Bond Pricing
P
B
C t
=
=
Price of the bond 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%
P
B

40
 t
20 

1
1
( 1 .
03 ) t

1000

1
( 1 .
03 )
20
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%
P
B

40 * t
20 

1
1
( 1 .
03 ) t

1000 *
1
( 1 .
03 )
20
For t
T 

1
( 1
1
 r ) t use annuity factor formula :
1 r

 1


1

1 r

T

 t
20 

1
1
( 1 .
03 ) t

1
.
03


1


1
1 .
03

20



14 .
877
P
B


40 * 14 .
877
1 , 148 .
77

1000 * .
5537

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?
P
B

40
 t
19 

1
1
( 1 .
03 ) t

1000

1
( 1 .
03 )
19
Annuity Factor

1
.
03


1


1
1 .
03

19



14 .
324
P
B


40 * 14 .
324
1 , 143 .
26

1000 * .
5703
HPR

1 , 143 .
26

40
1 , 148 .
77

1

.
03

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

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 Yield
Coupon Bonds
Current Yield

C
P
• 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
Yield Measure Annualization
Yield to Maturity i

F

P
P

365 days
Yield on a Discount
Basis or Bank Discount Yield i db

F

P
F

360 days

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 substitutes
Implication: Expected Return on bonds of different maturities are equal
For n-period bond: y nt
= y t
+ y t+1
+ y t+2 n
+ ... + y t+(n –1)
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, l nt
, to hold long-term bonds
Results in following modification of Expectations Theory y nt y t
+ y e t+1
+ y e t+2
= n
+ ... + y e t+(n –1)
+ l nt

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)