Bond Prices and
Yields
Slide 1
Comm 324 – W. suo
Bond Characteristics
 Face or par value
 Coupon rate

Zero coupon bond
 Compounding and payments
 Indenture
 Issuers
Slide 2
Comm 324 – W. suo
Provisions of Bonds






Secured or unsecured
Registered or bearer bonds (Canada)
Call provision
Convertible provision
Retractable and extendible (putable) bonds
Floating rate bond
Slide 3
Comm 324 – W. suo
Bond Pricing
T
PB   C t t
t 1 (1 r )
PB =
Ct =
T =
r =

ParValueT
(1  r )T
price of the bond
interest or coupon payments
number of periods to maturity
the appropriate semi-annual discount rate
 Quoted price vs Cash Price (or “dirty price”)

Slide 4
Accrued interest, day-count convention
Comm 324 – W. suo
Solving for Price: 10-yr, 8%
Coupon Bond, FV = $1,000
Ct
P
T
r
= 40 (SA)
= 1000
= 60 periods
= 5% (SA)
60
1
1,000
PB  40  

t
60
(1

0.05)
(1

0.05)
t 1
PB = $810.71
Slide 5
Comm 324 – W. suo
Yields





Yield to maturity
Yield to first call
Bond Equivalent Yield
Effective Annual Yield
Current Yield (Annual Interest/Market Price)
Slide 6
Comm 324 – W. suo
Yield to Maturity Example
35

t
t 1 (1 r )
20
950

10 yr Maturity

1000
T
(1 r )
Coupon Rate = 7%
Price = $950
Solve for r = semiannual rate
r = 3.8635%
Slide 7
Comm 324 – W. suo
Yield Measures
Bond Equivalent Yield
3.86% x 2 = 7.72%
Effective Annual Yield
(1.0386)2 - 1 = 7.88%
Current Yield (Annual Interest/Market Price)
$70 / $950 = 7.37 %
Slide 8
Comm 324 – W. suo
Realized Yield versus YTM
 Reinvestment Assumptions
 Holding Period Return



Slide 9
Changes in rates affects returns
Reinvestment of coupon payments
Change in price of the bond
Comm 324 – W. suo
Holding-Period Return:
Single Period
I  ( P1  P0 )
HPR 
P0
where
I = interest payment
P1 = price in one period
P0 = purchase price
Slide 10
Comm 324 – W. suo
Holding-Period Example
CR = 8% ;
YTM = 8%;
Semiannual Compounding
In 6M the rate falls to 7%;
N=10 years
P0 = $1000
P1 =$1068.55
40  (1068.55  1000)
HPR 
1000
HPR = 10.85% (semiannual)
Slide 11
Comm 324 – W. suo
Realized Compound Yield vs.
YTM
 Requires actual calculation of reinvestment income
 Solve for the Internal Rate of Return using the
following:


Slide 12
Future Value: sale price + future value of coupons
Investment: purchase price
Comm 324 – W. suo
Example
 Two-year bond selling at par, 10% coupon paid
once a year. First coupon is reinvested at 8%. Then:
FV  1,100  100 1.08  1, 208
P  (1  y)2  1, 208
y (realized )  (1.208)0.5  1
Slide 13
Comm 324 – W. suo
Price Paths of
Coupon Bonds
Price
Premium bond
1,000
Discount bond
0
Slide 14
Maturity date
Time
Comm 324 – W. suo
Zero-Coupon Bonds
and Taxation Issues
 For constant yields, discount bond prices rise over
time and premium bond prices decline over time
 Original issue discount bonds’ price appreciation
(based on constant yield) is taxed as ordinary
income
 Price changes stemming from yield changes are
taxed as capital gains if the bond is sold
Slide 15
Comm 324 – W. suo
Example: Tax

30-year bond with 4% coupon rate, issued at an 8% YTM; if sold one year later, when
YTM=7%, for a 36% income tax and a 20% capital gains tax:
P0=549.69;
P1(8%)=553.66;
P1(7%)=631.67
Income tax  (553.66  549.69)  0.36 
40  0.36  15.83
CG tax  (631.67  553.66)  0.20  15.6
Total tax  15.83  15.6  31.43
After  tax income  40 
 (631.67  549.69)  31.43  90.55
Rate of return  90.55 / 549.69  16.5%
Slide 16
Comm 324 – W. suo
Default Risk and Ratings
 Rating companies


Moody’s Investor Service
Standard & Poor’s

Canadian Bond Rating Service (CBRS)
 Rating Categories


Slide 17
Investment grade
Speculative grade
Comm 324 – W. suo
Factors Used by Rating
Companies
 Methods are proprietary
 Accounting ratios





Coverage ratios
Leverage ratio
Liquidity ratios
Profitability ratios
Cash flow to debt
 Other qualitative factors
Slide 18
Comm 324 – W. suo
Financial Ratios by Rating
Class
US Industrial LT Debt,
1997-1999 Medians
AAA
A
BBB
B
EBIT interest coverage
17.5
6.8
3.9
1.0
EBITDA interest coverage
21.8
9.6
6.1
2.0
Funds flow/total debt (%)
105.8
46.1
30.5
9.4
Free operating CF/debt (%)
55.4
15.6
6.6
(4.6)
Return on capital (%)
28.2
19.9
14.0
7.2
Operating income/sales (%)
29.2
18.3
15.3
11.2
LT debt/capital (%)
15.2
32.5
41.0
70.7
Total debt/capital (%)
26.9
40.1
47.4
74.6
Slide 19
Comm 324 – W. suo
Protection Against Default




Sinking funds
Subordination of future debt
Dividend restrictions
Collateral
Slide 20
Comm 324 – W. suo
Overview of Term
Structure of Interest Rates
 Relationship between yield to maturity and maturity
 Information on expected future short term rates can
be implied from yield curve
 The yield curve is a graph that displays the
relationship between yield and maturity
 Three major theories are proposed to explain the
observed yield curve
Slide 21
Comm 324 – W. suo
Important Terms







Bond yields
Spot rates
Forward rates
Yield curve
Term structure or pure yield curve
Structure of forward rates
Using observed rates to predict future rates
Slide 22
Comm 324 – W. suo
Yield Curves
Yields
Upward Sloping
Flat
Downward Sloping
Maturity
Slide 23
Comm 324 – W. suo
Measuring the term structure
- The bootstrapping method
 Derive spot rates from bond yields of varying
maturities
 Treat each coupon as a mini-zero coupon bond
 Use bonds of progressively longer maturities,
starting from T-bills
 “Clean price” method and “dirty price” method
Slide 24
Comm 324 – W. suo
Building zero curve:
Boot-strapping
 Example: T-bills: 6 month with yield of 4%; One year with
yield of 5%
 18 month 5% coupon bond traded at $990
 2 year 6% coupon bond traded at par
This implies y1=2%, y2=5%, y3=2.8664%, y4=3.02%
Spot rate:
0.5
1
1.5
2
4.04%
5%
5.81%
6.13%
Slide 25
Comm 324 – W. suo
Example
 Observe prices and yields on August 17, 2004; find
the spot rate for December 1, 2005
 Observed yields: 3.90%, 4.04% for 6M and 12M,
respectively
 Observed clean price for 6% bond expiring on
December 1, 2005: $1002.29
 Dirty price = clean price + (time elapsed in
semesters) x coupon
Slide 26
Comm 324 – W. suo
Bootstrapping example (cont.)
2.5
1,002.29 
 30  1,035.4
6
3.5
30 
30
6



3.5/12
9.5/12
(1  0.039)
(1  0.0404)
1,030

(1  y3 )15.5/12
 Solving, we find y3=2.08%, or 4.16% annually
Slide 27
Comm 324 – W. suo
Using Spot Rates to price
Coupon Bonds
 A coupon bond can be viewed as a series of zero
coupon bonds
 To find the value, each payment is discounted at the
zero coupon rate
 Once the bond value is found, one can solve for the
yield
 It’s the reason for which similar maturity and
default risk bonds sell at different yields to maturity
Slide 28
Comm 324 – W. suo
Sample Bonds
A
Maturity
Coupon Rate
Par Value
Cash flow in 1-3
Cash flow in 4
B
4 years
4 years
6%
8%
1,000
1,000
60
80
1,060
1,080
Assuming annual compounding
Slide 29
Comm 324 – W. suo
Calculation of Price Using Spot
Rates (Bond A)
Period
1
.05
60
PV of Cash
Flow
57.14
2
.0575
60
53.65
3
.063
60
49.95
4
.067
1,060
817.80
Total
Slide 30
Spot Rate
Cash Flow
978.54
Comm 324 – W. suo
Calculation of Price Using Spot
Rates (Bond B)
Period
1
.05
80
PV of Cash
Flow
76.19
2
.0575
80
71.54
3
.063
80
66.60
4
.067
1,080
833.23
Total
Slide 31
Spot Rate
Cash Flow
1,047.56
Comm 324 – W. suo
Solving for the YTM
Bond A
 Bond Price
 YTM
= 978.54
= 6.63%
Bond B
 Price
 YTM
= 1,047.56
= 6.61%
Slide 32
Comm 324 – W. suo