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)