3.
Consider the following monthly cash flows (see the diagram below):
Cash flows of an amount X are made for months 1, 3, 5, …, 17 and 19 (the ten odd-
numbered months) and cash flows of an amount Z are made for months 2, 4, 6, …, 18
and 20 (the ten even-numbered months). The APR is 6% and is compounded on a
monthly basis. What is the present value of these cash flows today if X = $2,000 and
Z = - $700?
A)
12,311
B)
12,406
C)
25,569
D)
25,664
E)
32,955
Solution:
B
The monthly interest rate is 0.5% but since the X’s cash flows are made every two
months, we need to calculate the 2-months equivalent interest rate:
I
2m
=%
0025
.
1
1
%)
5
.
0
1
(
2
=
−
+
=
r
The present value of the Z’s cash flows is given by:
Using your calculator:
I
2m
= 1.0025%, n=10, PMT = -700, FV=0, COMP PV
PVz
0
= -$6629.02 at t=0
(Since fist payment begins at t=2 and “i" is calculated for every
2 month period, and last payment is at t=20)
And the present value of the X’s is given by:
Since X begins at t=1, using your calculator for a regular annuity will give PV at
t =-1
:
I
2m
= 1.0025%, n=10, PMT = 2000, FV=0, COMP PV
PVx
--1
= -$18,940.07 at t= -1
(Since fist payment begins at t=1 and “i" is calculated for
every 2 month period, and last payment is at t=19, you are really calculating PV of an
annuity at t= -1)
To adjust for PVx at t=0-> 18,940.07 x (1.005)
1
= $19,034.77
The total present value (Z + X) is equal to:
75
.
405
,
12
$
02
.
629
,
6
$
77
.
034
,
19
$
=
−
=
PV
Today
1
2
3
4
19
20
X
Z
X
Z
X
Z