Examples on Computing Present Value and Yield to Maturity
Examples on Computing Present Value and Yield to Maturity
(Econ 121: Mishkin Chapter 4 Materials)
Instructor: Chao Wei
A Useful Formula:
a
+
a2
+
a3
+
...
+
an
=
a
- an+1 1-a .
(1)
Special Case: When 0 < a < 1, and n ,
a + a2
+ a3
+ ... + a
=
a 1 - a.
(2)
Example 1 Calculate the present value for the following payments:
1.
$500 two years from now when the interest rate is 5%:
500 (1+0.05)2
.
2. $100 every three years for 12 years when the interest rate is 10%::
PV
=
(1
100 + 0.1)3
+
(1
100 + 0.1)6
+
(1
100 + 0.1)9
+
(1
100 + 0.1)12
=
100
1 1.13
+
1 1.16
+
...
+
1 1.112
.
We can
apply
the
formula
in equation
(1) by
recognizing
that
a
=
1 1.13
and
n = 4 in this case. Applying the formula, we have
PV
= 100 ?
1 1.13
-
1 1.13
1
-
1 1.13
4+1
= 205.85.
3. $100 every three years for 12 years when the interest rate is 10%, plus $50 bonus at the end of 12 years.
PV
=
100 (1 + 0.1)3
+
100 (1 + 0.1)6
+
100 (1 + 0.1)9
+
100 (1 + 0.1)12
+
(1
50 + 0.1)12
= 221.78
Example 2 Suppose you buy a $1000 face-value coupon bond with a coupon rate of 10%, a maturity of 4 years,
1
1. Suppose you purchase the bond at a price of $1000, what is the yield to maturity?
First write down the formula for yield to maturity:
1000
=
1000 (1
? +
10% i)
+
1000 ? 10% (1 + i)2
+
1000 ? 10% (1 + i)3
+
1000 ? 10% (1 + i)4
+
1000 (1 + i)4
i = 10%
2. Suppose the purchase price is $800, what is the yield to maturity?
800
=
1000 (1
? +
10% i)
+
1000 ? 10% (1 + i)2
+
1000 ? 10% (1 + i)3
+
1000 ? 10% (1 + i)4
+
1000 (1 + i)4
i = 17.34%
3. Suppose the purchase price is $1200, what is the yield to maturity?
1200
=
1000 (1
? +
10% i)
+
1000 ? 10% (1 + i)2
+
1000 ? 10% (1 + i)3
+
1000 ? 10% (1 + i)4
+
1000 (1 + i)4
i = 4.43%
2
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