UGBA 103 Summer 2008
Avinash Verma
Yield to Maturity as a Measure of Return for Bonds
Y
IELD
TO
M
ATURITY
AS
A
M
EASURE
OF
R
ETURN
FOR
B
ONDS
A
Coupon Bond
makes a fixed payment, known as
Coupon
(denoted
C
), every period
(usually 6 months in the US) during the course of its life (denoted as
n
periods), and a
balloon payment at the end of its life. The balloon payment is called the Face Value or
the Maturity Value of the Bond, and is denoted
M
. The quantity
C
/
M
is known as the
coupon rate of the bond.
Applying our Fundamental Valuation Equation (Price equals sum of PV of future
expected cash flows) to these cash flows, and assuming that the cash flows are free of
default risk (true when the Bond is issued by the US Treasury), the price of the bond,
P
,
will be:
(
29
(
29
(
29
n
n
n
n
r
M
r
C
r
C
r
C
P
+
+
+
+
+
+
+
+
=
1
1
...
1
1
2
2
1
If we buy an asset for
P
0
dollars, hold it for
n
periods (usually years), and then sell it for
P
n
dollars, our per period return,
R
n
,
is unambiguously defined to be:
1
1
0

≡
n
n
n
P
P
R
In the example above, cash flows occurred only at two points in time, the beginning and
the end of the holding period. In the presence of intermediate cash flows (such as
coupons), it is not unambiguously clear how the return should be measured.
One possible measure of return is the yield to maturity (YTM), denoted
y
. YTM is
defined to be the rate discounted at which the future cash flows of the asset (here a
coupon bond) equal its price. In symbols:
(
29
(
29
(
29
(
29
(
29
n
n
n
n
y
M
y
y
C
y
M
y
C
y
C
y
C
P
+
+
+

=
+
+
+
+
+
+
+
+
=
1
1
1
1
1
1
...
1
1
2
The equation above cannot be solved analytically for
y
. It has to be solved by trial and
error.
Now, among other things, we are interested in how the percentage coupon rate of the
bond,
C
/
M
, compares with its YTM for bonds that sell at a premium (
P
>
M
), at par
(
P
=
M
), and at a discount (
P
<
M
).
Algebraically, for premium bonds, we are interested in:
© Avinash Verma
1