Bonds – Valuation of Equity and Fixed Income

Bonds Valuation

1. 4. Bonds – Valuation

2. ◼Zero coupon bond (sell at discount)
Invoice price (sale price)
◼actual price paid when buying the bond
◼Invoice price = quoted price + accrued interest
Indenture
◼A contract between the issuer and the bondholder
◼Specifies: coupon rate, par value, maturity, and bond
provisions
Face or par value (principal)
Coupon rate
◼
Bond Characteristics valueface
($) payments coupon annual
rate Coupon =

3. Secured or unsecured
Call provision (valuable to the issuer)
Convertible provision (valuable to the bondholder)
Retractable (puttable) bonds (valuable to the
bondholder)
Extendible bonds (valuable to the bondholder)
Floating vs. fixed-rate bonds
Provisions of Bonds

4. T
T
t
t
t
B
y
F
y
C
)1()1(1 ++
+
=
= B = price of the bond
C
t = interest or coupon payments (per
coupon period)
F = face value of the bond
T = number of periods to maturity (in
coupon periods)
y = appropriate yield to maturity (per
coupon period)
Bond Pricing

5. We have:
C
t = 40 (SA)
F = 1000
T = 60 (SA) periods
y = 5% (SA)
Solving for Price – An Example$810.71
)05.01(
000,1
)05.01(
40
60
60
1
=
+
+
+

=
=t
t
B
Consider a 30-year bond, with an 8% annual coupon
rate, making payments semiannually. The bond’s
par value is $1,000, and the quoted (annual) yield
to maturity is 10%.

6. Bond prices and market interest rates have an
inverse and nonlinear relationship
When interest rates get very high the value of
the bond will be very low (high discount rate
=> low PV)
When rates approach zero, the value of the
bond approaches the sum of the cash flows
Bond Prices and Interest Rates

7. Coupon
Rate
Price
Yield to
Maturity
Prices and Interest Rates
Face
Value

8. Premium vs. Discount Bonds
Coupon
Rate
(c)
Price (B)
Yield to
Maturity
(y)
Face
Value (F)
Premium Bonds
(B>F, and y<c)
Discount Bonds
(B<F, and y>c)
Par Bonds
(B=F, and y=c)

9. Price Paths of Coupon Bonds
(“Pull to Par”)
Discount bond
Premium bond
Price
F
Maturity
Date (T)
0
Time

10. Yield to Maturity
Most often quoted (as simple interest)
Is the promised rate of return based on the current
market price if:
◼bond is held to maturity
◼coupons are reinvested at the same rate
The interest rate that makes the present value of
the bond’s payments equal to its price
Solve the bond price formula for y:)1()1(1 yy
C
T
T
t
t
t
F
B
++
+=
=

11. Yield to Maturity – An Example)1(
1000
)1(
35
950
20
20
1 yyt
t
++
+=
=
Consider a 10-year bond, with a 7% annual
coupon rate, making payments semiannually.
The bond’s par value is $1,000, and its quoted
market price is $950. Find the bond’s yield to
maturity.
Solve for y (the semiannual rate):
y = 3.8635% (s.a rate)

12. Yield Measures
Bond Equivalent Yield (is the quoted yield)

3.86% x 2 = 7.72%
Effective Annual Yield
(1.0386)
2
- 1 = 7.88%
Current Yield (=Annual Interest/Market Price)
(2 x $35) / $950 = 7.37 %

13. Yield Measures…
Note the following relationships:
For Par Bonds:

Coupon Rate = Current Yield = YTM
For Discount Bonds:
Coupon Rate < Current Yield < YTM
For Premium Bonds:
YTM < Current Yield < Coupon Rate

14. Callable Bonds
The company (issuer) has an option to buy back (call)
the entire bond issue for a call price.
It is profitable for the issuing company to call the
bond when the market price of the bond exceeds the
call price. The difference (bond price – call price) is
the issuer’s gain, and
the bondholder’s loss
Because of the potential loss for the bondholder, a
callable bond must be cheaper than a straight bond.
In other words, a callable bond yield is greater than a
simple bond yield (difference = call premium)
The price of a callable bond is capped - a rational
bondholder will never agree to pay for the bond a
price higher than the call price.

15. Prices and Yields for Callable Bonds
Price (B)
Yield to
Maturity (y)
Straight Bond
Callable Bond
Call
Price

16. Yield to Call
There is always a chance that the bond will be called
before its stated maturity
Yield to call may be more relevant to bondholders than
yield to maturity (especially if market price is close to
the call price)
Solve the following formula for y
c
:c
Tc
c
T
t
tc
t
y
CP
y
C
B
)1()1(
1
+
+
+
=
=
where: y
c
= YTC, CP = call price, and T
c = expected
time of call

17. Yield to Call – An Example
Consider a 10-year callable bond, with an 18% coupon rate,
making payments semiannually. The bond’s par value is
$1,000, and its current quoted market price is $1,200.
The bond indenture indicates the following call schedule:
Period (years) Call price
5 - 7.5 $1,180
7.5 - 8 $1,120
8 - 9 $1,060
9 - 10 $1,000
Find the bond’s yield to first call (YTFC) (call in 5 years = 10
s.a. periods)10
10
1
)1(
180,1
)1(
90
200,1
c
t
tc
yy +
+
+
=
=
=> y
c
= 7.38% and YTFC = 2xy
c
= 14.76%