Chapter 5
Valuing Bonds
Chapter 5 Topic Overview
Bond Characteristics
Annual and Semi-Annual Bond Valuation
‹ Reading Bond Quotes
‹ Finding Returns on Bonds
‹ Bond Risk and Other Important Bond
Valuation Relationships
‹
‹
Bond Characteristics
z
z
z
z
z
Face (or Par) Value = stated face value that
is the amount the issuer must repay, usually
$1,000
Coupon Interest Rate
Coupon (cpn) = Coupon Rate x Face Value
Maturity Date = when the face value is
repaid.
This makes a bond’s cash flows look like this:
Characteristics of Bonds
• Bonds pay fixed coupon (interest)
payments at fixed intervals (usually
every 6 months) and pay the face
value at maturity.
cpn/2 cpn/2 cpn/2 cpn/2 … cpn/2+par
0
1
2
...
n
Bonds
WARNING
The coupon rate IS NOT the discount rate used in
the Present Value calculations.
The coupon rate merely tells us what cash flow the bond will
produce.
Since the coupon rate is listed as a %, this misconception is quite
common.
Bond Pricing
The price of a bond is the Present Value of
all cash flows generated by the bond (i.e.
coupons and face value) discounted at the
required rate of return.
PV =
cpn
cpn
(cpn + par )
+
+....+
1
2
(1 + r ) (1 + r )
(1 + r ) t
Bond Valuation
z
Discount the bond’s cash flows at the
investor’s required rate of return.
z
z
z
the coupon payment stream (an annuity).
the face (par) value payment (a single sum).
PV = cpn (PVAF r, t) + par /(1+r)t
cpn
0
cpn
1
cpn+par
2
...
t
Bond Valuation Example #1
‹
‹
Duff’s Beer has $1,000 par value bonds outstanding
that make annual coupon payments. These bonds
have an 8% annual coupon rate and 12 years left to
maturity. Bonds with similar risk have a required
return of 10%, and Moe Szyslak thinks this required
return is reasonable.
What’s the most that Moe is willing to pay for a
Duff’s Beer bond?
0
80
80
80
...
1000
80
1
2
3
...
12
P/Y = 1
12 = N
10 = I/Y
1,000 = FV
80 = PMT
CPT PV = -$863.73
Note: If the coupon rate < discount rate, the
bond will sell for less than the par value: a
discount.
Let’s Play with Example #1
‹ Homer
Simpson is interested in buying
a Duff Beer bond but demands an 8
percent required return.
‹ What is the most Homer would pay for
this bond?
80
80
80
...
1000
80
1
2
3
...
12
0
P/Y = 1
12 = N
8 = I/Y
1,000 = FV
80 = PMT
CPT PV = -$1,000
Note: If the coupon rate = discount rate, the
bond will sell for its par value.
Let’s Play with Example #1
some more.
‹ Barney
(belch!) Gumble is interested in
buying a Duff Beer bond and demands
on a 6 percent required return.
‹ What is the most Barney (belch!) would
pay for this bond?
0
80
80
80
...
1000
80
1
2
3
...
12
P/Y = 1
12 = N
6 = I/Y
1,000 = FV
80 = PMT
CPT PV = -$1,167.68
Note: If the coupon rate > discount rate, the
bond will sell for more than the par value: a
premium.
Bond Prices and Interest Rates
have an inverse relationship!
($)M arket Value
Bond Values for 8% Annual Coupon Bonds
1400
1200
1000
800
600
400
200
0
12-yr Bond
0%
2%
4%
6%
8%
10%
12%
Required Return
Bonds with Semiannual
Coupons
‹
Double the number of years, and divide
required return and annual coupon by 2.
VB = annual cpn/2(PVAFr/2,2t) + par /(1+r/2)2t
Semiannual Example
‹
‹
‹
A $1000 par value bond with an annual coupon rate
of 9% pays coupons semiannually with 15 years left
to maturity. What is the most you would be willing to
pay for this bond if your required return is 8% APR?
Semiannual coupon = 9%/2($1000) = $45
15x2 = 30 remaining coupons
0
45
45
45
...
1000
45
1
2
3
...
30
P/Y = 1
15x2 =30 = N
8/2 = 4 = I/Y
1,000 = FV
90/2 = 45 = PMT
CPT PV = -$1,086.46
Bond Yields
z
z
Current Yield - Annual coupon payments
divided by bond price.
Yield To Maturity - Interest rate for which the
present value of the bond’s payments equal
the price.
Bond Yields
Calculating Yield to Maturity (YTM=r)
If you are given the price of a bond (PV) and
the coupon rate, the yield to maturity can be
found by solving for r.
PV =
cpn
cpn
(cpn + par )
+
+....+
1
2
(1 + r ) (1 + r )
(1 + r ) t
Yield to Maturity Example
$1000 face value bond with a 10% coupon
rate paid annually with 20 years left to
maturity sells for $1091.29.
‹ What is this bond’s yield to maturity?
‹
-1091.29 100
0
1
100
100
2
3
...
...
1000
100
20
P/Y = 1
-1091.29 = PV
20 = N
1,000 = FV
100 = PMT
CPT I/Y = 9% = YTM
What is the bond’s current yield?
Current yield = annual cpn/price = $100/$1091.29 = 9.2%
Note: a bond’s current yield will be in between its coupon rate
and YTM