Bond Pricing

PB = Price of the bond

Ct = interest or coupon payments

T = number of periods to maturity

r = semi-annual discount rate or the semi-annual yield to maturity

Price of 8%, 10-yr. with yield at 6%

)03.1(

11000

)03.1(

140 20

20

1

t

tBP

Coupon = 4%*1,000 = 40 (Semiannual)

Discount Rate = 3% (Semiannual)

Maturity = 10 years or 20 periods

Par (Face) Value = 1,000

Calculate Price 8%, 10-yr. with yield at 6%

T

T

tt

ttB

rrr

P

1

11

1 :formulafactor annuity use

)1(

1For

)03.1(

1*1000

)03.1(

1*40

1

20

20

1

77.148,1

5537.*1000877.14*40

877.1403.1

11

03.

1

)03.1(

120

20

1

PB

tt

Holding Period Return What is the price of the bond in 6 months time (ex-coupon), and

what was the holding period return over that time?

)03.1(

11000

)03.1(

140 19

19

1

t

tBP

03.177.148,1

4026.143,1HPR

26.143,1

5703.*1000324.14*40

324.1403.1

11

03.

1FactorAnnuity 19

PB

Premium and Discount Bonds

Premium Bond– Coupon rate exceeds yield to maturity– Bond price will decline to par over its

maturity Discount Bond

– Yield to maturity exceeds coupon rate– Bond price will increase to par over its

maturity

Prices and Yield to Maturity

Price

Yield

YTM is discount rate that sets PV of bond cash flows equal to current market price

Yield to Maturity

Yield to maturity = interest rate that equates today’s value with present value of all future payments

Need Today’s Price or ‘Value’ to find implicit interest rate

Like Internal Rate of Return

Current YieldCoupon Bonds

P

CYieldCurrent

• Is better approximation to yield to maturity, nearer price is to par and longer is maturity of bond• Change in current yield always signals change in same direction as yield to maturity

Yields on Discount Instruments

daysF

PFi

daysP

PFi

db

360

365

Yield to Maturity

Yield on a DiscountBasis

or Bank Discount Yield

Yield Measure Annualization

Bond Page of the WSJ and other Financial Press Jan 23, 2003

Bond Page of the WSJ contd. Jan 23, 2003

Alternative Measures of Yield

Yield to Call– Call price replaces par– Call date replaces maturity

Holding Period Yield (actual return)– Considers actual reinvestment of

coupons– Considers any change in price if the bond

is held less than its maturity Realized Compound Yield

– Reinvestment rate of coupons

Default Risk

Agency Assessment– Coverage ratios– Leverage ratios– Liquidity ratios– Profitability ratios– Cash flow to debt

Company’s Protection Against– Sinking funds– Subordination of future debt– Dividend restrictions– Collateral

Term Structure of Interest Rates

Relationship between yields to maturity and maturity Yield curve - a graph of the yields on bonds relative to

the number of years to maturity– Usually Treasury Bonds– Have to be similar risk or other factors would be

influencing yields

Expectations Hypothesis

Key Assumption: Bonds of different maturities are perfect substitutesImplication: Expected Return on bonds of different maturities are equal

For n-period bond:

yt + yt+1 + yt+2 + ... + yt+(n–1)ynt = n

In words: Interest rate on long bond = average short rates expected to occur over life of long bond

Numerical example:One-year interest rate over the next five years 5%, 6%, 7%, 8% and 9%:

Interest rate on two-year bond:(5% + 6%)/2 = 5.5%

Interest rate for five-year bond:(5% + 6% + 7% + 8% + 9%)/5 = 7%

Interest rate for one to five year bonds:5%, 5.5%, 6%, 6.5% and 7%.

Liquidity Premium Theory

Key Assumption: Bonds of different maturities are substitutes, but are not perfect substitutes

Implication: Modifies Expectations Theory with features of Segmented Markets Theory

Investors prefer short rather than long bonds must be paid positive liquidity (term) premium, lnt, to hold long-term bonds

Results in following modification of Expectations Theory

yt + yet+1 + ye

t+2 + ... + yet+(n–1)ynt = + lnt n

Relationship Between the Liquidity Premium and Expectations Theories

Theories of Term Structure

Expectations – Long term rates are a function of expected future

short term rates– Upward slope means that the market is expecting

higher future short term rates– Downward slope means that the market is

expecting lower future short term rates Liquidity Preference

– Upward bias over expectations– The observed long-term rate includes a risk

premium

Innovations in the Bond Market

Reverse floaters Asset-backed bonds Pay-in-kind bonds Catastrophe bonds Indexed bonds

– TIPS (Treasury Inflation Protected Securities)