Bond Prices and Yields

Comm 324 – W. suoSlide 1

Bond Prices Bond Prices and Yieldsand Yields


Face or par value Coupon rate

Zero coupon bond Compounding and payments Indenture Issuers

Bond Characteristics


Secured or unsecured Registered or bearer bonds (Canada) Call provision Convertible provision Retractable and extendible (putable) bonds Floating rate bond

Provisions of Bonds


1 (1 )(1 )

TTtB Tt

t

ParValueCPrr

PB = price of the bond

Ct = interest or coupon payments

T = number of periods to maturityr = the appropriate semi-annual discount rate Quoted price vs Cash Price (or “dirty price”)

Accrued interest, day-count convention

Bond Pricing


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 t

t

P


Yields

Yield to maturity Yield to first call Bond Equivalent Yield Effective Annual Yield Current Yield (Annual Interest/Market Price)


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%


Yield Measures

Bond Equivalent Yield3.86% x 2 = 7.72%

Effective Annual Yield(1.0386)2 - 1 = 7.88%

Current Yield (Annual Interest/Market Price)$70 / $950 = 7.37 %


Realized Yield versus YTM

Reinvestment Assumptions Holding Period Return

Changes in rates affects returns Reinvestment of coupon payments Change in price of the bond


Holding-Period Return: Single Period

whereI = interest paymentP1 = price in one periodP0 = purchase price

0

01 )(P

PPIHPR


Holding-Period Example

CR = 8% ; YTM = 8%; N=10 yearsSemiannual Compounding P0 = $1000In 6M the rate falls to 7%; P1 =$1068.55

HPR = 10.85% (semiannual)

40 (1068.55 1000)1000

HPR


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


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


Price Paths of Coupon Bonds

Price

1,000

Maturity date0

Discount bond

Time

Premium bond


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


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 15.615.83 15.6 31.43

40(631.67 549.69) 31.43 90.55

90.55 / 549.69 16.5%

CG taxTotal taxAfter tax income

Rate of return


Rating companies Moody’s Investor Service Standard & Poor’s

Canadian Bond Rating Service (CBRS)

Rating Categories Investment grade Speculative grade

Default Risk and Ratings


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


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


Sinking funds Subordination of future debt Dividend restrictions Collateral

Protection Against Default


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


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


Yields

Maturity

Upward Sloping

Downward Sloping

Flat

Yield Curves


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


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 parThis 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%


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


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 1,035.46

3.530 306(1 0.039) (1 0.0404)

1,030(1 )y


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


Sample Bonds

Assuming annual compounding

A BMaturity 4 years 4 yearsCoupon Rate 6% 8%Par Value 1,000 1,000Cash flow in 1-3 60 80Cash flow in 4 1,060 1,080


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


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


Solving for the YTM

Bond A Bond Price = 978.54 YTM = 6.63%

Bond B Price = 1,047.56 YTM = 6.61%

Bond Prices and Yields

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