Figure 1:
0
1
2
$20,000
$20,000
0
Cash Flow
Time Period
1.a
The present value of these cash flows, and thus
the amount we must
invest today, is:
$20
,
000
1
.
04
2
+
$20
,
000
1
.
04
4
=
$
35,587.21
The discount rate follows from the 8% yieldtomaturity (
R
= 0
.
08) and
semiannual compounding (
m
= 2), implying
i
= 4%.
1.b
Scenario 1: Unchanged Interest Rates.
If we spend $35,587.21 on 4year zeros
today, this means we own bonds with a total face value (or par amount) of:
$35
,
587
.
21
×
(1
.
04)
8
= $48
,
703
.
55
At the end of year 1, we need $20,000 for the first payment. Thus we need
to sell some fraction of our bond holdings; that fraction is determined by the
market in the following manner:
par
(1
.
04)
6
= $20
,
000
(8)
We are just solving the zero formula in reverse to determine the par value
we must sell in the market to receive $20,000 in year 1.
The solution is
par
= $25
,
306
.
38.
After the sale, we are left with bonds with a total face value of $48
,
703
.
55

$25
,
306
.
38 = $23
,
397
.
17 At the end of year 2, we need to make another pay
ment of $20,000. Solving a similar equation to (8):
par
(1
.
04)
4
= $20
,
000
(9)
3