comm 324 – w. suo slide 1. comm 324 – w. suo slide 2 face or par value coupon rate zero coupon...
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Comm 324 – W. suoSlide 1Slide 1
Bond Prices Bond Prices and Yieldsand Yields
Comm 324 – W. suoSlide 2Slide 2
Face or par value Coupon rate
Zero coupon bond
Compounding and payments Indenture Issuers
Bond Characteristics
Comm 324 – W. suoSlide 3Slide 3
Secured or unsecured Registered or bearer bonds (Canada) Call provision Convertible provision Retractable and extendible (putable) bonds Floating rate bond
Provisions of Bonds
Comm 324 – W. suoSlide 4Slide 4
1 (1 )(1 )
TTtB Tt
t
ParValueCPrr
PB = price of the bond
Ct = interest or coupon payments
T = number of periods to maturity
r = the appropriate semi-annual discount rate
Quoted price vs Cash Price (or “dirty price”) Accrued interest, day-count convention
Bond Pricing
Comm 324 – W. suoSlide 5Slide 5
Ct = 40 (SA)P = 1000T = 60 periodsr = 5% (SA)
PB = $810.71
Solving for Price: 10-yr, 8% Coupon Bond, FV = $1,000
60
601
1 1,00040
(1 0.05) (1 0.05)B tt
P
Comm 324 – W. suoSlide 6Slide 6
Yields
Yield to maturity Yield to first call Bond Equivalent Yield Effective Annual Yield Current Yield (Annual Interest/Market Price)
Comm 324 – W. suoSlide 7Slide 7
Yield to Maturity Example
20
1
35 1000950(1 )(1 )
Ttt rr
10 yr Maturity Coupon Rate = 7%
Price = $950
Solve for r = semiannual rate
r = 3.8635%
Comm 324 – W. suoSlide 8Slide 8
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 %
Comm 324 – W. suoSlide 9Slide 9
Realized Yield versus YTM
Reinvestment Assumptions Holding Period Return
Changes in rates affects returns Reinvestment of coupon payments Change in price of the bond
Comm 324 – W. suoSlide 10Slide 10
Holding-Period Return: Single Period
whereI = interest payment
P1 = price in one period
P0 = purchase price
0
01 )(
P
PPIHPR
Comm 324 – W. suoSlide 11Slide 11
Holding-Period Example
CR = 8% ; YTM = 8%; N=10 years
Semiannual Compounding P0 = $1000
In 6M the rate falls to 7%; P1 =$1068.55
HPR = 10.85% (semiannual)
40 (1068.55 1000)
1000HPR
Comm 324 – W. suoSlide 12Slide 12
Realized Compound Yield vs. YTM
Requires actual calculation of reinvestment income Solve for the Internal Rate of Return using the
following: Future Value: sale price + future value of coupons Investment: purchase price
Comm 324 – W. suoSlide 13Slide 13
Example
Two-year bond selling at par, 10% coupon paid once a year. First coupon is reinvested at 8%. Then:
1,100 100 1.08 1,208FV
2(1 ) 1,208P y
0.5( ) (1.208) 1y realized
Comm 324 – W. suoSlide 14Slide 14
Price Paths of Coupon Bonds
Price
1,000
Maturity date0
Discount bond
Time
Premium bond
Comm 324 – W. suoSlide 15Slide 15
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
Comm 324 – W. suoSlide 16Slide 16
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%)=631.67Income tax (553.66 549.69) 0.36
40 0.36 15.83
(631.67 553.66) 0.20 15.6
15.83 15.6 31.43
40
(631.67 549.69) 31.43 90.55
90.55 / 549.69 16.5%
CG tax
Total tax
After tax income
Rate of return
Comm 324 – W. suoSlide 17Slide 17
Rating companies Moody’s Investor Service Standard & Poor’s
Canadian Bond Rating Service (CBRS)
Rating Categories Investment grade Speculative grade
Default Risk and Ratings
Comm 324 – W. suoSlide 18Slide 18
Methods are proprietary Accounting ratios
Coverage ratios Leverage ratio Liquidity ratios Profitability ratios Cash flow to debt
Other qualitative factors
Factors Used by Rating Companies
Comm 324 – W. suoSlide 19Slide 19
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
Comm 324 – W. suoSlide 20Slide 20
Sinking funds Subordination of future debt Dividend restrictions Collateral
Protection Against Default
Comm 324 – W. suoSlide 21Slide 21
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
Overview of Term Structure of Interest Rates
Comm 324 – W. suoSlide 22Slide 22
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
Comm 324 – W. suoSlide 23Slide 23
Yields
Maturity
Upward Sloping
Downward Sloping
Flat
Yield Curves
Comm 324 – W. suoSlide 24Slide 24
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
Comm 324 – W. suoSlide 25Slide 25
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%
Comm 324 – W. suoSlide 26Slide 26
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
Comm 324 – W. suoSlide 27Slide 27
Bootstrapping example (cont.)
Solving, we find y3=2.08%, or 4.16% annually
3.5/12 9.5/12
15.5/123
2.51,002.29 30 1,035.4
63.5
30 306(1 0.039) (1 0.0404)
1,030
(1 )y
Comm 324 – W. suoSlide 28Slide 28
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
Comm 324 – W. suoSlide 29Slide 29
Sample Bonds
Assuming annual compounding
A B
Maturity 4 years 4 years
Coupon Rate 6% 8%
Par Value 1,000 1,000
Cash flow in 1-3 60 80
Cash flow in 4 1,060 1,080
Comm 324 – W. suoSlide 30Slide 30
Calculation of Price Using Spot Rates (Bond A)
Period Spot Rate Cash Flow PV of Cash Flow
1 .05 60 57.14
2 .0575 60 53.65
3 .063 60 49.95
4 .067 1,060 817.80
Total 978.54
Comm 324 – W. suoSlide 31Slide 31
Calculation of Price Using Spot Rates (Bond B)
Period Spot Rate Cash Flow PV of Cash Flow
1 .05 80 76.19
2 .0575 80 71.54
3 .063 80 66.60
4 .067 1,080 833.23
Total 1,047.56
Comm 324 – W. suoSlide 32Slide 32
Solving for the YTM
Bond A Bond Price = 978.54 YTM = 6.63%
Bond B Price = 1,047.56 YTM = 6.61%