Chapter 9
Debt Instruments
Quantitative Issues
Learning Objectives
A.
B.
C.
D.
E.
F.
Bond Valuation
Yield Measures
Duration
Managing Bond Portfolios
Term Structure
Factors affecting Prices/Yields
Five Bond Pricing Theorems
A. Bond prices move inversely to changes in interest rates
B. The longer the maturity of a bond, the more price
sensitive the bond
C. The price sensitivity of bonds to changes in interest rates
increases as maturity increases, but at a decreasing rate
D. Bonds with lower coupons are more price sensitive
E. Yield decreases have a greater impact on bond prices
than similar yield increases
TI BA II Plus (Pro) - BOND PRICES
A. The price of a bond (PB) is a combination of a
present value of an annuity (the present value of
the coupons to be received) and the present value
of the face (par) value of the bond.
B. PB = $Coupon * PVIFA + $Face * PVIF
N
CFn
PB  
n
n 1 (1  k b )
for n  1, 2, 3, ..., N
COMPUTING BOND PRICES using the BA II Plus
A. Example: Suppose we have a bond paying a 12%
coupon rate ($120), paid semi-annually. The bond
matures in 20 years and has a face value of $1,000. If
the current market rate (YTM) is 9%, how much should
this bond sell for (price)?
B. In this type of problem we will use all five TVM keys;
[ N ] [ I/Y ] [ PV ] [ PMT ] [ FV ]
COMPUTING BOND PRICES using the BA II Plus
C. The bond pays coupons (interest) twice a year
(semi-annual): We set the periods per year (P/Y)
and (C/Y) to 2.
D. The 12% coupon rate ( $120 per year) is paid in
two [PMT=] $60 installments.
E. The bond will have a maturity (face, par) value
[FV] of $1000.00.
F. The current market rate is [I/Y] 9% (the required
YTM for bonds in this risk class).
COMPUTING BOND PRICES using the BA II Plus
G. BA II PLUS Solution
1. ENTER 20 [2nd] [N], [N]
2. ENTER 9 [I/Y]
3. ENTER 60 [PMT]
4. ENTER 1000 [FV]
5. PRESS [CPT] [PV]
N = 40.00
I/Y = 9.00
PMT = 60.00
FV = 1,000.00
PV =
-1,276.02
We would have to pay $1,276.02 to buy this bond
today.
COMPUTING BOND PRICES using the BA II Plus
H. What if the current YTM is 8.5%?
1. Enter 8.5, press [I/Y]
2. Press [CPT] [PV]: PV = - 1,333.85
3. Clearly, a lower YTM results in a higher price.
A. What if the current YTM is 9.5%?
1.
2.
3.
Enter 9.5, press [I/Y]
Press [CPT] [PV]: PV = - 1,222.04
Clearly, a lower YTM results in a higher price.
Bond prices move inversely to changes in interest rates
Yield Measures
A. Coupon Rate = Annual Coupon ÷ $1000
B. Current Yield = Annual Coupon ÷ Price
C. Yield To Maturity = actual rate earned on bond if
held to maturity (same concept as IRR)
1. To compute YTM: CLR TVM, then
a. Set P/Y value (2 or 4 are typical)
b. Enter N, [-]Price (PV), interest PMT, FV ($1000 typical)
c. Compute I/Y
2. Text Example (p9.9): P/Y = 2, N = 12, PV = -950,
PMT = 20, FV = 1000. CPT I/Y = 4.9741
Yield to Maturity
A. Yield to maturity is the rate at which a bond’s
cash flows are discounted
B. Changes as market interest rates change
C. Yield to maturity and coupon rate (CR)
1. If P(b) < F, then YTM > CR (discount bond)
2. If P(b) = F, then YTM = CR
3. If P(b) > F, then YTM < CR (premium bond)
Actual Return & Yield to Maturity
A. If you buy a bond and hold it until maturity, will
your actual return equal the bond’s yield to
maturity?
1. No, unless you can reinvest the coupons at the yield
to maturity rate
2. If reinvestment rate is less than YTM, actual return
will be less than YTM
3. If reinvestment rate is greater than YTM, actual return
will be greater than YTM
Yield Relationships (Fig 9-2)
Computing Yield to [First] Call
A. Bonds may be issued “Callable” during periods
of high interest rates. The “callable” feature
allows the issuer to recall the bonds and reissue
the debt at lower rates.
1. Recalls typically require a premium to be paid.
2. Example: Suppose you are considering buying a 10%
coupon bond (paid quarterly) recallable in 3 years at
107.5 (a premium of $75 in addition to any accrued
interest). The FV = 1075 and the N value would be 3
* P/Y.
3. The bond currently sells for $1464.07. What is the
YTFC (yield to first call)? YTM = 6%
Computing Yield to [First] Call
3.
4.
5.
6.
7.
8.
N = 3 * 4 = 12
PV = - 1464.07
PMT = 25
FV = 1075
[CPT] [I/Y] = 2.3133
This is obviously not a good deal.
1. Right off the bat – you’re taking a $389.07 capital loss.
2. The $75 early call premium is insufficient to cover the
expected capital loss.
3. The YTFC (2.3133%) is less than the current YTM (6%).
4. You’re better off buying a new issue bond.
Assessing Interest Rate Risk
A. Bond Price Volatility
1. Maturity effect: longer a bond’s term to maturity,
greater percentage change in price for given change in
interest rates
2. Coupon effect: lower a bond’s coupon rate, greater
percentage change in price for given change in
interest rates
3. Yield-to-maturity effect: For given change in interest
rates, bonds with lower YTM have greater percentage
price changes than bonds with higher YTM – all other
things equal.
Assessing Interest Rate Risk
A. A bond’s interest rate risk is defined as the sensitivity of
price to a change in YTM.
B. Which bond is more price sensitive?
1.
2.
Bond A: 10% coupon, 10 year maturity
Bond B: 5% coupon, 5 year maturity
C. We can’t say without some sort of summary measure of
interest rate risk
D. Such a measure is called duration
Duration
A. Duration measures the amount of time before the
investor receives the “average” dollar from a bond
B. Duration is a function of a bond’s coupon rate, time
to maturity and yield to maturity
C. Duration:
1.
2.
3.
Increases as the coupon rate decreases
Increases as the time to maturity increases
Increases as yield to maturity decreases
D. The longer the duration of a bond, the more sensitive
its price to a given change in interest rates.
Duration
E. Formulas for Duration
T
CFt * t

t
(
1

Y
)
D  tT1
CFt

t
t 1 (1  Y )
T
CFt
P0  
t
(
1

Y
)
t 1
Note: Exponents in Eq. 9-5
should be t not T
Denominator above is also
equal to the current price
(DCF)
Duration
F. Uses of Duration
1. Price volatility index
a.
2.
Larger duration statistic, more volatile price of bond
Immunization
a.
Interest rate risk minimized on bond portfolio by maintaining portfolio
with duration equal to investor’s planning horizon
G. Principal Characteristics
1.
2.
3.
4.
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
Direct relationship between maturity and duration
Duration
A. Modified Duration
1. Adjusted measure of duration used to estimate a
bond’s interest rate sensitivity
2. D* = D  (1 + YTM)
% Chg in price of bond = –D x % Chg in YTM
% Chg in price of bond = – D* x [Chg in YTM]
Interest Rate Risk
A. Price Risk
1. Risk of existing bond’s price changing in response to
unknown future interest rate changes
a. If rates increase, bond’s price decreases
b. If rates decrease, bond’s price increases
B. Reinvestment Rate Risk
1. Risk associated with reinvesting coupon payments at
unknown future interest rates
a. If rates increase, coupons are reinvested at higher rates than
previously expected
b. If rates decrease, coupons are reinvested at lower rates than
previously expected
Bond Portfolio Immunization
A. Strategies a Function of Needs
1. If a single time horizon goal, purchasing zero-coupon
bond whose maturity corresponds with planning
horizon
2. If multiple goals, purchasing series of
zero-coupon bonds whose maturities correspond with
multiple planning horizons
3. 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.
Managing Bond Portfolios
A. Bond Swaps
1. Technique for managing bond portfolio by selling
some bonds and buying others
2. Possible benefits achieved:
a.
b.
c.
d.
tax treatment
yields
maturity structure
trading profits
Managing Bond Portfolios
A. Types of Swaps
1.
2.
3.
4.
Substitution swap (tax loss issues)
Inter-market spread swap (transports vs. utilities)
Pure-yield pick-up swap
Rate anticipation swap: Yield expectations
B. Portfolio Structure
1. Bullet Portfolio (one maturity date)
2. Bond ladders (equally distributed dollar allocations
over time)
3. Barbells (varying maturities)
a. Allocations to shortest-term and longest-term holdings
Term Structure of Interest Rates
A. Typical: rising to right (see Fig 9-5, pp 9.35)
1. Normal
2. Inverted
3. Flat
B. Theories of Term Structure
1.
2.
3.
4.
Expectations (spot vs. forward rates – see pp 9.38ff)
Liquidity Preference (premiums for longer terms)
Market Segmentation (effects of supply & demand)
Preferred Habitat (maturity preference)
Factors Affecting Bond Yields
A. General credit conditions: Credit conditions affect all
yields to one degree or another.
B. Default risk: Riskier issues require higher promised
yields.
C. Coupon effect: Low-coupon issues offer yields that are
partially taxed as capital gains.
D. Marketability: Actively traded issues tend to be worth
more than similar issues less actively traded.
E. Call protection: Protection from early call tends to
enhance bond’s value.
F. Sinking Fund Requirements: reduce probability of
default