculative
•B2          B       B         B
•B3          B-      B-        B-
•Caa1        CCC+    CCC       CCC     Substantial Risk
•Caa2        CCC     -         -       In Poor Standing
•Caa3        CCC-    -         -
•Ca          -       -         -       Extremely Speculative
•C           -       -         -       May be in Default
•-           -       DDD       -       Default
•-           -       DD        DD      Default
•-           D       D         -       Default
•-           -       -         DP      Default

27
Bond Ratings – Investment
Quality
 Moody’s Aaa and S&P AAA – capacity to pay is
extremely strong
 Moody’s Aa and S&P AA – capacity to pay is very strong
 Moody’s A and S&P A – capacity to pay is strong, but
more susceptible to changes in circumstances
 Moody’s Baa and S&P BBB – capacity to pay is
the firm’s ability to pay

28              28
Bond Ratings - Speculative

 Moody’s Ba, B, Caa and Ca
 S&P BB, B, CCC, CC
 Considered speculative with respect to capacity to pay.
The “B” ratings are the lowest degree of speculation.
 Moody’s C and S&P C – income bonds with no interest
being paid
 Moody’s D and S&P D – in default with principal and
interest in arrears

29              29
Issuer: Government and Agencies
 Treasury Securities
 Federal government debt
 T-bills – pure discount bonds with original maturity of
one year or less
 T-notes – coupon debt with original maturity between
one and ten years
 T-bonds coupon debt with original maturity greater
than ten years
 Municipal Securities
 Debt of state and local governments
 Varying degrees of default risk, rated similar to
corporate debt
 Interest received is tax-exempt at the federal level
30
Example 7.4
 A taxable bond has a yield of 8% and a
municipal bond has a yield of 6%
 If you are in a 40% tax bracket, which bond
do you prefer?
 8%(1 - .4) = 4.8%
 The after-tax return on the corporate bond is
4.8%, compared to a 6% return on the municipal
 At what tax rate would you be indifferent
between the two bonds?
 8%(1 – T) = 6%
 T = 25%                                         31
Zero-Coupon Bonds
 Make no periodic interest payments (coupon rate =
0%)
 The entire yield-to-maturity comes from the difference
between the purchase price and the par value
 Cannot sell for more than par value
 Sometimes called zeroes, or deep discount bonds.
 Treasury Bills and Treasury strips are good examples
of zeroes.

32
Pure Discount Bonds: Example
 At the 1-year treasury auction the issue sells
for a price of 90.
 What is the annual rate you would earn if you
purchased this bond on the issue and held it
until maturity?
 Rate = (P1 – P0) / P0
 Rate = (100/90 – 1) = 0.11   (effective rate!)

33
Pure Discount Bonds: Example 2
 At the 6 month treasury auction the issue sells for a price of
95.
   What is the (effective) annual rate if you purchase and hold
this bond until maturity?

 For 6 months: r= (100/95 – 1)=0.05263
 Annualization:
 5.263% x 2 = 10.526% (Simple yield)
 1.052631/m -1 = 0.108 => 10.8% (Compound Yield)
 Where m=0,5

34
Pure Discount Bonds: Example 3
 At the 2-year treasury auction the issue sells for a
price of 81.
 What is the annual rate you would earn if you
purchased this bond on the issue and held it until
maturity?
   HPR for 2 years: r= (100/81 – 1)=0.23456
   Annualization:
 HPR/n = 23.456 / 2 = 11.728% (Simple yield)
 (1+EAR)2= (1+HPR) and EAR=(1+HPR)1/2 -1
 EAR = (1.23456)1/2 - 1 = 0.1111 or 11.11%
(Compound Yield)
 Compare simple yield vs compound yield?
35
Floating-Rate Bonds
 Coupon rate floats depending on some index value
 Examples – adjustable rate mortgages and inflation-
 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”

36        36
Other Bond Types

 Disaster bonds
 Income bonds
 Convertible bonds
 Put bonds
 There are many other types of provisions that
can be added to a bond and many bonds have
several provisions – it is important to recognize
how these provisions affect required returns

37           37
Bond Markets (Second Hand)
 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 company or municipal
issues
 Treasury securities are an exception

38
Treasury Quotations
   Quotation:
 8.00   Nov 21     128:07 128:08 5 5.31%
   Coupon rate 8% and Matures in November 2021
   Bid price is 128 and 7/32% of par value.
   7/32= 0.21875 of 1 percent
   128 7/32 = 128.21875%.
   Bid price is 1.2821875*1000= \$1282.1875.
   Ask price is 128 and 8/32% of par value or 128.25%.
   Ask price is 1.2825*1000=\$1282.50 .
   The difference between the bid and ask prices (\$0.3125) is
money.
   Ask price changed by 5/32 of 1 percent from the previous
day: 0.15625 of 1% or \$1.5625 for a \$1000 worth of T-bond.
   The yield is 5.31% based on the ask price.
39
Clean vs. Dirty Prices
 Clean price: quoted price
 Dirty price: price actually paid = quoted price plus
accrued interest
 Example: Consider T-bond in previous slide, assume
today is July 15, 2007
 Number of days since last coupon = 61
 Number of days in the coupon period = 182
 Accrued interest = (61/182)(.04*1000) = 13.4
 Clean price = 1282.50
 Dirty price = 1282.50 + 13.4 = 1295.9
 So, you would actually pay \$ 1295.9 for the bond
40
Inflation and Interest Rates

 Nominal rate of interest= quoted rate of
 Real rate of interest= shows change in
 The nominal rate of interest includes our
desired real rate of return plus a
compensation for inflation.

41
The Fisher Effect
 The Fisher Effect defines the relationship
between real rates, nominal rates and inflation
 (1 + R) = (1 + r*)(1 + h), where
 R = nominal rate
 r* = real rate
 h = expected inflation rate
 R = r* + h + r*h
 Approximation
 R = r* + h

42
Example 7.6
 If we require a 10% real return and we expect
inflation to be 8%, what is the nominal rate?
 R = (1.1)(1.08) – 1 = .188 = 18.8%
 Approximation: R = 10% + 8% = 18%
 Because the real return and expected inflation
are relatively high, there is significant
difference between the actual Fisher Effect
and the approximation.

43
Term Structure of Interest Rates

 Term structure is the relationship between time to
maturity and yields, all else equal
 It is important to recognize that we pull out the effect of
default risk, different coupons, etc. For ex. Yields on
treasury issues.
 Yield curve – graphical representation of the term
structure
 Normal – upward-sloping, long-term yields are higher than short-
term yields
 Inverted – downward-sloping, long-term yields are lower than
short-term yields

44                44
Example of Yield Curve

 Below is the Treasury yield curve for 2/11/03

45
Figure 7.6 – Upward-Sloping Yield Curve

46
The shape of the Treasuries yield curve

The yields on Treasuries depend upon:
 Real rate of interest – opportunity cost of
deferred consumption in real terms.
 Expected inflation – investors must be
compensated for anticipated loses in
 Maturity risk premium – investors demand
compensation for their interest rate risk
exposure.

47
Decomposition of Yields to Maturity
 Corporate bonds face additional risks: credit risk +
liquidity risk

 Credit risk: The risk that coupons and the principal
may not be paid off.
 A bond’s credit risk is often captured by its bond
rating.
 Bonds issuers pay credit rating firms to rate their
debt.
 Liquidty risk: The more thinly traded a bond, the

48
 There is a specific formula for finding bond
 PRICE(Settlement,Maturity,Rate,Yld,Redemption,
Frequency,Basis)
 YIELD(Settlement,Maturity,Rate,Pr,Redemption,
Frequency,Basis)
 Settlement and maturity need to be actual dates
 The redemption and Pr need to given as % of par
value
 Click on the Excel icon for an example
49
 End..

50

   On 15/08/1995, a bond with 9 month maturity (15/05/1996) is
bought at a price that would yield 84.70% annual. Six months
later, on 13/02/1996 the same T-Bond is sold at a price that
would yield 71% annual. What would be your return over the
investment period should you sell the bond on 13/02/1996?

Number of days to maturity: 274
Price = 100,000 / (1+ 0.8470*274/365) = 61,131 TL
   Selling price on 13/02/1996:
Number of days to maturity: 92
Selling Price = 100,000 / (1+ 0.71*92/365) = 84,820 TL

51
   Periodic rate of return (over 274 – 92 = 182 days):
(84,820 – 61,131) / 61,131 = 23,689 / 61,131 = 0.3875
or 38.75%

   Annual rate of return (simple):
   0.3875*365/182 = 0.7771 or 77.71%

   Annual rate of return (compounded):
   Investment duration is 182 days and the period rate of
return is 0.3875.
   EAR = (1.3875)^365/182 – 1 = 0.9274 or 92.74%.
52
Turkish Treasury Bills - Example

 Example:
 Here is a line from Reuter page on March 19, 2007:
 Value:                   19March07
 Maturity:                04June07
 Average Price:           95
 Simple Yield:            24.95
 Compounded Yield: 27.52
 Now, let us verify how these values are computed:

53
Turkish Treasury Bills (contn’d)
 Number of days from 19 March 07 to 04 June 07 = 77
days.
 Period rate of return over (77/365) years :
 r = (Face / PV) - 1 =(100 / 95) - 1 = 0.05263

 Annual Simple Rate = 0.05263 * (365/77) = 0.2495
 Compounded yield:
 EAR = ( 1 + 0.05263)(365/77) – 1 = 0.27523
 Conclusion: 24.95% vs 27.5%
54

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