Exercises 11.1
Identify the type of annuity, deferred period, annuity period, and number of payments for the investment
and payments in Problems 1 and 2:
Exercise 11.1, Solution 1:
a.
$500 is deposited in a savings account at the end of each month for 3 years but the first deposit is
made 5 months from now
•
The payments form an ordinary deferred annuity and the annuity period is 3 years.
•
Payments are made at the end of each payment period (monthly). Therefore,
n
= 36.
•
Payments start 5 months from now. Therefore, the ordinary annuity term starts 4 months from
now (one payment interval before the first periodic payment). Therefore, the deferred period
is 4 months.
b.
$500 is deposited in a savings account at the beginning of every 6 months for 5 years and 6
months and the first deposit is made in 2 years
•
The payments form a deferred annuity due and the annuity period is 5 years and 6 months
•
Payments are made at the beginning of each payment period (semiannually). Therefore,
n
= 2
× 5 + 1 = 11.
•
Payments start 2 years from now. Therefore, the term for the annuity due starts at the same
time. Therefore, the deferred period is 2 years.
Exercise 11.1, Solution 3:
This is an ordinary simple deferred annuity as:
•
Payments are made at the end of each payment period (quarterly)
•
Compounding period (quarterly) = payment period (quarterly)
•
The deferred period is 1 year and 9 months.
n
= 4 payments/year × 5 years = 20 quarterly payments.
j
= 12% = 0.12,
m
= 4
= 0.03 quarterly.
Step 1: Calculate the present value of the annuity (
PV
annuity
)
= $148,774.7486…
N
I/Y
P/Y
C/Y
PV
PMT
FV
20
12
4
4
?
10,000
0
From the calculator computations shown, we get the
PV
= 148,774.7486
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View Full DocumentStep 2: Calculate the present value of this amount at the beginning of the deferred period
(
PV
def
)
Deferred period
n
=
m
×
t
= 4 × (1 year and 9 months) = 4 × 1.75 = 7 quarterly periods
PV
def
= PV
annuity
(1 +
i
)

n
= 148,774.7486…(1 + 0.03)

7
= $120,967.4852… = $120,967.49.
N
I/Y
P/Y
C/Y
PV
PMT
FV
7
12
4
4
?
0
148,774.748
6...
From the calculator computations shown, we get the
PV
= 120,967.4852.
Therefore, the business would have to invest an amount of $120,967.49.in the highgrowth fund.
Exercise 11.1, Solution 5:
This is an ordinary general deferred annuity as:
•
Payments are made at the end of each payment period (monthly)
•
Compounding period (quarterly) ≠ payment period (monthly)
•
The deferred period is 6 years and 11 months.
n
= 12 payments/year × 10 years = 120 monthly payments.
j
= 8% = 0.08,
m
= 4
= 0.02 quarterly.
i
2
= (1 +
i
)
c
– 1 = (1 + 0.02)
(4/12)
– 1 = 0.006622…monthly.
Step 1: Calculate the present value of the annuity (
PV
annuity
)
= 3000
= $247,833.4192…
N
I/Y
P/Y
C/Y
PV
PMT
FV
120
8
12
4
?
3000
0
From the calculator computations shown, we get the
PV
= 247,833.4192
Step 2: Calculate the present value of this amount at the beginning of the deferred period
(
PV
def
)
Deferred period
n
=
m
×
t
= 12 × (6 year and 11 months) = 12 × 6.916666.
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 Winter '12
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 Time Value Of Money, Future Value, Perpetuity, Period, BGN

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