Chapter 9
Debt Instruments
Quantitative Issues
Pricing a Bond
 T coupon 
PAR
P0  

t 
T
 t 1 1  Y   1  Y 
where P0 = price of bond today
T = maturity of the bond
Y = appropriate discount rate
PAR = par or face value of the bond
Bond Prices with
Semiannual Payments
  coupon  

 2T 
PAR
2 


P0  

t
 t 1  Y    Y  2T

1     1  
2   
2


•
•
•
Divide coupon payment by two
Multiply maturity of bond by two.
Divide discount rate by two
Bond Yields & Rates
•
•
•
•
•
•
Coupon rate (nominal yield)
Current yield (coupon / price)
Yield to maturity (YTM = IRR)
Realized compound yield to maturity (RCYTM)
Yield to First (earliest) Call
Realized return
ABC Example
•
•
•
•
•
Coupon:
Par Value:
Maturity:
Callable:
Price:
$40 per year
$1,000
6 years
in 3 years @ $1040
$950
Coupon Rate
• Stated dollar return of fixed-income
investment
• Equals annual interest payments divided by
par value
Current Yield
• Bond’s coupon rate divided by current
market price
OR
• Stock’s indicated dividend rate divided
by per-share price
Yield to Maturity
• Measure of bond yield that takes into account capital gain
or loss, as well as coupon payments
• Discount rate that would make present value of bond’s
cash flows (payments plus face value at maturity) equal
purchase price of bond
T
P0  
t 1
C
1  Y
t

PAR
1  Y
T
where C = the coupon payment
Yield Relationships
Yield to Call
P0 
Tc
coupon
 1  Y 
t 1
c
t

call price
1  Yc T
40
40
40
1040
950 



2
3
1  Yc  1  Yc  1  Yc  1  Yc 3
where Tc = time to earliest call
Yc = yield to first call
• Almost identical to YTM, except
– Call price replaces par value
– Time to call replaces term to maturity
Realized Rate (Yield)
• Ex post rate of return or yield from
investment (internal rate of return)
where TH = holding period
YH = realized rate of return
Bond Price Volatility
• Bond prices and interest rates inversely related
• Maturity effect: longer a bond’s term to maturity,
greater percentage change in price for given
change in interest rates
• Coupon effect: lower a bond’s coupon rate, greater
percentage change in price for given change in
interest rates
• Yield-to-maturity effect: For given change in
interest rates, bonds with lower YTMs have
greater percentage price changes than bonds with
higher YTMs – all other things equal
Which Bond’s Price Is Most Volatile?
• Bond X: 25 years to maturity, 10% coupon
rate, and a 6% YTM
• Bond Y: 10 years to maturity, 2% coupon
rate, and a 6% YTM
• Bond Z: 17.5 years to maturity, 6% coupon
rate, and a 4% YTM
Answer
• Based on maturity effect, it would be X
• Based on coupon effect, it would be Y
• Based on yield-to-maturity effect, it would
be Z
Duration
• Weighted average amount of time until
present value of bond’s purchase price
repaid to the investor
• Based on time-weighted present value of
bond’s principal and interest payments
divided by the bond’s price
• Used as measure of bond’s sensitivity to
interest rate changes
Formula for Duration
T
D
t x Ct
 1  Y 
t 1
t
P0
Where P0 = price of the bond today
Y = yield to maturity
Ct = cash flow in period t (coupon, principal or both)
T = term to maturity
Equation 9-6
• Insert Equation
1  Y 1  Y   T  C  Y 
D

T
Y
C 1  Y   1  Y


• Where
Y = yield to maturity
C = coupon rate
T = term to maturity
Uses of Duration
• Price volatility index
– Larger duration statistic, more volatile price of
bond
• Immunization
– Interest rate risk minimized on bond portfolio
by maintaining portfolio with duration equal to
investor’s planning horizon
Major Characteristics of Duration
• Duration of zero-coupon bond equal to term
to maturity
• Duration of coupon bond always less than
term to maturity
• Inverse relationship between coupon rate
and duration
(continued)
Major Characteristics of
Duration (continued)
• Inverse relationship between yield to
maturity and duration
• Direct relationship between maturity and
duration
Modified Duration
• Adjusted measure of duration used to estimate a
bond’s interest rate sensitivity
D* = D  (1 + YTM)
% Chg in price of bond = –D x % Chg in YTM
% Chg in price of bond = – D* x [Chg in YTM]
Convexity
Portfolio Duration
• Market value weighted average of durations
of individual securities in the portfolio
Components of Interest
Rate Risk
• Price Risk
• Reinvestment Rate Risk
Price Risk
• Risk of existing bond’s price changing in
response to unknown future interest rate
changes
– If rates increase, bond’s price decreases
– If rates decrease, bond’s price increases
Reinvestment Rate Risk
• Risk associated with reinvesting coupon
payments at unknown future interest rates
– If rates increase, coupons are reinvested at
higher rates than previously expected
– If rates decrease, coupons are reinvested at
lower rates than previously expected
Immunizing a Portfolio
• If a single time horizon goal, purchasing
zero-coupon bond whose maturity
corresponds with planning horizon
• If multiple goals, purchasing series of
zero-coupon bonds whose maturities
correspond with multiple planning horizons
(continued)
Immunizing a Portfolio (continued)
• Assembling and managing bond portfolio
whose duration is kept equal to planning
horizon
Note: this strategy involves regular
adjustment of portfolio because duration of
portfolio will change at SLOWER rate than
will time itself
Bond Swaps
• Technique for managing bond portfolio by selling
some bonds and buying others
• Possible benefits achieved:
–
–
–
–
tax treatment
yields
maturity structure
trading profits
Types of Swaps
• Substitution swap
– Tax swap
• Intermarket spread swap
• Pure-yield pick-up swap
• Rate anticipation swap
Strategies for Managing
a Bond Portfolio
• Bullet Portfolio
– Entire portfolio is placed in one maturity
• Bond ladders
– Equally distributed dollar allocations over time
• Barbells
– Majority of dollar allocations in shortest-term
and longest-term holdings
Yield Curve or Term Structure
•
•
•
•
Vertical axis: yield to maturity
Horizontal axis: term to maturity
Bonds of like quality
Always based on Treasuries
Shapes of Yield Curve
• Rising: Most common (used to be
only one observed)
• Falling: Next most common
• Humped
• Flat: Rare
Types of Yield
Curves
Theories of the Yield Curve
• Unbiased expectations
– Long-term rates reflect market’s expectation of
current and future short-term rates.
• Preferred habitat
– Significantly more attractive rates can induce
investors and borrowers out of their preferred
maturity structures
(continued)
Theories of Yield Curve
(continued)
• Market Segmentation:
– Yields reflect supply and demand for each
maturity class.
• Liquidity Preference:
– Borrowers are risk averse and demand premium
for buying long-term securities
– Yield curves tend to be upward sloping.
(continued)
Theories of the Yield Curve
(continued)
• Preferred habitat
– Significantly more attractive rates can induce
investors and borrowers out of their preferred
maturity structures
• Unbiased expectations
– Long-term rates reflect market’s expectation of
current and future short-term rates.
Factors Affecting Bond Yields
• General credit conditions: Credit conditions affect
all yields to one degree or another.
• Default risk: Riskier issues require higher
promised yields.
• Term structure: Yields vary with maturity
• Duration: Weighted average amount of time until
present value of purchase price is recouped.
• Coupon effect: Low-coupon issues offer yields
that are partially taxed as capital gains.
(continued)
Factors Affecting Bond Yields (continued)
• Seasonings: Newly issued bonds may sell at slight discount
to otherwise-equivalent established issues.
• Marketability: Actively traded issues tend to be worth
more than similar issues less actively traded.
• Call protection: Protection from early call tends to enhance
bond’s value.
• Sinking fund provisions: Sinking funds reduce probability
of default, thereby tending to enhance bond’s value.
• Me-first rules: Bonds protected from diluting effect of
additional borrowings are generally worth more than
otherwise-equivalent unprotected issues.